Digital Filters
27
symmetric impulse response
antisymmetric impulse
response
a symmetric impulse response is such that
h(n) = h(N - n), n = [0, . . . , N ] ,
(15)
and an antisymmetric impulse response is such that
h(n) = -h(N - n), n = [0, . . . , N ] .
(16)
It is possible to show that the symmetry (or antisymmetry) of the impulse
response is a sufficient condition to ensure the linearity of phase. This property
is important to ensure the invariance of the shape of signals going through the
filter. For instance, if a sawtooth signal is the input of a linear-phase lowpass
filter, the output is still a sawtooth signal with rounded corners.
In order to prove that symmetry is a sufficient condition for phase linearity
for an N -th order FIR filter (with N odd integer), we write the transfer function
as
H(z) = h(0) + . . . + h(
N - 1
2
)z
-
N -1
2
+ h(
N - 1
2
)z
-
N +1
2
+ . . . + h(0)z
-N
=
N -1
2
n=0
h(n) z
-n
+ z
-N +n
.
(17)
The frequency response can be expressed as
H() =
N -1
2
n=0
h(n) e
-jn
+ e
j(-N +n)
=
N -1
2
n=0
h(n)e
-j
N
2
e
-j(n-
N
2
)
+ e
j(n-
N
2
)
(18)
= e
-j
N
2
2
N -1
2
n=0
h(n) cos ((n -
N
2
)) .
In the latter term we have isolated the phase contribution from a (real) weighted
sum of sinusoidal functions. The phase contribution is a straight line having slope
-N/2, as we have already seen in the special case of the first-order averaging
filter (5). Where the real term changes sign there are indeed 180
phase shifts, so
that we should more precisely say that the phase is piecewise linear. However,
phase discontinuities at isolated points do not alter the overall constancy of
group delay, and they are nevertheless irrelevant because at those points the
magnitude is zero.
The same property of piecewise phase linearity holds for antisymmetric im-
pulse responses and for even values of N .
At this point, we are going to introduce a very useful FIR filter. It is linear
phase and it has order 2 (i.e., length 3). The averaging filter (5) was also a linear
phase filter, but it is not possible to change the shape of its frequency response
without giving up the phase linearity. In fact, filters having form H(z) = h(0) +
h(1)z
-1
can have linear phase only if h(0) = ħh(1), and this force them to
have a magnitude response such as that of fig. 1 or like its high-pass mirrored