EE3414 Homework #1 Solution



EE3414 Homework #2 Solution

1. For the signal [pic], plot its waveform, and then illustrate the resulting samples with the following sampling intervals:

a. [pic] b. [pic] c. [pic]

For each case, also sketch the reconstructed continuous time signal from the samples using linear interpolation (i.e. connecting samples by straight lines). Based on your sketch, determine the fundamental frequency of the reconstructed signal. In which case you had aliasing distortion? What is the minimal sampling frequency and the corresponding sampling interval needed to avoid aliasing?

Solution:

See figures on the next page.

We had aliasing distortion in both part b and part c. To avoid aliasing, the minimum sampling frequency should be greater than[pic] :[pic]2*1=2 Hz. The corresponding sampling interval is [pic] sec.

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[pic]

2. For the sample signal and each of the sampling periods given in Prob. 1, illustrate the spectrum of the original signal and the spectrum of the sampled signal. Assuming you apply the ideal low-pass filter for reconstruction, what is the frequency of the reconstructed signal? Does your solution conform to the answers you got for Prob. 1?

Solution:

See figures on the next page.

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[pic]

[pic] Page 5

[pic]

4. The figure below is part of a speech waveform. As can be seen, it has a nearly periodic structure, and within each period, there exists many fine ripples. The frequency associated with the basic period is called the pitch, the shortest fine ripple determines the highest frequency in the signal.

a. Estimate its fundamental period by determining the length of a basic cycle. Also, estimate the maximum frequency by determining the length of a shortest ripple.

b. Based on your estimate of the highest frequency, determine what is the minimal sampling rate for this signal. (Note that this signal is actually already a discrete time signal obtained by sampling a continuous time signal. The sampling frequency used was f_s=22050Hz. The horizontal axis of the plot indicates the sample index k. Hence the actual time interval between two indices is the sampling interval \Delta=1/f_s=1/22050 sec. )

c. Indicate on the blown-up plot, what would be the remaining samples if you down sample this signal by a factor of 10 (i.e., you take every 10th sample). Then illustrate the reconstructed signal by connecting the samples with straight lines. Is the fine structure of the original signal retained after this down sampling?

[pic]

a. Looking at the waveform plot, the first 4 cycle occupy 300 samples. Therefore, each cycle takes about 75 samples. The fundamental period is the time corresponding to 1 cycle, therefore[pic]sec.

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The pitch frequency (i.e. fundamental frequency) is [pic]

The shortest ripple occurs around the sample 2500. There are two ripples in about 30

samples. So the shortest ripple takes about 15 samples, with time [pic]sec. The corresponding maximum frequency is [pic]

b. According to Nyquist sampling theorem, sampling rate [pic], so the minimal sampling rate for this signal is [pic]

c. Figure is shown as following. And the fine structure isn’t retained after this down sampling. This is because the new sampling rate (22050/10=2205 KHz) is less than the minimal sampling rate derived above.

[pic]

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