Chapter 1 : Introduction



DEPARTMENT OF ECE

SUBJECT: DIGITAL SIGNAL PROCESSING CODE:CS 2403

SEM.: VII DEPT.:CSE

PART - A (Q&A)

UNIT I SIGNALS AND SYSTEMS

Basic elements of DSP – concepts of frequency in Analog and Digital Signals – sampling theorem – Discrete – time signals, systems – Analysis of discrete time LTI systems – Z transform – Convolution (linear and circular) – Correlation.

UNIT II FREQUENCY TRANSFORMATIONS

Introduction to DFT – Properties of DFT – Filtering methods based on DFT – FFT Algorithms Decimation – in – time Algorithms, Decimation – in – frequency Algorithms – Use of FFT in Linear Filtering – DCT.

UNIT III IIR FILTER DESIGN

Structures of IIR – Analog filter design – Discrete time IIR filter from analog filter – IIR filter design by Impulse Invariance, Bilinear transformation, Approximation of derivatives – (HPF, BPF, BRF) filter design using frequency translation

UNIT IV FIR FILTER DESIGN

Structures of FIR – Linear phase FIR filter – Filter design using windowing techniques, Frequency sampling techniques – Finite word length effects in digital Filters

UNIT V APPLICATIONS

Multirate signal processing – Speech compression – Adaptive filter – Musical sound processing – Image enhancement.

PART A (Q&A)

UNIT I SIGNALS AND SYSTEMS

Q. 1. What is DSP?

 Ans. DSP is defined as changing or analysing information which discrete sequences of numbers.

Q. 2. What are the limitations of digital signal processing? 

Ans. The digital signal processing systems have many advantages. Even though there are certain disadvantages as follows

1. Bandwidth limitations : In case of DSP, if input signal is having wide bandwidth then it demands for high speed ADC. This is because to avoid aliasing effect, the sampling rate should be atleast twice the bandwidth. Thus such signals require fast digital signal processors. But always there is a practical limitation in the speed of processors and ADC.

2. System complexity : The digital signal processing system makes use of converters like ADC and DAC. This increases the system complexity compared to analog systems. Similarly in many applications the time required for this conversion is more.

3. Power Consumption: A typical digital signal processing chip contains more than 4 lakh transistors. Thus power dissipation is more in caps systems compared to analog systems.

4. DSP systems are expensive as compared to analog system.

Q. 3. What are the applications of DSP.

 Ans. The applications of DSP are given below

1. Image processing like pattern recognition, animation, robotic vision, image enhancement.

2. Instrumentation and control like spectral analysis, noise reduction, data compression.

3. Speech/Audio like speech recognition, speech synthesis, equalisation.

4. Biomedical like scanners ECG analysis, patient monitoring

5. Telecommunication like in echo cancellation, spread spectrum and data communication.

6. Military like Sonar processing, radar processing, secure communication.

7. Consumer applications like digital audio and video, power like monitor.

8.  Automotive applications like vibration analysis, voice commands and cellular telephones.

9. Industrial applications like rabotics and CNC, power line monitors.

Q. 4. Define terms ‘signal’ and ‘system’?

 Ans. A ‘signal’ may be defined as a physical quantity which varies with time, space or any independent variable Example — voltage, current A ‘system may be defined as a combination of devices and networks or subsystem chosen to do a desired action Example Electrical N/W, mechanical system 

Q 5 Write the major classification of signals’ 

Ans. There are various types of signals Every signal is having its own characteristic The processing of signal mainly depends on the characteristics of that particular signal So classification of signal is necessary Broadly the signal are classified as follows

1 Continuous and discrete time signals

2. Continuous valued and discrete valued signals.

3. Periodic and non periodic signals.

4 Even and odd signals

5. Energy and power signals:

6 Deterministic and random signals

7. Multichannel and multidimensional signals. 

Q. 6. Explain sampling function of sinc function. 

 Ans In mathematics, the sinc, function, denoted by sinc(x) and sometimes as Sa (x),

has two definitions, In digital processing and information theory, the normalized sinc function is commonly defined by

[pic]

In mathemetics, the historical unnormalized sinc function is defined by

[pic]

In both cases, the value of the function at the removable singularity at zero, usually calculated by l’Hospital rule, is something specified explicitly as the limit value 1 The sinc function is analytic everywhere. 

Q. 7. What are energy and power signals? 

Ans. The energy E of a signal x(n) is defined as

[pic]

The energy of a signal can be finite or infinite. If E is finite [pic] then x(n)

is called an energy signal.

