Student: Stanley

**ELEG333**

**Signals and Systems**

**Final Exam**

**Spring 2020**

Note:

The exam is open book, open notes. You can use any help you need except the help of another person (directly or indirectly).

The exam has to be uploaded on canvas by Friday, May 8, 10 AM. Exams submitted after this time will lose 5 points per hour (late). File can be uploaded only in word/ pdf format.

There will be a help session on Wednesday (5/6/20) from 10 am – 11 am on Zoom. Bring your questions there.

Label all graphs and write a short description of the plot below each plot describing what you are observing. Presentation of plots count!

Two separate recordings of equal length are made of a periodic pulse train being transmitted down a noisy channel. The following table shows the recorded values of the sampled voltages. Determine the amount of lag between the two recordings and the period of the waveform. You can use MATLAB for this problem. Use Cross-correlation (or any other technique) to get your answer. Attach the plots (graphs).

Record 1:

6.02 -5.98 7.92 -7.96 -0.78 -8.34 9.22 -2.65 -3.7 9.51

5.53 3.5 -3.18 -8.85 8.21 1.69 -0.06 6.65 -8.00 -9.21

-0.78 7.27 -5.98 -3.97 9.11 4.23 2.99 -1.85 -5.27 3.81

6.62 -2.64 2.08 -5.91 -3.58 -1.65 3.64 -8.19 -3.50 4.84

7.25 2.93 -4.42 -8.21 3.61

Record 2:

8.93 -7.20 -0.82 3.23 1.44 5.43 -9.88 -1.13 0.79 9.83

-8.73 4.64 -8.49 -4.66 -8.84 5.55 -8.24 -0.37 2.71 4.63

1.88 -0.92 -5.33 9.01 9.23 -3.70 5.08 -0.72 -5.08 -2.60

9.67 -8.55 -3.08 4.18 8.11 0.74 -3.87 -4.09 8.03 6.91

-9.87 -3.62 -8.29 -5.80 -7.04

The desired amplitude response of a certain band-pass FIR filter can be stated as:

H(*f*) = 1 for 250 £ *f* £ 750 Hz

= 0 elsewhere

The sampling rate is 2 KHz, and the order of the filter should be N = 15. Using FIR1 command in MATLAB, generate the 16 coefficients. Plot the frequency response of the filter to verify that it satisfies the specifications. Look and comment on both the phase and magnitude plot. Attach the program (commands) and plots.

Generate a signal:

x = 2*cos(0.1*pi*n) +3*sin(0.5*pi*n) + 2*cos(0.9*pi*n); (use n = 0:1000)

Assume sampling frequency Fs = 2000 Hz

Plot this signal in time domain and frequency domain.

Find the frequencies of the peaks in the frequency domain plot.

Now filter the signal x using the command:

y = filter(B, 1, x);

Plot this signal in time and frequency domain

Find the frequency of the peak in the frequency domain.

Did the filter work?

Problem 3.

A digital signal contains two frequency components: w_{1} = 0.1p, and w_{2} = 0.2p ; the different components have amplitudes a_{1} = 1 and a_{2} = 5.

- Generate such a sequence, which contains 1000 points (exactly 1000 points); plot the first 100; can you identify two distinct frequencies?
- If you were to calculate the 1000-point FFT, X(k), at what values of k would these frequency components appear? Show your calculations.
- Use Matlab to calculate and plot (magnitude of the) the 1000 point FFT (plot only the first 500 points, since the rest are redundant); do the frequencies appear at predicted locations?

Re-do problem 4 above using the frequencies w_{3} = 0.2p and w_{4} = 0.202p. What is the primary difference between these two problems?

Re-do problem 4 above using the frequencies w_{5} = 0.2p and w_{6} = 0.2001p. What is the primary difference between this problem and the previous ones?

Problem 6

For the following filter specifications:

Gain > -1dB for w < 0.1p, gain < -20 dB for 0.4 p < w < p

- Design a butterworth filter. Write down the order and coefficients of this filter. Check the corner frequencies using the “freqz” command to see if the filter satisfies the specifications.
- Design a ChevyshevI filter. Write down the order and coefficients of this filter. Check the corner frequencies using the “freqz” command to see if the filter satisfies the specifications.
- Design an elliptical filter. Write down the order and coefficients of this filter. Check the corner frequencies using the “freqz” command to see if the filter satisfies the specifications.
- Compare the three filters in (a), (b) and (c). Comment on the differences (both magnitude and phase responses).

Show all commands used and attach all plots. Label all plots.