Question 1:

Consider two digital sequences:

Here, M is a positive integer, and is a value in the interval

a) Using the time-domain approach, determine closed-form expression of the cross-correlation sequence r_xy(l).
b) Using the z-domain approach, determine the closed-form expression of the cross-correlation sequence r_xy(l).
c) Suppose that M = 10 and a = 0.8. Write MATLAB code to compute and plot the cross-correlation sequence r_xy(l). Compare the results with the answers in part (a) and (b).

Question 2:

Consider two template signals: s₁(n) and sin(0.48Πn) and s₂(n) = -cos(0.1Πn)a digital communication link using frequency shift keying (FSK), the signals and represent the two binary numbers 1 and 0, respectively. Suppose the binary sequence is transmitted down a channel that is corrupted by Gaussian noise with a mean of zero and a variance of 0.36.

a) Construct and plot the received signal x(n) = s(n) + w(n), where s(n) is the signal representing the binary sequence and is the Gaussian noise sequence.

b) Compute and plot the correlation sequences r_s1s2(l), r_s0s0(l), r_s1s0(l) and r_s2s0(l) . Compute and plot the cross-correlation sequences and of the received signal with the template signals and s₂(n).

c) Determine a procedure for detecting and identifying the received symbols.