Many signals that posses infinite energy, have a finite average power. The average

power of a discrete time signal x(n) is defined as

[pic]

If E is finite, P = 0. On the other hand, If E is infinite, the average power may be

either finite or infinite. If P is finite (and non zero), the signal is called a power signal. 

Q. 8. Differentiate between linear-Nonlinear system. 

Ans. A system is called linear, if superposition principle applies to that system. This

means that linear system may be defined as one whose response to the sum of the

weighted inputs is same as the sum of the weighted responses.

Let us consider two systems defined as follows.

[pic]

Here x1(t) is the input or excitation and y1(t) is its output or response and

[pic]

Here x2 (t) is the input or excitation and y2(t) is its output or response

Then for a linear system

[pic]

Where a1 and a2 are constants.

Linearity property for both continuous time and discrete time systems may be written

as for continuous time system

[pic]

For discrete time system

[pic]

For any non-linear system, the principle of super-position does not hold true and

equations (3) and (4) are not satisfied.

Few examples of linear system are filters, communication channels etc. 

Q. 9. Define periodic and non periodic signals Give an example in each case. 

Ans A periodic signal repeats after fixed period But non-periodic signal never repeats

Periodic signal like x(t) sin wt and Non periodic signal like [pic]A discrete time signal is periodic, if its frequency can be expressed as a ratio of two integers i.e.

[pic]

Here k and N are integer and N is the period of discrete time signal 

Q 10. What is scaling of discrete time signals?

 

Ans Scaling of discrete time signals is divided into two parts

 Time Scaling Operations

As the name indicates, time scaling operations are related to the change in time scale. There are two types of time scaling operations.

• Down scaling (Compression)

Up scaling (Expansion)

(n) Amplitude Scaling Operation

As the name indicates, in case of amplitude scaling operations, amplitude of signal is changed Different amplitude scaling operations are as follows

• Upscaling (Amplification)

• Downscahng (Attenuation)

• Addition

• Multiplication 

Q 11 What is the difference between static and dynamic discrete time signals? 

Ans. There can be static and dynamic discrete time systems but cannot be signals 

Q 12. Define a discrete time unit sequence functions 

Ans. A discrete time unit signal is denoted by U(n) Its value is unity for all positive values of n. That means its value is one for n 0. While for other values of n, its value is zero.

[pic]

In form of sequence it can be written as

[pic]

Graphically it is represented as shown below

        [pic] 

Q. 13. Define a discrete time unit ramp function. 

Ans. A discrete time unit ramp function is denoted as Ur (n) and it is defined as

[pic]

Figure below shows the graphical representation of a discrete unit ramp function.

[pic] 

Q. 14. Define transfer function of a system. 

Ans. A system may be defined as a set of elements or functional blocks which are connected together and produces an output in response to an input signal. The response of the system depends upon transfer function of the system.

Mathematically it is defined by

[pic]

[pic]

Where x(t) is input or excitation

y(t) is 0/P or response

h(t) is transfer function of the system.

[pic]

Q. 15. State the necessary and sufficient condition for stability of LTI systems 

Ans. LTI system is stable if its impulse response is absolutely summable i e

 

[pic]

Here h(k)= h(n) is the impulse response of LTI system Thus equation (1)  give the

 condition of stability in terms of impulse response of the system.

Now the stability factor is denoted by ‘s’.

        [pic] 

Q. 16. What are the constraints on the transfer function if it were to represent a causal LTI system? 

Ans. If h(n) is the response of released LTI system to a unit impulse applied at n = 0, it follows that h(n) = 0 for n < 0 is both a necessary and a sufficient condition for causality Hence on LTI system is causal if and only if its impulse response is zero for negative values of n. 

Q 17. Define LTI system’ 

Ans. If a system has both the linearity and time in varience properties, then the system is called as linear time m varient (LTI) system

[pic]

 

Q 18. What are the conditions for the region of convergence of a causal LTI system? 

Ans. A discrete time LTI system is causal if and only if the ROC of its transfer function is the extension of a circle, in including infinite

A discrete time LTI systems which has a rational transfer function H(z) will be causal if and only if.

(z) The ROC is the extension of a circle outside the outermost pole and

(ii) Units H(z) expressed as a ratio of polynomials in z, the order of the numerator should be smaller than order of denomenator. 

