EE2003
Circuit Theory
Chapter 18
Fourier Transform
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
1
Fourier Transform
Chapter 18
18.1 Definition of the Fourier Transform
18.2 Properties of the Fourier Transform
18.3 Circuit Applications
2
18.1 Definition of Fourier Transform (1)
• It is an integral transformation of f(t) from
the time domain to the frequency domain F(w)
• F(w) is a complex function; its magnitude is
called the amplitude spectrum, while its phase
is called the phase spectrum.
Given a function f(t), its fourier transform
denoted by F(w), is defined by

F (w )   f (t )e

 jwt
dt
3
18.1 Definition of Fourier Transform (2)
Example 1
Determine the Fourier transform of a single
rectangular pulse of wide t and height A, as
shown below.
A rectangular pulse
4
18.1 Definition of Fourier Transform (3)
Solution:
t /2
F (w )  
t / 2
Ae jwt dt
A  jwt t / 2

e
t / 2
jw
2 A  e jwt / 2  e  jwt / 2 



w 
2j

 At sin c
wt
2
Amplitude spectrum of
the rectangular pulse
5
18.1 Definition of Fourier Transform (4)
Example 2:
Obtain the Fourier transform of the “switchedon” exponential function as shown below.
6
18.1 Definition of Fourier Transform (5)
Solution:
 at

e
,
 at
f (t )  e u (t )  
 0,
Hence,
F (w )  


f (t )e
 jw t
t0
t0

dt   e  jate  jwt dt


  e ( a  jw ) t dt


1
a  jw
7
18.2 Properties of Fourier Transform (1)
Linearity:
If F1(w) and F2(w) are, respectively, the Fourier
Transforms of f1(t) and f2(t)
F a1 f1 (t )  a2 f 2 (t )  a1F1 (w)  a2 F2 (w)
Example 3:
F sin( w0t ) 
  

1
F e jw0t  F e  jw0t  j  (w  w0 )   (w  w0 )
2j
8
18.2 Properties of Fourier Transform (2)
Time Scaling:
If F (w) is the Fourier Transforms of f (t), then
1 w
F  f (at )  F ( ), a is a constant
a
a
If |a|>1, frequency compression, or time expansion
If |a|<1, frequency expansion, or time compression
9
18.2 Properties of Fourier Transform (3)
Time Shifting:
If F (w) is the Fourier Transforms of f (t), then
F  f (t  t0 )  e  jwt0 F (w )
Example 4:
 j 2w
e
F e (t 2)u (t  2) 
1  jw


10
18.2 Properties of Fourier Transform (4)
Frequency Shifting (Amplitude Modulation):
If F (w) is the Fourier Transforms of f (t), then


F f (t )e jw0t  F (w  w0 )
Example 5:
1
1


F f (t ) cos(w0t )  F (w  w0 )  F (w  w0 )
2
2
11
18.2 Properties of Fourier Transform (5)
Time Differentiation:
If F (w) is the Fourier Transforms of f (t), then
the Fourier Transform of its derivative is
 df

F  u (t )  jwF ( s )
 dt

Example 6:
1
 d  at

F  e u (t )  
 dt
 a  jw
12
18.2 Properties of Fourier Transform (6)
Time Integration:
If F (w) is the Fourier Transforms of f (t), then
the Fourier Transform of its integral is
t
F (w )


F  f (t )dt 
F (0) (w )
 

jw
Example 7:
1
F u (t ) 
  (w )
jw
13
18.2 Properties of Fourier Transform (7)
Reversal:
If F(w) is the Fourier Transforms of f (t), then
reversing f(t) about the time axis reverses F(w)
about frequency.
L f (t )  F (w)  F * (w)
Example 8:
F 1  F u(t )  u(t )  2 (w)
14
18.2 Properties of Fourier Transform (8)
Duality:
If F(w) is the Fourier Transforms of f (t), then the
Fourier transform of F(t) is 2f(-w).
F  f (t )  F (w)

F F (t )  2f (w)
Example 9:
t
If f (t )  e , then
2
F (w )  2
w 1
Duality
property
If F(t) 
2
then
2
t 1
F 2f (w )  2e
w
15
18.2 Properties of Fourier Transform (9)

• It is defined as y(t )   x( )h(t   )d or y(t )  x(t ) * h(t )
• If X(w), H(w) and Y(w) are the Fourier transforms
of x(t), h(t), and y(t), respectively, then
Y (w)  F h(t ) * x(t )  H (w) X (w)
• In the view of duality property of Fourier
transforms, we expect
Y (w )  F h(t ) x(t ) 
1
H (w ) * X (w )
2
16
18.3 Circuit Applications (1)
• Fourier transforms can be applied to circuits with nonsinusoidal excitation in exactly the same way as phasor
techniques being applied to circuits with sinusoidal
excitations.
Y(w) = H(w)X(w)
• By transforming the functions for the circuit elements into
the frequency domain and take the Fourier transforms of
the excitations, conventional circuit analysis techniques
could be applied to determine unknown response in
frequency domain.
• Finally, apply the inverse Fourier transform to obtain the
response in the time domain.
17
18.3 Circuit Applications (2)
Example 10:
Find v0(t) in the circuit shown below for
vi(t)=2e-3tu(t)
18
18.3 Circuit Applications (3)
Solution:
2
The Fourier tr ansform of the input signal is Vi (w ) 
3  jw
V (w )
1
The transfer function of the circuit is H (w )  0

Vi (w ) 1  j 2w
Hence,
V0 (w ) 
1
(3  jw )(0.5  jw )
Taking the inverse Fourier tr ansform gives v0 (t )  0.4(e 0.5t  e 3t )u (t )
19