Digital Filters
29
dc frequency
(a)
(b)
0
1
2
3
0
0.2
0.4
0.6
0.8
frequency [rad/sample]
magnitude
0
1
2
3
-3
-2.5
-2
-1.5
-1
-0.5
0
frequency [rad/sample]
phase [rad]
Figure 8: Frequency response (magnitude (a) and phase (b)) of the length-3
linear phase FIR filter with coefficients a
0
= 0.17654 and a
1
= 0.64693
frequencies, we can easily get a magnitude response larger than one at zero
frequency (also called dc frequency). Especially in signal processing flowgraphs
having loops it is often desirable to normalize the maximum value of the magni-
tude response to one, in such a way that amplifications generating instabilities
can be avoided. Of course, it is always possible to rescale the filter input or
output by a scalar that is reciprocal to H(0) = a
1
+ 2a
0
so that the response
is forced to be unitary at dc
. Instead of drawing the pole-zero diagram of the
filter, let us represent the contours of the logarithm of the magnitude of the
transfer function, evaluated on the complex plane in a square centered on the
origin (see fig. 9). The effects of the double pole in the origin and of the zeros
z = -0.29695 and z = -3.36754 are clearly visible. A filter such as (8) has been
proposed as part of an algorithm for synthesis of plucked string sounds [104].
-4
-2
0
2
4
-4
-2
0
2
4
Magnitude of the Transfer Function
Figure 9: Magnitude of the transfer function [in dB] of an order-2 FIR filter on
the complex z plane
5
The reader is invited to reformulate the system (22) with
1
= 0 and
2
= . This
corresponds to setting the magnitude at dc and Nyquist rate.