Suppose we are given a finite-length sequence h[n](it could be part  of an infinite-length impulse response from a discrete system that has  been windowed) and would like to obtain a rational approximation for it. This means that if H(z) = Z[h[n]], a rational approximation of it  would be H(z) = B(z)/A(z), from which we get

H(z) A(z) = B(z)

Letting

for some choice of M and N, equations from H(z) A(z) = B(z) should allow us to find the M + N €“ 1 coefficients {a_k b_k}.

(a) Find a matrix equation that would allow us to find the coefficients  of B(z) and A(z).

(b) Let h[n] = 0.5ⁿ (u[n] ˆ’ u[n ˆ’ 101]) be the sequence we wish  to obtain a rational approximation and let B(z) = b₀ while  A(z) = a₀ + a₁z^ˆ’1 , find the equations to solve for the coefficients  {b₀, a₀, a₁}.

Transcribed Image Text:

M-1 N-1 -k Ba) Συ,", Α()-1+ Σα . A(3)-1+ Σa k=1 k=0