Signal Processing Lab - SJTU
Contents
2D FFT 20
ALIASING 23
AMPLITUDE MODULATION 28
AMPLITUDE VS POWER SPECTRUM 32
AUTOCORRELATION 35
BITS VS RESOLUTION 40
CEPSTRUM 43
CLIPPING 47
CONSTANT PERCENTAGE BANDWIDTH FILTER 50
CONVOLUTION 55
CONVOLUTION FREQUENCY 62
CROSSCORRELATION 65
DECONVOLUTION 69
DELTA FUNCTION 72
EVEN AND ODD FUNCTIONS 75
FFT BANDWIDTH 79
FFT EVEN - ODD 83
FFT LINEARITY 87
FILTER RESPONSE TIME 90
FIR FILTER (Finite Impulse Response) 93
FREQUENCY AVERAGING 97
FREQUENCY MODULATION 102
FREQUENCY SHIFT 106
IIR FILTERS (Infinite Impulse Response) 110
LEAKAGE 118
LIN – LOG FREQUENCY SCALES 122
LOW PASS FILTER 125
MEDIAN FILTER 130
ORBITS 134
ORDER TRACKING 138
PARSEVAL’S THEOREM 143
PHASE IN TIME AND FREQUENCY 147
PICKET FENCE EFFECT 150
RESONANCE 155
RIDING & BEATING 159
RMS – PEAK -CREST 162
SIGNAL DIFFERENTIATION 165
SIGNAL INTEGRATION 168
SINGLE POLE FILTER 172
SQUARE & SINC FUNCTIONS 176
STROBOSCOPE 180
TIME DOMAIN AVERAGING (TWO SIGNALS) 183
TIME DOMAIN AVERAGING NOISE 187
TIME SCALING FFT 191
TIME SHIFTING FFT 195
TIME VS FREQUENCY 198
TOTAL HARMONIC DISTORTION 202
TRANSFER FUNCTION 205
TRANSIENTS 208
TRANSMISSIBILITY 211
WAVES & SPECTRA 215
WINDOWS FOR FREQUENCY ANALYSIS 220
WINDOWS AMPLITUDE 224
WINDOWS COMPARISON 227
WINDOWS OVERLAPPING 229
WINDOWS: NOISE FLOOR 234
WINDOWS RESOLUTION 237
2D FFT
Many important signal processing problems involve the processing of multidimensional signals. All the properties of signals and systems can be extended to the multidimensional case. The two-dimensional discrete Fourier transform is a very useful tool to analyze two-dimensional signals such as photographs and seismic array data.
A two-dimensional discrete Fourier transform can be implemented by using a one-dimensional transform first on the rows and then on the columns or vice versa.
Figure 1 shows an example of the two-dimensional Fourier transform. Part (a) shows the picture of Jean Baptiste Fourier. Part (b) shows the magnitude plot of the two-dimensional Fourier transform of (a). Part (c) shows the phase plot of the 2D Fourier transform of (a).
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Figure 1. (a) Picture of Fourier, (b) magnitude spectrum and (c) phase spectrum.
Figure 2 shows a square function. This function is the one used in the simulator.
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Figure 2. 3D square function.
The Simulator
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1. Delay Y: Slide control for the delay of the pulse in the Y coordinate.
2. Delay X: Slide control for the delay of the pulse in the X coordinate.
3. Width X: Slide control for the width of the pulse in the X coordinate.
4. Width Y: Slide control for the width of the pulse in the Y coordinate.
5. Image Graph: display of the signal in time or space domain.
6. Magnitude of 2D FFT: Intensity graph display of the magnitude of the 2D-FFT of the Signal.
7. Return to Menu: Return to main menu
8. Show Help: General description of this VI.
Practice with the simulator
1. Set different values for width and delay for the two-dimensional square function and observe the magnitude 2D FFT plot.
2. Set both widths of the signal to 1. Can you tell what happen with the 2D FFT plot? (Clue: Delta function).
Related Topics
• FFT Bandwidth
• FFT Even – Odd
• FFT Linearity
• Frequency shift
• Picket Fence Effect
• Scaling Time
• Time shift
• Time Vs. Frequency
ALIASING
“False low-frequency signals produced in a data sampling process when the sampling rate is less than twice the frequency of the highest frequency component contained in the sample”[1]
Aliasing is the phenomenon where in effect a high frequency component takes on the identity of a lower frequency. It occurs because the time function was not sampled at a sufficiently high rate.
In figure 1, a 4 Hz signal is being read, using a 5 Hz sampling rate. When the collection of sampled points is plotted, a 1 Hz signal is obtained as a result of aliasing.
To avoid aliasing the sampling rate must be at least two times bigger than the highest frequency component of the sampled signal. Harry Nyquist[2] discovered this rule that is called the Nyquist sampling rule.
[pic]
Figure 1. Aliasing effect in signal sampling.
In order to acquire a signal that has a rich spectrum of frequencies and there is the need of avoiding aliasing, these steps had to be followed:
1. Select the signal’s highest frequency that it’s going to be analyzed. For example: 5000Hz.
2. Use an analog anti-aliasing filter to cut the frequencies higher than the frequency selected in the prior step. Figure 2 shows the response of a 5000Hz filter. It doesn't totally filter frequencies over 5000Hz and those frequencies will produce aliasing.
3. Use a sampling speed larger than two times the upper limit frequency. It is recommended to use a sampling rate 2.56 times the higher frequency. For the example the sampling frequency will be 12.800 Hz (2.56 x 5000).
4. Acquire 2n samples at the sampling frequency rate. The number of samples is a power of two as a requirement of the fast Fourier transform (FFT). For the example the number of samples will be 210=1024.
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Figure 2. Anti-aliasing filter response.
5. Apply the FFT to the sample.
6. Take the first 400 lines of the spectrum (frequencies from 0 to 5000 Hz) and discard the last 112 lines (frequencies from 5000 to 6400 Hz). This last interval contains aliasing and the attenuated data. See figure 2.
Aliasing can be seen in western movies where a slowing stagecoach wheel will appear to rotate backward, stop, and the rotate forward at a decreasing speed as the stagecoach comes to halt. The backward rotation is caused by a film-framing speed that is slower than the time required for a spoke to rotate into the position occupied by the adjacent spoke when the previous frame was exposed.
The most important application of aliasing is the strobe light used to stop high-speed motion. The strobe-flashing rate is the measuring speed. When the flash rate equals the rotational speed, there seems to be no motion at all. By varying the flash rate slightly, the motion can be moved forward or backward in slow motion.
The Simulator
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1. Frequency of the Signal to be Acquired: Slide control of the frequency of the signal to be acquired.
2. Cycles: Digital indicator of the frequency of the signal to be acquired.
3. Sampling Frequency: Slide control of the sampling frequency.
4. Sampling Freq./Signal Freq.: Digital indicator of the ratio sampling frequency / signal frequency.
5. ALIASING: Warning lamp for aliasing. When there is aliasing the lamp turns red.
6. Return to Menu: Return to Main Menu
7. Show Help: General description of this VI.
8. Original and Acquired signals: Graph of the original signal to be acquired and the acquired signal in the time domain.
9. Signal Spectrum and acquired Signal Spectrum: Amplitude spectrum of the original signal to be acquired, and amplitude spectrum of the signal acquired.
Practice with the simulator
1. Put the Frequency of the signal to be Acquired in 10 Hz.
2. Move the sampling frequency slider to a High value. See what happens with the plots.
3. Move the sampling frequency slider to a Low value. See what happens with the plots and the Aliasing lamp.
4. Move the sampling frequency and try to obtain a 0.5, 1, 1.5, 2 and 2.5 in the Sampling Freq./Signal Freq. indicator. See what happens with the plots.
Related Topics
• Clipping
• Stroboscope
• Waves & spectra
AMPLITUDE MODULATION
“Modulation: The modification of one signal by another in either amplitude or frequency. Amplitude modulation produces a variation in amplitude plus side-bands around the carrier at the frequency of the modulating signal”[3]
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Figure 1. Amplitude modulation example.
“Ordinarily, the transmission of a message signal (be it in analog or digital form) over a band-pass communication channel (e.g., telephone line, satellite channel) requires a shift of the range of frequencies contained in the signal into other frequency ranges suitable for transmission, and a corresponding shift back to the original frequency range after reception. For example, a radio system must operate with frequencies of 30kHz and upward, whereas the message signal usually contains frequencies in the audio frequency range, so some form of frequency-band shifting must be used for the system to operate satisfactorily. A shift of the range of frequencies in a signal is accomplished by using modulation, defined as the process by which some characteristic of a carrier is varied in accordance with a modulation wave. The message signal is referred to as the modulating wave, and the result of the modulation process is referred to as the modulated wave. At the receiving end of the communication system, we usually require the message signal to be recovered. This is accomplished by using a process known as demodulation, or detection, which is the inverse of the modulation process”[4].
Amplitude modulation is often confused with beating, but both phenomena’s are very different.
The Simulator
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1. Frequency of Tone: Slide control of the frequency of the modulating signal.
2. Frequency of Carrier Signal: Slide control of the frequency of the carrier signal.
3. Modulation Factor: Slide control of the modulation factor.
4. Overmodulation: Overmodulation warning lamp.
5. Return to Menu: Return to main menu
6. Show Help: General description of this VI.
7. Modulated Signal: Modulated signal in time domain.
8. Spectrum of Modulated Signal: Modulated signal in frequency domain.
Practice with the simulator
1. Set the frequency of modulating signal in 0 Hz and carrier signal in 50 Hz.
2. Gradually increase the frequency of the modulating signal and watch the plots.
3. Set the frequency of modulating signal in 10 Hz and gradually increase and decrease the frequency of the carrier signal.
4. Increase the modulation factor and observe the amplitude of the signal in the time and frequency domain.
5. Set the frequency of modulating signal in 10 Hz and gradually move the frequency of the carrier signal to values under 10 Hz.
Related Topics
• Frequency Modulation
• Ridding and Beating
AMPLITUDE VS POWER SPECTRUM
The amplitude spectrum is the result of applying the FFT to a time signal. It displays all the frequencies contained in the signal and their respective amplitude value. The power spectrum is obtained by squaring the amplitude spectrum.
The power spectrum is used to analyze the power of the signal. For example, if the amplitude spectrum shows the current in a signal, the power spectrum shows the power it dissipates.
As shown in figure 1, the power spectrum enhances those frequencies with a large amplitude value .
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Figure 1. Power and amplitude spectra of the same signal.
The Simulator
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1. Signal: Graph of the signal in the time domain.
2. Wave Type: Menu ring to select the type of signal.
3. Frequency: Slide control of the frequency of the signal.
4. Cycles: Digital indicator of the number of cycles in the time window.
5. Return to Menu: Return to main menu
6. Show Help: General description of this VI.
7. Power Spectrum: Graph of the power spectrum of the signal.
8. Amplitude Spectrum: Graph of the amplitude spectrum of the signal.
9. Phase: Slide control and indicator of the phase of the signal.
Practice with the simulator
1. Try different wave-types and frequencies and compare the spectra plots.
2. See what happen with the high amplitude frequencies and with the low amplitude frequencies.
Related Topics
• Waves & Spectra
AUTOCORRELATION
The autocorrelation function is a tool to compare a signal with itself. It makes a copy of a signal and superimposes it over the original signal at each point across it, calculating a value that indicates how well both signals fit with each other. The autocorrelation plot shows the level of correlation in the points that were analyzed.
The autocorrelation value goes from –1 to +1. A value of +1 indicates perfect correlation. A value of zero indicates that there is no correlation between the signals compared. A value of –1 indicates an inverse correlation.
Autocorrelation plot is symmetrical to the center (0), and its value is always +1 because of the comparison of the signal with itself in the same point.
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Figure 1. Original signal (a), signal comparison at 2 shows high correlation (b), signal comparison at 1 shows low correlation(c) and autocorrelation plot (d).
The most useful applications of the autocorrelation are:
– To detect periodicity on a signal. If the autocorrelation plot is periodic, then the signal is periodic.
– To eliminate noise from a signal. See figure 2.
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Figure 2. Using autocorrelation to eliminate noise from a signal.
The autocorrelation of a time series at lag L is given by:
[pic]
Where L: Lag
n: number of samples.
Y: vector of samples
Yi: sample in the position i
The Simulator
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1. Type of signal: Menu ring to select the signal type.
2. Frequency of the Signal: Slide to control the frequency of the signal.
3. Noise Amplitude: Slide to control the amplitude of the white noise to be added to the signal.
4. Return to Menu: Return to Main Menu
5. Pause: Stop the execution in order to examine the graphs in detail
6. Show Help: General description of this VI
7. Signal + White Noise: Time domain graph of the signal plus noise to be autocorrelated.
8. Autocorrelation: Autocorrelation of the Signal + Noise in the time domain.
Practice with the simulator
1. Set the Noise Amplitude to 0 in order to try different signals and appreciate a clean autocorrelation plot.
2. Gradually increase the Noise Amplitude and see how the autocorrelation plot is affected.
Try different signals, frequencies and Noise Amplitudes and compare the differences between the autocorrelation plots.
Related Topics
• Crosscorrelation
BITS VS RESOLUTION
The conversion of an analog signal to a digital signal is a basic need for signal processing using digital instruments like the personal computer (PC).
Most sensors produce analog signals that vary when the measured variable changes. For example, a thermocouple that measures temperatures from 0 to 100ºC and produces a signal from 0 to 5V. To analyze the signal produced by the thermocuple on a computer, it must be digitalized because computers only work with discrete values.
“The resolution in analog-digital conversion is the number of bits that an analog to digital converter (ADC) uses to represent the analog signal. The higher the resolution, the higher the number of divisions the voltage range is broken into, and therefore, the smaller the detectable voltage change. Figure 1 shows a sine wave and its corresponding digital image as obtained by a 3-bit ADC. A 3-bit converter divides the analog range into 23 = 8 divisions. Each division is represented by a binary code between 000 and 111. Cleary the digital representation is not a good representation of the original analog signal because information was lost in the conversion. By increasing the resolution to 16 bits, however, the number of codes from ADC increases from 8 to 65,536, so you can obtain an extremely accurate digital representation of the analog signal.
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Figure 1. Digitized Sine Wave with 3-bits Resolution”[5].
Dynamic Range
The dynamic range is the ratio between the largest and the smallest values that the ADC can read. The ratio can be expressed in voltage decibels. The equation for the dynamic range of an ADC is given by:
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where n is the number of bits.
The Simulator
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1. Bits: Digital indicator of the number of bits in the A/D conversion.
2. A/D Bits: Slide control of the number of bits in the A/D conversion.
3. Resolution: Resolution of the A/D conversion.
4. Minimum voltage: Minimum voltage expressed in milivolts.
5. Return to Menu: Return to main menu
6. Show Help: General description of this VI.
7. Acquired Signal: Signal acquired in the time domain. The box 1 is enlarged in Zoom 1.
8. Zoom 1: Enlarged view of Box 1.
9. Zoom 2: Enlarged view of Box 2.
10. Zoom 3: Enlarged view of Box 3.
Practice with the Simulator
1. Try different resolutions sliding the bits control and compare the acquired signal and its respective zoom plots.
2. Compare the dynamic range values for each resolution.
CEPSTRUM
“Cepstrum analysis is the name given to a range of techniques all involving functions which can be considered as a spectrum of a logarithmic spectrum. In fact, the cepstrum was first defined as far back as 1963 as the power spectrum of the logarithmic power spectrum. It was proposed at that time as a better alternative to the autocorrelation function for the detection of echoes in seismic signals. Presumably because it was a spectrum of a spectrum, the authors coined the word cepstrum by paraphrasing spectrum and at the same time proposed a number of other terms derived in a similar manner. A list of the most common is as follows:
– Cepstrum from Spectrum
– Quefrency from Frequency
– Rahmonics from Harmonics
– Lifter from Filter
– Gamnitude from Magnitude
– Saphe from Phase”[6].