Q. 19. State sampling theorem.

Ans. A continuous time signal x(t) can be completely respresented in its sampled form and recoverd back from the sample form if the sampling frequency  

[pic]

   where ‘W’ is the maximum frequency of the continuous time signal x(t)

 Q. 20. Convolve {1,3,1) and (1,2,2,).

 Ans.

[pic]

[pic]

[pic]

 y (n) is output of the convolution.

 Q. 21. What is the difference between stable astable system?

 Ans.

 [pic]

[pic]

 Q. 22. Differentiate time variant from time invariant system. 

Ans. A system is called time invariant if its input output characteristics do not charge

with time. A LTI discrete time system satisfies boths the linearity and the time invariance properties.

To test if any given systems is time invariant, first apply an arbitrary sequence x (n) and find y (n).

y (n) = T [x (n)]

Now delay the input sequence by k samples and find output sequence denote it as. y(n,k) T[x(n-k)]

Delay the output sequence by k samples denote it as

[pic]

 

For all possible values of k, the systems is the invariant on the other hand

[pic]

Even for one value of k, the system is time variant.

the output.

Even for one value of k, the system is time variant.

[pic]

[pic]

 Q. 23. What are symmetric and asymmetric signals?

Ans. An even signal is that type of signal which exhibits symmetry in the time domain This type of signal is identical about the origin Mathematically, an even signal must satisfy the following condition.

For a continuous-time signal, x (t) = x (— t)

For a discrete-time signal, x (n) x (— n)

Figure shows continuous-time and discrete-time even signals.

Similarly, an odd signal is that type of signal which exhibits anti-symmetry. This type of signal is not identical about the origin Actually, the signal is identical to its negative Mathematically, an odd signal must satisfy the following condition

[pic]

For a continuous-time signal, x (t) = x (- t)

 For a discrete-time signal, x (n) — x (— n)

Figure shows continuous-time and discrete-time odd signals.

[pic]

[pic]

 Q. 24. What is the frequency response of a discrete LTI system? Derive the frequency response of a system whose impulse response is given by h(n) = a” u(n —1) for (a) 0) becomes unbounded for z = and zn (when n > 0) becomes unbounded for z = 0.

[pic]

 Q. 35. What is the relationship between Z transform and the Discrete Fourier transform?

 Ans. Let us consider a sequence x(n) having z-transforrn with ROC that includes the

[pic]

unit circle. If X(z) is sampled at the N equally spaced points on the unit circle. If X(z) is

sampled at N equally spaced pomts on the unit circle.

[pic]

We obtain

[pic]

Expression is (2) identical to the Fourier transform X(w) evaluated at the N. equally  spaced. Frequencies

[pic]

If the sequence x(n) has a finite duration of length N or less, the sequence can be

recovered from its N-point DFT. Hence its Z-transform is uniquely determined by its N-point DFI’. Consequently, X(z) can be expressed as a function of the DFT {X(k)} as

follows

[pic]

When evaluated on the unit circle (3) yields the Fourier transform of the finite duration sequence in terms of its DFT in the form:

[pic]

This expression for Fourier transform is a polynomial interpolation formula for X(w)

expressed in terms of the, values {x(k)) of the polynomial at a set of equally spaced

discrete frequencies

[pic]

[pic]

 Q. 36. What are the application’s of z-transform?

 Ans. 1. z-transform is an important tool in the analysis of signals and linear time invarient systems.

2. It is used for the analysis of discrete time systems in frequency domain which in generally more efficient than time domain analysis.

3. It is used for filtering process.

4. Causality of discrete time LTL system.

5. Stability of discrete time LTI system.

6. Determination of poles and zeros of rational z-transform.

[pic]

 

[pic]

 Q. 37. What are the conditions for the region of convergence of a non causal LTI system.

 Ans. The condition for non-causal of discrete time LTI system is that the impulse response of a causal discrete time LTI system js given as

[pic]

This means that h (n) is two sided.

Also, transfer function H(z) is the z-transform of h

(n). The ROC of H (z) of non-causal discrete time LTI

system is the entire z-plane except [pic]

[pic]

[pic]

 Q. 38. State and prove convolution property of z transform?

 Ans. The convolution property for Z-transforms is very important for systems

analysis and design. In words : The transform of the convolution is the product of the transforms.

[pic]

Where [pic] denotes convolution (in this case, discrete-time convolution).

Proof. This is somewhat easier (and more general) to prove for noncausal sequences.

