Page 137 - DSP Integrated Circuits

122 Chapter 4 Digital Filters

4.2.4 Complementary FIR Filters

In many applications the need arises to split the input signal into two or more fre-
quency bands. For example, in certain transmission systems for speech, the speech
signal is partitioned into several frequency bands using a filter bank. A filter bank
is a set of bandpass filters with staggered center frequencies which cover the
whole frequency range. The first and the last filter are a lowpass and a highpass
filter, respectively. The filtered signals are then processed individually to reduce
the number of bits that have to be transmitted to the receiver where the frequency
components are combined into an intelligible speech signal.
A special case of band splitting filters, a lowpass and a highpass filter, is
obtained by imposing the following symmetry constraints. Let H(z) be an even-
order (N = odd) FIR filter. The complementary transfer function H c is defined by

A solution to Equation (4.11) is

The two transfer functions are complementary. Note that the attenuation is
6.02 dB at the crossover angle. We will later show that these two filters can be
effectively realized by a slight modification of a single FIR filter structure.
When one of the filters has a passband, the other filter will have a stopband
and vice versa. The passband ripple will be very small if requirements for the
other filter's stopband is large. For example, an attenuation of 60 dB (82 ~ 0.001) in
the stopband of the complementary filter corresponds to a ripple of only 0.00869
dB in the passband of the normal filter.
An input signal will be split by the two filters into two parts such that if the
two signal components are added, they will combine into the original signal except
for a delay corresponding to the group delay of the filters. A filter bank can be real-
ized by connecting such complementary FIR filters like a binary tree. Each node in
the tree will correspond to a complementary FIR filter that splits the frequency
band into two parts.

4.3 FIR FILTER STRUCTURES

An FIR filter can be realized using either recursive or nonrecursive algorithms
[16, 18]. The former, however, suffer from a number of drawbacks and should not
be used in practice. On the other hand, nonrecursive filters are always stable and
cannot sustain any type of parasitic oscillation, except when the filters are a part
of a recursive loop. They generate little round-off noise. However, they require a
large number of arithmetic operations and large memories.

4.3.1 Direct Form
Contrary to IIR filters, only a few structures are of interest for the realization of
FIR filters. The number of structures is larger for multirate filters, i.e., filters with
several sample frequencies. One of the best and yet simplest structures is the
direct form or transversal structure, which is depicted in Figure 4.5.