The main idea of cepstrum is to reduce to a single component a fundamental frequency and its harmonics.
To see how cepstrum works see figure 1. A signal X (figure 1-a) is converted to its frequency domain by a Fourier transform. The frequency domain plot (figure 1-b) shows the fundamental frequency of the signal X 5Hz and its odd harmonics 15Hz, 25Hz, 35Hz, and so on. The plot from figure 1-c is obtained calculating the logarithm of the Fourier transform of signal X. The result is that the difference of amplitudes between the fundamental frequency and its harmonics is reduced and the result is a periodic signal with a period of 10Hz (separation between harmonics). If this new signal is considered as a time function, and its spectrum is obtained (figure 1-d), the result will be a fundamental quefrency of 100 ms and other quefrencies because the signal from figure 1-c is not sinusoidal.
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Figure 1. Simulation of the cepstrum process.
Cepstrum is used principally to detect periodic structures in signals, like families of harmonics or sidebands with uniform spacing. It is ideally suited to the analysis of complex signals such as generated by gearboxes.
The Simulator
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1. Signal: Graph display of signal in time domain.
2. Log. of Power Spectrum: Graph display of the logarithm of the power spectrum of the signal.
3. Return to Menu: Return to main menu
4. Pause: Stop the execution in order to examine the graphs in detail.
5. Show Help: General description of this VI.
6. Power Spectrum: Graph display of the power spectrum of the signal.
7. Cepstrum: Graph display of the magnitude of the cepstrum of the signal.
8. Frequency: Slide control and digital display for the frequency of the signal.
9. Wave Type: Menu to select wave type.
Practice with the simulator
1. Try different signal frequencies and compare the plots.
Related Topics
Waves & Spectra
CLIPPING
“The truncation or flattening of the positive and/or negative portions of the signal, normally caused by overloading electronic circuits and machinery problems”[7].
“In certain situations, false or misleading signals can be present. For example, when a sinusoid is clipped (as occurs when the input to the real-time analyzer is overloaded slightly), it can cause a string of harmonics. The amplitude of the harmonics is normally quite low”[8].
The clipping effect makes that a sinusoid wave behaves like a square waveform with harmonics in the spectrum.
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Figure 1. Clipped signal in time and frequency domain.
The Simulator
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1. Signal: Graph of the signal in the time domain.
2. Clipping Level: Slide control of the clipping level.
3. Frequency: Slide control of the frequency of the signal.
4. Phase Indicator: Digital indicator of phase.
5. Cycles: Digital indicator of the total number of cycles of the signal in the time window.
6. Phase: Slide control of the phase of the signal in degrees.
7. Return to menu: Return to main menu.
8. Show Help: General description of this VI.
9. Frequency Spectrum: Graph of the amplitude spectrum of the signal.
10. Phase Spectrum: Phase spectrum of the signal in the frequency domain.
Practice with the simulator
1. Set an integer number for the signal cycles setting to zero the fine control.
2. Move the clipping level control to its upper position and observe the spectrum. Observe the fundamental frequency.
3. Gradually move down the clipping level control and observe the plots. See what happens with the fundamental frequency.
4. Observe what happens with the fundamental frequency when the clipping is lower than zero.
5. Move the fine control to obtain a fractional number of cycles and observe the plots.
Related Topics
• Aliasing
• Total Harmonic Distortion
CONSTANT PERCENTAGE BANDWIDTH FILTER
A few years ago frequency analyzers had two problems to calculate a spectrum using the Fourier transform: first, the digital processors embedded in frequency analyzers were too slow calculating the Fourier transform (FT). Second, the algorithm for the calculation of the FT was not efficient. Because of that situation analyzers used analog filters to calculate spectra.
Analyzers used two kinds of filters: constant absolute bandwidth filter and constant relative (percentage) bandwidth filter. Both filters are band-pass. The difference is that the constant bandwidth filter gives uniform resolution and separation on a linear frequency scale and the constant percentage filter gives uniform resolution on a logarithmic frequency scale.
Band pass filters have a range of frequencies that pass through the filter. For the constant bandwidth filter this range is a constant value of hertz (For example 6Hz). For the constant percentage bandwidth filter this range is a percentage of the center frequency (For example 10%).
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Figure 1. Difference between constant absolute bandwidth and constant percentage bandwidth filters.
As shown in figure 1, a constant percentage bandwidth filter keeps its resolution in all the frequencies of the spectrum at logarithmic scale. The constant bandwidth filter has very good resolution at high frequencies and not so good at low frequencies.
The process to obtain the spectrum from a time signal with band-pass filters consists of using as many filters as lines of resolution are wanted. See the next example:
Suppose that one need a 20 line resolution spectrum, containing frequencies between 0 and 1000 Hz.
Using constant bandwidth filters is very easy: Just separate the filters by the bandwidth. In this case each line of resolution will represent 50 Hz (1000Hz / 20 lines of resolution). In this case the first line of the spectrum is going to represent a range of frequencies between 0 and 50 Hz, the second line is representing frequencies between 50 and 100Hz, and so on.
Using constant percentage bandwidth filter is a little bit complex. See the next example:
– First obtain the percentage of separation between frequencies calculating the 20th root of 1000. In this case the result is 1,41 approximately.
– Signal must be passed through a filter for each value of 1,41n for n = 1 to 20. So the first filter is in 1,41 Hz, the second filter is in 1,412 = 1,98 Hz, and so on.
– Measure the signal’s amplitude obtained with each filter. Plot each amplitude value in a logarithmic plot of amplitude vs. frequency. This is the spectrum of the signal.
This process seems to be very complicated and expensive because of the huge quantity of filters needed. But one filter can be used to filter in many frequency ranges, by changing the sampling rate. For example, if a filter is designed to pass through frequencies around 100Hz and a signal is sampled at half speed, the filter is going to pass through frequencies around 50Hz.
Comparing the resolution of the first line of both spectra (0-50Hz for constant bandwidth filters and 0-1,41Hz for constant percentage bandwidth filters) is easy to see which one has better resolution in low frequencies. With high frequencies the effect results inverted. See this effect graphically in figure 2.
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Figure 2. Comparison of the spectra obtained with constant absolute bandwidth filters
and constant percentage bandwidth filters.
The FT behaves like a constant bandwidth filter, because it has uniform resolution in a linear frequency scale.
To calculate the central frequency (fc) of a (a) constant bandwidth filter and of a (b) constant percentage bandwidth filter use the following equations:
(a) [pic] (b) [pic]
where fl and fu are respectively the lower and upper frequencies of the bandwidth interval.
The Simulator
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1. Graph Constant Bandwidth: display of a constant bandwidth spectrum.
2. Return to Menu: Return to main menu
3. Run Cycle: Control button to run cycle
4. Show Help: General description of this VI
5. Constant % Bandwidth: Graph display of a constant percentage bandwidth spectrum.
6. Bandwidth: : Digital control for the percentage of bandwidth (0.5% to 23 %).
7. Frequency Bins: Digital indicator of the number of frequency bins.
Practice with the simulator
1. Set the percentage bandwidth of the filter in 3%.
2. Run an execution cycle and compare the response of both techniques in low and high frequency using the zoom tool.
3. Change the percentage bandwidth of the filter to higher and lower values between the range of 0,5 to 23,0% and compare the responses, running a few cycles.
4. Use a percentage bandwidth of 0,63% to obtain the same number of lines than the constant bandwidth spectrum (512).
Related Topics
• FFT Bandwidth
• Filter Response Time
• IRR Filter
• Low Pass Filter
• Median Filter
• Single Pole Filter Step By Step
CONVOLUTION
Convolution between two time functions f(t) and h(t) is defined as:
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It is symbolically represented as:
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“The convolution procedure can be summarized as:
1. Folding: Take the mirror image of h(() about ordinate axis ⇒ h(-().
2. Displacement: Shift h(-() by the amount of t.
3. Multiplication: Multiply the shifted function h(t-() by f(().
4. Integration: Area under the product of h(t-() and f(() is the value of the convolution at time t.”[9]
The convolution function obeys the commutative property, where f(t) * h(t) = h(t) * f(t).
To understand how convolution works see figure 1 where a function f(t) is convolved with a shifted delta function (see related topic Delta function).
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Figure 1. Convolution with a Delta function.
It can be said that the general effect of convolving a function with a delta function is to shift its origin to the origin of the delta function.
Figure 2 shows the convolution of two different functions (2-a) and (2-b). The (2-c) part shows the result of applying the four steps mentioned before at every t. Finally the (2-d) part shows the sum of all the functions in figure 2-c.
[pic]
Figure 2. Convolution of two time functions.
“One major application of this relationship is to the case where f(t) represents an input signal to a physical system and h(t) the impulse response of the system. g(t) will then be the output of the system”[10].
[pic]
Figure 3. Impulse function convolution.
Convolution Theorem
Possibly the most important and powerful tool in modern scientific analysis is the relation between convolution and Fourier transform.
The convolution theorem is defined as:
If f(t) has the Fourier transform F(f) and h(t) has the Fourier transform H(f), then f(t)*h(t) has the Fourier transform F(f)H(f).
FT [ f(t) * h(t) ] = F(f) H(f)
In simpler words a convolution in the time domain is equivalent to a multiplication in the frequency domain.
See figure 4 for a graphical representation of the Convolution Theorem. Observe that in this case the convolution is done in the frequency domain.
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Figure 4. Graphical example of the frequency convolution theorem.
The Simulator
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1. Signal X: Graph of the signal X in the time domain.
2. Signal Y: Menu ring to select the type of signal Y.
3. Width: Width of pulse in signal Y.
4. Delay: Delay of pulse in signal Y.
5. Return to Menu: Return to main menu
6. Show Help: General description of this VI.
7. Signal Y: Graph of the signal Y in the time domain.
8. Convolution X * Y: Graph of the convolved signals in the time domain.
Practice with the simulator
1. Signal X is composed by many delta functions and its being convolved with impulse function Y. Try different properties of signal Y like the signal type (square or sinc), the impulse width and the impulse delay and observe the results in the convolution plot.
Related Topics
• Convolution frequency
• Deconvolution
• Delta function
CONVOLUTION FREQUENCY
The convolution theorem says that when two functions are convolved in the time domain, their respective Fourier transforms get multiplied as shown in the following equation:
FT [ f(t) * h(t) ] = F(f) H(f)
See the Convolution topic for a detailed explanation.
This simulator is intended to analyze the behavior of the frequency domain when the convolution is done in time domain.
The Simulator
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Signal Y Frequency Domain: Graph display of signal Y in the frequency domain.
Signal Y: Control to choose signal Y. It has two choices : Square and Sinc.
Signal Y frequency: Slide control for the frequency of signal Y.
Signal X Frequency Domain: Graph display of signal X in the frequency domain.
Convolution X*Y Frequency Domain: Graph display of convolution X*Y in frequency domain.
Return to Menu: Return to main menu
Show Help: General description of this VI.
Practice with the simulator
1. Remember that the plots are showing the spectrum of signals X and Y. Signal X is a pulse function in time and frequency domains. Signal Y can be a square or a sinc function.
2. Set signal Y to square and try different width values. See how the convolution spectrum behaves.
3. Now set the signal Y to sinc and use different values of delta t, comparing the results in the convolution spectrum.
Related Topics
• Convolution
CROSSCORRELATION
“The correlation coefficient is a measure of linear association between two variables. Values of the correlation coefficient are always between -1 and +1. A correlation coefficient of +1 indicates that two variables are perfectly related in a positive linear sense, a correlation coefficient of -1 indicates that two variables are perfectly related in a negative linear sense, and a correlation coefficient of 0 indicates that there is no linear relationship between the two variables”[11].
“Correlation: A measure of similarity between two dynamic signals accomplished in the time domain”[12].
The crosscorrelation function compares two different signals calculating a correlation value for every point of one signal compared with the other. As shown in figure 1, two functions are being compared. In the match position 15 the functions are very similar to each other and the correlation plot indicates a value closer to +1. A different situation occurs in the match position 35 where the functions differ from each other. In this case the correlation value is closer to zero.
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Figure 1. Crosscorrelation between two functions.
The cross-correlation function is used extensively in pattern recognition and signal detection. It allows comparing two signals to discover a relation between them. Radar uses crosscorrelation to detect different objects. The radar has a library of previously recorded echoes from different aircrafts. When an unknown echo is detected, it uses crosscorrelation to compare it with all the patterns in the library. Crosscorrelation process determines which pattern is the one that fits better with the unknown echo, and discovers what kind of aircraft is the one that is producing the unknown signal.
If we designate the two series as Y1i and Y2i and define n* as the number of overlapped positions between the two chains, the cross-correlation for match position m is:
[pic]
Where n*: number of overlapped positions.
Y1: series 1 vector of samples
Y2: series 2 vector of samples
The Simulator
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1. Signal 1: Menu to select type of signal as Signal 1.
2. Frequency of Signal: Slide to control frequency of Signal 1.
3. Noise Amplitude: Slide to control white noise amplitude in Signal 1.
4. Signal 2: Menu to select type of signal as Signal 2.
5. Frequency of Signal: Slide to control frequency of Signal 2.
6. Noise Amplitude: Slide to control white noise amplitude in Signal 2.
7. Return to Menu: Return to main menu.
8. Pause: Stops the execution in order to examine the graphs in detail.
9. Show Help: General description of this VI.
10. Signal 1 + White Noise: Signal 1 plus white noise in time domain.
11. Signal 2 + White Noise: Signal 2 plus white noise in time domain.
12. Crosscorrelation: Crosscorrelation of Signal 1 + noise and Signal 2 + noise in time domain.
Practice with the simulator
1. Set the Frequencies of the both signals at the same value. See how the crosscorrelation plot shows a clear waveform.
2. Change the signal type (sine, square…) and see how the crosscorrelation plot keeps a clear waveform.
3. Add noise to the signals and see how the crosscorrelation eliminates it.
4. Gradually change the signal 1 frequency and see the crosscorrelation plot.
5. Try different kinds of signals with different frequencies and watch the results in the crosscorrelation plot.
Related Topics
Autocorrelation
DECONVOLUTION
Deconvolution is the inverse function of convolution. If g(t) = f(t) * h(t), g(t) deconvolved with f(t) equals h(t) and g(t) deconvolved with h(t) equals f(t).
Let’s see an example of a deconvolution application:
There is a system with a photocell that reads bar codes. The photocell is passed across a black line drawn over a white paper and it responses as shown in figure 1-a. When the photocell is passed across a bar code, the result obtained is as shown in figure 1-b. The function in figure 1-b is the result of a convolution between a square function (bar code) and the photocell response (figure 1-a). So the result of reading a bar code (figure 1-b) can be deconvoled with the photocell response function (figure 1-a) to obtain the original bar code shown in figure 1-c.
[pic]
Figure 1. Deconvolution application.
The Simulator
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1. Original Signal X: Menu ring to select the original signal.
2. Frequency: Slide control and digital indicator of the frequency of the original signal.
3. Weight Y: Menu ring to select the weight signal.
4. Width: Indicator of the witdh of the signal.
5. Width: Slide control and digital indicator of the width of the weight signal.
6. Return to Menu: Return to main menu
7. Show Help: General description of this VI.
8. Original Signal X: Graph of the original signal in time domain.
9. Original Signal Y: Graph of the weight signal in time domain.
10. Adquired Signal (Convolution X * Y): Graph of the acquired signal in time domain. This is the convolution of original and weight signals.