[pic]

[pic]

 Q.39. State the correlation property of two sequence in z-domain. Give its ROC.

 Ans. Correlation is a measure of the degree to which two signals are similar. The

correlation of two signals is divided into two.

1. Cross correlation (b) Auto correlation

[pic]

[pic]

 Q. 40. Find out the Z-transform for the following discrete time sequence.

[pic]

 Ans.  

        [pic]

 

[pic]

[pic]

 Q. 41. Determine to z-transform of the following signal and sketch the pole zero

pattern:

[pic]

 Ans.                                

        [pic]

 

[pic]

[pic]

 Q. 42. Find z transform of

[pic]                                                

 Ans. We have standard z-transform pair.

[pic]

[pic]

[pic]

Q. 43, What are the various methods to find out inverse z transform?

 Ans. (a) Cauchy Rihemen’s theorem

(b) Long division method.

(c) Partial function.

[pic]

 [pic]

Q. 44. Find the z-transforms of  [pic]

 Ans.         [pic]

The X (z) is finite for all values of because

[pic]

The ROC is entire z-.plane.

[pic]

 Q. 45. Determine the system function

[pic][pic]

 Ans. Taking z-transform of both sides.

 [pic]

[pic]

 

Q. 46. Determine the pole-zero plot for the system described by difference equation

[pic]

 Ans. Taking z-transform of both sides.

 [pic]

The ROC & pole zero plot shown in Fig. below

[pic]

From the following figure, we can observe the followmg

1.ROC of the system function include unit circle.

2. ROC of the system function cannot have any poles.

[pic]

 Q. 47. Determine the signal x(n) whose z-transform is given by

[pic]

 Ans. By taking the first derivative of X (z), we obtain

[pic]

[pic]

 Q. 48. What is the relation between z transform and Laplace transform?

 Ans. The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of

[pic]

Where T = 1/fs is the sampling period (in units of time e.g. seconds) and fs is the

sampling rate (in samples per second or hertz)

Let

[pic]

be a sampling impulse traln (also called a Dirac comb) and

[pic]

be the continuous-time representation of the sampled x(t).

[pic]

The Laplace transform of the sampled signal [pic]

[pic]

This is precisely the definition of the unilateral Z-transform of the discrete function x[n]

[pic]

Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal:

 

[pic] The similarly between the z and Laplace transforms is expanded

upon in the theory of time scale calculus.

[pic]

 Q. 49. Find out the Z-transform for the following discrete time sequence [pic]

[pic] 

 Ans.

[pic]

UNIT II FREQUENCY TRANSFORMATIONS

Q. 1. Define DFT.

 Ans. It is a finite duration discrete frequency sequence which is obtained by sampling one period of fourier transform. Sampling is done ‘N’ equally spaced points over the period extending from [pic]. The DFT of discrete sequence x(n) is denoted by x(k) and it is given by.

[pic]

where k = 0, 1, 2 N—I.

[pic]

 Q. 2. Define the Discrete Time Fourier Transform.

 Ans. The Discrete Time Fourier Transform [pic]  of a discrete line signal x(n) is expressed as

[pic]

DTFT is periodic units period [pic] . So any interval of length [pic]  is sufficient for the

complete specification of the spectrum. Generally, we draw the spectrum in the fundamental internal [pic]

[pic]

 Q. 3. What is the linearity property of DTFT?

 Ans. If

[pic]

According to definition of DTFT

[pic]

Comparing each summation term with definition of DTFT then we can write

[pic]

[pic]

 Q. 4 Explain the symmetry properties of DFTs which provide basis for fast algorithms.

 Ans. Most approaches for improving the efficiency of computation of DFT, exploits

the symmetry and periodicity property of[pic]  i.e.

[pic]

 

[pic]

 Q.5 Exploit of these two basicproperty results in computational efficient algorithms which are collectively known as FFT algorithms. Q. 6. What is zero padding in DFT?

 Ans. The process of lengthening a sequence by adding zero valued samples is called

appending with zeros or zero padding. This is done to equate linear convolution units circular convolutions in case of DFT.

[pic]

 Q. 6. What is the importance of FF1’?

 Ans. Fast Fourier Transform (FFT) is to decompose successively the N-point DFT computation into computations of smaller size DFT’s and to take advantage of the periodicity and symmetry properties of the complex number [pic] Such decompositions, if properly carried out, can result in a significant surving in the computational complexity given by the total number of multiplications and the total number of additions needed to compute all N DFT samples. The total no. of complex multiplications is reduced to [pic]w.r.t. DFT and the total no. of complex additions is [pic]

[pic]

 Q. 7. What is the quantization error in the direct computation of DFT?

 Ans. The quantization errors in the direct computation of DFT is in particular the effect of round off errors due to the multiplications performed in the DFT with fixed point arithmetic, for example.

