Problem 1

Consider two digital sequences:

x(n) = aⁿ, if M ≤ n ≤ N

           0, Otherwise

y(n) = bⁿu(n)

Here, M and N are two positive integers, and a and b are two values in the interval (0,1).

a) Using the time-domain approach, determine the closed-form expression of the cross-correlation sequence r_xy(l), i.e. r_xy as a function of l.

b) Using the z-domain approach, determine the closed-form expression of the cross-correlation sequence r_xy(l).

c) Suppose that M = 5, N = 20, a = 0.9, and b = 0.8. Use MATLAB to compute and plot the cross-correlation sequence r_xy(l). Use plots to compare the results with the answers of part (a) and (b).

Problem 2

Consider the template signal that is being transmitted: s(n) = {1, -1,0, -1,1, -1,1}. The received signal x(n) is a delayed and noise-corrupted version of s(n):

x(n) = as (n - D) + w(n)

where w(n) is a zero-mean white noise sequence, and D is the time delay measured in number of samples.

a) Assume that D = 4, a = 0.8, and the white noise has variance equal to 0.25. Generate and plot the signal x(n) for 0 ≤ n ≤20.

b) Compute and plot the cross-correlation r_xs(l) and determine the time delay of x(n).

c) Compute and plot the impulse response h(n) of the matched filter.

d) Compute and plot the response y(n) of the matched filter to x(n). Compare y(n) to rxs(l). Can you determine the time delay of s(n) from the output y(n) of the matched filter?

Attachment:- Report_Template.rar