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LJNEAR TIME-INVARIANT SYSTEMS
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( b ) Sequences h [ k ] ,x [ k l and h [ n - k ] , x [ k ] h [ n- k l for different values of n are sketched in Fig. 2-24. From Fig. 2-24 we see that x [ k J and h [ n - k ] do not overlap for n < 0 and n > 5, and hence y [ n ] = 0 for n < 0 and n > 5. For 0 5 n 1 5, x [ k ] and h [ n - k ] overlap. : Thus, summing x [ k ] h [ n- k ] for 0 s n 2 5, we obtain
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2.31. If x , [ n ] and x 2 [ n ] are both periodic sequences with common period N,the convolution of x , [ n ] and x 2 [ n ] does not converge. In this case, we define the periodic convolution of x , [ n ] and x 2 [ n ] as
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Show that f [ n ] is periodic with period N.
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Since x , [ n ] is periodic with period N, we have
Then from Eq. (2.138) we have
Thus, f [ n 1 is periodic with period N.
2.32. The step response s [ n ] of a discrete-time LTI system is given by
s [ n ]= a n u [ n ]
Find the impulse response h [ n ] of the system.
O<a<l
CHAP. 21
LINEAR TIME-INVARIANT SYSTEMS
From Eq. (2.41) the impulse response h[nl is given by h[n]
=s[n] =
- s[n
- 11 = crnu[n] - a"-'u[n
- 11
{S[n] + a n u [ n - 11) - a n - ' u [ n - 11
= 6[n]
- (1 - a)crn-'u[n - 11
PROPERTIES OF DISCRETE-TIME LTI SYSTEMS
2.33. Show that if the input x[n] to a discrete-time LTI system is periodic with period N , then the output y[n] is also periodic with period N .
Let h[n] be the impulse response of the system. Then by Eq. (2.39) we have
Let n = m + N . T h e n
+N] = k=
h[k]x[m
+N - k] = C
h[k]x[(m - k )
+N ]
Since x[n] is periodic with period N, we have x[(m-k) + N ] =x[m-k] Thus, y[m
+N ] =
h[k]x[m
- k ] = y[m]
which indicates that the output y[n] is periodic with period N.
2.34. The impulse response h[n] of a discrete-time LTI system is shown in Fig. 2-26(a). Determine and sketch the output y[n] of this system t o the input x[n] shown in Fig. 2-26(b) without using the convolution technique.
From Fig. 2-26(b) we can express x[n] as x[n]
= 6[n
- 21 - S[n - 41
Fig. 2-26
LINEAR TIME-INVARIANT SYSTEMS
[CHAP. 2
Since the system is linear and time-invariant and by the definition of the impulse response, we see that the output y [ n ] is given by y[n] = h [ n -21 - h [ n -41 which is sketched in Fig. 2-27.
Fig. 2-27
2.35. A discrete-time system is causal if for every choice of no the value of the output sequence y [ n ] at n = n o depends on only the values of the input sequence x [ n ] for n I no (see Sec. 1.5D). From this definition derive the causality condition (2.44) for a discrete-time LTI system, that is,
From Eq. (2.39) we have
Y[.]
h[klx[n- k l
Note that the first summation represents a weighted sum of future values of x [ n ] . Thus, if the system is causal, then
CHAP. 2 1
LINEAR TIME-INVARIANT SYSTEMS
This can be true only if
h [ n ]= 0
Now if h[n]= 0 for n < 0 , then Eq. (2.139) becomes
which indicates that the value of the output y[n] depends on only the past and the present input values.
2 3 . Consider a discrete-time LTI system whose input x [ n ] and output y [ n ] are related by .6
Is the system causal
By definition (2.30) and Eq. (1.48)the impulse response h [ n ]of the system is given by
By changing the variable k
+ 1 = m and by Eq. (1.50)we obtain
n+ 1
h [ n ]= 2-("+')
S [ m ] = 2-("+"u[n+ 1 ]
( 2.140)
From Eq. (2.140) we have h [ - 11 = u[O]= 1
+ 0. Thus, the system is not causal.
2 3 . Verify the BIBO stability condition [Eq. ( 2 . 4 9 ) ]for discrete-time LTI systems. .7
Assume that the input x [ n ] of a discrete-time LTl system is bounded, that is,
Ix[n]l l k l
all n
(2.141)
Then, using Eq. (2.351, we have
Since lxIn - k)l i k , from Eq. (2.141). Therefore, if the impulse response is absolutely summable, that is,
we have
l y [ n ] l ~ k , K =<km ~
and the system is BIB0 stable.
2 3 . Consider a discrete-time LTI system with impulse response h [ n ] given by .8
Is this system causal ( b ) Is this system BIBO stable
LINEAR TIME-INVARIANT SYSTEMS
[CHAP. 2
Since h [ n ] = 0 for n < 0, the system is causal. ( b ) Using Eq. (1.91) (Prob. 1,191, we have
a a Therefore, the system is B I B 0 stable if l1 < 1 and unstable if l1 2 1.
SYSTEMS DESCRIBED BY DIFFERENCE EQUATIONS
239. The discrete-time system shown in Fig. 2-28 consists of one unit delay element and one scalar multiplier. Write a difference equation that relates the output y [ n ] and the input x [ n ] .
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