Question

consider a causal and stable LTI system with rational transfer function h(z). whose corresponding impulse response begins at n= 0. Furthermore, H(1) = 5/4. The poles of H(z) are pk = 1/√2 exp (j(2k-1)/4 π ) for k = 1, 2, 3, 4. The zeros of H(z) are all at z=0. Let g(n) = jⁿh(n). The value of g(8) equals_____.

Answer 1

g(8) equals the value of h(8) as j^8 equals to 1. To find g(8), we need to calculate the system impulse response h(n) at n=8 considering the poles, and then apply the transformation g(n) = j^n h(n).

Step-by-step explanation:

The value of g(8) for the given Linear Time-Invariant (LTI) system with specified poles and zeros can be found using the system's impulse response h(n) and alteration by a factor of j^n. Since all poles are on the unit circle and the zeros are at the origin, we can derive h(n) from the transfer function H(z), and subsequently find g(n).

To calculate g(8), we need to take into account that h(8) will be affected by the system's poles, and then multiply by j^8. Since j^8 = 1 (as j is the imaginary unit and j^4 = 1), g(8) will have the same magnitude as h(8) but potentially a different phase due to the multiplication by j terms at earlier indices. The value of H(1) being 5/4 is used to normalize the system's gain.