Solved EC516 HOMEWORK 11

Description

Problem 11.1 (20 points)
(A) Consider the 5-point signal given as 𝑥[𝑛] = 𝑢[𝑛] − 𝑢[𝑛 − 5], where 𝑢[𝑛] is the unit
step. Let 𝑋*[𝑛, 𝜔) be the TDFT of 𝑥[𝑛] with respect to analysis window 𝑤[𝑛] = 𝛿[𝑛].
Also let 𝑓1[𝑚] = 𝑤[𝑚]𝑥[𝑛 + 𝑚] denote the short-time section at each time n that is a
signal as a function of m.
(a) Sketch magnitude and phase of 𝑋(𝑒67).
(b) Sketch the short-time sections 𝑓89[𝑛], 𝑓:[𝑛], and 𝑓9:[𝑛]. Justify your answers.
(c) Sketch 𝑋*[𝑛,
;
<
) as a function of n. Justify your answer
(d) Sketch 𝑋*[0, 𝜔) as a function of 𝜔. Justify your answer.
(B) Consider the 5-point signal given as 𝑥[𝑛] = 𝑢[𝑛] − 𝑢[𝑛 − 5], where 𝑢[𝑛] is the unit
step. Let 𝑋*[𝑛, 𝜔) be the TDFT of 𝑥[𝑛] with respect to analysis window 𝑤[𝑛] = 1 for all
n.
(a) Sketch the short-time sections 𝑓89[𝑛], 𝑓:[𝑛], and 𝑓9:[𝑛]. Justify your answers.
(b) Sketch 𝑋*[𝑛,
<;
? ) as a function of n. Justify your answer
(c) Sketch 𝑋*[𝑛,
@;
? ) as a function of n. Justify your answer
(d) Sketch 𝑋*[0, 𝜔) as a function of 𝜔. Justify your answer.
Problem 11.2 (20 points)
a) Let 𝑋*[𝑛, 𝜔) denote the TDFT of a signal 𝑥[𝑛] with respect to an analysis window 𝑤[𝑛].
Use the filtering view of the TDFT to argue that the TDFT of 𝑥[𝑛 − 𝑛:] is 𝑋*[𝑛 − 𝑛:, 𝜔).
HINT: A filter is a time-invariant system.
b) Let 𝑋*[𝑛, 𝜔) denote the TDFT of a real-valued signal 𝑥[𝑛] with respect to a real-valued
analysis window 𝑤[𝑛]. Use the Fourier transform view of the TDFT to argue that
𝑋*[𝑛, 𝜔) = 𝑋*
∗ [𝑛, −𝜔)
c) Using the filtering view of the TDFT, construct a counterexample to show that the TDFT
of the convolution of two signals is not necessarily equal to the product of their
individual TDFTs.
Problem 11.3 (20 points)
Let 𝑔[𝑛] be a 128-point signal whose Parametric Signal Model is given as 𝐻(𝑧) = E
9F∑ HI JK
ILJ MNI.
Show that the values of 𝐻 O𝑒6
PQI
PRS T for 𝑘 = 0,1, ⋯ ,255 can be calculated by taking the DFT of
some signal (using, say, an FFT algorithm) and taking the reciprocal of those DFT values.
Problem 11.4 (20 points)
Consider the following situation:
In this situation, 𝑦[𝑛] = 𝑥[𝑛] ∗ ℎ[𝑛] and ℎ[𝑛] is known to be minimum phase, while 𝑥[𝑛] is
known to be maximum phase. The windowing operation uses 𝑤[𝑛] = 𝑢[−𝑛] where 𝑢[𝑛] is the
unit step.
Show that the final output after the inverse cepstrum operation is 𝛼𝑥[𝑛], where 𝛼 is some constant.
Problem 11.5 (20 points)
(a) What shift and what amplitude scaling would you perform on the signal
such that the resulting signal has a z-transform that satisfies the
restricted model for computing the complex cepstrum. Justify your answer.
(b) Suppose . Let be a signal such that ,
where and are the respective complex cepstra of and . Sketch .
Justify your answer.
xc[n] = −2δ[n + 2]+ 4δ[n +1]
xb[n] = (0.5)
n
u[n] gb[n] gˆb[n]+ xˆb[n] = − −(0.5)
n
n
u[n −1]
gˆb[n] xˆb[n] gb[n] xb[n] gb[n]
Experiential DSP 11 (100 points)
(A) Use MATLAB to record your voice while you say the word “sad” using a sampling rate of
16 KHz. Splice off the beginning portion from the recorded signal so that the resulting
signal 𝑥[𝑛] has no silence preceding the sound of the “s” in “sad.” Use MATLAB to plot
the signal 𝑥[𝑛].
(B) Let 𝑤[𝑛] = 𝑢[𝑛] − 𝑢[𝑛 − 256] and let 𝑓1[𝑚] = 𝑤[𝑚]𝑥[𝑛 + 𝑚].
(a) Use MATLAB to calculate and plot 𝑓:[𝑚] for 0 ≤ 𝑚 < 256.
(b) Determine an integer value 𝑛: such that 𝑓1K[𝑚] falls in a portion of 𝑥[𝑛: + 𝑚]
during which the “a” sound is being spoken during “sad.” Use MATLAB to calculate
and plot 𝑓1K[𝑚] for 0 ≤ 𝑚 < 256. You should observe that this signal has a
somewhat periodic structure.
(c) Plot (using MATLAB) the magnitude of the 512-point DFT of 𝑓1K[𝑚] you observed in
the previous part. How is this plot consistent with the fact that the corresponding
short-time section has a somewhat periodic structure.