Description
Objectives:
Understand different concepts of Fourier transform along with its applications.
Prerequisites:
DTFT, DFT and circular convolution.
Important notes:
Initially each student needs to execute DFT using FFT MATLAB command. Then create
your own DFT function and again execute all following problems using your own created
function. At the end all need to compare the output, obtain using FFT MATLAB
command and your own function. Verify answer using IFFT MATLAB command wherever
required.
Problems
1. Consider analog sinusoidal wave ( ) ( ) is sampled at twice the Nyquist
rate for full one period. Find out its DFT. If x(t) is sampled at Nyquist rate for one period
what will be its DFT?. Instead of one period if two full periods are sampled then how the
result is affected in both cases?
Write a program to compute the DFT of given sinusoidal wave by specifying amplitude,
frequency, sampling frequency and number of periods to obtain DFT from user. Plot
magnitude spectrum with horizontal axis indicating the analog frequency. Assume the
phase of the sinusoidal wave is zero.
2. Use MATLAB to compute DFT of following sequences and verify the answer by finding
IFFT command. Display output sequences on command window and also obtain
magnitude and phase spectrum.
a) x(n) = {1,2,3,4}
b) x(n) = ( ) ( ) for N=4 and N= 8
SEAS Winter 2020
2
3. Convolution Application using DFT-IDFT
a. Write a MATLAB program to find circular convolution of two sequences using
DFT-IDFT based approach.
x1(n) = {1,-1,-2,3,-1} and x2(n) = {1,2,3}
x1(n) = {1, 2, 1, 2} and x2(n)= {3,2,1,4}
b. Write a program to perform linear convolution using above DFT-IDFT approach
based circular convolution.
c. Plot and Verify above sequences for (a) and (b)
4. Application of Fourier transform in Audio signal processing Application (Transform and
Noise removal)
a. Explore following commands
Audioread
length
audiowrite
fftshift
filter
butter (Butterworth filter) [i.e [b,a] = butter(3, [0.3 0.7],’bandpass’)]
b. Take an audio file “sample_sound.wav” from shared folder. Apply different
commends which are mentioned above on given audio file to find sampling
frequency, frequency range of audio input file and normalized frequency. Plot
the normalized audio signal. Take Fourier transform to plot and analyze
magnitude spectrum of the same.
c. Now generate one noise signal inform of sinusoidal wave and add it to audio file
which you have read. Generate noisy audio file and save it. By applying
Butterworth equation find different filter coefficients. Once filter coefficients has
been received, apply filter on noisy audio file using generated filter coefficients.
Generate the filtered output audio file and save it. Plot Filtered Signal. Now
listen original file, noisy audio file and filtered output audio file one by one.
