By Monson Hayes
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Additional info for Schaum's Outline of Digital Signal Processing
Example text
6) Linear, shift-varying, unstable, noncausal, invertible. ( c ) Linear, shift-invariant, stable. noncausal, invertible. (d) Nonlinear, shift-invariant, stable, causal, invertible. ( 0 ) Nonlinear, shift-invariant, stable, noncausal, noninvertible. 51 No. Consider the system y(n) = x(n)cos(nr/2). 53 (a) Yes. (6) No. ( c ) No. 54 The sequence values. beginning at index n = -I, are y(n) = {2,0, -4, 1,6,0, I , -1, -2,6,3). 57 (a) s(n) = 1. (6) Any finite a . ( c ) la[ < 1. n h(k). With h(n) = u(n) - u(n - 6) the step response is k=-cc (6) h(n) = s(n) - s(n - 1).
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.
For example, ((3))g = 3, ((12))g = 4, and ((-6))d = 2. If x(n) = (i)%in(nn/2)u(n), make a sketch of (a) x((n))3 and (b)x((n - 2))3. (a) We begin by noting that ((n))3, for any value of n, is always an integer in the range [O, 21. In fact, because ((n))3 = ((n 3k)h for any k , + Therefore, x((n))3 is periodic with a period N = 3. It thus follows t h a t ~ ( ( n )is ) ~formed by periodically repeating the first three values of x(n) as illustrated in the figure below: (b) The sequence x((n - 2))3 is also periodic with a period N = 3, except that the signal is shifted to the right by no = 2 compared to the periodic sequence in part (a).