11. Deconvolved Signal: Graph of the deconvolved signal (acquired signal deconvolved with weight signal).
Practice with the simulator
1. Set the original signal X in Square and the Signal Y in Hanning.
2. Observe how the acquired signal is rounded in the edges because of the Signal Y function and how the Deconvolved signal is squared again.
3. Try different signals and frequencies and observe how well deconvolution works in each case.
Related Topics
• Convolution
• Delta Function
DELTA FUNCTION
The Dirac[13] delta function, denoted by ((t), is defined as having zero amplitude everywhere except at t=0, where it is infinitely large in such way that it contains unit area under its curve.
The amplitude spectrum of the delta function ( (t-(), is the same for every (. But the complex spectrum of the function rotates over the frequency axis like a spiral.
[pic]
Figure 1. Delta function in the time domain (a) and in the frequency domain (b)
The delta function can be defined by two properties:
1 ) ( ( t ) = 0, t ( 0
2 ) [pic]
This function is used to analyze more complex functions, separating them into simple functions and using the principle of superposition.
The Simulator
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1. Delay: Time delay indicator of the delta pulse.
2. Delta Pulse: Delta function in time domain.
3. Wait: Control of the velocity to change the delay of the delta function.
4. Show Help: General description of this VI.
5. Pause: Stops the execution in order to examine the graphs in detail.
6. Return to Menu: Return to main menu.
7. Amplitude Spectrum: Amplitude spectrum of delta function.
8. 3D View of the FFT 3-D: representation of the complex FFT transform of the delta function.
The dots represent the tips of the complex vectors and the red line is the loci of their tails.
9. 3D View of the FFT 3-D: representation of the complex FFT transform of the delta function.
Each line is a representation of a complex vector, and frequency increases to the right.
Practice with the simulator
1. See how no matter where the pulse is located, the amplitude spectrum remains the same.
2. Watch the 3D plots. The red one shows the vector ending points. The yellow one shows the entire vectors.
3. Pause the execution of the simulator and use the zoom tool in different parts of the plots.
Related Topics
• Convolution
• Deconvolution
EVEN AND ODD FUNCTIONS
An even function is one that seems to be reflected in the Y-axis. Mathematically speaking, an even function is every function that obey this property:
[pic]
The following are examples of even functions:
[pic]
The Fourier transform of an even function is real and even. The imaginary part of the frequency domain is equal to zero.
An odd function obeys to this property:
[pic]
The following are examples of odd functions:
[pic]
The Fourier transform of an odd function is odd and imaginary. The real part of the frequency domain is equal to zero.
Every function can be converted to a sum of an even and an odd function as shown in figure 1.
[pic]
Figure 1. Division of a function into even and odd components. Function a = b + c.
The causal function in figure 1-a is divided in two functions: the even function shown in figure 1-b and the odd function shown in figure 1-c.
The Simulator
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1. Signal: Graph of signal in time domain.
2. Show Help: General description of this VI.
3. Return to Menu: Return to main menu
4. Frequency of signal: Slide control and digital indicator of the frequency of the signal.
5. Phase of signal: Slide control and digital indicator of the phase of the signal.
6. Even or Odd: Slide indicator of even or odd signal.
7. Real part of FFT: Graph of real part of Fourier transform of the signal.
8. Imaginary part of FFT: Graph of imaginary part of Fourier transform of the signal.
Practice with the simulator
1. Set the phase of the signal in 90º and –90º to obtain even functions.
2. Move the phase to 0º and 180º to obtain odd functions.
3. Compare the plots.
Related Topics
• FFT Even Odd
• Waves & Spectra
FFT BANDWIDTH
“The term bandwidth originates from the use of bandpass filters, which have the property of passing only that part of the total power whose frequency lies within a finite range (the bandwidth). The concept can be understood from considerations of the so-called “ideal filter” whose power transmission characteristics are illustrated in figure 1. This filter transmits, at full power, all components lying within its passband of width B and attenuates completely all components at other frequencies.
[pic]
Figure 1. Ideal filter.
The concept of bandwidth can also be extended to mean the degree of frequency uncertainty associated with measurement. This applies directly to the case of the ideal filter, in the sense that the frequency of a transmitted component can only be said to lie somewhere in the bandwidth”[14]
When a signal is sampled a collection of data points is obtained. Applying the fast Fourier transform to the sample generates a new collection of data (spectrum) containing half of the points of the sample. Each data point contained in the spectrum is called Bin. Each bin represents an interval of frequencies and this interval is called bandwidth. A narrower interval indicates a better resolution spectrum. A spectrum where each bin represents one hertz (1Hz bandwidth) has better resolution than a spectrum where each bin represent 4 hertz (4Hz bandwidth).
Bandwidth: In an FFT analyzer, the real bandwidth (frequency resolution) is equal to: (frequency span / number of lines) x window equivalent noise bandwidth.
Figure 2 shows the difference between two spectra. The spectrum in figure 2-a has 512 bins of resolution and a bandwidth of 1Hz. Because the spectrum shows frequencies between 0 and 512 Hz, each bin represents 1 hertz. The spectrum in figure 2-b has 64 bins of resolution and a bandwidth of 8 Hz. This means that each bin from the spectrum represents a range of 8 hertz.
[pic]
Figure 2. FFT Bandwidth comparison.
The Simulator
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1. Signal: Graph display of the signal in time domain.
2. Milliseconds to wait: Slide control for the time between displays ( Milliseconds ).
3. Frequency Spectrum: Graph display of the amplitude spectrum of the signal.
4. Frequency Spectrum: Graph display of the amplitude spectrum of the signal with a variable bandwidth.
5. FFT Size: Menu control for the size of the FFT.
6. Bandwidth: : Digital indicator of bandwidth.
7. Return to Menu: Return to main menu
8. Pause: Stop the execution in order to examine the graphs in detail.
9. Show Help: General description of this VI.
Practice with the simulator
1. Set different FFT Size values (number of data samples that enter to the FFT function) and compare the spectra.
2. Reduce the execution speed or use the Pause button to analyze the plots.
3. Use the zoom tools to view specific frequencies in the spectra.
Related Topics
• 2D-FFT
• Constant Percentage Bandwidth Filter
• FFT Even – Odd
• FFT Linearity
• Frequency shift Time shift
• Picket Fence Effect
• Scaling Time
• Time Vs. Frequency
FFT EVEN - ODD
A signal can be Real, Imaginary or Complex (Real and Imaginary). It can also have the property of being even or odd. When the Fourier transform is applied to a signal, the properties of the transformed signal depend on the properties of the input signal.
The following table shows some properties of the Fourier transform for complex functions.
Time Domain Frequency Domain
Real Real part even, imaginary part odd.
Imaginary Real part odd, imaginary part even.
Real even, imaginary odd Real
Real odd, imaginary even Imaginary
Real and even Real and even
Real and odd Imaginary and odd
Imaginary and even Imaginary and even
Imaginary and odd Real and odd
Complex and even Complex and even
Complex and odd Complex and odd
[pic]
Figure 1. Even and odd functions with their respective real and imaginary Fourier transform.
The Simulator
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1. Frequency of Signal: Slide control and digital indicator of the frequency of the signal.
2. Real part of Signal: Menu ring to select the real part of the signal (zero, odd, even, real).
3. Imaginary part of Signal: Menu ring to select the imaginary part of the signal (zero, odd, even, imaginary).
4. Return to Menu: Return to main menu
5. Show Help: General description of this VI.
6. Real part of Signal: Real part of signal in time domain.
7. Real part of FFT: Real part of signal in frequency domain.
8. Type of Signal: Indicator of type of signal.
9. Imaginary part of Signal: Imaginary part of signal in time domain.
10. Imaginary part of FFT: Imaginary part of signal in frequency domain.
11. Type of Signal: Indicator of type of signal
Practice with the simulator
1. Use the menu rings real part of signal and imaginary part of signal and try different kind of signals and compare its spectra.
Related Topics
• 2D-FFT
• Even & odd functions
• FFT Bandwidth
• FFT Linearity
• Frequency Effect
• Picket Fence
• Scaling Time Shift
• Time Shift
• Time Vs. Frequency
• Waves & Spectra
FFT LINEARITY
Linearity is one of the Fourier transform properties. It enunciates that if two signals are added in the time domain, their respective amplitude spectrums will be added in the frequency domain.
If x(t) and y(t) have the Fourier transforms X(f) and Y(f), respectively, then the sum x(t) + y(t) has the Fourier transform X(f) + Y(f).
F[x(t) + y(t)] = F[x(t)] + F[y(t)]
Figure 1 clearly illustrates the linearity property.
[pic]
Figure 1. Fourier transform linearity property.
The Simulator
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1. Frequency of signal 1: Slide control and digital indicator of signal 1 frequency.
2. Frequency of signal 2: Slide control and digital indicator of signal 2 frequency.
3. Return to Menu: Return to main menu
4. Show Help: General description of this VI.
5. Signal 1 & Signal 2: Signal 1 and 2 in the time domain.
6. Spectra of Signal 1 & Signal 2: Signal 1 and 2 in the frequency domain.
7. Signal 1 + Signal 2: Signal 1 + Signal 2 in the time domain.
8. Spectrum of Signal 1 + Signal 2: Signal 1 + Signal 2 in the frequency domain.
Practice with the simulator
1. Set different frequency values for signals 1 and 2. Compare the amplitude spectra plots.
2. Set equal frequency values for signals 1 and 2, and watch how the amplitude spectrum of the sum of the signals is twice higher than the other spectrum.
Related Topics
• 2D-FFT
• FFT Even - Odd
• FFT Bandwidth
• Frequency Shift
• Picket fence Effect
• Scaling Time
• Time Vs. Frequency
• Time Shift
• Waves & Spectra
FILTER RESPONSE TIME
Every filter, no matter its type or characteristics, takes a time to respond when a signal is passed through it because it needs to perform calculations over the input signal in order to do the filtering.
During the response time the filter output contains errors, that gradually disappear until the expected response appears on the output.
Figure 1 shows an example of filter response time. Observe how the amplitude of the frequency starts at zero and gradually reaches a higher value until it turns stable.
[pic]
Figure 1. Filter Response Time.
The filter response time is inversely proportional to the bandwidth of the filter. For example, a band reject filter used to filter frequencies from 90 to 110Hz takes less time to response than a filter with a bandwidth from 98 to 102Hz.
The Simulator
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1. Filter type: Menu to change the type of filter.
2. Lower cutoff frequency: Slide control and digital indicator of the lower cutoff frequency.
3. Side band attenuation (dB): Indicator of the side band attenuation in dB.
4. Order: Control and digital indicator of the order of the filter.
5. Return to Menu: Return to main menu
6. Show Help: General description of this VI.
7. Signal: Sine signal 100 Hz. Time domain.
8. Filtered signal: Filtered signal. Time domain.
9. Topology: Selector of filter type.
10. Pass Band Ripple: Control for the pass band ripple.
11. Upper cutoff frequency: Slide control and digital indicator of the upper cutoff frequency.
Practice with the simulator
1. Try different filter types varying the value of the cut frequency and observe the filter response.
2. Compare the filter responses by using different filter orders.
Related Topics
• Constant Percentage Bandwidth Filter
• IRR Filter
• Low Pass Filter
• Median Filter
• Single Pole Filter Step By Step
FIR FILTER (Finite Impulse Response)
The FIR filter is also known as non recursive filter, convolution filter or moving-average filter since the exit of the filter can be expressed like one convolution:
[pic]
where x represents the entrance sequence to be filtered, Y represents the output and h represents the coefficients of filter.
In this type of filters there is no recursion: The exit depends only on the entrance and not on previous values of the exit. The answer is a weighted sum of previous and present values of the input response. Is has finite duration, so if the input stays at 0 during n periods the output also will be 0.
FIR Filters have linear phase response: the output answer has a constant delay, so these filters are always stable. The disadvantage of FIR filters is that for some particular specifications they need an filter order greater than a IIR filter.
Comparison IIR - FIR
Although IIR filters are good, they have some drawbacks like not being able to take advantage of the FFT in the implementation, because to do this a finite number of points is necessary
Some advantages of FIR filters with respect to IIR filters are:
- Linear phase response
- Implementation in a recursive and non recursive way
- Implementation using the FFT
- Non recursive FIR filters, are always stable (its response has constant delay)
One disadvantage of the FIR filter is that a high number of points N is required to come near to an ideal filter.
The Simulator
[pic]
1. Filter Type: Menu selector of type of filter.
2. Taps: Digital control for the number of taps of the filter. Increasing taps will improve the filter's ability to attenuate frequency components in its stopband and to reduce ripple in its passband.
3. Window: Menu control for the type of window applied.
Smoothing windows decrease ripple in the filter passband and improve the filter's ability to attenuate frequency components in the filter stopband.
4. Magnitude Display: Switch to select linear or logarithmic magnitude display.
5. Low frequency cutoff: Slide control for the low cutoff frequency. The low cutoff frequency fl must observe the Nyquist criterion 0( fl ( 0.5 fs when fs is the sampling frequency.
6. High frequency cutoff: Slide control for the high cutoff frequency. The simulator ignores this parameter when filter type is lowpass or highpass.
7. Return to menu: Return to main menu.
8. Show Help: General description of this simulator.
9. Magnitude: Frequency domain graph of the magnitude response of the filter.
10. Phase: Graph of the Phase of the filter in frequency domain.
Practice with the Simulator
1. Compare the different windows for the same specifications.
2. For the same type of window change the value of Taps and watch the poles.
3. For the same type of filter change the value of the cut off frequency and observe the behavior.
4. Compare the plots in linear and logarithmic scale.
Related Topics
• IRR Filter
• Median Filter
• Filter Response Time
• Constant Percentage Bandwidth Filter
• Single Pole Filter Step By Step
FREQUENCY AVERAGING
“In dynamic signal analyzer, digitally averaging several measurements improves statistical accuracy and reduces the level of random asynchronous components”[15].
Signal averaging is a technique usually used to obtain a noiseless signal. It can be applied in both domains: time and frequency. The time domain averaging process is more complex because it needs to use a trigger in order to capture the signal at the same phase value on each sample. The frequency domain average is very simple because it just averages the spectra never worrying about the phase. All the spectra averaged must have the same sampling rate and number of bins.
Figure 1 shows how the frequency averaging eliminates the unpredictable behavior of noise turning it into a constant line covering all frequencies in the spectrum.
Frequency averaging doesn’t take noise components near to zero as time domain averaging does. But time averaging remove all the frequencies not synchronized with the trigger no matter if those frequencies contain important information of the signal.
[pic]
Figure 1. Frequency averaging.
The Simulator
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Signal in Time Domain: Graph display of the signal in time domain.
Signal Frequency: Slide control for the frequency of the signal.
Noise Amplitude: Slide control for the noise amplitude.
Number Of Averages: Slide control for the number of frequency domain averages.
Amplitude Spectrum:. Graph display of the signal in frequency domain.
Averaged Amplitude Spectrum: Graph display of the averaged signal in frequency domain.
Pause: Stop the execution in order to examine the graphs in detail.
Return to Menu: Return to main menu
Show Help: General description of this VI.
10. Wave Type: Menu to select the type of signal.
11. Restart Averaging: Button to restart averaging.
Practice with the simulator
1. Set the noise amplitude at 60.
2. Try different values for the number of averages and observe the spectra.
3. Increase and reduce the noise amplitude and see how the averaged spectrum noise content varies.
Related Topics
• Time Domain Averaging two signals
• Windows Overlap
FREQUENCY MODULATION
Frequency modulation (FM) is a nonlinear modulation process. Consequently, unlike amplitude modulation, the spectrum of a FM wave is not related in a simple manner to that of the modulating wave.
The purpose of FM is to send information over a carrier signal by varying its frequency with a modulating signal.
If both signals are pure sine tones, then the spectrum produced consists of the carrier frequency fc plus pairs of sidebands equally spaced about the carrier at a frequency distance equal to the modulating frequency fm.
