Signals and Systems - Electrical Engineering

642 C H A P T E R 11: Introduction to the Design of Discrete Filters

the receiver be ideally equal within a time delay and a constant attenuation factor. To achieve this,

the transfer function of an ideal communication channel should equal that of an all-pass filter with

a linear phase.

Indeed, if the output of the transmitter is a discretized baseband signalx[n] and the recovered signal

at the receiver isαx[n−N 0 ], for an attenuation factorαand a time delayN 0 , ideally the channel is

represented by a transfer function

H(z)=

Z(αx[n−N 0 ])

Z(x[n])

=αz−N^0 (11.5)

The constant gain of the all-pass filter permits all frequency components of the input to appear in the

output. The linear phase simply delays the signal, which is a very tolerable distortion.

To appreciate the effect of linear phase, consider the filtering of a signal

x[n]= 1 +cos(ω 0 n)+cos(ω 1 n) ω 1 = 2 ω 0 n≥ 0

using an all-pass filter with transfer functionH(z)=αz−N^0. The magnitude response of this filter is

α, and its phase is linear, as shown in Figure 11.3(a). The steady-state output of the all-pass filter is

yss[n]= 1 H(ej^0 )+|H(ejω^0 )|cos(ω 0 n+∠H(ejω^0 ))+|H(ejω^1 )|cos(ω 1 n+∠H(ejω^1 ))

=α[ 1 +cos(ωo(n−N 0 ))+cos(ω 1 (n−N 0 ))]=αx[n−N 0 ]

which is the input signal attenuated byαand delayedN 0 samples.

Suppose then that the all-pass filter has a phase function that is nonlinear, for instance, the one in

Figure 11.3(b). The steady-state output would then be

yss[n]= 1 H(ej^0 )+|H(ejω^0 )|cos(ω 0 n+∠H(ejω^0 )+|H(ejω^1 )|cos(ω 1 n+∠H(ejω^1 ))

=α[1+cos(ω 0 (n−N 0 ))+cos(ω 1 (n−0.5N 0 ))]6=αx[n−N 0 ]

In the case of the linear phase each of the frequency components is delayedN 0 samples, and thus

the output is just a delayed version of the input. On the other hand, in the case of a nonlinear phase

the frequency component of frequencyω 1 is delayed less than the other two frequency components,

creating distortion in the signal so that the output is not a delayed version of the input.

FIGURE 11.3

(a) Linear and (b) nonlinear phase.

θ(ω)

ω 0 ω 1 π ω

−π −N 0 ω 0

(b)

ω 0

θ(ω)

−N 0 ω 0

−N 0 ω 1

ω 1 π ω

−π

(a)