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A causal linear shift-invariant discrete-time system has a system function
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H (z) =
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0.72-')(I - j2z-')(I j2z-') (1 - 0.82-')(I 0 . 8 ~ - I )
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(a) Find a minimum phase system function H,,,(z) and an allpass system function Hap(z) such that
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(b) Find a minimum phase system function Hmi,(z)and a linear phase system function Hlp(z) such that
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Let x ( n ) be a real-valued minimum phase sequence. Find another real-valued minimum phase sequence y(n) such that x(0) = y(0) and y(n) = Ix(n)l. Find two different real-valued sequences that satisfy the following constraints: I,
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2. x(0) = Oandx(1) > 0. I X ( ~ J " =~$ ) ~
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5.60 If a feedback system of the form G ( z ) = K is used to compensate the system
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for what values of K will the closed-loop system be stable 5.61 For the system
H(z) = I
+ 1 .2zr1+ 1.5z-'
find a feedback system of the form
G(z) = I
+ g(l)z-I + g(2)z-2
that will move the poles of H ( z ) to z = 0.5 and 5.62
= -0.5.
Find the closed-loop system function of a feedback network with
H(z) =
and G ( z ) =
- iz-I.
CHAP. 51
TRANSFORM ANALYSIS OF SYSTEMS
Answers to Supplementary Problems
Unstable. Yes.
Yes.
(a) Yes. (b) No.
Yes.
None.
H ( z ) = 0.4 -t0 . 8 ~ -- 0.5z-' '
+ 0 . 2 r 3+ z - ~ .
b(k) = a ( p --k ) fork = 0 , 1 ,
.. . , p - 1 , and b ( p ) = 1 .
Hmin(z) FlR with each zero having even order (i.e., Hmin(z) G 2 ( z ) is = where G ( z )is a minimum phase system).
2 H ( z )= 1 - az-I'
(a)and ( b )have minimum phase but ( c ) does not.
TRANSFORM ANALYSIS OF SYSTEMS
[CHAP. 5
x , ( n ) = S(n - 1) - fS(n
2 ) and x2(n) = aS(n - I )
g ( l ) = - 1.2 and g(2) = -2.
2 -2
(h) Q ( z ) = '
-1 + A z - 2 I + "- I- kz-2 iz
6
The DFT
6.1 INTRODUCTION
In previous chapters, we have seen how to represent a sequence in terms of a linear combination of complex exponentials using the discrete-time Fourier transform (DTFT) and how the sequence values may be used as the coefficients in a power series expansion of a complex-valued function of z. For finite-length sequences there is another representation, called the discrete Fourier transform (DFT). Unlike the DTFT, which is a continuous function of a continuous variable, w , the DFT is a sequence that corresponds to samples of the DTFT. Such a representation is very useful for digital computations and for digital hardware implementations. In this chapter, we look at the DFT, explore its properties, and see how it may be used to perform such tasks as digital filtering and evaluating the frequency response of a linear shift-invariant system.
6.2 DISCRETE FOURIER SERIES
Let K(n) be a periodic sequence with a period N:
Although, strictly speaking,K(n) does not have a Fourier transform because it is not absolutely summable, it can be expressed in terms of a discrete Fourier series (DFS) as follows:
which is a decomposition of K(n) into a sum of N harmonically related complex exponentials. The values of the discrete Fourier series coefficients, $(k), may be derived by multiplying both sides of this expansion by e - j Z n n l l Nsumming over one period, and using the fact that the complex exponentials are orthogonal: ,
The result is
Note that the DFS coefficients are periodic with a period N:
Equations (6.1) and (6.2) form a DFS pair, and we write
EXAMPLE 6.2.1
Let us find the discrete Fourier series representation for the sequence
THE DlT
[CHAP. 6
where
x(n) =
O(n<5
else
Note that P(n) is a periodic sequence with a period N = 10. Therefore, the DFS coefficients are
which, for 0 5 k 5 9 , may be simplified to
k=O x(k) = k odd
2(k + N ) = f ( k )
k even
The DFS coefficients for all other values of k may be found from the periodicity of ~ ( k ) :
A notational simplification that is often used for the DFS is to define
WN + e - ~ 2 n / N for the complex exponentials and write the DFS pair as follows:
The discrete Fourier series has a number of useful and interesting properties. A few of these properties are described below.
Linearity
The DFS pair satisfies the property of linearity. Specifically, ifPl(n) and i 2 ( n ) are periodic with period N, the DFS coefficients of the sum are equal to the sum of the coefficients for .f (n) and f z(n) individually,
Shift
If a periodic sequence I ( n ) is shifted, the DFS coefficients are multiplied by a complex exponential. In other words, if z ( k ) are the DFS coefficients for P(n), the DFS coefficients for y(n) = 2(n - no) are
f (k) = w f 0 2 ( k )
Similarly, if P (n ) is multiplied by a complex exponential,
the DFS coefficients of i ( n ) are shifted:
Y (k) = $(k + k,,)
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