In Figure E-35 are some pole-zero plots of transfer functions of systems of the general form in which A = 1, the z’s are the zeros and the p’s are the poles.

Question

(a) Which one(s) have a magnitude frequency response that is nonzero at ? = 0 ?

(b) Which one(s) have a magnitude frequency response that is nonzero as ? ? ??

(c) There are two which have a bandpass frequency response (zero at zero and zero at infinity). Which one is more underdamped (higher Q)?

(d) Which one has a magnitude frequency response whose shape is closest to being a bandstop filter?

(e) Which one(s) have a magnitude frequency response that approaches K / ?6 at very high frequencies (K is a constant)?

(f) Which one has a magnitude frequency response that is constant?

(g) Which one has a magnitude frequency response whose shape is closest to an ideal lowpass filter?

(h) Which one(s) have a phase frequency response that is discontinuous at ? = 0?

Answer 1

(a) Which one(s) have a magnitude frequency response that is nonzero at ? = 0 ?
A,D,E,F The ones which do not have a zero at zero.

(b) Which one(s) have a magnitude frequency response that is nonzero as ? ? ??

(c) There are two which have a bandpass frequency response (zero at zero and zero at infinity). Which one is more underdamped (higher Q)?
C B and C are both bandpass. The poles in C are closer to the ? axis.

(d) Which one has a magnitude frequency response whose shape is closest to being a bandstop filter?
F To be a bandstop filter there must be zeros on the ? axis. None of these filters has zeros on the ? axis but F has two complex conjugate zeros near the ? axis.

(e) Which one(s) have a magnitude frequency response that approaches K / ?6 at very high frequencies (K is a constant)?
E To have this kind of response there must be 6 more finite poles than finite zeros. Only E has that quality.

(f) Which one has a magnitude frequency response that is constant?
A For a constant magnitude response the number of finite zeros and poles must be the same and the vector lengths from them to the ? axis must always be the same for a matched pair of a zero and a pole.
(g) Which one has a magnitude frequency response whose shape is closest to an ideal lowpass filter?
D A is all pass. B and C are bandpass. E is not easily classified but it has a peaked response near 4 radians/s and is therefore not a good approximation to a lowpass filter. F is almost a bandstop filter.

(h) Which one(s) have a phase frequency response that is discontinuous at ? = 0?
B,C To have a discontinuous phase at ? =0 , there must be a zero at zero.