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LINEAR TIME-INVARIANT SYSTEMS
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we can express y(t ) as
=tu(t)
- ( t - 2)u(t - 2) - ( t - 3)u(t -3) + ( r - 5)u(t - 5 )
which is plotted in Fig. 2-7.
Fig. 2-7
( b ) Functions h ( r ) , X ( T ) and h(t - 71, x ( r ) h ( t - 7 ) for different values of t are sketched in Fig. 2-8. From Fig. 2-8 we see that x ( r ) and h(t - 7 ) do not overlap for t < 0 and t > 5, and hence y ( t ) = 0 for t < 0 and t > 5. For the other intervals, x ( r ) and h(t - T ) overlap. Thus, computing the area under the rectangular pulses for these intervals, we obtain
which is plotted in Fig. 2-9.
Let h ( t ) be the triangular pulse shown in Fig. 2-10(a) and let x ( t ) be the unit impulse train [Fig. 2-10(b)] expressed as
Determine and sketch y ( t ) = h ( t )* x( t ) for the following values of T: ( a ) T = 3, ( b ) T = 2, ( c ) T = 1.5.
CHAP. 21
LINEAR TIME-INVARIANT SYSTEMS
1 1-2
Fig. 2-8
LINEAR TIME-INVARIANT SYSTEMS
[CHAP. 2
Fig. 2-9
Fig. 2-10
Using Eqs. (2.59) and (2.91, we obtain
( a ) For T = 3, Eq. (2.69) becomes
which is sketched in Fig. 2-1 l(ah ( b ) For T = 2, Eq. (2.69) becomes
which is sketched in Fig. 2-1 1(b). ( c ) For T = 1.5, Eq. (2.69) becomes
which is sketched in Fig. 2-ll(c). Note that when T < 2 , the triangular pulses are no longer separated and they overlap.
CHAP. 21
LINEAR TIME-INVARIANT SYSTEMS
Fig. 2-11
If x , ( t ) and x 2 ( t ) are both periodic signals with a common period To, the convolution of x ,( t ) and x 2 ( t ) does not converge. In this case, we define the periodic conaolution of x , ( t ) and x 2 ( t ) as
( a ) Show that f ( t ) is periodic with period To. ( b ) Show that f ( t )=
la'T 0 x , ( ( l x 2 ( ta
7 )d~
(2.71)
for any a . ( c ) Compute and sketch the periodic convolution of the square-wave signal x ( t ) shown in Fig. 2-12 with itself.
LINEAR TIME-INVARIANT SYSTEMS
[CHAP. 2
Fig. 2-12
Fig. 2-13
CHAP. 2 1
LINEAR TIME-INVARIANT SYSTEMS
( a ) Since x 2 ( t )is periodic with period To, we have x 2 ( t + To - 7 ) = x Z ( t- T ) Then from Eq. (2.70) have we
Thus, f ( t ) is periodic with period To. ( b ) Since both x l ( r ) and x , ( i ) are periodic with the same period To,x1(7)x2(t T ) is also periodic with period To. Then using property (1.88) (Prob. 1.17), we obtain
for an arbitrary a. We evaluate the periodic convolution graphically. Signals x ( r ) , x(t - T ) , and x ( r ) x ( t - T ) are sketched in Fig. 2-13(a), from which we obtain
f(t)=
0 < t 5 T0/2
- To)
T0/2 < t ITo
+ T o ) =f ( t )
which is plotted in Fig. 2-13(b).
PROPERTIES OF CONTINUOUS-TIME LTI SYSTEMS
T h e signals in Figs. 2-14(a) and ( b ) are the input x ( t ) and the output y ( t ) , respectively, of a certain continuous-time LTI system. Sketch the output t o the following inputs: ( a ) x ( t - 2); ( b ) i x ( t ) . ( a ) Since the system is time-invariant, the output will be y(t - 2) which is sketched in Fig. 2-14(~). ( b ) Since the system is linear, the output will be fy(t) which is sketched in Fig. 2-14(d).
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