Page 177 - DSP Integrated Circuits

162 Chapter 4 Digital Filters

4.19 BIRECIPROCAL LATTICE WAVE DIGITAL
FILTERS

Significant simplifications of the digital filter algorithm can be made if the magni-
tude functions I Sn I ^ and I $21' ^ have certain symmetric properties around n/2.
Lattice wave digital filters of this type are called bireciprocal filters. The attenua-
tion is always 3.01 dB at coT = n/2. These filters are also referred to as half-band
IIR filters. Note that the attenuation in the passband depends on the stopband
attenuation. Half-band filters are highly efficient for decimation and interpolation
in multirate signal processing systems. The primary motivation for using half-
band filters is the dramatically reduced computational complexity.

EXAMPLE 4.12
Use the program just mentioned [11], or MATLAB, to design a bireciprocal low-
pass filter of Cauer type, with A maxreq = 0.05 dB, f creq = 150 kHz, A mj nre<7 = 73 dB,
f sreq = 230 kHz, and f sampie = 800 kHz.
For bireciprocal filters we have, due to the inherent antisymmetry,

where

and

The passband ripple is extremely small. Hence, the sensitivity in the passband
can, according to Equation (4.54), be expected to be very low.
Now, bireciprocal filters are defined by the constraint

which implies the following symmetry constraint for the passband angle and stop-
band angle:

The corresponding relation between the stopband and passband edges in the
f-domain is

Hence, we get