The strength of the sidebands depends on the MODULATION INDEX, which is the ratio of the amplitude to the frequency of the modulating signal. Note that the amplitude of the modulating signal equals the maximum frequency deviation Δf of the resulting wave. That is, the stronger the amplitude of the modulating signal, the greater the number of sidebands which contribute significantly to the spectrum.
FM has the characteristic to contain very low noise level compared to the one contained in AM. Anyone who has tuned an FM receiver has probably noticed the “quieting” of background noise of FM reception. This characteristic makes wideband FM preferable to AM for high-quality transmission.
[pic]
Figure 1. Frequency modulation example.
An example of FM is the violinist's vibrato, where the length of a string (and therefore the resulting pitch) is rapidly altered by a fast oscillating movement of the finger and wrist.
The vibration produced by some mechanical systems is sometimes frequency modulated. For example, the tooth gear wearing or the excessive play in the bearings holding it, can produce frequency modulation.
The Simulator
[pic]
1. Frequency of Carrier: Slide control of carrier's frequency.
2. Frequency of Tone: Slide control of modulating signal's frequency.
3. Amplitude Spectrum of Modulated Signal: Modulated signal in frequency domain.
4. Modulation Index: Slide control for the modulation index.
5. Return to Menu: Return to main menu
6. Show Help: General description of this VI.
7. Frequency Modulated Signal: Modulated signal in time domain.
Practice with the simulator
1. Set the frequency of modulating signal to 2,00 Hz, the modulation index to 0,5 and watch the amplitude spectrum plot. Use the zoom tools in order to appreciate it better.
2. Gradually move the frequency of modulating signal (integer part) to different values and compare the results.
3. Now use fractional values in the frequency of the modulating signal and see how the spectrum lost the symmetry.
4. Use different values of the modulating index and compare the plots obtained.
5. Move the frequency of carrier signal and see how the modulation pattern is displaced trough the spectrum.
Related Topics
• Amplitude Modulation
• Ridding and Beating
FREQUENCY SHIFT
The frequency shifting consist of moving the frequency of a signal to a different value.
To shift a frequency the time function must be multiplied by a cosine function. After this the amplitude spectrum of the signal is going to have two frequencies: one equal to the original frequency plus the cosine frequency and the other equal to the original frequency minus the cosine frequency.
For example: If a signal of 100Hz is multiplied by a cosine signal of 20Hz, the result is a signal with two frequencies: 80Hz and 120Hz.
Figure 1 shows first a signal and its spectrum. Then it shows two cases of the signal multiplied by a cosine function and its spectrum. Observe how the frequency is shifted.
[pic]
Figure 1. Frequency shifting in FFT.
This process is commonly known as modulation.
The Simulator
[pic]
1. Signal Frequency: Slide control and digital indicator and control of signal frequency.
2. Cosine Frequency (Fk): Slide control and digital indicator and control of cosine frequency.
3. Return to Menu: Return to main menu
4. Show Help: General description of this VI.
5. Signal: Signal in time domain.
6. Signal * COS (2p Fk t): Signal multiplied by COS (2*pi Fk t) in time domain.
7. Signal Spectrum: Signal in frequency domain.
8. Spectrum of Signal * COS( 2p Fk t ): Signal multiplied by COS (2*pi Fk t) in frequency domain.
Practice with the simulator
1. Set the signal frequency control to 100Hz and the cosine frequency to 0Hz. Observe that the signal is not affected.
2. Gradually move the cosine frequency and watch the spectra.
3. If the cosine frequency is higher than the signal frequency the left component in the spectrum is going to be reflected because of the aliasing effect.
Related Topics
• 2D-FFT
• FFT Bandwidth
• FFT Even - Odd
• FFT Linearity
• Picket Fence Effect
• Scaling Time
• Time Shift
• Time Vs. Frequency
• Waves & Spectra
IIR FILTERS (Infinite Impulse Response)
This type of filter is recursive, this means that the exit depends not only on the present input but in addition depends on previous values of the output (filter with feedback).
The general equation that characterizes IIR filters, is:
[pic]
Where Nb is the number of forward coefficients ( bj ) and Na is the number of reverse coefficients ( ak ).
The impulse response of a IIR filter has an infinite duration.
IIR Filters produce a general distortion of phase. This means that the phase is not linear with the frequency. The order of a IIR filter is smaller than the one of a filter FIR for a same application.
Order of a digital filter
The order of a digital filter (or number of poles), can be defined as the number of the previous inputs of information used to calculate the present exit.
Attenuation
The attenuation is the lost in dB in which a signal incurs when it goes through a filter.
Butterworth filters
This type of filter is also known as flat and it is monotonic in the Pass-Band and the Stop-Band.
Figure 1 shows the answer of an ideal low pass filter (continuous line) and the answers of three types of Butterworth filters (dashed lines). As the attenuations become more pronounced, we can see they come nearer to the ideal filter.
[pic]
Figure 1. Butterworth filters with different attenuation values.
When the order (N) of the filter increases the filter tends to come near to an ideal filter, the response in the PassBand tends to be flatter. This can be observed in figure 2.
[pic]
Figure 2. Dependence of Butterworth magnitude characteristic on the to order N
Chebyshev Filter Type I
The Chebyshev filter presents better characteristics than the Butterworth in pass-band. In the Stop-Band, Butterworth and Chebyshev have good behavior at high frequencies, but their characteristics disappear when decreasing the frequency. The transition of passband to stopband is faster than for the butterworth filter.
Ripple are the oscillations in amplitude in the response of a filter and it is measured in dB. Figure 3 shows a signal with Equiripple in passband. The filters Chebychev and Elliptic, have Equiripple characteristic, which means that the differences between the tips and valleys of the response of amplitude in passband are equal. The Butterworth and Bessel filters do not present ripple.
The Chebyshev filter type I has equiriple characteristics in the Pass-Band and it has monotonic decay in the Stop-Band (figure 3).
[pic]
Figure 3 Chebyshev Type I Filter
Chebyshev Filter Type II
The Chebyshev filter type II presents characteristics equiriple in the StopBand and monotonic in the PassBand (figure 4). In the Chebyshev filter Type II, stopband does not approximate to zero as quickly as the Chebyshev filter type I, but the absence of the undulation in passband is an important advantage.
[pic]
Figure 4. Chebyshev Type II Filter
Elliptical Filter
The elliptical approximation presents a better behavior in the Stop-Band, since the elliptical filters reduce to the minimum the width of the transition of passband to stopband. The elliptical approximation presents equiriple in the Pass-Band and in the Stop-Band. See figure 5.
[pic]
Figure 5. Elliptic Filter.
Bessel Filter
The Bessel filter, also called the Thomson filter, presents maximally flat time delay in passband but its selectivity is poor. The filter shows an excellent performance in the dominion of time, this allows that the filtered signals maintain their waveform in the passband frequencies. The Bessel filters require a higher order than other filters generally to obtain a good attenuation in stopband. See figure 6.
[pic]
Figure 6. Bessel Filter
The Simulator
[pic]
1. Filter Design: Selector of filter design.
2. Filter Type: Selector of filter type.
3. Ripple dB: Digital control for the ripple in the pass band. It is not applicable to all the filter designs.
4. Attenuation dB Filter: Digital control for the attenuation in decibels.
5. Order: Digital control of the order of the filter. Increasing the Order will improve the filter's ability to attenuate frequency components in its stopband and to reduce ripple in its passband.
6. Magnitude Display: Selector of magnitude display: Linear or Log.
7. Lower Cut-Off Frequency:. Control and digital indicator of the lower cut-off frequency.
8. Upper Cut-Off Frequency:. Control and digital indicator of the higher cut-off frequency.
9. Return to Menu: Return to main menu
10. Show Help: General description of this VI.
11. Magnitude Frequency domain: magnitude response of the filter
12. Phase: Phase of the filter response in the frequency domain.
Practice with the Simulator
1. Compare for the same specifications different designs of filters.
2. For a same design of filter change the order and observe the behavior.
3. Try different filter attenuations and compare the responses.
4. Observe the behavior in linear and logarithmic scale.
Related Topics
1. Constant Percentage Bandwidth Filter
2. Filter Response Time
3. Low Pass Filter
4. Median Filter
5. Single Pole Filter Step By Step
LEAKAGE
“An error introduced into the FFT process caused by using finite-length time blocks that to not match at the ends. Its effects are a smearing of the frequency lines at lower amplitudes. This error is minimized by the use of window functions”[16].
The fast Fourier transform (FFT) is calculated over a finite collection of points representing a signal and it is assumed that this sample of points its constant and continuous over time. When a signal sample contains a non-integer number of cycles and the sample is joined with a copy of it, the last point of the first copy do not coincides with the first point of the second copy and a distortion in the amplitude spectrum occurs because of the discontinuities at the boundaries. This effect is known as spectral leakage.
The amount of leakage depends on the amplitude of the discontinuity, a larger one causing more leakage.
Figure 1 shows how FFT assume the signal when a sample is taken. The sampling at the left does not have any leakage because the signal sample begins and ends in the same part of the wave. But the sample at the right begins and ends in different parts of the cycle and the FFT assumes a signal with discontinuities.
[pic]
Figure 1. Comparison of two signal samples: the first one has no leakage but the second one has.
Leakage effect can be avoided by the use of Windows.
The Simulator
[pic]
1. Frequency: Slide control of the frequency of the signal.
2. Phase: Phase spectrum of the sampled signal.
3. Cycles: Digital indicator of the total number of cycles in the time window.
4. Phase: Slide control of the phase of the signal.
5. Return to Menu: Return to main menu.
6. Show Help: General description of this VI.
7. Signal: Signal in time domain.
8. Spectrum: Amplitude spectrum of the sampled signal.
9. Signal Sampled: Graph showing the jump or discontinuity between two time windows.
Practice with the simulator
1. Put the coarse slide in 10 and the fine slide in 0 in order to obtain a 10 Hz (cycles) signal. See the peak at 10 Hz in the spectrum.
2. Gradually move the fine slide, and see what happens with the peak in the spectrum. See the jump in the signal sampled graph.
Related Topics
• Picket Fence Effect
• Total Harmonic Distortion
• Waves & spectra
• Windows
LIN – LOG FREQUENCY SCALES
There are two ways to obtain a spectrum from a signal: using the Fourier transform (FT) and using filters. The spectrum obtained with FT has a constant bandwidth. The spectrum obtained with filters can have constant or constant percentage bandwidth, depending on the filter’s bandwidth.
In a spectrum with constant bandwidth each bin has the same bandwidth. In a spectrum with constant percentage bandwidth each bin has a bandwidth that is a percentage of the center frequency of the filter.
The simulator shows a spectrum from 0 to 10.000Hz in a linear scale and in a logarithmic scale. The spectrum in linear scale is going to have 400 lines of resolution (bins) where each bin bandwidth is 25Hz (10.000Hz / 400 bins). The spectrum in logarithmic scale is going to have the same number of bins. The bandwidth of the first bin is equal to:
[pic]Hz
In general the bandwidth of the bin i is given by the following equation:
[pic]
The logarithmic scale has better resolution displaying the lower frequencies, than the linear scale. The linear scale has better resolution displaying the higher frequencies, than the logarithmic scale.
The logarithmic frequency scale is very used in predictive maintenance because in one spectrum it can be displayed a wide range of frequencies.
The Simulator
[pic]
1. Linear frequency scale - Constant bandwidth = 25Hz: Linear frequency scale. Constant bandwidth = 25Hz
2. Logarithmic frequency scale - Constant percentage bandwidth = 2,32%: Logarithmic frequency scale. Constant percentage bandwidth = 2,32%
3. Show Help: General description of this VI.
4. Return to Menu: Return to main menu.
5. Pause: Stop the execution in order to examine the graphs in detail.
Practice with the simulator
1. Compare the linear scale plot and the logarithmic scale plot in the lower and higher frequencies using the zoom tools.
Related Topics
• Constant Percentage Bandwidth Filter
• Waves & Spectra
LOW PASS FILTER
The filters are circuits that allow the passage of a certain frequency band while they attenuate all the signals that are not included within this band
A filter can attenuate or suppress the components of each frequency by any amount. The filters are specified in terms of the components of the wave that allows to pass. The most common types of filters are: low - pass, high - pass, pass band and band reject.
An ideal low pass filter (figure 1) passes all frequencies below Fc and stops all frequencies above Fc.
[pic]
Figure 1. Ideal Low Pass Filter.
The frequency at Fc is denominated cut frequency.
The filters Low-pass (figure 2), let pass all the frequencies below an specific frequency (Fc).The continuous line corresponds to the graph of an ideal low pass filer; the dashed line correspond to a real low pass filter.
[pic]
Figure 2 Low Pass Filter
Figure 2b shows the frequency response graph for the ideal and for a real high pass filter.
[pic]
Figure 3. High Pass Filter
Figure 3 shows the frequency response of a band-pass filter. The filter pass all frequencies between FL and FU and stops all the frequencies outside this interval.
[pic]
Figure 4. Band Pass Filter
The notch filter is the inverse of the band-pass filter: it stops the frequencies between FL and FU and pass all the frequencies outside this interval. See figure 5.
[pic]
Figure 5. Notch Filter
The Simulator
[pic]
1. Signal Frequency: Control and digital indicator of the integer number of cycles of the sine signal in the window.
2. Noise Amplitude: Control and digital indicator of the white noise amplitude.
3. Low Pass Filter Frequency: Control and digital indicator of the corner frequency of the low pass filter.
4. Show Help: General description of this VI.
5. Return to Menu: Return to main menu.
6. Pause: Stop the execution in order to examine the graphs in detail.
7. Signal+Noise: Graph of the white noise + sine signal in time domain.
8. Filtered Signal: Graph of the filtered signal in time domain.
9. Noise: Graph of the white noise in time domain.
10. Spectrum Signal+Noise: Graph of the amplitude spectrum of sine signal + noise.
11. Filtered Spectrum: Graph of the amplitude spectrum of filtered signal.
Practice with the Simulator
1. Place the frequency of the signal in 5.
2. Place the amplitude of the noise in the low value and observe how it affects the
original signal.
3. Place the amplitude of the noise in the high value and observe how it affects the original signal.
4. Move the frequency of the filter to the values of 0, 5, 10, 15, 20 and observe the signal filtered in the time domain and its corresponding spectrum.
Related Topics
• Constant Percentage Bandwidth Filter
• Filter Response Time
• IRR Filter
• Median Filter
• Single Pole Filter Step By Step
MEDIAN FILTER
“If Y represents the output sequence Filtered Data, and if Ji represents a subset of the input sequence X centered about the ith element of x,
Ji = { xi-r, xi-r+1, ... , xi-1,xi,xi+1, ... , xi+r-1, xi+r} (where Ji contains 2r+1 elements)
and if the indexed elements outside the range of X equal zero, the median filter obtains the elements of Y using
yi = Median (Ji), for i = 0 to n-1.
where n is the number of elements in the input sequence X, and r is the filter rank”[17].
In the simulator one can identify the original signal (yellow signal), and the filtered signal (red signal). The original signal can be modified and it is possible to establish values for the amplitude, the noise, the delay and the width of the pulse.
The filter rank must be less than the width of the pulse, otherwise the filtered signal misses the pulse. Both situations are shown in figures 1 and 2. In figure 1, the filter rank is smaller than the pulse width. Figure 2 shows the opposite situation.
[pic]
Figure 1. Median Filter with rank equal to 5.
[pic]
Figure 2. Median Filter with rank equal to 15.
The Simulator
[pic]
1. Amplitude: Control and digital indicator of the pulse amplitude.
2. Width: Control and digital indicator of the width of the pulse.
3. Delay: Control and digital indicator of the delay of the pulse.
4. Noise Level: Control and digital indicator of the amplitude of the noise.
5. Filter Rank: Filter Rank controls the "width" of the median filter.
Increasing Filter Rank will reduce the number of noise spikes in the filtered signal.
