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both H ( z ) and G ( z )have minimum phase. However, the sum
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CHAP. 51
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TRANSFORM ANALYSIS OF SYSTEMS may have a zero anywhere in the z-plane by choosing the appropriate values for A and B . For example, because H(z) G(z) has a zero at 0.5B 0.75A -
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to place a zero at z = 2, we may set A = I and solve the following equation for B:
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A nonminimum phase causal sequence .r(n) has a z-transform
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For what values of the constant cr will the sequence y(n) = crnx(n) be minimum phase
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Multiplying a sequence by d'moves the poles and zeros radially by a factor of a:
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In order for Y ( z ) to be minimum phase, all of the poles and zeros must be inside the unit circle. Because the singularity (pole or zero) of X ( 2 ) that is the furthest from the unit circle is the zero at z = y ( n ) will be minimum phase if Iff1 <
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A causal linear shift-invariant system has a system function
Find a factorization for H(z) of the form
where Hmi,(z) has minimum phase. and Hap(z)is an allpass filter.
The system function H(z) has a nonminimum phase factor, (1 - 2 ~ - ~which may be written as the product of a ), minimum phase term and an allpass factor as follows:
Therefore, H(z) may be written as the product of a minimum phase system with an allpass system as follows:
A causal linear shift-invariant system has a system function
H (z) = Find a factorization for H ( z ) of the form
+ z-I)(2 - 3 z - I )
1iz-l
where Hmi,(z) has minimum phase, and Hlp(z)is a linear phase system.
TRANSFORM ANALYSIS OF SYSTEMS
[CHAP. 5
This system is not minimum phase because the factor ( 2 - 3 z - ' ) corresponds to a zero outside the unit circle at z = However, we may express this factor as the product of a minimum phase term with a linear phase term as follows. First, we reflect the zero about the unit circle and replace it with a pole:
Then, we multiply this term with a linear phase factor that has a zero at z =
$ and a zero at z = ::
Thus, the factorization for H ( z ) is
H ( z )=
3 + z-' (2z-' - 3)(2 - 32-') ( I - 4z-')(2z-l - 3 )
53 .1
Find a real-valued causal sequence with ~ ( 0> 0 and )
I X ( @ ' ) [ ~ = (1
+ a2)- 20 cos w
- .elW
We begin by expressing IX (ej'")12in terms of complex exponentials:
I X ( P ' " ) I ~ = (I+ a 2 )
UP-^^^
Replacing d" with z, and e-1'" with z - ' , we have
G ( z )= X ( Z ) X ( Z - ' ) = ( I + a 2 ) - az - u z - ' = ( I
- u z f l ) ( l- a z )
Therefore, a real-valued causal sequence with the given magnitude with .u(O) > 1 is
.r(n) = 6 ( n )- u6(n - I)
Find the minimum phase system that has a magnitude response given by
To solve this problem, we begin by expressing ( ~ ( e j ' " )in ~ l terms of complex exponentials as follows:
Replacing ei" by z, and e-JWby z-I, this becomes
The minimum phase system is then formed by extracting the poles and zeros that are inside the unit circle:
Use the initial value theorem to show that if hmi,(n) is a minimum phase sequence, and if h ( n ) is a causal sequence with the same Fourier transform magnitude, then
CHAP. 51
TRANSFORM ANALYSIS OF SYSTEMS
The initial value theorem states that for a causal sequence, the initial value may be found from the z-transform as follows: h(0) =: I % H(z) Let hmin(n) a minimum phase sequence, and let h ( n ) be the nonminimum phase sequence that is formed by be reflecting a zero from inside the unit circle at z = a to its conjugate reciprocal location at z = I/a*:
have the same Fourier transform magnitude. Because ( z - ' - a * ) / ( l - az-I) is an allpass filter. h ( n ) and hmin(n) Using the initial value theorem, we may compare the value of h(0) to h,l,i,,(0):
and because la1 < I, lh(0)I < lhm,,(0)l. Because the magnitude of Ih(O)l is reduced each time that a zero of Hmin(z) is flipped outside the unit circle,
Ih(0)l < Ihrnm(0)l
for any sequence h ( n ) that has a Fourier transform with the same magnitude as that of hmi.(n).
Prove the minimum energy delay property for minimum phase sequences. Let hm,,(n) be a minimum phase sequence, and let a k be a zero of Hmin(z). Then Hmin(z) be written as may
HmIn(z) = ( 1
-akz-'Fmin(z)
(5.23)
where G m i n ( zis another minimum phase sequence. Because Hm,,(z) is minimum phase, lakI < I . Let H ( z ) be the ) causal nonminimum phase sequence that is formed by replacing the zero at z = a k with a zero at z = ] / a ; :
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