A finite duration sequence  [pic] is defined as

[pic]

[x(n)] is a complex valued sequence. Assume that the real and imaginary components

of x(n) and [pic]are represented by b bits. Consequently the computation of the product

x(n) W requires four real multiplications. Each real multiplication is rounded from 2b bits to b bits and hence there are four quantization errors for each complex valued multiplication.

In the direct computation of the DFT, there are N complex valued multiplications for each point in the DFT. Therefore the total number of real multiplications in the computation of a single point in the DFT is 4N. Consequently there are 4N quantization errors.

[pic]

 Q. 8. What is the advantage of in-place computation?

 Ans. The main advantage of in-place computation is reduction in the memory size

in-place computation reduces the memory size.

[pic]

‘a’ & ‘b’ are inputs and ‘A’ and ‘B’ are outputs of butterfly. For anyone input ‘a’ and ‘b’ two memory locations are required for each. One memory location to store real part and other memory location to store imagining part. So for both inputs ‘a’ & ‘b’ = 2 + 2 = 4 memory location are required.

Thus outputs ‘A’ & ‘B’ are calculated by using the values ‘a’ & ‘b’ stored inmemory.

‘A’ & ‘B’ complex numbers, so 2 + 2 = 4 memory location are required.

Once the computation of ‘A’ & ‘B’ done then values of ‘a’ & ‘b’ are not required. Instead of storing ‘A’ & ‘B’ at other memory locations, there values are stored at the same place where ‘a’ & ‘b’ were stored. That means ‘A’ & ‘B’ are stored in the place of ‘a’ & ‘b’. This is called as in-place computation.

[pic]

Q. 9. Indicate the number of stages, the number of complex multiplications at each stage, and the total number of multiplications required to compute 64-point FFT using radix-2 algorithm.

 Ans.

 [pic]

[pic]

 Q. 10 Perform circular, convolution of two sequences

[pic]

 Ans. Circular convolution is

[pic]

[pic]

[pic]

[pic]

Q. 11. Write application of FFT algorithm.

 Ans. Linear if itering, correlation analysis and spectrum analysis are same important applications of FFT algorithm.

[pic]

 Q.12. Consider a complex sequence

[pic]

(a) Find the Fourier Transform X(cv). (b) Find the N-point DFT

 Ans. (a) Fourier transform of x (n) is given by.

 

[pic]

(b) N-point DFT is obtained if [pic] is replaced by [pic]

[pic]

[pic]

 Q. 13. Compute the DFT of sequence [pic]

 Ans.

[pic]

[pic]

 

[pic]

Q. 14.What is a decimation in time algorithm?

DIT algorithm is used to calculate the DFT of a N point sequence. Initially the N point sequence is divided into two N/2 point sequences Xeven (n) and Xodd (n). The N/2 point DFTs of these two sequences are evaluated and combined to give the N point DFT. Similarly the N/2 point DFTs can be expressed as a combination of N/4 point DFTs. This process is continued until left with 2 point DFT. This algorithm is called decimation in time because the sequence X(n) is often splitted into smaller sequences.

Q.15. Compute the DFT of x(n) =δ(n).

N-1 N-1

X(k) = ∑ x(n)WNKn = ∑ δ(n) WNKn = 1.

n=0 n=0

Q.16.What is meant by radix-2 FFT?

The FFT algorithm is most efficient in calculating N point DFT. If the number of point N can be expressed as a power of 2 ie N= 2M where M is an integer , then this algorithm is known as radix-2 FFT algorithm.

Q.17. What is decimation in frequency algorithm?

It is one of the FFT algorithms. In this the output sequence X(k) is divided into smaller subsequences, that is why the name decimation in frequency. Initially the input sequence is divided into two consisting of the first N/2 samples of X(n) and the last N/2 samples of X(n).The above procedure can now be iterated to express each N/2 point DFT as a combination of two N/4 point DFTs.This process is continued until we are left with 2 point and 1 DFT.

Q.18. What are the differences and similarities between DIF and DIT algorithms?

Differences:

For DIT the input is bit reversed while the output is in natural order, whereas for DIF the input is in natural order while the output is bit reversed.