Decreasing Filter Rank will improve the filter's ability to follow rapid changes in the input signal.
6. Time Domain: Graph display of the noisy pulse and the median filtered pulse.
7. Amplitude: Indicator of the amplitude of the filtered pulse.
8. Width: Indicator of the width of the filtered pulse.
9. Delay: Indicator of the delay of the filtered pulse.
10. Show Help: General description of this VI.
11. Pause: Stop the execution in order to examine the graphs in detail.
12. Return to Menu: Return to main menu.
13. Frequency Spectrum: Graph of the frequency spectrum of the pulse and the filtered pulse.
Practice with the Simulator
1. Place the control of amplitude of the signal in 10.
2. Modify the width of the pulse to 15.
3. Place the level of noise in values 0, 1, 2 and observe how it affects the original signal (yellow signal).
4. Move the control of Filter Rank to the values of 1, 5, 10, 15, 20 and observe the change in the filtered signal (red signal).
5. For each one of the previous steps you can observe the estimated parameters of the signal.
Related Topics
• Constant Percentage Bandwidth Filter
• Filter Response Time
• IRR Filter
• Low Pass Filter
• Single Pole Filter Step By Step
ORBITS
“Lissajous (Orbit): The pattern created when signals from X and Y transducers are combined. Conventionally used with X-Y shaft displacement probes to display dynamic motion of the shaft centerline”[18].
[pic]
Figure 1. Orbits analysis of an unbalanced disc.
“The orbit is a greatly magnified view of a shaft’s dynamic motion within its bearing. It is the best way to assess dynamic behavior such as eccentricity, stability, and the effects of preload, misalignment, and unbalance on machines with fluid film, hydrodynamic bearings”[19].
[pic]
Figure 2. Different type of orbits and its interpretation.
The Simulator
[pic]
1. Vibration Frequency: Slide control and digital indicator and control of the frequency of the vibration.
2. Vibration frequency / RPM: Digital indicator of the ratio: Vibration frequency / RPM
3. Noise Magnitude: Slide control of noise magnitude.
4. Return to Menu: Return to main menu
5. Oscilloscope: Oscilloscope display of the orbits.
6. Show Help: General description of this VI.
Practice with the simulator
1. Set VF/RPM at 1 and observe the circle in the plot. This happens because an object is vibrating at the same frequency than its rotation speed.
2. Gradually increase and decrease the VF/RPM value watching the plot.
3. Use integer values for VF/RPM to see completed orbits.
ORDER TRACKING
There are signals that contain frequencies varying their value in time. For example, when a rotating engine is powered off, its speed is going to decrease to zero rpm. If the rotational frequency is measured, its is gradually going to fall to zero. A spectrum of this situation will show a range of peaks from the rotational speed to zero. Figure 1-a shows a signal with a varying frequency and its spectrum.
A technique to have a better appreciation of the phenomena is to obtain spectra periodically and represent all the spectra in a waterfall or cascade plot like the one shown in figure 2.
[pic]
Figure 1. Constant VS variable signal sampling
[pic]
Figure 2. 3-Dimensional spectral map or Waterfall plot.
Order tracking is a tool that can be used when it is intended to analyze a signal at different speeds and it is desired to avoid that frequencies directly related to the speed, move across the waterfall plot.
Order tracking works adjusting the sampling rate keeping it proportional to the rotational speed. So if the sampling rate is 10 times the rotational speed, when the rotational speed is 50 Hz the sampling rate will be 500Hz; when the rotational speed is 6Hz the sampling rate will be 60Hz. Figure 1 shows the difference of sampling a signal (figure 1-b) with a constant rate (figure 1-a) and sampling a signal with a variable rate proportional to the signal frequency (figure 1-c).
With order tracking, frequencies related to the speed will remain still in the waterfall plot. Frequencies not related to the speed will move across the waterfall spectra.
Order tracking most important applications are:
- Identify frequencies affected by the speed of a system.
- Identify mechanical resonances in structures.
[pic]
Figure 3. Order analysis obtained during the rundown of a large turbo-generator.
The Simulator
[pic]
Time Domain Signal: Graph display of the signal in time domain. As time increases, amplitude and frequency decrease.
Spectrum: Spectrum of the signal sampled at regular intervals in time.
Tracking Order Spectrum: Spectrum of the signal using tracking.
Sampling Period: Slide to control the sampling period for the spectrum.
Sampling Interval: Slide to control the number of samples per cycle. Include a digital display and control.
Return to Menu: Return to main menu
Show Help: General description of this VI.
Practice with the simulator
1. Move the sampling period control to see the range of frequencies obtained in the spectrum (where order tracking is not being used).
2. Try different values for the samples/cycle control, to see how the order tracking spectrum behaves.
Related Topics
• Resonance
• Transmissibility
• Waves & Spectra
PARSEVAL’S THEOREM
If x(t) and X(() are a Fourier transform pair, then
[pic]
This expression, referred to as Parseval’s relation, follows from the direct application of the Fourier transform. The quantity on the left-hand side of the equation is the total energy in the signal x(t). Parseval’s relation says that this total energy may be determined either by computing the energy per unit of time (|x(t)|2) and integrating over all time, or by computing the energy per unit frequency (|X(()|2/2() and integrating over all frequencies.
This equation is expressed considering a continuous Fourier transform. For discrete functions, the relationship between power as computed in the time domain and as computed in the frequency domain is given by
[pic]
A simpler way to describe the Parseval’s theorem is saying that the signal energy is the same in time and frequency domains.
The Simulator
[pic]
1. Sum[x(n) x(n)]: Digital indicator of the Sum represented above. (Time domain).
2. 1/N Sum[|X(k)|^2]: Digital indicator of the Sum represented above. (Frequency domain).
3. Signal in Time Domain: Signal in time domain.
4. Return to Menu: Return to main menu
5. Pause: Stops the execution in order to examine the graphs in detail.
6. Show Help: General description of this VI.
7. Signal in Frequency Domain: Signal in frequency domain.
8. Input Signal: Menu for the input signal.
9. Frequency: Slide control for the frequency of the signal.
Practice with the simulator
1. See the energy value for different signals in time and frequency domain. Use the Pause button in order to compare the energy values.
Related Topics
• Waves & Spectra
PHASE IN TIME AND FREQUENCY
The amplitude spectrum of a signal is not affected by changes in the signal’s phase. The phase can be modified and the amplitude spectrum will remain the same. This happens because the amplitude spectrum is the resultant of the sum of the vectors contained in the Fourier transform real and imaginary parts. The real and imaginary parts of the Fourier transform change with the phase, but the resultant vector of both parts remains constant.
One case in which a phase change can affect an amplitude spectrum is the following:
If there are two signals with the same frequency and the sum of them is being acquired and if both signals have the same phase, their amplitudes will be positively added. But if one signal phase is 180º degrees away from the other, their amplitudes will be added negatively obtaining an amplitude of 0. In this specific case, different amplitude spectra will be obtained from different phase values.
The Simulator
[pic]
1. Sine 1 Frequency: Slide control for Sine 1 frequency.
2. Sine 2 Frequency: Slide control for Sine 2 frequency.
3. Sine 1 Phase: Slide control for Sine 1 phase.
4. Return to Menu: Return to main menu
5. Show Help: General description of this VI.
6. Signal = Sine 1 + Sine 2: Signal in time domain.
7. Spectrum of Signal: Signal in frequency domain.
Practice with the simulator
1. Set the sine 1 frequency in 2Hz and the sine 2 frequency in 4 Hz.
2. Move the sine 1 phase to different values and observe the plots.
3. Now make sine 1 frequency equal to sine 2 frequency.
4. Set the sine 1 phase to zero. See how the amplitude spectrum gets higher.
5. Now move the sine 1 phase to different values and watch the plots.
6. See what happens when the sine 1 phase is 180º.
Related Topics
• Waves & Spectra
PICKET FENCE EFFECT
In general, unless a frequency component coincides exactly with an analysis line, there will be an error in both the indicated amplitude and frequency.
Suppose a spectrum displaying frequencies from 0 to 400Hz with a resolution of 1 hertz. Frequencies with non-integer values are not going to be represented by one exactly bin of the spectrum; these frequencies are going to be represented by the two integers. For example, a frequency of 80,3Hz is going to be represented as a peak in 80Hz and a peak in 81Hz. The amplitude of each peak (80 and 81Hz) depends on the proximity with the fractional frequency (80,3Hz). In the example, the 80Hz peak is going to have higher amplitude than the 81Hz peak because it is closer to the 80,3Hz frequency.
[pic]
Figure 1. Amplitude and frequency compensation for Picket Fence Effect with Hanning weighting.
The upper part of figure 1 shows 3 situations: In part (a) no picket fence effect is happening. In part (b) a frequency is located exactly in the middle of two bins so both bins have the same amplitude ((dB=0). In part (c) a frequency is located between two bins, but it is closer to one of them. The difference between the bins ((dB) is not zero.
As shown in figure 1-b, the amplitude difference between two consecutive bins allows to calculate the value in hertz and amplitude of a frequency laying between them. The graph in figure 1-d is a curve designed to obtain the value of frequency and amplitude of a frequency component.
[pic]
Figure 2. Illustration of Picket Fence Effect.
The Simulator
[pic]
Window: Menu control to select the type of window applied to the signal.
Frequency: Slide control for the frequency of the signal from 100.0 to 101.0 Hz.
Phase: Slide control for the phase of the signal.
Frequency Hz: Digital indicator of the frequency of the signal.
L100 / L101: Digital indicator of the magnitude of frequency line 100 divided by the magnitude of frequency line 101.
Line Magnitude: Digital display for the magnitudes of the signal in frequency lines from 96 to 104 Hz.
Amplitude spectrum: Graph display of the amplitude spectrum of the signal.
Amplitude spectrum around 100 Hz: Graph display of the amplitude spectrum of the signal around 100 Hz.
Return to Menu: Return to main menu
Show Help: General description of this VI.
Practice with the simulator
1. Set the window control in None (uniform), try different frequency values and observe the plots and the numerical indicators.
2. Compare the 100 and 101Hz peak’s amplitude when the frequency is near to 100Hz, near 101Hz and when the frequency is 100,5Hz
3. Try different windows and frequencies comparing the plots and the numerical indictors.
4. Set the frequency at 100Hz and observe the noise introduced by each window because of its equivalent noise bandwidth value.
5. Move the phase control and observe how it affects the plots.
Related Topics
• 2D-FFT
• FFT Bandwidth
• FFT Even - Odd
• FFT Linearity
• Frequency shift
• Leakage
• Scaling Time
• Time shift
• Time Vs. Frequency
RESONANCE
“Natural frequencies excited by such forces as mass unbalance and its orders amplify vibration. This mechanism is called resonance when it occurs on a structure. The degree of amplification depends on the magnitudes of the force and damping as well as the proximity of the forcing frequency to the natural frequency. Either the forcing frequency (shaft speed) or a natural frequency (depend upon design) must be changed to solve the problem”[20].
All physical objects “ring” at certain frequencies when they are tapped with a hammer. These unique tones produced depend on the material stiffness, its shape, and its mass. These unique tones are called natural frequencies (Wn). When something produces a wave with a frequency near to the object’s natural frequency, the object gets excited and vibrates. If the object has no damping the resonance can destroy it.
[pic]
Figure 1. Amplitude and phase response at Resonance.
To understand the resonance effect, some concepts must be clarified:
Damping (C): Effect of converting energy into heat. It is measured in mass / time units.
Critical damping coefficient (Cc): The minimum viscous damping that will allow a displaced system to return to its original position without oscillation.
Quality factor (Q): A measure of the system’s damping. A large Q indicates a small value of damping. It is obtained by the equation Q = Cc / 2C.
Damping ratio ((): Ratio between damping and critical damping (C/Cc).
Some damping ratios of common materials are:
Material ( (C/Cc)
Steel 0,001
Concrete 0,01
Lead 0,02
Natural rubber 0,05
Butyl rubber 0,05 – 0,50
In electrical circuits, the resistor performs the same function as a damper does in mechanical systems.
At resonance the phase changes 180º degrees and the slope of the change is inversely proportional to the damping.
The Simulator
[pic]
1. C/Cc Slide control for the ratio C/Cc.
2. C/Cc ratio Digital indicator of the ratio C/Cc selected.
3. Q = Digital indicator of the quality factor or amplification factor.
4. Return to Menu Return to main menu
5. Show Help General description of this VI.
6. Maximum Value Indicator of the maximum value
7. Amplitude of forced vibration Graph of the amplitude ratio versus frequency ratio. The red square shows the maximum value.
8. Phase Graph of the phase angle between force and displacement versus frequency ratio.
Practice with the simulator
1. Try different values of the damping ratio (C/Cc) and observe the plots.
Related Topics
• Transmissibility
RIDING & BEATING
“Beat frequency: A variation in amplitude equal to the sum or difference of two closely spaced frequencies”[21].
Beats are two separate oscillating sources that interfere with each other. The interference causes an addition and subtraction of the two waveforms and produces a new waveform that has a frequency, which is the difference between the two source frequencies.
When a signal contains two frequencies very close to each other an effect called beating is produced. This interference is called beating because it generates beats at a frequency equal to the difference of the close frequencies. In figure 1-a, the beating effect can be observed.
[pic]
Figure 1. Beating (a) and riding (b)
In the opposite situation when the two frequencies are distant to each other, an effect called riding is produced. The name of riding is because the bigger frequency seems to be “riding” the lower frequency. See figure 1-b.
Beating can be easily observed on twin-engine airplanes. When the two propellers are not turning at the same speed, a vibration (and sound) that rise and fall is produced. Beating is very common on industries when machines with motors that rotate at similar velocities work closely. A small difference between the rotation speed of the motors produces the beating.
The Simulator
[pic]
1. Frequency Signal 1: Slide control of the integer number of cycles in the time window of the signal 1.
2. Frequency Signal 2: Slide control of the integer number of cycles in the time window of the signal 2.
3. Amplitude of Signal 1: Slide control of the amplitude of signal 1.
4. Amplitude of Signal 2: Slide control of the amplitude of signal 2.
5. Return to Menu: Return to main menu.
6. Show Help: General description of this VI.
7. Signal 1 + Signal 2: Plot of the time domain sum of signal 1 plus signal 2.
8. Amplitude Spectrum of Signal 1 + Signal 2: Frequency domain plot of signal 1 plus signal 2.
Practice with the simulator
1. Set the Frequency Signal 1 slide control value close to the Frequency Signal 2 value. Use high frequency values and equal amplitude values in both signals in order to appreciate better the beating effect. Recommended values: Freq. 1 = 38 Hz, Freq. 2 = 35 Hz and Amp. 1 & 2 = 8.
2. Move the Frequency Signal 1 slide control to 3 Hz and observe the riding effect.
3. Try different combinations of frequencies and amplitudes and see how the plots are affected.
Related Topics
• Amplitude Modulation
• Frequency Modulation
RMS – PEAK -CREST
The peak of a waveform is the absolute value from zero to the maximum value the waveform reaches. The peak to peak value is the absolute value from the maximum negative excursion to the maximum positive excursion and it is called amplitude.
The root mean square value (RMS) is used to measure the energy contained in a waveform. For sine waveforms it corresponds to 0.707 times the peak. The energy contained in a waveform is equivalent to the energy contained in a continuous DC signal with a value equal to RMS. For example, when an AC current is applied to a resistor, it will dissipate the same heat (energy) as if a DC current with a voltage equal to the AC RMS’s value is applied.
[pic]
Figure 1. Parameters of a waveform.
