THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS in VS .NET

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THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
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[CHAP. 4
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PROPERTIES OF THE Z-TRANSFORM
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4.8. Verify t h e time-shifting property (4.18), that is,
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By definition (4.3)
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By the change of variables m
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- no, we obtain
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Because of the multiplication by 2-"0, for no > 0, additional poles are introduced at r = 0 and will be deleted at z = w. Similarly, if no < 0, additional zeros are introduced at z = 0 and will be deleted at z = m. Therefore, the points z = 0 and z = oo can be either added to or deleted from the ROC by time shifting. Thus, we have
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where R and R' are the ROCs before and after the time-shift operation.
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CHAP. 41
THE Z-TRANSFORMAND DISCRETE-TIME LTI SYSTEMS
Verify Eq. (4.211, that is,
By definition (4.3)
A pole (or zero) at z = zk in X(z) moves to z = zoz,, and the ROC expands or contracts by the factor Izol. Thus, we have
4.10. Find the z-transform and the associated ROC for each of the following sequences:
(a) From Eq. (4.15)
S[n]
all z
Applying the time-shifting property (4.181, we obtain
( b ) From Eq. (4.16)
IZI> I
Again by the time-shifting property (4.18) we obtain
From Eqs. (4.8) and (4.10) anu[n] w By Eq. (4.20) we obtain an+ 'u[n
z-a z z-a
Izl> la1
+ I]
lal< lzl < m
(4.73)
( d l From Eq. (4.16)
THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
[CHAP. 4
By the time-reversal property (4.23) we obtain
( e ) From Eqs. (4.8) and (4.10)
anu[n]
Izl> la1
Again by the time-reversal property (4.23) we obtain
4.11. Verify the multiplication by n (or differentiation in z ) property (4.24), that is,
From definition (4.3)
Differentiating both sides with respect to
we have
Thus, we conclude that
4.12. Find the z-transform of each of the following sequences:
(a) x [ n ] = n a n u [ n ] ( b ) x [ n ] = nan- l u [ n ]
( a ) From Eqs. (4.8) and (4.10) anu[n] o z-a
I z I > la1
Using the multiplication by n property (4.24), we get
CHAP. 41
T H E Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
( b ) Differentiating Eq. (4.76)with respect to a, we have
Note that dividing both sides of Eq. (4.77)by a , we obtain Eq. (4.78).
4.13. Verify the convolution property (4.26), that is,
By definition (2.35)
Thus, by definition (4.3)
Noting that the term in parentheses in the last expression is the z-transform of the shifted signal x 2 [ n - k ] , then by the time-shifting property (4.18)we have
with an ROC that contains the intersection of the ROC of X , ( z ) and X , ( z ) . If a zero of one transform cancels a pole of the other, the ROC of Y ( z )may be larger. Thus, we conclude that
4.14. Verify the accumulation property (4.25), that is,
From Eq. (2.40)we have
Thus, using Eq. (4.16) and the convolution property (4.26),we obtain
with the ROC that includes the intersection of the ROC of X ( z ) and the ROC of the z-transform of u [ n ] .Thus,
THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
[CHAP. 4
INVERSE Z-TRANSFORM
4.15. Find the inverse z-transform of
X ( z ) = z 2 ( l - i2-')(1 -2-')(I
+22-7
3 +z-I
0 < lzl< 00
(4.79)
Multiplying out the factors of Eq. (4.79),we can express X ( z ) as X ( Z )= z 2 + t z Then, by definition ( 4 . 3 )
X ( z ) = x [ - 2 ] z 2 + x [ - 1 ] z + x [ o ]+ x [ 1 ] z - '
and we get
x [ n ] = { ...,O , l , $ , - 5 , 1 , 0 ,... }
4.16. Using the power series expansion technique, find the inverse z-transform of the following X ( 2):
( a ) Since the ROC is ( z ( >la(, that is, the exterior of a circle, x [ n ] is a right-sided sequence. Thus, we must divide to obtain a series in the power of z - ' . Carrying out the long division, we obtain
Thus, 1 - az-' and so by definition ( 4 . 3 ) we have
x[n]=O x[O]=1 n<O x[l]=a x[2]=a2 X(z)=
= 1+ a ~ - ' + a ~ z - ~ +
Thus, we obtain
x [ n ]= anu[n]
( 6 ) Since the ROC is lzl < lal, that is, the interior of a circle, x[n] is a left-sided sequence. Thus, we must divide so as to obtain a series in the power of z as follows, Multiplying both the numerator and denominator of X ( z ) by z , we have z
X(z)
CHAP. 41
THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
and carrying out the long division, we obtain -a-'z - a - 2 z 2 - a - 3 z 3 -
Thus,
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