Page 199 - DSP Integrated Circuits

184 Chapter 4 Digital Filters

4.14 An analog Cauer filter with A max = 0.09883 dB (p = 15%),A min = 67.09 dB
f c = 2 kHz, and f s = 4.7324 kHz has the following poles and zeros:

The filter is uniquely described by the notation C051525. Use this filter and
the bilinear transformation to design a digital lowpass filter with a passband
edge of 32 kHz when the sample frequency is 256 kHz.
(a) What are A max, A mi n, co cT, and (O ST for the digital filter.
(b) Sketch the pole-zero configuration for both the analog and digital niters.
4.15 Determine the poles for the lattice WDF in Examples 4.11 and 4.12.
Comment on the ordering of the poles in the two branches.
4.16 A third-order lowpass Butterworth filter has the transfer function H(z).
Sketch the pole-zero configuration as well as the magnitude response for a
digital filter obtained from the lowpass filter by the transformation z —> Z L
forL = 2and3.

4.17 Show that the reflection function, Equation (4.26), is an allpass function for a
reactance.
4.18 Derive the wave-flow graph for the two-port series and parallel adaptors
from the definitions of the incident and reflected waves.
4.19 Why is it not possible to realize bireciprocal wave digital filters of Chebyshev
I or II type?
4.20 A bireciprocal lattice wave digital filter of lowpass type is used to design a
symmetric bandpass (BP) filter. Hint: The well-known bandpass frequency
2
transformation z -> -2 is used.
(a) Determine the wave-flow graph for the lowpass filter.
(b) Determine the wave-flow graph for the BP filter.
(c) Determine the number of adaptors required in the BP filter.
(d) Determine the number of multiplications and additions in the BP filter.
(e) Sketch the magnitude functions for both filters.
4.21 Determine the Fourier transform for the new sequences that are derived
from the sequence x(n) according to:
(a) XI(H) = x(2n)

Also sketch the magnitude of the Fourier transform for the three sequences.

4.22 Determine and sketch the Fourier transform of a sequence x\(m) which is
wT
obtained from another sequence x(ri) with the Fourier transform X(eJ }
according to Equation (4.67).