The coefficient of kurtosis is a measurement of the peakedness of a signal. The coefficient of Kurtosis (α4)is obtained with the following equation:
[pic]
where m2 and m4 are the second and fourth moments about the mean of the samples.
The equations to obtain the moments m2 and m4 are:
[pic] [pic]
where n is the number of samples and [pic] is the mean of the samples.
The crest factor is also a measurement of the peakedness of the signal and it is obtained dividing the peak value by the rms value.
[pic]
The Simulator
[pic]
1. Signal: Menu to select the type of signal.
2. Signal graph: Time domain graph of the selected signal.
3. Show Help: General description of this VI.
4. RMS value: RMS value of the signal in the graph.
5. Peak value: Peak value of the signal in the graph.
6. Crest Factor: Crest factor value of the signal in the graph.
7. Kurtosis: Kurtosis value of the signal in the graph.
8. Return to Menu: Return to main menu
9. Pause: Stops the execution in order to examine the graphs in detail.
Practice with the simulator
1. Set the Signal control to Noise signal. Pause the execution and analyze the parameters of the signal.
2. Change the signal control and observe the parameters of different signals.
SIGNAL DIFFERENTIATION
Let x(t) be a signal with Fourier transform X((). Then, by differentiating both sides of the Fourier transform equation we obtain:
[pic]
[pic]
From the equation we can conclude that the differentiation in the time domain can be replaced by a multiplication by j( in the frequency domain. This multiplication of the Fourier transform X(() by the factor j( implies that differentiation of x(t) with respect to time enhances the high frequency components of the signal x(t).
The Simulator
[pic]
1. Signal: Graph display of signal in time domain
2. Differentiated Signal: Graph display of the differentiated signal in time domain.
3. Amplitude Spectrum of the Signal: Graph display of the amplitude spectrum of the signal.
4. Amplitude Spectrum of the Differentiated Signal: Graph display of the amplitude spectrum of the differentiated signal.
5. Show Help: General description of this VI.
6. Return to Menu: Return to main menu.
7. Pause: Stops the execution in order to observe the graph in detail.
Practice with the simulator
1. Observe different signals and its derivate in time and frequency domains.
2. Pause the execution and compare in detail the signals using the zoom tools. Compare the low frequencies and the high frequencies in both spectra.
Related Topics
• Signal Integration
• Time Vs Frequency
• Waves & Spectra
SIGNAL INTEGRATION
Since differentiation in time corresponds to multiplication by j( in the frequency domain, one might conclude that integration should involve division by j( in the frequency domain. This is indeed the case, but it is only one part of the picture. The precise relationship is:
[pic]
The impulse term on the right-hand side of the equation reflects the DC or average value that can result from the integration.
Division of the Fourier transform X(() by the factor j(, implies that integration of x(t) with respect to time suppresses the high-frequency components of x(t). As expected, this effect is the opposite of that produced by differentiation of g(t).
In vibration analysis when an acceleration signal is integrated the resultant signal is the velocity of the vibration and if the signal is integrated again, the signal obtained is displacement. Acceleration emphasizes high frequencies and displacement emphasizes low frequencies. See figure 1.
[pic]
Figure 1. The integration and double integration of acceleration
to obtain velocity and displacement respectively.
The Simulator
[pic]
1. Signal: Graph display of the signal.
2. Amplitude Spectrum of the Signal: Graph display of the amplitude spectrum of the signal.
3. Show Help: General description of this VI.
4. Return to Menu: Return to main menu
5. Pause: Stops the execution in order to examine the graphs in detail.
6. Integrated Signal: Graph display of the signal integrated in time domain.
7. Amplitude Spectrum of the Integrated Signal: Graph display of the amplitude spectrum of the integrated signal.
Practice with the simulator
1. Observe different signals and its integrate in time and frequency domains.
2. Pause the execution and compare in detail the signals using the zoom tools. Compare the low frequencies and the high frequencies in both spectra.
Related Topics
• Signal Differentiation
• Time Vs Frequency
• Waves & Spectra
SINGLE POLE FILTER
This simulator shows the basic structure for a low pass filter of a single pole. The structure is made up of a delay unit (delay Z-1), multipliers, adders and the constants A and B. In general, each delay unit of Z-1 in the structure of the filter defines a pole in the filter. This simulator shows how the filter works, step by step.
For each sample period, the new data sample entering to the filter is multiplied by the constant A and added to B times the previous output value which has been stored in the delay unit Z-1.
[pic]
Figure 1. Diagram for a single-pole filter
Observe that the sum of the constants A and B is always equal to one, with the purpose of not changing the amplification.
In the simulator the input signal is a sinusoidal signal whose frequency can be modified. The constant A is established by the assigned control, once established the constant A, the constant B takes a value that is equal to 1-A.
It is possible to follow the calculations by setting on the step by step switch.
The Simulator
[pic]
1. Input: Graph indicator of the Input signal in time domain.
2. Frequency: Digital indicator of the frequency of the signal.
3. Frequency: Slide control of the frequency of the input signal.
4. Show Help: General description of this VI.
5. Return to Menu: Return to main menu
6. Pause: Stops the execution in order to examine the graphs in detail.
7. A: Digital indicator for constant A.
8. A: Slide control for constant A.
9. B: Digital indicator for constant B. B equals one minus A.
10. Output: Graph of the output signal in time domain.
11. Peak to Peak value: Digital indicator of peak to peak value of the filtered signal.
12. Sample: Digital indicator of the number of the sample of the input signal processed.
13. Input: Digital indicator of input value.
14. 1 Value: Digital indicator of value at point 1.
15. 2 Value: Digital indicator of value at point 2.
16. 3 Value: Digital indicator of value at point 3.
17. Output: Digital indicator of output value.
18. Time between steps: Slide control to slow down the process in order to observe it step by step.
19. Step by step: Switch to turn on/off the step by step behavior of the simulator.
20. Next: Shows the next state (step) of the calculation.
21. Signal In Frequency: Graph of the signal spectrum before filtering.
22. Signal Out Frequency: Graph of the signal spectrum after filtering.
Practice with the Simulator
1. Place the frequency of the signal of entrance in 10.
2. Modify the value of the constant A and place it in 0.1, observe how the value of constant A change the value of the constant B.
3. Move the value of the constant A to the values 0.001, 0.010, 0.1 and 1; observe the change in the output.
4. Place the value of frequency of the signal of entrance 20, 30, 40 and 50;observe the change in the output.
5. Set the step by step switch at on and using the next button analyze the values of the filter at each step.
Related Topics
• Low Pass Filter
• IRR Filter
• Median Filter
• Filter Response Time
• Constant Percentage Bandwidth Filter
SQUARE & SINC FUNCTIONS
Consider the signal
[pic]
which is depicted in figure 1-a. This function is named as the sinc function. Its Fourier transform is shown in figure 1-b.
[pic]
Figure 1. Sinc and square functions.
Another way to express the sinc function is:
[pic]
Now consider the rectangular pulse signal
[pic]
as shown in Figure 1-b. The Fourier transform of this signal is shown in Figure 1-a.
Comparing both cases an interesting relationship can be seen. The sinc function is the Fourier transform of the rectangular pulse and vice versa. This is a consequence of the duality property for Fourier transforms.
Both functions (sinc and square) are very important for the Fourier analysis when the discrete Fourier transform needs to be applied to a finite interval of a signal. Doing this is like multiplying the signal by a uniform window (see Windows simulator). This uniform window is a square function and has its own spectrum in the frequency domain: a sinc function, so when the uniform window multiplies the signal, the signal spectrum is convolved with the window spectrum as shown in figure 2. Part (a) of figure 2 plots the signal and its spectrum. Part (b) is the uniform window and its spectrum. Part (c) shows the result of multiplying the signal by the uniform window, in other words, using a portion of the original signal.
[pic]
Figure 2. Result of the truncation of a signal.
The previous explanation shows the importance of the square function, because every time the Fourier transform is applied to a signal interval, it is equivalent of multiplying the signal by a square function. This multiplication in time domain is equivalent to a convolution in frequency domain. (see Convolution simulator)
If the width of the square function is infinite the spectrum will be a thin line in 0 Hz and if the width is short the spectrum will be like the one shown in figure 1.
The Simulator
[pic]
1. Sinc - square: Button to select Sinc or Square function.
2. Width: Control and digital indicator of the width of the pulse.
3. Return to Menu: Return to main menu
4. Show Help: General description of this VI.
5. Time Domain: Signal in the time domain.
6. Amplitude Spectrum: Signal in the frequency domain.
Practice with the simulator
1. Use the Function switch to plot a sinc or a square function and see the Amplitude spectrum plot of each function.
2. Modify the width of the functions and observe the plots.
Related Topics
• Waves & Spectra
• Windows
STROBOSCOPE
“A good example of aliasing is using a strobe light to stop high-speed motion. The strobe flashing rate is the measuring speed. When the flash rate equals the rotational speed, you can stop the motion. This is analogous to a constant value. By varying the flash rate slightly, you can make the motion move forward or backward in slow motion. This is the phantom image of low-frequency data due to high-speed motion when the measuring speed is too slow”[22].
The stroboscope is a lamp that can be adjusted to flash at a desired frequency. It is very useful to measure the rotational speed of an object adjusting the frequency of the strobe-light to the value in which the rotating object seems to be still.
It also allows to analyze the behavior of a high-speed cyclic process in slow motion permitting an exhaustive analysis of the movements involved.
You can see an industrial application of the strobe light on a machine that prints color stickers over a ribbon where each primary-color ink must be adjusted individually. Using a strobe light over the ribbon flashing at the same speed which stickers flow, makes that the ribbon looks still, allowing the machine operator to adjust the colors while the machine is producing the stickers.
The Simulator
[pic]
1. Stroboscope: Stroboscope simulator.
2. Show Help: General description of this VI.
3. Return to Menu: Return to main menu.
4. Bar Rotation: Slide control for the frequency of the bar rotation.
5. CPS Strobe / RPS Bar: Ratio CPM of the Strobe / RPM of the Bar
6. RPS Bar / CPS Strobe: Ratio RPM of the Bar / CPM of the Strobe.
7. Strobe Frequency: Flashing frequency of the strobe
Practice with the simulator
1. Set the strobe frequency in 100 and press the Run Cycle button. The bar will remain still because the bar is rotating at 100 RPM.
2. Try different strobe frequencies like 99, 101, 50, 200 and 20. Compare the behavior of the bar in the plot.
3. Use the milliseconds / frame control to adjust the velocity of the plot depending on your processor speed and video configuration. The ideal value is the smallest one that lets you see the entire bar rotating.
4. Can you explain why at a strobe frequency of 200 there are two images of the bar?
Related Topics
• Aliasing
TIME DOMAIN AVERAGING (TWO SIGNALS)
Time domain averaging is a technique to eliminate undesirable frequencies from a signal, in order to appreciate better a special frequency.
When a signal sample is taken, every frequency contained in the sample has a different phase. If several samples are taken from the same signal, it is expected that every sample contain the same group of frequencies, but each one with a different phase value in every sample. Averaging several samples of the same signal will produce that the amplitudes of the different frequencies trend to zero, because of the sum of the frequencies at different phases. An extreme situation is when a frequency with phase p, is averaged with the same frequency with a phase value equal to p + (. The result is that the signal completely disappears. See the next example:
[pic]
The averaging process will not affect a frequency that keeps its phase value in every sample of the signal, so this frequency (and its harmonics) will remain clear and strong, while the rest of frequencies in the signal trend to disappear. In figure 1-a the first 3 left plots are the same, but each one has a different phase. The fourth plot is the average of the 3 upper signals. This average is a signal with the same frequency, but with a lower amplitude. In the right case (b) the 3 samples have the same phase and the average plot shows that the process affected neither the amplitude nor the frequency.
[pic]
Figure 1. Average of a frequency in different phases (a) and in the same phase (b).
Using time domain averaging can be easily analyzed a frequency that is “lost” in a rich spectrum of frequencies.
In order to use this technique a trigger must indicate the acquiring system when to take a sample of the signal to keep constant the phase value of the elected frequency.
The Simulator
[pic]
1. Signal Frequency: Slide control and digital indicator and control of the frequency of the signal (red).
2. Number of averages: Slide control and digital indicator and control of the number of averages.
3. Delay in Seconds: Slide control and digital indicator and control of the time delay between plot actualization.
4. Show Help: General description of this VI.
5. Return to Menu: Return to main menu
6. Run cycle: Press this button to run a cycle.
7. Signal 1 & Signal 2: Graphs of the signal we are interested in (red) and a disturbing signal (yellow).
8. Average (Signal 1 + Signal 2): Time domain graph of the two signals sum after average.
9. Averaged Signals: Graphs of the time domain averaged signals. The averages are synchronized with the red signal.
10. Spectrum of averaged (Signal 1 + Signal 2): Frequency domain graph of the two signals sum after average.
11. Spectrum Signal 1 & Signal 2: Frequency spectrum of signal 1 and signal 2.
Practice with the simulator
1. Set the Signal Frequency to 10 Hz, the Number of averages to 10 and the Delay in seconds to 2 seconds.
2. Press the Run Cycle button and watch the plots. See how the disturbing signal (yellow) disappears from the plots while the averaging process run.
3. The disturbing signal’s frequency is 25,6 Hz. Use a close Signal Frequency (27Hz) in order to see the beating effect and how it disappears because of the averaging.
Related Topics
• Frequency Averaging
• Ridding and beating
• Time Domain Averaging Noise
• Windows Overlap
TIME DOMAIN AVERAGING NOISE
“Synchronous time averaging is accomplished by averaging a series of phase-referenced time domain signals. The result is usually transformed into the frequency domain upon completion of the averaging process. Synchronous components are greatly enhanced; non-synchronous components and noise tend to zero.”[23]
The time domain average process helps to remove from a signal the noise that is appearing randomly. The noise is eliminated by the averaging process because of its randomly nature.
Figure 1 shows a signal with too much noise. It appears like there may be a sine wave imbedded in there, but it is not easy to discern it.
[pic]
Figure 1. Signal with noise.
If the signal is averaged in the time domain without a trigger, then the result is as shown in figure 2.
[pic]
Figure 2. Signal with noise averaged without using a trigger.
But when a trigger is used, the result is completely different. A very definite sine wave now emerges after the averaging process, as shown in figure 3.
[pic]
Figure 3. Signal with noise averaged using a trigger.
In general, noise is the range of unwanted frequencies contained in a signal. Signals entering the measuring system that do not represent the variable being measured.
The noise elimination allows a better accuracy in the analysis.
The Simulator
[pic]
1. Signal & Noise: Graph of the signal (Red) and the noise (Yellow) in the time domain.
2. Signal Frequency: Slide control and digital indicator and control of the frequency of the signal.
3. Number of averages: Slide control and digital indicator and control of the number of averages.
4. Delay in Seconds: Slide control and digital indicator and control of the delay in seconds between averages. Slow down the process to see how the averaged signal is changing.
5. Signal + Noise: Graph of the signal and noise added in time domain.
6. Averaged Signal: Graph of the averaged signal + noise in time domain.
7. Spectrum Signal + Noise: Amplitude spectrum of signal + noise.
8. Spectrum of Averaged Signal: Amplitude spectrum of signal + noise averaged.
9. Return to Menu: Return to main menu
10. Run cycle: Press this button to run a cycle.
11. Show Help: General description of this VI.
Practice with the simulator
1. Set the Number of averages to 1.
2. Press the Run Cycle button.
3. Compare the Signal + Noise plot and its Spectrum with the Averaged Signal and its Spectrum.
4. Gradually increment the Number of averages and repeat the steps 2 and 3.
5. Try with different frequencies.
Related Topics
• Frequency Averaging
• Time Domain Averaging Two Signals
• Windows Overlap
TIME SCALING FFT
[pic]
Figure 1. Time scaling property.
