Experiment-3
AIM:
Write a MATLAB program to implement and verify the following DFT (Discrete Fourier Transform) properties:
1. Linearity
2. Time Reversal
3. Time Shifting
4. Convolution
Software Used:
MATLAB
Theory:
The DFT possesses several important properties that simplify the analysis and processing of discrete signals. In
this experiment, we focus on verifying the following:
1. Linearity:-
Linearity Property of DFT states that the DFT of a linear combination of signals is equal to the
same linear combination of their DFTs.
If
𝒙𝟏 (𝒏) → 𝑿𝟏 (𝒌), 𝒙𝟐 (𝒏) → 𝑿𝟐 (𝒌)
Then for constants a and b,
𝒂𝒙𝟏 (𝒏) + 𝒃𝒙𝟐 (𝒏) → 𝒂𝑿𝟏 (𝒌) + 𝒃𝑿𝟐 (𝒌)
2. Time Reversal:-
This property shows that flipping a sequence in time produces a flipped spectrum in frequency.
If
𝒙(𝒏) → 𝑿(𝒌)
Then,
𝒙(− 𝒏) → 𝑿(− 𝒌)
3. Time Shifting:-
If a sequence is shifted in time, its DFT is multiplied by a complex exponential factor. Thus, time-
domain shifting results in a linear phase shift in the frequency domain.
If
𝒙(𝒏) → 𝑿(𝒌)
Then,
𝟐𝝅
𝒙(𝒏 − 𝒏𝟎 ) → ⅇ−𝒋 𝑵 𝒌𝒏𝟎 𝑿(𝒌)
4. Convolution:-
The convolution property relates circular convolution in the time domain to multiplication in the
frequency domain.
Code:
clc;
clear;
close all;
x1_n = [1 2 3 4];
fprintf('First Signal');
disp(x1_n)
x2_n = [4 3 2 1];
fprintf('Second Signal')
disp(x2_n);
N = max(length(x1_n),length(x2_n));
x1_n_padded = [x1_n, zeros(1,N-length(x1_n))];
x2_n_padded = [x2_n, zeros(1,N-length(x2_n))];
% 1. Linearity
X1 = fft(x1_n, N);
X2 = fft(x2_n, N);
a = 2; b = 3;
lhs = fft(a*x1_n + b*x2_n, N);
rhs = a*X1 + b*X2;
disp('--- Linearity Property ---');
disp([lhs.' rhs.']); % two column vectors for comparison
% 2. Time Reversal
x_rev = [x1_n(1) fliplr(x1_n(2:end))];
X_rev = fft(x_rev);
X1_rev_expected = [X1(1) fliplr(X1(2:end))];
disp('--- Time Reversal Property ---');
disp([X_rev.' X1_rev_expected.']);
% 3. Time Shifting
k = 2;
x1_n_shifted = circshift(x1_n_padded, [k 0]);
fprintf('The time shifted signal is')
disp(x1_n_shifted)
X1_k_shifted = fft(x1_n_shifted);
phase_shift = exp(-1j*2*pi*(0:N-1)*k/N);
X_k_expected = X1 .* phase_shift;
disp('--- Time Shifting Property ---');
disp([X1_k_shifted.' X_k_expected.']);
% 4. Convolution Property
x_conv = cconv(x1_n, x2_n, N);
X_conv = fft(x1_n, N) .* fft(x2_n, N);
lhs_conv = fft(x_conv, N);
disp('--- Convolution Property ---');
disp([lhs_conv.' X_conv.']);
Result:
First Signal 1 2 3 4
Second Signal 4 3 2 1
--- Linearity Property ---
50.0000 + 0.0000i 50.0000 + 0.0000i
2.0000 - 2.0000i 2.0000 - 2.0000i
2.0000 + 0.0000i 2.0000 + 0.0000i
2.0000 + 2.0000i 2.0000 + 2.0000i
--- Time Reversal Property ---
10.0000 + 0.0000i 10.0000 + 0.0000i
-2.0000 - 2.0000i -2.0000 - 2.0000i
-2.0000 + 0.0000i -2.0000 + 0.0000i
-2.0000 + 2.0000i -2.0000 + 2.0000i
The time shifted signal is 3 4 1 2
--- Time Shifting Property ---
10.0000 + 0.0000i 10.0000 + 0.0000i
-2.0000 + 2.0000i 2.0000 - 2.0000i
-2.0000 + 0.0000i -2.0000 - 0.0000i
-2.0000 - 2.0000i 2.0000 + 2.0000i
--- Convolution Property ---
1.0000 + 0.0000i 1.0000 + 0.0000i
0.0000 + 0.0800i 0.0000 + 0.0800i
-0.0400 + 0.0000i -0.0400 + 0.0000i
0.0000 - 0.0800i 0.0000 - 0.0800i
Conclusion:
Linear convolution using for loop in MATLAB was successfully implemented and verified with the direct
convolution method.
