Let T represent a continuous-time LTI system. Then show that in Visual Studio .NET

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1.44. Let T represent a continuous-time LTI system. Then show that
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T{es'} = ks'
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where s is a complex variable and h is a complex constant.
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Let y ( t ) be the output of the system with input x ( t ) = e". Then
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T { e S t )= y ( t )
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Since the system is time-invariant, we have
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T(es('+'(l)) y ( I =
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for arbitrary real t o . Since the system is linear, we have
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T { ~ ~ ( ' + '=IT{eS' e s ' ~ } = e " ~ T { e S ' )= e S ' ~ y ( t ) I ))
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Hence,
y(r +I,) = e H 0 y ( t )
CHAP. 11
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 discrete-time 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 time-invariant, 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. 1-41 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. 1-41
1.47. Express the sequences shown in Fig. 1-42 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. 1-42
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 continuous-time system with the input-output relation
Determine whether this system is ( a ) linear, ( b ) time-invariant, ( c ) causal.
Consider a continuous-time system with the input-output relation
Determine whether this system is ( a ) linear, ( b ) time-invariant.
( a ) Linear; ( b ) Time-varying
Consider a discrete-time system with the input-output relation y [ n ] = T { x [ n ] )= x 2 [ n ] Determine whether this system is ( a ) linear, ( b ) time-invariant. Ans. ( a ) Nonlinear; ( b ) Time-invariant
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 time-varying 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. 1-43. Determine if each of the following systems is invertible. If the system is invertible, give the inverse system.
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