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For the system described by Eq. (2.55)the impulse response h[n] is given by in .NET framework
For the system described by Eq. (2.55)the impulse response h[n] is given by Scanning Quick Response Code In .NET Framework Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Draw QR Code 2d Barcode In VS .NET Using Barcode creation for .NET Control to generate, create Quick Response Code image in .NET framework applications. Note that the impulse response for this system has finite terms; that is, it is nonzero for only a finite time duration. Because of this property, the system specified by Eq. (2.55) is known as a finite impulse response (FIR) system. On the other hand, a system whose impulse response is nonzero for an infinite time duration is said to be an infinite impulse response (IIR) system. Examples of finding impulse responses are given in Probs. 2.44 and 2.45. In Chap. 4, we will find the impulse response by using transform techniques. QR Recognizer In Visual Studio .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. Printing Barcode In .NET Framework Using Barcode generator for VS .NET Control to generate, create bar code image in VS .NET applications. Solved Problems
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Make Code 128A In None Using Barcode printer for Font Control to generate, create ANSI/AIM Code 128 image in Font applications. GTIN  12 Creation In VS .NET Using Barcode generator for Reporting Service Control to generate, create UPC Symbol image in Reporting Service applications. CHAP. 21
LINEAR TIMEINVARIANT SYSTEMS
( b ) Let x ( t ) * h , ( t ) =f , ( t ) and h , ( t ) * h 2 ( t ) =f2(t). Then
Substituting A = a  T and interchanging the order of integration, we have
Now, since
we have
Thus, Show that
x(t) * u(t  to)= Ito
( a ) By definition ( 2 . 6 ) and Eq. (1.22) we have
( b ) By Eqs. ( 2 . 7 ) and (1.22)we have
By Eqs. ( 2 . 6 ) and (1.19) we have
since u(t  7 ) = 7<t 7>t' LINEAR TIMEINVARIANT SYSTEMS
[CHAP. 2
( d l In a similar manner, we have
x ( t )* u(t  t,,) = x ( r ) u ( t 7  t o ) dr
Let y ( r ) = x ( r ) * h ( t 1. Then show that
By Eq. ( 2 . 6 )we have
and Let r
A . Then
T =A
+ t , and Eq. (2.63b)becomes
Comparing Eqs. ( 2 . 6 3 ~and (2.63~1, see that replacing ) we Thus, we conclude that obtain Eq. ( 2 . 6 3 ~ ) . in Eq. ( 2 . 6 3 ~by r  r ) ,  r,, The input x ( t ) and the impulse response h ( t ) of a continuous time LTI system are given by
(a) Compute the output y ( t ) by Eq. ( 2 . 6 ) . ( b ) Compute the output y ( t ) by Eq. (2.10). ( a ) By Eq. ( 2 . 6 ) Functions X ( T ) and h(t  r ) are shown in Fig. 24(a)for t < 0 and t > 0. From Fig. 24(a) we see that for t < 0, x ( r ) and h(t  T ) do not overlap, while for t > 0, they overlap from T = 0 to T = I . Hence, for t < 0, y ( t ) = 0. For t > 0, we have Thus, we can write the output y ( t ) as
CHAP. 21
LINEAR TIMEINVARIANT SYSTEMS
( b ) By Eq. (2.10) Functions h ( r ) and x(t  7)are shown in Fig. 24(b) for t < 0 and t > 0. Again from Fig. 24(b) we see that for t < 0, h(7)and x(t  7 ) do not overlap, while for t > 0, they overlap from 7 = 0 to r = t . Hence, for t < 0, y ( t ) = 0. For t > 0, we have Thus, we can write the output y ( t ) as
which is the same as Eq. (2.64). F g 24 i.
LINEAR TIMEINVARIANT SYSTEMS
[CHAP. 2
Compute the output y(t for a continuoustime LTI system whose impulse response h ( t ) and the input x ( 0 are given by By Eq. ( 2 . 6 ) y ) ( t )* h )= 1 x ( r ) h ( t r ) dr
Functions ~ ( 7and h(t  7 ) are shown in Fig. 25(a) for t < 0 and t > 0. From Fig. 25(a) we ) see that for t < 0, X ( T ) and h(t  7 ) overlap from 7 =  w to 7 = t , while for t > 0, they overlap from 7 = 01 to 7 = 0. Hence, for t < 0, we have y(r) e'Te'VT)  rn
= e  a ' G 2 a ' d r = eat 2a
(2.66~) For t > 0, we have
Fig. 25 CHAP. 21
LINEAR TIMEINVARIANT SYSTEMS
) Combining Eqs. ( 2 . 6 6 ~and (2.6681, we can write y ( t ) as
y(t) 1 euIrl
CY>O
which is shown in Fig. 2  S b ) . Evaluate y ( t ) = x ( t ) * h ( t ), where x ( t ) and h ( t ) are shown in Fig. 26, ( a ) by an analytical technique, and ( b ) by a graphical method. Fig. 26 ( a ) We first express x(t and h ( t ) in functional form:

