FIR Filter Design Questions and Answers

Answer: A

The correct matching is P-3, Q-1, R-2, S-4.

Read each plot by asking where the poles boost and the zeros suppress the response on the unit circle (z = 1 is zero frequency, z = -1 is half the sampling rate):

• Plot 3: a pole near z = 1 and a zero at the origin, so low frequencies are emphasised: low pass (P).
• Plot 1: a zero at z = 1 kills the DC response and a pole near z = -1 boosts the highest frequencies: high pass (Q).
• Plot 2: a complex pole pair at an intermediate angle with a double zero at the origin peaks the response in a middle band: band pass (R).
• Plot 4: all four poles at the origin and only zeros elsewhere, so the impulse response is finite: FIR (S).

The other codes swap the low pass and high pass plots or give the FIR label to the band pass plot, which has poles away from the origin and so an infinite impulse response.

Answer: B

FIR filters are non-recursive, with the present output depending on present and past input samples only.

A finite impulse response filter computes

\[ y[n] = \sum_{k=0}^{M} b_{k}\, x[n-k] \]

so its output uses no past outputs. Without feedback its impulse response ends after M + 1 samples, and the filter is always stable.

A filter whose output also depends on past outputs is recursive; that is the IIR (infinite impulse response) structure. So the options combining recursive with inputs only, or non-recursive with past outputs, contradict the definitions.