Fourier Transform
Examples
1
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example 1
• By hand, do the following:
– Find complex conjugate of a = 1 - 2j
– Find magnitude of b = -4 + 5j
– Find product of a*b
2
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example 2
• By hand, do the following:
– Find polar representation of -2 – 3j
– Find Cartesian representation of 5e-j3
• Repeat the previous exercises using Matlab (see
functions conj, abs, angle)
3
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example 3
• Find the integration of the continuous impulse function with
∫ (2t
∞
−∞
∫
∞
−∞
2
+ 1)δ (t )dt
sin (2πt )δ (t − 1)dt
• Find the summation of the discrete impulse function with
∞
−πx
e
3
∑ δ ( x)
x = −∞
∞
∑ cos(2πx + π / 2)δ ( x + 1 / 2)
x = −∞
4
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example 4
• Find the Fourier transform of the one-sided “box” or “rectangle”
function
A 0 ≤ t ≤ T
f (t ) = 
 0 otherwise
The magnitude of F(u)
is the same as the
result when the box
was centered at the
origin; it is just
multiplied by a constant
exponential term (a
phase shift)
5
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example 5
• In Matlab, create a series of numbers f(0), f(1), ..., f(M-1).
• Find the Discrete Fourier Transform (DFT) of the series, directly
from the definition:
F (u )
=
M −1
∑
for u 0,1,..., M − 1
f ( x)e − j 2π ux / M=
x =0
x = 0:M-1;
u = 0:M-1;
% Create an input series f(1), … ,f(M)
% :
for n=1:M
F(n) = 0;
for m = 1:M
F(n) = F(n) + f(m) * exp(-j*2*pi*u(n)*x(m)/M);
end
end
6
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example 5 (continued)
• Show that applying the inverse DFT to F yields the original
series f
1 M −1
j 2πux / M
(
)
for x = 0,1,, M − 1
f ( x) =
F
u
e
∑
M u =0
for n=1:M
f2(n) = 0;
for m = 1:M
f2(n) = f2(n) + F(m) * exp(j*2*pi*u(n)*x(m)/M);
end
end
7
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example 5 (continued)
• Repeat example, using Matlab’s fft, ifft functions
clear all
close all
M = 10;
x = 0:M-1;
u = 0:M-1;
% Create an input series f(1), … ,f(M)
f = rand(1,M)
for n=1:M
F(n) = 0;
for m = 1:M
F(n) = F(n) + f(m) * exp(-j*2*pi*u(n)*x(m)/M);
end
end
disp(F)
for n=1:M
f2(n) = 0;
for m = 1:M
f2(n) = f2(n) + F(m) * exp(j*2*pi*u(n)*x(m)/M);
end
end
disp(f2)
F2 = fft(f);
disp(F2)
f3 = ifft(F2);
disp(f3)
Colorado School of Mines
8
Department of Electrical Engineering and Computer Science
Example 6
• Show that the 1D discrete Fourier transform is a
linear operation.
• Linearity:
𝑎𝑎𝑓𝑓1 𝑥𝑥 + 𝑏𝑏𝑓𝑓2 𝑥𝑥 ↔ 𝑎𝑎𝐹𝐹1 𝑢𝑢 + 𝑏𝑏𝐹𝐹2 𝑢𝑢
=
F (u )
M −1
∑
for u 0,1,..., M − 1
f ( x)e − j 2π ux / M=
x =0
First show that af(x) <-> aF(u), then show f1(x)+f2(x) <-> F1(u)+F2(u)
9
Colorado School of Mines
Department of Electrical Engineering and Computer Science