Group Delay and Phase Delay in Filters

Summary. In this chapter, we explore the way in which a filter’s output is delayed

with respect to its input. We show that if the filter’s phase is not linear, then the

filter delays signals with different carrier frequencies by different amounts.

Keywords. group delay, phase delay, bandpass signals, carrier frequency.

27.1 Group and Phase Delay in Continuous-time Filters

Suppose that one has a narrow-band bandpass signal, y(t). That is, suppose

that y(t) can be written as

y(t) = e2πjFctx(t)

where X(f) = 0 for all |f| > B, B << Fc , and Fc is the signal’s carrier

frequency.

Consider the Fourier transform. representation of x(t)

x(t) = _ B

−B

e2πjftX(f) df

= _ B

−B

|X(f)|ej_ X(f)e2πjft df.

We find that x(t) is composed of sinusoids of the form

|X(f)|ej_ X(f)e2πjft, |f| ≤ B.

Similarly, we find that y(t) is composed of sinusoids of the form

|X(f)|ej_ X(f)e2πj(f+Fc)t, |f| ≤ B.

186 27 Group Delay and Phase Delay in Filters

When the components of the function y(t) pass through a filter whose frequency

response is H(f), then the components of the resulting function, which

we call z(t), are

|H(Fc + f)|ej_ H(Fc+f)|X(f)|ej_ X(f)e2πj(f+Fc)t, |f| ≤ B.

Let us define the function ΘH(f) ≡ _ H(f). For relatively small f we can

approximate ΘH(Fc + f) by ΘH(Fc) + Θ_

H(Fc)f. Also, assuming that the

magnitude of the filter response does not change much for small f, we can

approximate |H(Fc +f)| by |H(Fc)|. (This “0th-order” approximation is reasonable,

because a small enough error in the magnitude can only produce a

small error in the final estimate of the signal.)We approximate the constituent

components of z(t) by

|H(Fc)|ej(ΘH(Fc)+Θ

_

H(Fc)f)|X(f)|ej_ X(f)e2πj(f+Fc)t, |f| ≤ B.

Rewriting this, we find that the constituent components of z(t) are

|H(Fc)|ejΘH(Fc)e2πjFct|X(f)|ej_ X(f)ejΘ

_

H(Fc)f e2πjft, |f| ≤ B.

This, in turn, can be written as

|H(Fc)|e2πjFc(t+ΘH(Fc)/(2πFc))|X(f)|ej_ X(f)e2πjf

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