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( d ) X ( z ) does not exist.
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Show that if x [ n ] is a left-sided sequence and X ( z ) converges from some value of z, then the ROC of X ( z ) is of the form
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where rmin is the smallest magnitude of any of the poles of X ( z ) .
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Proceed in a manner similar to Prob. 4.5.
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( a ) State all the possible regions of convergence. ( b ) For which ROC is X ( z ) the z-transform of a causal sequence Ans. ( a ) 0 < lzl < 1 , l < lzI< 2,2 < I z k 3, lzl> 3 ( b ) lz1>3
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Verify the time-reversal property (4.23), that is,
Hint: Change n to - n in definition (4.3).
CHAP. 41
T H E Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
Show the following properties for the z-transform.
( a ) If x [ n ]is even, then X ( z L ' )= X ( z ) . ( b ) If x [ n ] is odd, then X ( z - ' ) = - X ( z ) . ( c ) If x [ n ]is odd, then there is a zero in X ( z ) at z = 1. Hint: ( a ) Use Eqs. (1.2) and (4.23). ( b ) Use Eqs. (1.3) and (4.23). ( c ) Use the result from part ( b ) .
4.46. Consider the continuous-time signal
Let the sequence x [ n ]be obtained by uniform sampling of x ( t ) such that x [ n ]= x(nT,), where T, is the sampling interval. Find the z-transform of x [ n ] .
Derive the following transform pairs:
(sin n 0 n ) u [ n ]
(sin R o ) z z 2 - (2cos R o ) z + 1
lzl> I
Hint: Use Euler's formulas.
e' and use Eqs. (4.8) and (4.10) with a = . 0* '
Find the z-transforms of the following x [ n ] :
(a) (b) (c) (dl
x [ n ]= ( n - 3)u[n- 31 x[nl = ( n - 3)u[n] x[nl = u [ n ]- u [ n - 31 x[nl = n{u[nl- u [ n - 33)
Am. ( a )
z - ~
( 2 - 1)
, , lz1> 1
THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
[CHAP. 4
Using the relation
a n u [ n ]H
Izl> la1
find the z-transform of the following x [ n ] :
( a ) x [ n ]= nun-'u[n]
( b ) x [ n ] = n(n - l ) a " - 2 u [ n ] ( c ) x [ n l = n(n - 1 ) . . . ( n - k
Hint:
+l)~"-~u[n]
Differentiate both sides of the given relation consecutively with respect to a .
Using the z-transform, verify Eqs. (2.130) and (2.131) in Prob. 2.27, that is,
Hint: 4.51.
Use Eq. (4.26) of the z-transform and transform pairs 1 and 4 from Table 4-1.
Using the z-transform, redo Prob. 2.47.
Hint:
Use Eq. (4.26) and Table 4-1.
Find the inverse z-transform of
X ( z ) = ea/' Hint:
1.4> 0
Use the power series expansion of the exponential function e r .
Using the method of long division, find the inverse z-transform of the following X ( z ) :
CHAP. 41
THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
4.54. Using the method of partial-fraction expansion, redo Prob. 4.53.
Ans.
( a ) x [ n ]= (1 - 2 9 4 - n - 11 (6) x [ n ] = - u [ n ] - 2"u[-n - 11 (c) ~ [ n ]( - 1 + 2")u[n] =
4.55. Consider the system shown in Fig. 4-9. Find the system function H ( z ) and its impulse response hbl.
H(z)=
1 - 72
, -, , h [ n l =
Fig. 4-9
4.56. Consider the system shown in Fig. 4-10.
( a ) Find the system function H ( z ) . ( b ) Find the difference equation relating the output y [ n ] and input x [ n ] .
Fig. 4-10
T H E Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS
[CHAP. 4
Consider a discrete-time LTI system whose system function H(z)is given by
( a ) Find the step response s[n]. ( b ) Find the output y[n]to the input x [ n ]= nu[n].
Ans.
( a ) s[n]= [2 - ( ~ ) " ] u [ n ] ( 6 ) y[nl = 2[(4)"+ n - llu[nl
Consider a causal discrete-time system whose output y[n]and input x [ n ]are related by y [ n ] - : y [ n - 1 1 + i y [ n -21 = x [ n ] ( a ) Find its system function H ( z ) . ( b ) Find its impulse response h[n].
(a) H(z)=
, lzl> 2 ( z - $ ) ( z- 3) ( b ) h [ n ]= [3($)"- 2(f)"]u[n]
Using the unilateral z-transform, solve the following difference equations with the given initial conditions. ( a ) y [ n ]- 3y[n - 11 = x [ n ] with x [ n ]= 4u[n],y [ - 11 = 1 , ( b ) y[n]- 5y[n - 11 + 6y[n - 2 = x [ n ] with x [ n ]= u[n],y [ - I ] = 3, y [ - 2 ] = 2 1 ,
(a) y[n]=-2+9(3)",nr -1 ( b ) y[n]= + 8(2In - Z(3ln, n 2 -2
Determine the initial and final values of x [ n ]for each of the following X ( z ) :
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