Assignment Details:
1. (a) Find the convolution between x[n] = 3u[n] – 3u[n-9] and w[n] = (0.4)^n u[n]. Show the detailed steps of your working.
1. (b) Find the output y[n] for n = 2 to 6 for an LTI system with CCDE y[n] = 3x[n] + x[n-1] + 2y[n-1] – y[n-2], given that the input is x[n] = [2↑-3″ 5″], and that y[-1] = 0,y[-2] = 1. Show the detailed steps of your working.
2. The system function of a system is H(z) = (1-7z^(-1) + 5z^(-2)) / (1 + 0.5z^(-1) )(1-5z^(-1) + 6z^(-2) ).
(a) Find all possible ROCs for this system.
(b) For each ROC from 2.(a), find the impulse response h[n] of the system.
(c) For each ROC from 2.(a), find if the system is (i) causal, and (ii) stable.
3. (a) The system function of a filter is H(z) = -2 + z^(-1) – 2z^(-2). Find the system function of a zero-phase filter with the same magnitude response as H(z).
(b) Consider a single pole at z = 0.8e^2j [system function H(z) = 1 / (1 – 0.8e^2j z^(-1)) ].
(i) Find its magnitude response.
(ii) Find the maximum value of the magnitude response.
(iii) Find the frequency ω where the magnitude response is maximum.
(c) A linear phase filter has 2 zeros. If one zero is at z = -4, find the other zero.
4. (a) Give an example of two different system functions (two different filters/LTI systems) that have the same magnitude response.
(b) Give an example of two different system functions that have the same phase response.
(c) Give an example of two different system functions that have the same group delay but have different phase responses.
(d) Give an example of two different system functions that have the same ROC.