“The time scaling Fourier transform property is well-known in many fields of scientific endeavor. As shown in figure 1 time scale expansion corresponds to frequency scale compression. Note that as the time scale expands, the frequency scale not only contracts but the amplitude increases vertically in such a way as to keep the area constant. This is a well-known concept in radar and antenna theory”[24].
The Simulator
[pic]
Original Signal: Graph indicator of the Signal in time domain
Zoom of Original Signal: Graph indicator of the central portion of the signal in time domain
Impulse Width: Slide control and digital indicator and control of the width of the sine wave cycle in time domain
Time Scale: Digital indicator of the relative time scale.
Spectrum of Signal Frequency Domain: Graph indicator of the spectrum of the signal
Zoom Spectrum of Signal Frequency Domain: Graph indicator of the first part of the frequency spectrum of the signal.
Return to Menu: Return to main menu
Show Help: General description of this VI.
Practice with the simulator
1. Gradually increase the impulse width from a lower value 10 to the higher value 512 and observe the spectra. Use the zoom tools to analyze the plots in detail.
Related Topics
• 2D-FFT
• FFT Bandwidth
• FFT Even – Odd
• FFT Linearity
• Frequency Shift
• Picket Fence Effect
• Time Shift
• Time Vs. Frequency
• Waves & Spectra
TIME SHIFTING FFT
Time shifting is a property of the Fourier transform. It occurs when the phase of a signal is modified. The phase modification alters the magnitude of the vectors on the real and imaginary spectra, but it doesn’t alter the amplitude spectrum.
Figure 1 shows a signal with different phase values and its corresponding real and imaginary spectra.
[pic]
Figure 1. Time shifting in FFT.
For specific phase values, the real and imaginary spectra display specific situations:
Phase Real spectrum Imaginary spectrum
0o & 180o Zero Positive and negative
90o Positive Zero
270o Negative Zero
The Simulator
[pic]
1. Signal: Signal in time domain.
2. Frequency of Signal: Slide control and digital indicator and control of signal's frequency.
3. Phase of Signal: Slide control and digital indicator and control of the phase of the signal (time shift in time domain).
4. Show Help: General description of this VI.
5. Return to Menu: Return to main menu
6. Frequency Spectrum: Graph of the frequency spectrum of the signal.
7. Wave Type: Menu selector for the type of signal.
Practice with the simulator
1. Move the Phase of the signal control to different values and observe the plots. Try phase values like 0º, 45º, 90º, -90º, 180º.
Related Topics
• 2D-FFT
• FFT Bandwidth
• FFT Even - Odd
• FFT Linearity
• Frequency Shift
• Picket Fence Effect
• Scaling Time
• Time Vs. Frequency
• Waves & Spectra
TIME VS FREQUENCY
Choosing an appropriated sampling rate is a crucial matter to measure a signal. Figure 1 shows a signal containing two frequencies (2Hz and 34Hz) sampled at two rates (10Hz and 100Hz) but containing the same number of samples: 10. In figure 1-a sampling is done at low speed (10Hz) during 1000 milliseconds. The low speed sampling allows having a good definition of the 2Hz component but misses the 34Hz frequency. In Figure 1-b the high sampling rate (100Hz during 100 milliseconds) allows to capture the 34Hz frequency but there is not enough data to represent the 2Hz frequency with precision.
[pic]
Figure 1. Resolution in time and frequency domain.
If the spectra of both signals are obtained, each spectrum is going to have 5 bins because only 10 samples where obtained. The spectra of the low rate sampling will show frequencies from 0 to 5Hz and each bin is going to represent 1Hz equal to divide the maximum frequency 5Hz by 5 bins. The 2Hz frequency is going to be precisely represented by bin 2.
The spectra of the high rate sampling will show frequencies from 0 to 50Hz and each bin is going to represent a range of 10Hz equal to divide the maximum frequency 50Hz by 5 bins. Bin 1 represents frequencies between 0 and 10Hz, bin 2 represents frequencies between 10 and 20Hz, and so on. The 2Hz frequency is not going to be precisely represented because its is going to be included in the range (1-10Hz) of bin 1. From that range it is not possible to know which frequencies are included there. The 34Hz frequency is going to be included in bin 4 (30 to 40Hz). Even that this frequency is not going to be precisely represented, it is going to be represented contrary to the low sampling rate spectrum, which doesn’t show it at all.
If a signal is sampled at high frequency the time plot has a high resolution and the spectrum has a low resolution and conversely if a signal is sampled at low frequency the time plot has a low resolution and the spectrum has a high resolution.
In this example both sampling methods have failures. A better solution is to sample 100 points at a high speed (100Hz). By doing this, both frequencies are going to be precisely represented in the spectrum
The Simulator
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1. Show Help: General description of this VI.
2. Return to Menu: Return to main menu
3. Pause: Stops the execution in order to examine the graphs in detail.
4. Signal: Time domain graph of signal sampled at 10,000 point/sec.
5. Amplitude Spectrum of the Signal: Amplitude spectrum of the signal sampled at 10,000 point/sec.
6. Signal: Time domain graph of signal sampled at 1,000 point/sec.
7. Amplitude spectrum of the signal: Amplitude spectrum of the signal sampled at 1,000 point/sec.
Practice with the simulator
1. Pause the execution of the simulator to appreciate better the plots.
2. Use the zoom tools to compare the plots in time and frequency domain.
3. Which sampling rate has better resolution in the time domain? Which in the frequency domain?
Related Topics
• 2D-FFT
• FFT Bandwidth
• FFT Even – Odd
• FFT Linearity
• Frequency shift
• Picket Fence Effect
• Scaling Time
• Time shift
• Waves & Spectra
TOTAL HARMONIC DISTORTION
“Harmonic: A frequency component that is an integer (whole number) multiple of a fundamental (reference) frequency”[25].
To determine the amount of nonlinear distortion that a system introduces, it is necessary to measure the amplitudes of the harmonics that were introduced by the system relative to the amplitude of the fundamental frequency. Harmonic distortion is a relative measure of the amplitudes of the harmonics as compared to the amplitude of the fundamental frequency. If the amplitude of the fundamental frequency is A1, and the amplitudes of the harmonics are A2 (second harmonic), A3 (third harmonic), ...An (Nth harmonic), then the total harmonic distortion (THD) is given by the equation:
[pic]
The percentage total harmonic distortion (% THD) is: %THD = 100 THD
Harmonics can be produced in at least two ways: First, by an event that repeats several times during each revolution. Second, by a distortion of a pure sine wave, like the clipping phenomena.
[pic]
Figure 1. Waveform with white noise, clipping and leakage (time and frequency domains).
Figure 1 shows a signal that contains white noise, clipping and leakage. The clipping produces the harmonics, the white noise increases the harmonics amplitude and the leakage spreads part of the fundamental frequency energy across the spectrum increasing the noise and harmonics amplitude.
The Simulator
[pic]
1. Signal: Graph display of the of the signal in time domain.
2. Clipping Level: Slide control of the clipping level.
3. Noise Amplitude: Slide control of the noise amplitude.
4. Signal Frequency: Slide control for the frequency of the signal, includes a digital display and control.
5. Window: Menu control to select the type of window applied to the signal.
6. Spectrum of The Signal: Graph display of the power spectrum of the signal.
7. % THD + Noise: Digital indicator of the percent total harmonic distortion plus noise present in the signal
8. % THD: Digital indicator of the percent total Harmonic distortion present in the signal.
9. Harmonic Frequencies: Digital indicator of Harmonic frequencies.
10. Harmonic Amplitudes: Digital indicator of Harmonic amplitudes.
11. Return to Menu: Return to main menu
12. Pause: Stop the execution in order to examine the graphs in detail.
13. Show Help: General description of this VI.
Practice with the simulator
1. Set the Fundamental frequency to an integer value, the noise amplitude to zero and take the clipping level to its upper position. Observe the plots and the %THD and %THD + noise values.
2. Try different situations adding noise and clipping and using integer and fractional fundamental frequencies. Compare the plots and the %THD and %THD + noise values.
Related Topics
• Clipping
• Leakage
TRANSFER FUNCTION
Transfer function or frequency response function represent the ratio of output-input in the frequency domain of a system.
The majority of systems receives an input signal and produces an output signal as response. The function that turns the input signal into the output signal is called transfer function. Figure 1 shows an input signal a(t), an output signal b(t) and the transfer function h(t) and its respective frequency domain A(f), B(f) and H(f).
[pic]
Figure 1. Graphical representation of the Transfer function.
For these functions it can be said that:
[pic]
[pic]
where * denotes convolution and · denotes multiplication.
These functions became very useful when the function h(t) is unknown, because it can be obtained deconvolving b(t) and a(t), or getting H(f) dividing B(f) by A(f) and calculating its inverse Fourier transform.
The Simulator
[pic]
Input Signal: Graph display of the input function in time
1. Input Delay: Slide control to select the delay of the impulse when a delta function (impulse) is selected as the input.
Input Signal Frequency: Slide control and digital control and indicator to select the maximum frequency of the chirp.
2. Menu input: Menu Control to select the input function.
3. Transfer Function Frequency domain: Graph display of the transfer function in frequency domain
4. Return to menu: Return to main menu.
5. Pause: Stop the execution in order to examine the graphs in detail.
6. Show Help: General description of this VI.
7. Output Signal: Graph display of the output function in time domain
8. Menu output: Menu Control to select the output function.
9. Output Signal Width: Slide control to select the width of the pulse in the output function when a square pulse is selected as output.
Output Cycles: Slide control and digital control and indicator to change the sine wave frequency.
10. Spectrum of Output Signal: Graph display of the spectrum of the output function in the frequency domain.
11. Spectrum of Input Signal: Graph display of the spectrum of the input function in the frequency domain.
Practice with the simulator
1. The simulator displays the functions: a(t), b(t), H(f) and B(f). Try different combinations of input – output functions and observe the behavior of the transfer function.
2. Change the parameters of the functions that can be configured
Related Topics
• Deconvolution
TRANSIENTS
Transients are non-stationary finite duration signals that begin and end in zero and they are analyzed as a whole.
Figure 1 shows different transients. The transient shown in figure 1-d is the most common transient. It represents the response of a simple structure when it receives a hit, usually with a hammer. The spectrum of this response shows the natural frequencies of the structure.
[pic]
Figure 1. Transients: (a) rectangular pulse, (b) half cosine pulse, (c) tone-burst and (d) response of a simple structure to a hammer blow.
Some considerations must be taken for transient signal analysis:
- Because transients begin and end in zero only the uniform window must be used.
- The entire transient must be sampled. That means that the sampling process must begin before the transient signal starts and end after the transient goes back to zero.
- The quantity of zeroes that the transient has at the begin and end affect the calculation of the energy in the transient.
The Simulator
[pic]
1. Transient Signal: Graph display for the transient in time domain.
2. Frequency: Slide control for the frequency of the transient.
3. Window: Menu to select the type of window.
4. Transient X Window: Graph display for the transient multiplied by a window.
5. Amplitude Spectrum: Amplitude spectrum of the transient.
6. Return to Menu: Return to main menu.
7. Show Help: General description of this VI.
8. Wave Type: Menu to select the type of wave.
Practice with the simulator
1. Select None from the window control and try different transient frequencies.
2. Try different windows and observe how windows affect the transient reducing the energy and amplitude in the spectrum.
Related Topics
• Waves & Spectra
• Windows
TRANSMISSIBILITY
“Transmissibility is the non-dimensional ratio of the response amplitude of a system in steady-state forced vibration to the excitation amplitude. The ratio may be one of forces, displacements, velocities, or accelerations”[26].
[pic]
Figure 1. Force transmissibility example.
Figure 1 shows an example of transmissibility of a vibration force produced by the rotor. The vibration force F is transmitted trough the springs to the floor with a force F1.
When the ratio between the vibration force frequency (() and the natural frequency of the system ((n) is closer to 1, resonance occurs. (n can be obtained from the equation:
[pic]
where k is the spring stiffness and m is the mass of the system.
If the example shown in figure 1 is at resonance, doing at least one of these changes can eliminate it:
1. Change the velocity of the rotor in order to change the vibration frequency (().
2. Change the springs using new springs with a different stiffness (k).
3. Change the mass (m) of the system adding or removing weights.
When the ratio (/(n is equal to [pic], the transmissibility equals 1 no matter the value of the damping ratio. If (/(n [pic], and the damping ratio is increased, transmissibility is increased too reaching a higher value of 1. If the damping ratio is reduced, transmissibility can reach a lower value of zero at a (/(n very large.
When the changes explained before are applied to a system in order to avoid resonance, it is desirable that the solution have a ratio of frequencies (/(n higher than [pic] where transmissibility can be leaded to zero.
The transmissibility equation is:
[pic]
where ( is the damping ratio and (r is the frequencies ratio ((/(n).
The Simulator
[pic]
1. C/Cc: Slide control and digital indicator and control of the ratio C/Cc.
2. Show Help: General description of this VI.
3. Return to Menu: Return to main menu
4. Transmissibility: Force transmissibility versus frequency ratio diagram. The red circle shows a fixed point in the graph.
5. Maximum value: Indicator of the maximum value in the graph.
Practice with the simulator
1. Try different values of the damping ratio (C/Cc) and observe how the transmissibility changes.
2. Can you tell why the orange spot doesn’t move when the damping ratio changes?
Related Topics
• Resonance
WAVES & SPECTRA
Every wave can be represented in two basic forms: time domain and frequency domain. Sometimes is simpler to understand a signal in the time domain and sometimes the frequency domain provides a simpler view.
The amplitude spectrum is the scalar representation of the Fourier transform plot. When the Fourier Transform is applied to a signal, a group of vectors is obtained. Each vector has a real and an imaginary component. The spectrum takes each one of those vectors and obtains its magnitude value. The other part of the spectrum is the phase that corresponds to the angle of every vector. Every magnitude in the amplitude spectrum had a corresponding phase, see figure 1. The phase plot is not used as much as the amplitude spectrum, but it is necessary to have both (amplitude and phase) in order to represent a signal back in the time domain.
[pic]
Figure 1. Vector representation.
The amplitude spectrum plot represents frequency Vs amplitude. There is an amplitude value for every frequency involved in the signal. An amplitude value of zero means that the frequency is not present in the signal. For a simple sine wave signal, the spectrum plots a peak at the frequency of the wave. See figure 2.
[pic]
[pic]
Figure 2. Different signals in time domain (a) and frequency domain (b).
The Fourier transform is defined by the expression:
[pic]
“The amplitude versus frequency or spectrum plot is the graphic display most widely used in machinery analysis”[27].
The Simulator
[pic]
1. Wave Type: Menu control to select the type of signal.
2. Frequency: Slide control for the frequency of he signal.
3. Phase: Phase of the spectrum. The vertical axis units are degrees.
4. Cycles: Indicator of the total number of cycles in the window. Integer plus fractional
5. Phase: Slide control of the phase of the signal in the window.
6. Return to Menu: Return to main menu.
7. Show Help: General description of this VI.
8. Signal: Signal in time domain.
9. Frequency Spectrum: Amplitude spectrum of the signal times window function.
Practice with the simulator
1. Set the Fine and the Phase slide to 0 in order to try different signals and appreciate a clean Spectrum.
2. See for example how a Square signal is made of many sine waveforms.
3. Move the Frequency slide to a non integer number of cycles in order to see the leakage effect in the spectrum and phase plots.
