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Problems 463
1. Determine the value of K for the system function in (8.73).
2. Using the K value from part 1, determine and plot the log-magnitude response of the
comb filter for L =6.
3. Describe the shape of your plot in part 2.
P8.9 (Adapted from [19]) As discussed in Chapter 1, echos and reverberations of a signal x(n)
can be obtained by scaling and delaying, that is,
y(n) =
∞∑
α k x(n − kD) (8.74)
where D is a positive integer for minimum delay and α k >α k−1 > 0.
1. Consider the IIR comb filter given by
k=0
H(z) =
1
1 − az −D (8.75)
Determine its impulse response. Explain why this filter can be used as a reverberator.
2. Consider the cascade of three allpass comb filters
H(z) = zD 1
− a 1
1 − a 1z −D 1 × zD 2
− a 2
1 − a 2z −D 2 × zD 3
− a 3
1 − a 3z −D 3
(8.76)
which can be used as a practical digital reverberator. Compute and plot the impulse
response of this reverberator for D 1 = 50, a 1 =0.7; D 2 = 41, a 2 =0.665; and D 3 = 32,
a 3 =0.63175.
3. Repeat part 2 for D 1 = 53, a 1 =0.7; D 2 = 40, a 2 =0.665; and D 3 = 31, a 3 =0.63175.
How does the shape of this reverberator different from the one in part 2? Which is a
good reverberator?
P8.10 Consider the 1st-order allpass system function given by
H(z) = a + z−1
, 0