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Problem 1. (30 points) Find the power of
x[n] = cos(−0.2πn − 0.4π) ~
X∞
l=−∞
δ[n + 2 − 10l]
where ~ denotes periodic convolution.
Problem 2. (40 points) Assume you are given the DTFT X(Ω) of a signal x[n] that starts at 0
and has length N1. You sample X(Ω) at N2 points (as you need to store it digitally), and keep these
N2 samples
X(
2πk
N2
), k = 0, . . . , N2 − 1
Unfortunately because you did not know what N1 was, you used N2 < N1. You now try to apply
IDFT to recover your original signal x[n].
(a) (25 points) Show that X(
2πk
N2
) is the DFT of a signal y[n] (of length N2), whose periodic
extension is yps[n] = P+∞
`=−∞ x[n − `N2]. That is, y[n] is an “aliased in time” version of x[n].
(Hint: You can either start from the expression for DTFT X(Ω), get DFT via sampling and
express this as the DFT of y[n]. Or you can start from yps[n] and show that its N2-point DFT
applied to the interval {0, 1, …, N2} is indeed equal to X(
2πk
N2
)).
(b) (15 points) Assume that x[n] = a
nu[n], 0 < a < 1, and thus X(Ω) = 1
1−ae−iΩ . Assume that
we take N samples of X(Ω), what will yps[n] be?
You may find the infinite geometric summation formula useful:
Xa
i=−∞
r
−i =
r
−a
1 − r
, for |r| < 1.
Problem 3. (30 points) z-transform: Determine the z-transform of
x[n] = (
0.2
nu[n] n is even
0.1
nu[n] n is odd
1
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