Problem 1:

A periodic signal, x(t) is shown below.

A = 10, T = 4 Sec.

• Write a MATLAB script to plot the signal, using enough points to get a smooth curve.

• Compute the Fourier series coefficients for the signal (if you can find them in the text, that is ok). Plot the single-sided or double-sided spectra for each signal. Include enough frequencies in the plots to adequately represent the frequency content of the signals.

• Plot partial sums of the Fourier series for the signal x(t) (terms 1 through N in the infinite series). Decide what N should be to get a get a good reproduction of the corresponding x(t).

Problem 2:

Consider the following two signals:

a) Plot these two signals on the same figure. Use a time axis of [-5,5] and proper sampling period. Add labels, etc

b) Derive (mathematically) the Fourier Transforms X1(ω) and X2(ω) of these two signals

c) Write a program to compute the Fourier Transforms numerically, and compare with b)

c) Plot the magnitude spectrum for the signals on the same figure.

d) Discuss your results and explain the relationship between the spectra X1(ω) and X2(ω)

Problem 3:

Consider a given system with the following transfer function:

H(s) = (s+1)/500s²(s+1)

a. Use impulse to find and plot the system impulse response h(t).

b. Use pzmap to generate a pole-zero plot for H(s).

c. Explain from the above plot whether the system is stable or not

d. Plot the signal

X(t) = e^-t Cos(5Πt)u(t)

For t=[0,5] using time resolution of .002 sec.

e. Use lsim to find the system response y(t) and plot it

Problem 4:

In this problem, we will examine the aliasing effect. In particular, we will examine the effect of different sampling rates when sampling the following continuous-time signal:

x(t) = 10Cos(2Π50t)

A. We wish to examine the frequency content that a reconstructed signal xr1 (t) would have, if we sampled x(t) with sampling frequency f1=1000 Hz (ts1=1/f1).

Plot x(t) over the time interval 0 to 1 sec., Call the signal vector as xr1.

Can you see x(t)? if not plot only first 100 values.

What is the length N of the signal x(t)?

Use the commends below to plot the spectrum of x(t), explain the commands and the plot, what do the peaks correspond to?

X=fftshift(fft(x,N));

f=linspace(-1,1-1/N,N)/(2*ts1);

plot(f,abs(X))

B. Repeat A with sampling sampling frequency f2=1000 Hz (ts2=1/f2).

C. Comment on the values of f1 and f2, how would you choose the correct sampling frequency?

What is the Nyquist frequency for this example? Try different values of the sampling frequency and examine the effect on the spectrum of the signal.