Consider the system described by the input-output relationship y[n] = a1 y[n - 1
ID: 1832912 • Letter: C
Question
Consider the system described by the input-output relationship
y[n] = a1 y[n - 1] + a2 y*[n - 1] + x[n] ,
where * indicates the complex conjugate of a complex number. We know that the complex
exponentials z n are eigenfunctions of LTI systems, and now we will see if the same is true for
this system.
(a) Assuming that y[n] = 0 for n < 0 , compute y1[n] for n = 0,1 where x1[n] = [n] .
(b) Assuming that y[n] = 0 for n < 0 , compute y2 [n] for n = 0,1 where x2 [n] = C[n] and C
is a complex number.
(c) Using your results from (a) and (b), show that the system is not linear by comparing
Cy1[n] to y2 [n] . Briefly explain your reasoning.
(d) Suppose that x[n] = z^n , for some complex z . Is x[n] an eigenfunction of the system? In
other words, is there some scalar H ( z ) such that y[n] = H ( z ) z^n, where H ( z ) is not a
function of n? Justify your answer. (You can do this using proof by contradiction.
Assume the hypothesized eigenfunction relationship, and plug it into the difference
equation. If the hypothesis leads to a contradiction, then the hypothesis must be false.
Otherwise, you can prove H ( z ) exists by solving for it.)
Explanation / Answer
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