Hence, the system is time-invariant. in .NET

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Hence, the system is time-invariant.
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Fig. 1-33
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SIGNALS AND SYSTEMS
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Fig. 1-34 Unit ramp function.
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( c ) Let x ( t ) = k l u ( t ) ,with k , # 0. Then
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where r ( t ) = t u ( t ) is known as the unit ramp function (Fig. 1-34). Since y ( t ) grows linearly in time without bound, the system is not B I B 0 stable.
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1.34. Consider the system shown in Fig. 1-35. Determine whether it is (a) memoryless, ( b ) causal, ( c ) linear, ( d ) time-invariant, or ( e ) stable.
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From Fig. 1-35 we have
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= T { x ( t ) }= x ( t ) cos
Since the value of the output y ( t ) depends on only the present values of the input x ( t ) , the system is memoryless. Since the output y ( t ) does not depend on the future values of the input x ( t ) , the system is causal. Let x ( t ) = a , x ( t ) + a 2 x ( t ) .Then
y(t)
= T ( x ( t ) }=
[ a , x l ( t )+ a 2 x 2 ( t ) ] w,t cos
= a , x , ( t )cos w,t
+ a 2 x 2 ( t )cos w,t
= " l ~ , ( t+ f f 2 ~ 2 ( t ) ) Thus, the superposition property (1.68) is satisfied and the system is linear.
Fig. 1-35
SIGNALS AND SYSTEMS
[CHAP. 1
( d l Let y , ( t ) be the output produced by the shifted input x , ( t ) = x ( t -to). Then y , ( t ) = T { x ( t- t o ) )= x ( t - t,-,)c~s wct But Hence, the system is not time-invariant. ( e ) Since Icos w,tl s 1, we have Thus, if the input x ( t ) is bounded, then the output y ( t ) is also bounded and the system is B I B 0 stable.
1-35. A system has the input-output relation given by
y = T { x )= x 2
Show that this system is nonlinear.
2 T ( x , + x 2 )= ( x , + x 2 ) = x : +xS + 2 x , x 2
+ T { x , }+ T { x , ) = x : + x 2,
Thus, the system is nonlinear.
1.36. The discrete-time system shown in Fig. 1-36 is known as the unit delay element. Determine whether the system is ( a ) memoryless, ( b ) causal, ( c ) linear, ( d ) time-
invariant, o r (e) stable.
(a) The system input-output relation is given by
Since the output value at n depends on the input values at n - 1, the system is not memoryless. ( b ) Since the output does not depend on the future input values, the system is causal. (c) Let .r[n]=cw,x,[n] a,x2 [ n ] .Then + y [ n ] = T { a , x , [ n ]+ a 2 x 2 [ n ]= a , x , [ n - 11 + a 2 x 2 [ n 11 ) =~
I Y I
+ a2y2[nI ~ I
Thus, the superposition property (1.68) is satisfied and the system is linear. ( d ) Let ~ , [ n ] the response to x , [ n ]= x [ n - n o ] . Then be y I [ n ]= T ( x , [ n ] ) x , [ n - 11 = x [ n- 1 - n o ] = and y [ n - n o ] = x [ n - n, - 11 = x [ n - 1 - n o ] = y , [ n ] Hence, the system is time-invariant.
xlnl
Ui nt
delay
ylnl = xln-I] b
Fig. 1-36 Unit delay element
CHAP. 11
SIGNALS AND SYSTEMS
Since
l y [ n ] I = l x [ n - 111 I k
if I x [ n ] l s k for all n
the system is BIB0 stable.
1.37. Find the input-output relation of the feedback system shown in Fig. 1-37.
,-+y-lT
Unit delay
I I I
I I I I
From Fig. 1-37 the input to the unit delay element is x [ n ] - y [ n ] . Thus, the output y [ n ] of the unit delay element is [Eq. (1.111)l
Rearranging, we obtain
Thus the input-output relation of the system is described by a first-order difference equation with constant coefficients.
1.38. A system has the input-output relation given by
Determine whether the system is (a) memoryless, ( b ) causal, ( c ) linear, ( d ) time-invariant, or ( e ) stable.
( a ) Since the output value at n depends on only the input value at n , the system is memoryless. ( b ) Since the output does not depend on the future input values, the system is causal. (c) Let x [ n ] = a , x , [ n l + a z x , [ n ] . Then
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