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Let T represent a continuoustime LTI system. Then show that in Visual Studio .NET
1.44. Let T represent a continuoustime LTI system. Then show that QR Code Reader In .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET framework applications. QR Code JIS X 0510 Maker In Visual Studio .NET Using Barcode creation for .NET framework Control to generate, create QR Code JIS X 0510 image in .NET framework applications. T{es'} = ks' QR Code Decoder In VS .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET applications. Creating Barcode In .NET Using Barcode generation for .NET framework Control to generate, create barcode image in Visual Studio .NET applications. where s is a complex variable and h is a complex constant.
Scan Bar Code In .NET Using Barcode scanner for .NET Control to read, scan read, scan image in .NET framework applications. Encoding QR In Visual C# Using Barcode printer for VS .NET Control to generate, create Quick Response Code image in .NET applications. Let y ( t ) be the output of the system with input x ( t ) = e". Then
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SIGNALS AND SYSTEMS
Setting t = 0, we obtain
to) =y(0)eS1" Since to is arbitrary, by changing to to t,' we can rewrite Eq. (1.118) as y ( t ) = y(0)eS1 hes' = or where A = ~ ( 0 ) . T ( e S ' } AeS' = 1.45. Let T represent a discretetime LTI system. Then show that
T{zn) hzn = where z is a complex variable and A is a complex constant.
Let y[n]be the output of the system with input x [ n ]= z n . Then Tbnj=ybl Since the system is timeinvariant, we have T { z n + " ~ jy [ n + n o ] = for arbitrary integer no. Since the system is linear, we have Hence, Setting n = 0, we obtain Since no is arbitrary, by changing no to n, we can rewrite Eq. (1.120) as y [ n 3 = y[O]zn= Azn or T ( z n )= Azn where A = y[O]. In mathematical language, a function x ( . ) satisfying the equation is called an eigenfunction (or characteristic function) of the operator T , and the constant A is called the eigenvalue (or characteristic value) corresponding to the eigenfunction x(.). Thus Eqs. (1.117) and (1.119) indicate that the complex exponential functions are eigenfunctions of any LTI system. y [ n + n o] = z n O y [ n ] Supplementary Problems
Express the signals shown in Fig. 141 in terms of unit step functions. t Ans. ( a ) x ( t ) =  [ u ( t )  u(t  2)1 2 (6) ~ ( t= u(t + 1) + 2u(t)  u(t  1 )  u(t  2)  ~ (t3) ) SIGNALS AND SYSTEMS
[CHAP. 1
Fig. 141 1.47. Express the sequences shown in Fig. 142 in terms of unit step functions.
Am. ( a ) x[nl=u[n]u[n  ( N + I)] ( b ) x[n]= u[n  11 (c) x[nl = u[n + 2  u[n  41 1 (4 Fig. 142 CHAP. 11
SIGNALS AND SYSTEMS
Determine the even and odd components of the following signals: (a) xe(t) = i , xo(t) = i sgn t 1 (b) x,(t) = cos wot, xJt) 1 sin
(c) x,[nl = jcos n,n, xo[nl = sin Ron ( d l xe[nl = S[nI, xo[nl = 0 Let x(t) be an arbitrary signal with even and odd parts denoted by xe(t) and xo(t), respectively. Show that Hint: Use the results from Prob. 1.7 and Eq. (1.77). Let x[n] be an arbitrary sequence with even and odd parts denoted by x,[nl and xo[n], respectively. Show that Hinr: Use the results from Prob. 1.7 and Eq. (1.77). Determine whether or not each of the following signals is periodic. If a signal is periodic, determine its fundamental period. ( a ) x(r) = cos 2r ( g ) x[nl=cos  cos (h) x[n] = cos Am.
( 1 ("4") (2 . ( y) +
 2cos( ) (b) (dl (f (h) Periodic, period Periodic, period Periodic, period Periodic, period
= .rr =2 =8
( a ) Periodic, period = .rr
(c) Nonperiodic (el Nonperiodic ( g ) Nonperiodic
SIGNALS AND SYSTEMS
[CHAP. 1
Show that if x [ n ] is periodic with period N, then
Hint: See Prob. 1.17.
( a ) What is S(2t) ( b ) What is S[2n] (a) S(2t)=$(t) ( b ) S[2nl= S[n] Show that
sy I) Hint: = S1(t) Use Eqs. (1.101) and (1.99). Evaluate the following integrals: Ans.
( a ) sin t
( b ) 1 for t > 0 and 0 for t < 0; not defined for t (c) 0 ( d ) .sr
Consider a continuoustime system with the inputoutput relation
Determine whether this system is ( a ) linear, ( b ) timeinvariant, ( c ) causal.
Consider a continuoustime system with the inputoutput relation
Determine whether this system is ( a ) linear, ( b ) timeinvariant.
( a ) Linear; ( b ) Timevarying
Consider a discretetime system with the inputoutput relation y [ n ] = T { x [ n ] )= x 2 [ n ] Determine whether this system is ( a ) linear, ( b ) timeinvariant. Ans. ( a ) Nonlinear; ( b ) Timeinvariant CHAP. 11
SlGNALS AND SYSTEMS
Give an example of a system that satisfies the condition of homogeneity ( 1 . 6 7 ) but not the condition of additivity (1.66). Ans.
Consider the system described by
Give an example of a linear timevarying system such that with a periodic input the corresponding output is not periodic. Ans. y [ n ] = T { x [ n ] = m [ n ] ) A system is called invertible if we can determine its input signal x uniquely by observing its output signal y. This is illustrated in Fig. 143. Determine if each of the following systems is invertible. If the system is invertible, give the inverse system.

