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Is it true that all memoryless systems are shift-invariant Consider the linear shift-invariant system described by the first-order linear constant coefficient difference equation
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y ( n ) = uy(n - I )
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Determine the conditions (if any) for which this system is stable. Suppose that two systems, SIand SZ, connected in parallel. are
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( a ) If both S, and Sz are linear, shift-invariant, stable. and causal, will the parallel connection always be linear,
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shift-invariant, stable. and causal
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(h) If both S, and Sz are nonlinear. will the parallel connection necessarily be nonlinear
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(c) If both S I and S2 are shift-varying, will the parallel connection necessarily be shift-varying
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Find the convolution of the two sequences
s ( n ) = 6(n - 2 ) - 26(n
+ 36(n - 6) h ( n ) = 2S(n + 3 ) + S(n) + 26(n - 2 ) + 6(n
The unit sample response of a linear shift-invariant system is
h ( n ) = 36(n - 3)
+ 0.5S(n - 4 ) + 0.26(n - 5 ) + 0 . 7 6 ( n .
Find the response of this system to the input x ( n ) = u(n - I). A linear shift-invariant system has a unit sample response
h(n)= u(-n)
Find the output if the input is
A-(n)= ( i ) " r c ( n )
The step response of a system is defined as the response of the system to a unit step u ( n ) .
( a ) Let s ( n ) be the step response of a linear shift-invariant system. Express s ( n ) in terms of the unit sample response h ( n ) , and find s ( n ) when h ( n ) = u ( n ) u(n 6 ) .
(b) Derive an expression for h ( n ) in terms of s(n) and find the unit sample response For a system whose step response is
The unit sample response of a linear shift-invariant system is shown below.
CHAP. 11
SIGNALS AND SYSTEMS
( a ) Find the response of the system to the input u(n - 4). ( 6 ) Repeat for x ( n ) = ( - l ) " u ( n ) .
If x ( n ) = ( i ) " u ( n - 2 ) and h ( n ) = 2"u(-n
5 ) , find the convolution y ( n ) = x ( n ) h(n).
Given three sequences, h(n), g(n), and r ( n ) , express g(n) in terms of r ( n ) if
Let h ( n ) = a n u ( n )and x ( n ) = bnu(n). Find the convolution y ( n ) = x ( n ) * h ( n ) assuming that a # 6 . If x ( n ) = anu(n),find the convolution y ( n ) = x ( n ) * x(n). The input to a linear shift-invariant system is the unit step, x ( n ) = u ( n ) , and the response is y ( n ) = S(n). Find the unit sample response of this system. If h(n) = A6(n) +(f )"u(n) is the unit sample response of a linear shift-invariant system, and s ( n ) is the step response s(n) = 0. (the response of the system to a unit step), find the value of the constant A so that lim,,, The unit sample response of a linear shift-invariant system is
Find the response of the system to the complex exponential x ( n ) = exp(jnrr/4). Evaluate the convolution of the sequence x ( n ) = n(i)"cos(rrn) with the unit step, h ( n ) = u(n). Let
n(0.5)" 05n 55 n c O
and h(n) = ej%"u(-n). If y ( n ) = x ( n ) * h(n), what is the numerical value of y(-2) Given
and h ( n ) = S(n - 2 ) S(n - 3 ) 6(n - 4), at what value of n will the convolution y ( n ) = x ( n ) * h ( n ) attain its maximum value, and what is this maximum value A linear system has a response h k ( n ) = S(2n - k ) to the unit sample 6(n - k). Find the response of the system to the input x ( n ) = u(n). Consider the interconnection of three linear shift-invariant systems shown in the figure below.
x(n)
hAn)
h~(n)
+ (+> = +
~ ( n )
h3(n)
SIGNALS AND SYSTEMS
If hl(n) = u(n - 2), hn(n) = nu@) and h3(n) = 6(n
[CHAP. 1
- 2). find the unit sample response of the overall system.
Difference Equations
Consider the linear shift-invariant system described by the LCCDE y(n) = -iy(n - I) +2x(n) Find the response of this system to the input x(n) = 2 n = 0 , 2 , 4 , 6 ,... otherwise
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