Although this problem set may seem long, note that problems 9.2, 9.3,
and 9.4 require VERY short answers.
Problem 9.1 The decimation-in-time FFT algorithm was developed in
Section 9.3 for radix 2, i.e., N=2n. A similar approach
leads to a radix-3 algorithm when N=3n.
Draw a flow graph for a 9-point decimation-in-time
algorithm using a 3x3 decomposition of the DFT.
Problem 9.2 Oppenheim and Schafer, problem 7.3, part a only
Problem 9.3
Suppose we design a discrete-time filter using the impulse invariance
technique with a CT lowpass filter as a prototype. The
prototype filter has a cutoff frequency of
Wc=2p(1000) rad/sec and the impulse invariance
transformation uses T=0.2 ms.
Assume that aliasing effects are negligible. What is the cutoff
frequency wc of the resulting discrete-time filter?
Problem 9.4 We wish to design a discrete-time lowpass filter using the bilinear
transformation on a continuous-time ideal lowpass filter. Assume that
the continuous-time prototype has a cutoff frequency
Wc=2p(2000) rad/sec and we choose the bilinear
transformation parameter T=0.4 ms. What is the cutoff frequency
wc of the resulting discrete-time filter?
Problem 9.5 Design a single-pole lowpass discrete-time filter with a 3-dB
bandwidth of 0.2p using the bilinear transformation applied to the
analog filter
Hc(s)=
Wc
s+Wc
where Wc is the 3-dB bandwidth of the analog filter.
(a)
Design the filter, i.e. determine H(z).
(b)
Sketch a structure for realizing this filter.
Problem 9.6 Suppose that you have a causal continuous-time filter defined by the
system function:
Hc(s)=
s+0.1
(s+0.1)2+9
.
(a)
Use impulse invariance to determine the system function,
H(z) of a discrete-time filter.
(b)
Sketch the pole-zero plot and the frequency response magnitude
of the resulting discrete-time filter.
Problem 9.7 Consider designing a discrete-time filter with the system
function H(z) from a continuous-time filter with rational system
function Hc(s) by the transformation
H(z)=Hc(s)
½ ½
s=(1-z-2)/(1+z-2)
.
(9.1-1)
(i)
What contour does the jW axis in the s-plane map
to in the z-plane? Justify your answer.
(ii)
Determine the mapping between W and w
defined by this transformation. Sketch W as a function of
w.
(iii)
Suppose that the continuous-time filter is a stable
lowpass filter with passband frequency response such that
1-d1£ |Hc(jW)|£ 1+d1 for
|W| £ 1.
If the discrete-time system is obtained by the transformation in
Equation 9.1-1 above, determine the values of w in the interval
|w|£p for which
1-d1£ |H(e
jw
)|£ 1+d1 .
Does this transformation provide a reasonable way to design DT lowpass
filters from analog lowpass filter prototypes?