FALL 22 EC516 Problem Set 07
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FALL 22 EC516 Problem Set 07
Due: Sunday November 13 (Before Midnight) on Blackboard Learn
Please Note that Problem Set 08 will also be due on November 13
Please Note that Test 02 on November 16 will cover up to Problem Set 08
Problem 7.1
Let X" [n, 业) be the TDFT of x[n] = (−1). with respect to analysis window w[n] that is specified as w[n] = u[n − 1] − u[n − 2].
a) Sketch X" [n, 0) as a function of n . Justify your answer.
b) Sketch |X" [0, 业)| as a function of 业. Justify your answer.
Problem 7.2
Let G" [n, 业) = (0.5).78u[n − 1] (for all 业) be the TDFT of a signal g[n] = (0.5). u[n] with respect to some analysis window w[n] .
a) Determine and sketch the analysis window w[n] . Justify your answer.
b) Determine the numerical value |G: [0, 几)| where G: [n, 业) is the TDFT of the signal g[n] = (0.5). u[n] with respect to analysis window v[n] = u[n] − u[n − 3]. Justify
your answer.
Problem 7.3
Let H" [n, 业) be the TDFT of ℎ[n] with respect to analysis window w[n] = u[n − 1] − u[n − 5] a) Is it true that ℎ[n] = H" [n, 业)d业? Justify your answer.
b) Is it true that ℎ[n] = a Q H" [n, 业)eWX d业 for some non-zero finite constant, a . Justify your answer.
Problem 7.4
Consider an analysis window w[n] with z transform w(z) = 1 + z 78 + z 7\ + z 7] . This analysis window is used to compute the discrete TDTFT X" [n, k] of a signal x[n] using a temporal sampling factor of L = 1 and a frequency sampling factor of M = 2.
a) Can x[n] be synthesized from X" [n, k] using the FBS algorithm? Justify your answer.
b) Assuming x[n] = (−1). , is there any value of k such that X" [n, k] = 0 for all n?
Justify your answer.
c)
Problem 7.5
Let Q" [n, k] be the discrete TDFT (with temporal sampling factor L = 4 and a frequency sampling factor of M = 8) of a signal q[n] with respect to an analysis window w[n] that is specified as w[n] = ( 1/2). {u[n] − u[n − 5]}.
a) Determine and sketch a synthesis window f8 [n] that can be used to synthesize q[n] from Q" [n, k].
b) Determine and sketch a synthesis window fP [n] that is not the same as the window f8 [n] and that can also be used to synthesize q[n] from Q" [n, k].
2022-11-11