4. Move the Phase slide and see how to Phase plot is affected.
5. Try different signals, frequencies and phases values and compare the plots.
Related Topics
• 2D-FFT
• Aliasing
• Amplitude Vs Power Spectra
• FFT Bandwidth
• FFT Even - Odd
• FFT Linearity
• Frequency shift
• Leakage
• Picket Fence Effect
• Scaling Time
• Square & Sinc Functions
• Time shift
• Time Vs. Frequency
WINDOWS FOR FREQUENCY ANALYSIS
“A function designed to concentrate a sample time record in the center and reduce it at the ends to eliminate the effects of non-periodicity. Window functions represent a compromise between frequency resolution and amplitude accuracy”[28].
Windows solve de Leakage problem. The windows designed to eliminate the discontinuities, begin and end with zero or small values and increase toward the center of the sample. Windows are used multiplying them with the signal sample. See figure 1.
[pic]
Figure 1. Signal multiplied by a window function.
The most used window functions are Hanning, Hamming, Blackman-Harris, Exact Blackman, Flat Top, 4 Term B-Harris and 7 Term B-Harris. Windows are generally named as the person(s) who developed the functions that produce them.
When a window is used, frequencies and amplitudes are modified. A window designed to improve the signal frequency resolution deteriorates the amplitude accuracy and vice versa. There is only one window that does not modify the amplitude or the frequency at all: the Uniform window (also called rectangular window). It is produced by the function y = 1 for the interval and y=0 else where.
Every window has a constant associated, named the equivalent noise bandwidth. This value indicates the precision or resolution of frequencies in the spectrum. A lower value indicates more resolution.
Windows are commonly used to solve the leakage problem, but two-channel analyzers and impact testing structures use other windows like exponential and force decay.
The Simulator
[pic]
1. Frequency: Slide control for the frequency of the signal.
2. Coherent Gain: Digital indicator of the coherent gain of the selected window. It is the inverse of the scaling factor that was applied to the window.
3. Cycles: Digital indicator of the total number of cycles of the signal in the time window.
4. Phase: Slide control of the phase of the signal in degrees.
5. Window: Menu ring to select the window.
6. Equivalent Noise Bandwidth: Digital indicator of the equivalent noise bandwidth of the selected window.
7. Return to Menu: Return to main menu
8. Show Help: General description of this VI.
9. Signal: Graph of the signal in time domain.
10. Spectrum: Amplitude spectrum of the signal times the window function.
11. Signal X Window: Graph of the signal multiplied by the window function selected.
12. Phase: Phase spectrum of the signal times the window function.
13. Wave Type: Menu to select the type of signal.
Practice with the simulator
1. Put the coarse slide in 10 and the fine slide in 0.4 in order to obtain a 10,4 Hz (cycles) signal.
2. Change the menu window and compare the plots by using different windows.
3. Compare the constants of different windows.
4. Make a zoom in the spectrum near the 10,4 Hz peak. Compare the peak width by using no window, a Hanning window and a flat top window. Observe which window produces more resolution (less width) in the peak.
5. Move the sampling frequency slider to a High value. Observe what happens with the plots.
Related Topics
• Leakage
• Transients
• Windows
• Windows Amplitude
• Windows Comparison
• Windows Overlap
• Windows Resolution
• Windows: Noise floor
WINDOWS AMPLITUDE
One consideration to analyze before using windows is to compare the signal’s amplitude obtained with them. When a signal contains leakage, the amplitude gets reduced. Using windows recovers the amplitude value of peaks in the spectrum, but every type of window obtains a different amplitude value.
Figure 1 compares different peak’s weight and amplitude:
Peak Window used Contains Leakage Integer Cycles Amplitude Value
A None (uniform) No Yes ~7.0
B None (uniform) Yes No ~5.0
C Hamming Yes No ~6.0
D Flat Top Yes No ~7.0
[pic]
Figure 1. Windows amplitude comparison.
The Flat top window gives better amplitude in the peaks than the Hamming window, but the Flat Top has less resolution (see topic Windows Resolution).
The Simulator
[pic]
1. Signal: Signal in time domain.
2. Coarse: Slide control of the frequency of the signal (Integer number of cycles in the time window).
3. Fine: Slide control of the frequency of the signal (Fractional number of cycles in the time window).
4. Cycles / Window: Digital indicator of the total number of cycles per window.
5. Return to Menu: Return to main menu
6. Show Help: General description of this VI.
7. Window: Menu ring to select the window function.
8. Amplitude: Digital indicator of the amplitude in the spectrum.
9. Error: Digital indicator of the percentage error in the amplitude.
10. Signal X Window: Signal multiplied by the window functions selected in the time domain.
11. Spectrum Amplitude: spectrum of the signal multiplied by the window.
12. Phase: Graph indicator of the phase spectrum.
Practice with the simulator
1. Set a frequency with a fine value of 0 to avoid leakage and set the None window. Observe the amplitude value of 7,0711 with a 0% error.
2. Increase the fine control and see what happens with the amplitude value.
3. Use different windows and compare the amplitude and percentage of error values.
Related Topics
• Leakage
• Picket fence effect
• Windows
• Windows & Resolution
• Windows Comparison
• Windows Overlap
• Windows: Noise floor
WINDOWS COMPARISON
Windows are functions that play an important role in signal processing. The intention of this simulator is to compare different type of windows.
To learn about windows check the related topics listed at the end of this explanation.
The Simulator
[pic]
1. Time Domain: Graph display of sine wave1 + sine wave2 in time domain.
2. Sine 1 Frequency: Slide to control the frequency of sine wave 1.
3. Sine 2 Amplitude: Digital control of the amplitude of sine wave 1.
4. Sine 2 Frequency: Slide control of the amplitude of sine wave 2.
5. Sine 2 Amplitude: Digital control of the amplitude of sine wave 2.
6. Spectra of signal: Graph display of the spectra of the signal using two different windows.
7. Window 1: Menu to select the window A (plotted in yellow).
8. Window 2: Menu to select the window B (plotted in green).
9. Return to Menu: Return to main menu.
10. Show Help: General description of this VI.
Practice with the simulator
3. Select different characteristics of amplitude and frequency for the two signals and select different pairs of windows comparing their response in the frequency domain.
4. Compare window’s amplitude and resolution.
5. Test signals with and without leakage.
6. Use closer frequency values for signals 1 and 2, and zoom the spectrum to analyze the resolution.
Related Topics
• Leakage
• Picket fence effect
• Windows & Amplitude
• Windows & Resolution
• Windows Overlap
• Windows: Noise floor
WINDOWS OVERLAPPING
“Overlap: In block data acquisition used in FFT calculations, overlap defines the proportion of samples from the previous block that are included in the next block. With an overlap of 50%, one-half of the samples in a block will be obtained from the previous block. Overlap sampling is useful because sometimes the processing time required to collect a block of data is much longer than the time required to calculate an FFT. The improvement in statistical accuracy gained by averaging is determined by the amount of new data, incorporated in each block”[29].
The measurement of several data blocks from a signal allows one to obtain a collection of spectra that can be compared or averaged.
The sampling of 1024 point (samples) from a signal at a rate of 256 Hz takes 4 seconds. Obtaining 10 of those data blocks takes 40 seconds and 10240 samples are obtained. See figure 1-a.
If this process must be repeated many times (for example to measure one hundred motors in a plant), it is desirable to reduce the sampling time. A technique to reduce the sampling time is to overlap the samples. The overlap process is shown in figure 1-b.
[pic]
Figure 1. Comparison of sampling (a) without overlapping and (b) with 50% overlapping.
Observe in figure 1-b how the second data block takes half of the samples of the first data block. In this case the percentage of overlap is 50% and only 5632 samples are needed instead of 10240 samples, reducing the sampling time from 40 to 22 seconds.
The percentage of overlap can be adjusted depending on the needs of time reduction, but it is not recommendable to use a very high percentage of overlap because as the percentage of overlap is increased, the data blocks become more and more equal. For the case of 100% overlap all data blocks are identical.
Figure 2-a shows the result of applying a window function to each data block when overlapping is not being used. Observe that the samples where a window ends and the next window starts are being multiplied by zero or by very small values, so all the information contained in those points is wasted.
Figure 2-b shows the behavior of windows when the data blocks are being 50% overlapped. Windows are overlapped too and only the samples in the beginning of the first window and the points at the end of the last window are being missed. The rest of the points have the same importance in the analysis. This situation is desirable because with a higher or a lower percentage of overlap, a ripple appears in the upper part of the overlapping indicating that the samples have different degrees of importance for the analysis.
[pic]
Figure 2. Windows with 50% overlap.
Figure 3 shows an overlap greater than 50%. Observe the ripple mentioned before.
[pic]
Figure 3. Windows with more than 50% overlap.
The Simulator
[pic]
1. Overall Weighting Functions for Overlapping Windows: Graph of the overall weighting function.
2. Overlapped points: Slide indicator and digital indicator of the number of overlapped points.
3. Window: Menu to select the type of window.
4. % Overlap: Slide control and digital control and indicator for the percentage of overlap.
5. Return to Menu: Return to Main Menu
6. Pause: Stop the execution in order to examine the graphs in detail
7. Show Help: General description of this VI
Practice with the simulator
1. Select None window and set the percentage of overlap at 0.
2. Observe that the windows are not overlapped and the result is a straight line.
3. Change the window to Hanning and observe the same situation.
4. Gradually increase the percentage of overlap and see the plot. See what happens with the amplitude of the yellow line when the percentage of overlap is over 50%.
5. Set the window back in None and observe its behavior.
6. Try different windows and compare the results.
Related Topics
• Frequency Averaging
• Time Domain Averaging two signals
• Windows
• Windows & Amplitude
• Windows & Resolution
• Windows Comparison
• Windows: Noise floor
WINDOWS: NOISE FLOOR
Windows is a very useful tool for signal processing. However, there are a few considerations to analyze before using them. One of these considerations is related with the noise content in the signal. When the signal is passed trough a window function, the noise contained in the signal gets amplified.
Figure 1 compares the noise in two spectra. A Uniform window was applied to the first one and a Flat-top window was applied to the second one. Observe that the second plot has a higher value for a peak different than the fundamental frequency and also the rest of peaks reach higher values than the first plot.
Noise peaks get higher amplitude values but also get bigger bandwidth values. Both parameters increase the RMS value of noise
[pic]
Figure 1. Noise amplification produced by windows.
The noise output ratio of two windows is proportional to the square root of the ratio of the two window equivalent noise bandwidths.
Every window produces a different RMS value for noise, so this parameter must be considered when windows are used in signal processing systems.
The Simulator
[pic]
1. Original Signal: Signal in Time Domain. The signal is sum of a sinusoid and white noise.
2. Equivalent noise Bandwidth: Digital indicator of the equivalent noise bandwidth of the selected window.
3. Show Help: General description of this VI.
4. Return to Menu: Return to main menu
5. Coherent Gain: Coherent gain is the inverse of the scaling factor that was applied to the window.
6. RMS value of noise in the spectrum: Digital indicator of the RMS value of noise in the spectrum.
7. Coarse: Slide control for the number of cycles of the sinusoid in the window in integer values.
8. Fine: Slide control for the number of cycles of the sinusoid in the window in fractions of a cycle.
9. Cycles / Window: This is the total number of cycles per window (coarse plus fine).
10. Window Function: Menu control to select the window function.
11. Signal X Window: This is the signal (green) multiplied by the window function.
12. Spectrum: Amplitude spectrum of signal times window function. Only the first 150 points of the spectrum are shown in the graph. Notice the log scale in the vertical axis.
13. Phase: Graph of the phase spectrum.
Practice with the simulator
1. Try different window functions comparing the RMS value of noise in the spectrum.
Related Topics
• Leakage
• Picket fence effect
• Windows
• Windows & Amplitude
• Windows & Resolution
• Windows Comparison
• Windows Overlap
WINDOWS RESOLUTION
“The ability of an instrument to separate two closely spaced signals is called resolution. This ability to separate to signals is a direct function of the bandwidth of the band-pass filter used, or in the FFT, the width of a frequency bin”[30].
The equivalent noise bandwidth (ENB) of a window indicates the resolution that the window has. A uniform (rectangular) window has an ENB = 1 and a flat top window has an ENB = 2,97. This indicates that the uniform window has more resolution than the flat top window
[pic]
Figure 1. Windows resolution compared in the frequency domain.
Figure 1 shows the amplitude spectrum of a signal multiplied by four different windows. The signal is composed by two frequencies 48 and 52 Hz. Observe that the uniform and the Hanning windows allow to differentiate both frequencies in the spectrum. Observe also that the Flat top and the 7-term Blackman Harris windows show the two frequencies as a single peak.
The Simulator
[pic]
1. Signal 1 + Signal 2: Signal in time domain. It is a sum of two sine functions.
2. Window Constants:
-Equivalent Noise Bandwidth: Digital indicator of the equivalent noise bandwidth of the window function.
-Coherent Gain: Digital indicator of the coherent gain of the window function.
3. Return to Menu: Return to main menu
4. Show Help: General description of this VI.
5. Frequency Signal 1: Digital indicator of frequency of sine 1.
6. Frequency Signal 1: Slide control and digital indicator of frequency of Sine 1.
7. Frequency Signal 2: Digital indicator of frequency of sine 2.
8. Frequency Signal 2: Slide control and digital indicator of frequency of Sine 2.
9. Window: Menu ring to select the window function.
10. Signal X Window: Signal multiplied by the window function selected in time domain.
11. Spectrum Amplitude: spectrum of the signal multiplied by the window.
12. Original Signals: Graph of the original signals in time domain.
Practice with the simulator
1. Set frequencies for signals 1 and 2 very close. Separate them 1, 2, 3 and 4 Hz and try different windows.
2. Observe the equivalent noise bandwidth value for each window.
3. Which window has the better resolution? Which window has the worst resolution?
Related Topics
• Windows & Amplitude
• Windows Comparison
• Windows Overlap
• Windows: Noise floor
-----------------------
[1] MIT93, page 523.
[2] Nyquist, Harry (b. 1889, Nilsby, Sweden - d. 1976, Harlingen, Texas, U.S.)
American physicist and electrical and communications engineer, a prolific inventor who made fundamental theoretical and practical contributions to telecommunications. (Encyclopedia Britannica).
[3] MIT93, page 538.
[4] HAY89, page 260.
[5] The measurement and automation catalog 2001. National Instruments,page 246.
[6] RAN87, page 271.
[7] TAY94, page 344.
[8] TAY94, page 17.
[9] BRI74, page 51.
[10] RAN87, page 50.
[11] Encyclopedia Britannica
[12] MIT93, page 528.
[13] Dirac, Paul Adrien M. (b. 1902, Bristol, Gloucestershire, England – d. 1984, Tallahassee, Florida, U.S).
[pic]'()*FGHIOPQklmopqrst?‘ðìâìÞÖÞÖõ¬µ–õ‹|‹j|‹|ÃUõ¬µ)hÄ8µCJOJQJaJmHnHsH tHu[pic]#[14]?j}[pic]?hÄ8µU[pic]mHnHu[pic]j?hÄ8µU[pic]mHnHu[pic]He was a theoretical physicist known for his work in quantum mechanics and for his theory of the spinning electron. In 1933 he shared the Nobel Prize for Physics with the Austrian physicist Erwin Schrödinger. (Encyclopedia Britannica)
[15] RAN87 page 33.
[16] MIT93, page 524.
[17] WOW91, page 343.
[18] LabVIEW Online Reference
[19] MIT93, page 537.
[20] MIT93, page 118.
[21] ESH99, chapter 4 page 11.
[22] MIT93, page 525.
[23] WOW91, page 97.
[24] MIT93, page 156.
[25] BRI74, page 35.
[26] MIT93, page 535.
[27] HAR76, chapter 1 page 24.
[28] MIT93, page 142.
[29] MIT93, page 550.
[30] MIT93, page 540.
[31] FRA95, chapter 4, page 9.
-----------------------
Signal Processing Lab
“信号处理实验室”实验指导书 (英文)
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