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(b) With x(n) = anu(n),the z-transform is
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and the z-transform of the autocorrelation sequence is R,(z) =
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The autocorrelation sequence may be found by computing the inverse z-transform of R,(z). Performing a partial fraction expansion of R, (z), we have
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Thus, because the region of convergence is la\
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z < l/laJ,the inverse z-transform is
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In many disciplines, differential equations play a major role in characterizing the behavior of various phenomena. Obtaining an approximate solution to a differential equation with the use of a digital computer requires that the differential equation be put into a form that is suitable for digital computation. This problem presents a transformation procedure that will convert a differential equation into a difference equation, which may then be solved by a digital computer. Consider a first-order differential equation of the form
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where yu(0)=yo. Because numerical techniques are to be used, we will restrict our attention to investigating yu(t) at sampling instants nT where T is the sampling period. Evaluating the differential equation at t = n T , we have
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CHAP. 41
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THE z-TRANSFORM
From calculus we know that the derivative of a function y,(t) at t = nT is simply the slope of the function at t = n T . This slope may be approximated by the relationship
(a) Insert this approximation into the sampled differential equation above and find a difference equation that relates y(n) = y,(nT) and x ( n ) = x,(nT), and specify the appropriate initial conditions. ( b ) With x,(t) = u ( t ) and y,(O-) = I, numerically solve the differential equation using the difference equation approximation obtained above.
(c) Compare your approximation to the exact solution.
using the approximation
-Y&W
- [ y , ( n ~ ) - y,(nT - TI] T
we have
[ya(nT)
- ya(nT - T)] + crya(nT) = .r,(nT)
y,(O-) = yo
If we let y(n) = ya(nT) and x(n) = xa(nT),
With this becomes ~ ( n - ay(n )
1) = a T x ( n )
~ ( 0 = yo )
(b) Using the one-sided z-transform to solve this difference equation, we have
We must now derive the initial condition on y(n) at time n = - 1 from the initial condition at n = 0. From the difference equation, we have y(0) - ay(-I) = aTx(0) With y(0) = 1 and x(0) = 1, the initial condition becomes
With x,(t) = u(t) or x(n) = u(n),
Therefore. using the given initial condition, we have
THE 2-TRANSFORM
Performing a partial fraction expansion gives
[CHAP. 4
and we may find v(n) by taking the inverse z-transform:
Because this may be written as
(c) The solution to the differential equation is a sum of two terms. The first is the homogeneous solution, which is yh(t)= A e - O 1 where A is a constant that is selected in order to satisfy the initial condition ~ ( 0 - = 1. The second is the ) particular solution, which is
Thus, the total solution is
Evaluating this a1 time I = 0-,
we see that in order to match the initial conditions. we must have
If we compare this to the approximation in part ( h ) ,note that if T
C - ~ ~ ) , _ , T=
< I, <
(e )
aT -,I
+( Y T ) - ~
Supplementary Problems
z-'kansforms 4.41
Find the 2-transform of
CHAP. 41
THE z-TRANSFORM
The z-transform of a sequence x(n) is
If the region of convergence includes the unit circle, find the DTFT of x(n) at w = r / 2 .
Find the z-transform of each of the following sequences:
(a) x(n) = (--l)"u(n)
( b ) x(n) = ;u(n - I)
(c) x(n) = z cosh (crn)u(n)
Find the z-transform of the sequence
Find the z-transform of the sequence
How many different sequences have a z-transform given by
The sequence y(n) is formed from x(n) by
where X(z) = sinzr'. Find Y(z).
If x(n) is an absolutely summable sequence with a rational z-transform that has poles at z = f and z = 2, what can be said about the extent of x(n) (i.e., finite in length, right-sided, etc.)
Properties
A right-sided sequence x(n) has a z-transform X(z) given by
Find the values of x(n) for all n < 0. Use the z-transform to perform the convolution of the following two sequences:
Evaluate the following summation:
THE z-TRANSFORM
Find the value of x(0) for the sequence that has a z-transform
[CHAP. 4
A right-sided sequence has a z-transform
Find the index a ~ the value of the first nonzero value of x ( n ) . d
Inverse z-Transforms
Find the inverse z-transform of
4.55 4.56 4.57
Find the inverse z-transform of X(z) = coszp'. Assume that the ROC includes the unit circle, Izl = 1. Find the inverse z-transform of X(z) = e'. Assume that the ROC includes the unit circle, lzl = 1. Find the inverse z-transform of z5 - 3 X(z) = 1 - z-S lzl > I
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