11.2 Frequency-Selective Discrete Filters 643
Group Delay
A measure of linearity of the phase is obtained from thegroup delay function, which is defined as
τ(ω)=−
dθ(ω)
dω
(11.6)
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
θ(ω)=
{
−N 0 ω 0 < ω≤ω 0
−N 0 ω 0 ω 0 < ω≤π
for 0≤ω≤π, then we have that the group delay is
τ(ω)=
{
N 0 0 < ω≤ω 0
0 ω 0 < ω≤π
which is not constant.
11.2.2 IIR and FIR Discrete Filters
n A discrete filter with transfer function
H(z)=
B(z)
A(z)
=
∑M− 1
m= 0 bmz
−m
1 +
∑N− 1
k= 1 akz
−k=
∑∞
n= 0
h[n]z−n (11.7)
is calledinfinite-impulse responseorIIRsince its impulse responseh[n]typically has infinite length. It is
also calledrecursivebecause if the input of the filterH(z)isx[n]andy[n]is its output, the input–output
relationship is given by the difference equation
y[n]=−
N∑− 1
k= 1
aky[n−k]+
M∑− 1
m= 0
bmx[n−m] (11.8)
where the output recurs on previous outputs (i.e., the output is fed back).
n The transfer function of afinite-impulse responseorFIRfilter is
H(z)=B(z)=
M∑− 1
m= 0
bmz−m (11.9)
Its impulse response ish[n]=bn,n=0,...,M− 1 , and zero elsewhere, thus of finite length. This filter is
callednonrecursivegiven that the input–output relationship is given by
y[n]=
∑M
m= 0
bmx[n−m]=(b∗x)[n] (11.10)
or the convolution sum of the filter coefficients (or impulse response) and the input.