The DIF butterfly is slightly different from the DIT butterfly, the difference being that the complex multiplication takes place after the add-subtract operation in DIF.

Similarities:

Both algorithms require same number of operations to compute the DFT. Both algorithms can be done in place and both need to perform bit reversal at some place during the computation.

Q.19. Explain In-place computation.

To compute the elements p and q of the mth array , it is required to have elements in the p and q of the (m-1) array. If Xm(p) and Xm(q) are stored in the same register as Xm-1(p) and Xm-1(q) respectively ,it is possible to implement the above computation with only N array of complex storage registers. This kind of computation is commonly referred to as In-place computation.

Q.20.What are the applications of FFT algorithms?

The applications of FFT algorithms include

Linear filtering, Correlation, Spectrum analysis

Q.21.Calculate the number of multiplications needed in the calculation of DFT and FFT with 64 point sequence.

Ans.Number of complex multiplications required using direct computation is

N2 = 642 = 4096

Number of complex multiplications required using FFT is

(N/2) log N = ((64/2) log 64 = 192

speed improvement factor (4096/192) = 21.33.

Q.22. Define discrete linear convolution.

The discrete convolution of the two discrete variable function x(n) and h(n) is the

discrete variable function y(n) given by the summation

∞

y(n)= ∑ x(k) h(n-k)

k=-∞

Q.23. What are the properties of DIT FFT?

putation are done in place. Once a butterfly structure operation is performed on a

pair of complex numbers(a,b) to produce (A,B) there is no need to save the input pair

(a,b). Hence we can store the results(A,B) in the same location as(a,b).

2. Data x(n) after decimation is stored in reverse order.

Q.24.The direct computation of DFT of a sequence X(n) requires 4N2 real multiplications

and N(4N-2) real additions.

Q.25. The direct computation of DFT of a sequence X(n) requires N2 complex

multiplications and N(N-1) complex additions.

Q.26. What are the advantages of FFT algorithm?

Fast fourier transform reduces the computation time. In DFT computation, number of multiplication is N2 and the number of addition is N(N-1). In FFT algorithm, number of multiplication is only N/2(log2N) . Hence FFT reduces the number of elements (adder, multiplier Z &delay elements). This is achieved by effectively utilizing the symmetric and periodicity properties of Fourier transform.

UNIT III IIR FILTER DESIGN

Q. 1. What is frequency wraping n Bilinear transformation?

 Ans. The mapping of frequency from 1 to is approximately linear for small value of [pic]  For the higher frequencies, however the relation between Q x o becomes highly non-linear. This introduces the distortion in the frequency scale of digital filter relative to analog filter. This effect is known as wraping effect.

The influence of the wraping effect on the amplitude response can be demonstrated by considering on analog filter with no. of passband centered at regular derived digital filter has some numbers of passbands but the centre frequencies and the bandwidth of higher frequencies passband in digital domain tends to be reduced.

[pic]

[pic]

 Q. 2. What are the conditions for distortionless transmission?

 Ans. The conditions for distortionless transmission are given below.

1. Anti-aliasing filter must be used which is a low pass filter to remove high frequency noise contain in input signal. It avoids aliasing effect also.

2.Sample and hold circuit is used to keep the voltage level constant.

3.Output signal of Digital to analog converter is analog, that is a continuous signal. But it contain high frequency components. Such high frequency components are

understood. To remove these components reconstruction filter is used.

4.Amplifiers are used sometimes to bring the voltage level of input signal upto required level for distortionless transmission.

[pic]

 Q. 3. Show for 3rd order butterworth low pass filter the Location of its poles and zeroes in a s—plane.

 Ans.

[pic]

[pic]

 

Q.4. What are methods used to convert analog to digital filter?

Approximation of derivatives, Impulse invariant method & Bilinear transformation method.

Q.5. Write the pole mapping rule in Impulse invariant method?

A pole located at s = sp in the s plane is transferred into a pole in the z plane located at

Z = espTs

Q.6. What are the disadvantages of Impulse invariant method?

Although this method is useful for implementing LPF and HPF the method is unsuccessful for implementing digital filters for which |H(jw)| does not approach zero for large value of w such as the high pass filter .

Q.7. What are the advantages of Bilinear transformation method?

The Bilinear transform method provides non linear one to one mapping of the frequency points on the jw axis in the S plane to those on the unit circle in the Z plane.i.e Entire jw axis for - ( ................
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