Question

The transfer function of a digital filter is, H(z) - ao t a1z-1+ .avz-M (a) [5 points] For a Finite Impulse Response (FIR) filter, where ao is non-zero, which of the following must be true? (circle all that apply) i. There will be at least one non-zero value within b bM ii. There will be at least one non-zero value within a1 aN a0 fori 1,2,..N iv. bi = 0 for i = 1,2, M v.bo=1

Answer 1

A linear shift - invariant system can be described by using an Nth - order linear constant - coefficient difference equation of the form sigma_k = 0^N a_k y (n - k) = sigma_r = 0^M b_r x (n - r) (1) In this equation if N = 0, then we get y (n) = 1/alpha_0 [sigma_r = 0^M b_r x (n - r)] (2) This equation corresponds to an FIR system Taking the z - transform on both sides of eq (2), we get rightarrow y (z) = 1/a_0 sigma_r = 0^M b_r x (z) z^- r rightarrow H (z) = y (z)/x (z) = 1/a_0 sigma_r = 0^M b_r z^- r = 1/a_0 [b_0 + b_1 z^- 1 + b_2 z^- 2 + Ellipses + b_M z^- M] (3) This is the General form of the transfer function of an FIR digital filter \r\n The transfer function of the given digital filter is H (z) = b_0 + b_1 z^- 1 + Ellipses + b_M z^- M/a_0 + a_1 z^- 1 + Ellipses + a_N z^- N (4)