This document contains practice problems for a digital signal processing course. It includes 6 sections with multiple parts each: 1) Computing the discrete Fourier transform (DFT) and fast Fourier transform (FFT) of various sequences and plotting the results. 2) Computing the FFT of sequences and plotting the phase and magnitude. 3) Using a decimation in time FFT algorithm to find the DFT of sequences and plotting the results. 4) Questions about sampling an audio signal and computing the DFT and FFT. 5) Determining the circular convolution between two sequences. 6) Computing the circular convolution of two sequences using the DFT, along with related questions.

1. Helwan University
Faculty of Engineering
Department of Electronics, Communications & Computer
ELC 9421 Digital Signal Processing - Spring 2017
Sheet Eight
‫ن‬ ‫ا‬ ‫و‬ ‫ل‬ ‫ح‬ ‫ب‬ ‫ة‬ ‫س‬ ‫د‬ ‫ن‬ ‫ه‬ ‫ل‬ ‫ا‬ ‫ة‬ ‫ي‬ ‫ل‬ ‫ك‬–‫ن‬ ‫ا‬ ‫و‬ ‫ل‬ ‫ح‬ ‫ة‬ ‫ع‬ ‫م‬ ‫ا‬ ‫ج‬ Page 1 of 1
1) Compute the DFT of the following sequences:
a. 𝑥( 𝑛)  [1, 0, 1, 0]
b. 𝑥( 𝑛)  [𝑗, 0, 𝑗, 1]
c. 𝑥( 𝑛)  [1, 1, 1, 1, 1, 1, 1, 1]
d. 𝑥( 𝑛) = {0.5, 0.5, 0.5 ,0.5, 0, 0, 0, 0}
e. 𝑥[𝑛] = {1, 2, 2, 2, 1, 0, 0, 0}
f. 𝑥( 𝑛)  cos(0.25 𝜋 𝑛) 𝑎𝑛𝑑 𝑛  0, . . . , 7
g. 𝑥(𝑛)  0.9𝑛, 𝑛  0, . . . , 7
Then Plot the phase and magnitude of 𝑋(𝑘).
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2) Compute the FFT of the following sequences:
a. 𝑥( 𝑛)  [1, 0, 1, 0]
b. 𝑥( 𝑛)  [𝑗, 0, 𝑗, 1]
c. 𝑥( 𝑛)  [1, 1, 1, 1, 1, 1, 1, 1]
d. 𝑥( 𝑛) = {0.5, 0.5, 0.5 ,0.5, 0, 0, 0, 0}
e. 𝑥[𝑛] = {1, 2, 2, 2, 1, 0, 0, 0}
Then Plot the phase and magnitude of 𝑋(𝑘).
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3) Using the 8-point Decimation In Time FFT algorithm, find the DFT of:
a. 𝑥( 𝑛)  [1, 0, 1, 0]
b. 𝑥( 𝑛)  [𝑗, 0, 𝑗, 1]
c. 𝑥( 𝑛)  [1, 1, 1, 1, 1, 1, 1, 1]
d. 𝑥( 𝑛) = {0.5, 0.5, 0.5 ,0.5, 0, 0, 0, 0}
e. 𝑥[𝑛] = {1, 2, 2, 2, 1, 0, 0, 0}
Then Plot the phase and magnitude of 𝑋(𝑘).
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2. ‫ة‬ ‫ي‬ ‫ل‬ ‫ك‬‫ن‬ ‫ا‬ ‫و‬ ‫ل‬ ‫ح‬ ‫ب‬ ‫ة‬ ‫س‬ ‫د‬ ‫ن‬ ‫ه‬ ‫ل‬ ‫ا‬–‫ا‬ ‫ج‬‫م‬‫ة‬ ‫ع‬‫ن‬ ‫ا‬ ‫و‬ ‫ل‬ ‫ح‬ Page 2 of 2
4) An audio waveform x(t) is sampled at a rate of fs = 8000Hz for T = 0.5sec
resulting in a set of N samples.
a. Find number of samples N.
b. Find the frequency resolution.
c. Find the required number of operations that is needed to calculate the
DFT.
d. Find the required number of operations that is needed to calculate the FFT.
e. Find the saving in using FFT over DFT
f. What are the frequencies, associated with the following values of 𝑘 = 0, 1, 2.
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5) Determine the circular convolution between 𝑥(𝑛) and ℎ(𝑛), if
𝑥[𝑛] = {2, 5, −1, 4, 3, 0, 0, 0}
and
ℎ[𝑛] = {1, 3, 4, 1}
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6) Given 𝒙[𝒏] = 𝜹[𝒏] + 𝟐𝜹[𝒏 − 𝟏] + 𝜹[𝒏 − 𝟐] and 𝒉[𝒏] = 𝟑𝜹[𝒏 − 𝟐] − 𝟐𝜹[𝒏 − 𝟑].
a. Compute 𝑦[𝑛] = 𝑥[𝑛] ∗ ℎ[𝑛].
b. Compute the 4-point DFT 𝑋[𝑘] of 𝑥[𝑛].
c. Compute the 4-point DFT 𝐻[𝑘] of ℎ[𝑛].
d. Compute the circular convolution 𝑦𝑐[𝑛] between 𝑥[𝑛] and ℎ[𝑛], by two
different methods.
e. How many padding zeros are needed for x[n] and h[n] in order to find
their regular convolution y[n], using the DFT?
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Best Wishes,
Prof. Elsayed. M. Saad
Dr. Amr E. Mohamed