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H ( z )=
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I H(eJW)I I Hm,,(el'")i =
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Expressing Eqs. (5.23) and (5.24) in the time domain. we have
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hm,.(n) = gmin(n) - akgmln(n - 1 ) h ( n ) = gmin(n - 1) - a,l~mln(n)
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Now, let us evaluate the difference between the partial sums of ( h , , , ~ n ) / and [h(n)12: ~
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Expanding the square and canceling the common terms, this becomes
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which is greater than zero because \ak < 1. Therefore, 1
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TRANSFORM ANALYSIS OF SYSTEMS
[CHAP. 5
Because gmi,(n)is minimum phase, this procedure may be repeated for any remaining zeros in Gmi,(z).Therefore, it follows that any causal nonminimum phase sequence that has the same Fourier transform magnitude as Hmi,(z) will have a partial sum that is smaller than that for hmi,(n). Feedback
Suppose that we have an unstable second-order system
H(z) =
that we would like to stabilize with the feedback system shown below.
Find the system function of the closed-loop system, Q ( z ) ,and determine the values for the feedback gain K that result in a stable system.
The system function of the feedback network is
Therefore, this system will be stable if
which implies that
and if
which is automatically satisfied by the first condition, K
Let H ( z ) be an unstable system with
H(z) =
I 1 . 5 ~ - '- 3
(a) Using a feedback system of the form
G(z)= KZ-I
determine the values for the gain K , if any, that will stabilize this system.
(b) Repeat part (a)using a feedback system of the form
CHAP. 51
TRANSFORM ANALYSIS OF SYSTEMS
( a ) The system function of the feedback network is
Because the coefficient multiplying the term z - ~ larger than 1, this system will be unstable for all K . is
(b) With G ( i )= K z - ~ ,the closed-loop system function becomes
Q(z) =
+ H ( z ) . Kz-2
H(z)
I - 1.5~-I
+ (K - 3 ) z r 2
This system will not be stable unless
la(2)I = IK - 31 < I
which requires that
2<K<4
In addition, however, we must have
which requires that K z 3.5. Therefore, in order for this system to be stable, we must have
Let H ( z ) be a plant with a system function
Find a feedback system G ( z ) of the form
that will place a second-order pole in the closed-loop system at z = 1.
The system function of the closed-loop system is
To place a second-order pole at z = 1, we want to find c and d so that
The solution is c = -8.7 and d = 3.4.
TRANSFORM ANALYSIS O F SYSTEMS
[CHAP. 5
Supplementary Problems
The System Function
The input to a causal linear shift-invariant system is x(n) = u-17 - I ) The z-transform of the output of this system is
+( ~ U ( I , )
Find the system function H(z) of the filter.
A causal linear shift-invariant digital filler has a system function given by I
H (;) = (I
z-1)2(1 - az-'
+ $2-2)
Determine whether or not the filter is stable. The system function of a linear shift-invariant system is H ( z ) = el" If h ( n ) is a right-sided sequence, is this system stable' Let x ( n ) be a real-valued, causal sequence with a discrete-time Fourier transform X(cJ") = XR(el'*) jX,(eJ") If find x(n). The system function of a linear shift-invariant system is H(z) = log(l - f r - ~ ' ) Is this system causal Which of the following z-transforms could be Ihe system function of a causal system'
d (I ( a ) X(z) = -~ z - I ) ~ d z (3 - 22-')2
( z (>
The system function of a causal filter is H(z) = For what values of a will this filter be stable' A stable filter has a system function H(i) = such Find a stable and causal system G ( e J W ) that
+ UZ-' + 0 . 3 ~ - ~
(I I
3z-I)(l - \z-') O.2zr1 + 0 . 4 ~ ~ ~
If the frequency response of a stable linear shift-invariant system is real and even, will the inverse system be stable
CHAP. 51
TRANSFORM ANALYSlS OF SYSTEMS
Systems with Linear Phase
The system function of an FIR filter is H(z) = ( I
+ 0.22-' + 0 . 8 : ~ - ~ ) ~
Find a linear phase system that has a frequency response with the same magnitude. An FIR filter with generalized linear phase has the following properties: I.
h(n) is real, and h(n) = 0 for n c 0 and for n > 5.
xi=,(- = 0. l)"h(n)
3. 4.
H(z) is equal to zero at z = 0.7e~"'~.
zJ :
H(eJW)dw 4n. =
Find H(z). An FIR filter with a real-valued unit sample response has a group delay
t ( ~ )2 =
If the system function has a zero at z = j, and H(z)~;=,= 1, find h(n).
Let x(n) be a sequence that is equal to zero for n X(z) = 3(1 - 0.2:-')(I
and n > 5. If the z-transform of w(n) is
+ 0 . 5 ~ + 0.8z-')(l + 0 . 4 2 ' '
0.5~-~)
how many other sequences are equal to zero for n < 0 and n r 5, have the same initial value as x(n), and have the same phase
Allpass Filters
The system function of an FIR filter is
Find another causal FIR filter with h(n) = 0 for n > 4 that has the same frequency response magnitude.
A causal and stable allpass filter has a unit sample response that is real. The system function contains three poles, one of which is at z = 0.8. If H(z) has a zero at z = 2e'"I4, what is H(z) A linear shift-invariant system has a system function
If H(z) is an allpass filter, what is the relationship between the numerator coefficients h ( k ) and the denominator coefficients u(k)
Minimum Phase
What can you say about the poles and zeros of a minimum phase system H,,,(z) if there is an allpass system H,,(z) such that Hmm(z)Hap(z) = Hipl:z) where H,,(z) is a causal (generalized) linear phase system
TRANSFORM ANALYSIS O F SYSTEMS Find the minimum phase system that has a magnitude response given by
[CHAP. 5
Suppose that H ( z ) and G(z) are rational and have minimum phase. Which of the following also have minimum phase
(a) H - ' ( z ) , (b) H ( z ) / G ( z ) ,
H(z)
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