신호처리중간.pdf

Question 1 [40%]:
A function called autocorrelation for a real-valued, absolutely summable sequence x[n] is dened as X rxx [l] , x[n] x[n l]
n

Let X(z) be the z-transform of x[n]. a) Show that the z-transform of rxx [l] is given by Rxx (z) = X(z) X(z 1). b) Let x[n] = an u[n], |a| ` 1. Determine Rxx (z) and sketch its pole-zero plot and the ROC. c) Determine the autocorrelation rxx [l] for the x[n] in (b) above.

Question 2 [25%]:
The signal x[n] = {1, 2, 3, 4, 0, 4, 3, 2, 1} (with 0 corresponding to n = 0, i.e. x[0] = 0) has DTFT X(e|!). Without explicitly computing X(e|!), nd the following quantities: a) X(e| 0) b) X(e| ) R c) X(e|!)d! R d) |X(e|!)|2 d!

Question 3 [35%]:
For the following input-output pairs, determine whether or not there is an LTI system producing y[n] when the input is x[n]. If such a system exists, determine its magnitude and phase responses; otherwise explain why such a system is not possible. a) x[n] =
sin n/4 n

7! y[n] =

sin n/2 n

b) x[n…