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Consider a linear shift-invariant system with system function
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where la1 < 1.
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TRANSFORM ANALYSIS OF SYSTEMS
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[CHAP. 5
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(a) Find a difference equation to implement this system. (b) Show that this system is an allpass system (i.e., one for which the magnitude of the frequency response is constant).
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(c) H(z) is to be cascaded with a system G(z) s o that the overall system function is unity. If G(z) is to
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be a stable system, find the unit sample response, g(n).
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(a) Because cross-multiplying, we have ' l'(z)[l - az-'1 = ~ ( z ) [ z - - a*] Taking the inverse z-transform of both sides of the equation gives
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which is the desired difference equation.
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( b ) To show that this system is an allpass filter, note that the frequency response is
Therefore, the squared magnitude is
and H(ejw) is an allpass filter.
(c) The inverse system is
which has a pole at 2 = l/a" and a zero at z = a. Because la1 < I, the pole is outside the unit circle. Therefore, if g(n) is to be stable, the region of convergence must be lzl iI / l a l Thus, g(n) is the left-sided sequence
The system function of a causal FIR filter is
Find three other causal FIR filters with h(0) > 1 that have afrequency response with the same magnitude.
This filter has a pair of complex zeros and one real zero. The magnitude of the frequency response of this filter will not be changed if it is cascaded with an allpass filter [hat flips the zeros to their reciprocal location. Therefore, three other FIR filters that have the same magnitude response are
Note that each of these filters is causal with h(0) > I. The causality constraint along with the condition that h(0) > I prevents h(n) from being shifted or scaled by (- I ), two operalions that do not change the Fourier transform magnitude.
CHAP. 51
TRANSFORM ANALYSIS OF SYSTEMS
21 1
Let x ( n ) be a finite-length sequence that is zero for n < 0 and n > N. If x(n) is allowed to be complex. what is the maximum number of distinct finite-length sequences that have the same Fourier transform magnitude as x(n) Let X(z) be the z-transform of x ( n ) , which is of the form
Each zero may be reflected about the unit circle by multiplying by an allpass filter
without changing the magnitude of X(eJU). Because there are two possible locations for each of the N zeros. the number of distinct tinite-length sequences (ignoring delays and multiplication by a unit magnitude complex number) is 2'.
Show that the group delay of an allpass filter is nonnegative for all w . If a is real and Ial < I , the group delay of a filter that has a system function
which has a single pole at z = a, is (see Rob. 2.19)
Similarly, the group delay for a filter with the system function
which has a single zero at z = a, is r2(o) = -r,(o) Furthermore, if
the group delay is
Therefore, the group delay of a single allpass factor of the form
which, because IciI i1, is positive for all w . For complex roots, the allpass factors have the form
TRANSFORM ANALYSIS O F SYSTEMS Therefore, with cu = ale.^^, the frequency response is
(CHAP. 5
and the group delay is
which is nonnegative for all w .
Show that the phase of an allpass filter with h ( n ) real, if plotted as a continuous function of w , is nonpositive for all w .
The group delay is minus the derivative ot'the phase. Therefore, the phase is related to the group delay as follows:
Because the general form for the frequency response of an allpass filter is
then
= Thus, @,,(O) 0, and the positivity of q,(w)makes the phase nonpositive.
Minimum Phase
Suppose that H ( z ) and C(z) are rational and have minimum phase. Which of the following filters have minimum phase
(a) H ( z ) G ( z )
(b) H ( z )
+G(z)
If t l ( z ) and G ( z ) have minimum phase, neither H ( z ) nor G ( z )have any poles or zeros outside the unit circle. Because the poles and zeros of H ( z ) G ( z ) the union of the poles and zeros of H ( z ) and G(z),H ( z ) G ( z )will are not have any poles or zeros outside the unit circle and. therefore, has minimum phase.
(6) If H ( z )and G ( z ) have minimum phase. i t is no1 necessarily true that H ( z ) G ( z )will have minimum phase. We may show this by a simple counter example. If
G ( z )=
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