Linear Phase

11.2.1 Linear Phase

A filter changes the spectrum of its input in magnitude as well as in phase. Distortion in magnitude can be avoided by using an all-pass filter with unit magnitude for all frequencies. Phase distortion can

be avoided by requiring the phase response of the filter to be linear. For instance, when transmitting

a voice signal in a communication system it is important that the signals at the transmitter and at

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 signal x[n] and the recovered signal

at the receiver is αx[n −N 0 ], for an attenuation factor α and a time delay N 0 , ideally the channel is represented by a transfer function

Z(αx[n −N 0 ])

H(z) =

= αz −N 0 (11.5)

Z(x[n])

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 = 2ω 0 n ≥0 using an all-pass filter with transfer function H(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

y jω ss [n] = 1H(e ) + |H(e 0 ) | cos(ω 0 n + ∠H(e 0 )) + |H(e 1 ) | cos(ω 1 n + ∠H(e 1 ))

= α [1 + cos(ω o ( n −N 0 )) + cos(ω 1 ( n −N 0 )) ] = αx[n − N 0 ] which is the input signal attenuated by α and delayed N 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

j0

jω

ss [n] = 1H(e ) + |H(e 0 ) | cos(ω 0 n

+ ∠H(e jω 0 )

+ |H(e jω 1 ) | cos(ω

+ ∠H(e jω 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 delayed N 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 frequency ω 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.

(a)

(b)

11.2 Frequency-Selective Discrete Filters 643

Group Delay

A measure of linearity of the phase is obtained from the group delay function, which is defined as

The group delay is constant when the phase is linear. Deviation of the group delay from a constant indicates the degree of nonlinearity of the phase. In the above cases, when the phase is linear (i.e., for 0 ≤ ω ≤ π),

θ (ω) = −N 0 ω ⇒ τ (ω) = N 0

and when the phase is nonlinear or

for 0 ≤ ω ≤ π, then we have that the group delay is

which is not constant.