Fundamentals of
Electric Circuits
Chapter 18
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Overview
• In this chapter, the Fourier series is extended
to cover no-repeating signals.
• The concept of the Fourier transform: a
function that converts a signal from time
domain to frequency domain is introduced.
• The inverse transform, which converts from
frequency to time domain is also introduced.
• Properties of the transform are covered.
• Finally, a comparison between the Laplace
and Fourier transforms will be made.
2
Non Periodic Functions
• We have seen how Fourier series can
represent any periodic waveform.
• But, many signals of interest in
electronics are not periodic.
• Although these cannot be represented in
a Fourier series, they can be transformed
into frequency domain by use of
something called the Fourier transform.
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Stretching the Period
• One way to consider a nonperiodic function is to take a
periodic one and stretch the
period.
• Consider the periodic function
shown at the bottom of the
figure.
• If the period T→, then the
function becomes non-periodic.
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Effect on the Spectrum
• Here you can see the
impact of increasing
the period has.
• As the pulses are
spaced out more, the
peaks in the frequency
spectrum get closer
together
• The amplitude also
drops.
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Effect on the Spectrum II
• Note that the sum of the amplitudes does not
change.
• Ultimately what will happen is as the period
goes to infinity, the discrete spectrum will
become a continuous spectrum.
• The result is what is known as the Fourier
transform
F    F  f  t  


f t  e jt dt

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Inverse Fourier
• The Fourier transform is an integral
transform of f(t) from the time domain to the
frequency domain.
• In general, F(ω) is a complex function.
• Its magnitude is called the amplitude
spectrum.
• The phase is called the phase spectrum.
• There is also a inverse transform:
1
f  t   F 1  F   
2


F   e jt d

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Conditions on F(ω)
• The Fourier transform does not exist for all
functions.
• It only exists where the Fourier integral
converges.
• A sufficient, but not necessary condition is:

 f t  dt  

• An example of a function that would fail this
test is the unit ramp function.
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Example 1
Determine the Fourier transform of a single
rectangular pulse of wide t and height A, as shown
below.
A rectangular pulse
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Solution:
t /2
F ( )  
t / 2
Ae jt dt
A  jt t / 2

e
t / 2
j
2 A  e jt / 2  e  jt / 2 



 
2j

 At sin c
t
2
Amplitude spectrum of
the rectangular pulse
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Example 2:
Obtain the Fourier transform of the “switched-on”
exponential function as shown below.
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Solution:
 at

e
,
 at
f (t )  e u (t )  
 0,
Hence,

F ( )   f (t )e

 jt
t 0
t 0

dt   e  jat e  jt dt


  e ( a  j )t dt


1
a  j
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Linearity of F(ω)
• We will now establish some of properties of
the Fourier transform
• Linearity: If F1(ω) and F2(ω) are the Fourier
transforms of f1(t) and f2(t) then
F  a1 f1  t   a2 f 2  t    a1F1    a2 F2  
• Where a1 and a2 are constants.
• This simply states that the transform of a
linear combination of functions equals the
linear combination of each transform.
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Time Scaling
• In time scaling:
1  
F  f  at   F  
a a
• This shows that time expansion (|a|>1)
corresponds to frequency compression and
vice versa.
• In other words, imagine a pulse that
becomes shorter; the Fourier transform of
this broadens in frequency.
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Time Scaling II
• The effect of time scaling can be seen in this
example:
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Frequency Shifting
• In frequency shifting:
F  f  t  e j0t   F   0 
• A frequency shift in the frequency domain
adds a phase shift to the time function.
• This can also be seen as amplitude
modulation in the time domain.
• This has important consequences for
modulation, a common form of
communication.
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Time Differentiation
• In time differentiation:
F  f   t    j F  
• The transform of the derivative of f (t) is
obtained by multiplying the transform of f (t)
by jω.
• This can be generalized to the n’th
derivative:
n
 n


F  f  t     j  F  
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Time Integration
• In time integration:

 F  
F   f  t  dt  
  F  0    
 

j
• the transform of the integral of f (t) is
obtained by dividing the transform of f (t) by
jω and adding the result to the impulse term
that reflects the dc component F(0).
• Note that the upper bound of the integral is t.
• If it were not, the Fourier transform would be
of a constant.
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Reversal
• In reversal:
F  f  t    F     F *  
• This property states that reversing f (t) about
the time axis reverses F(ω) about the
frequency axis.
• This can be considered a special case of
time scaling if a=-1.
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Duality and Convolution
• Duality states
F  f  t    F    F  F  t    2 f   
• This expresses the symmetry property of the
Fourier transform.
• Convolution states:
Y    F  h  t  * x  t    H   X  
• An example of how this works is shown
below:
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Convolution
• Here is a graphical representation of
convolution:
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Fourier Transform Pairs
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More Fourier Transform Pairs
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Circuit Applications
• We can apply Fourier transforms to circuits with
non-sinusoidal excitations in exactly the same way
we apply phasor techniques to circuits with
sinusoidal excitations.
• Thus, Ohm’s law is still valid:
V    Z   I  
• We get the same expressions for impedances as in
phasor analysis
R 
R
L 
j L
C 
1
jC
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25
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Circuit Applications II
• Once transformed to frequency domain and
the transform of the excitations are taken,
the analysis of the circuit can proceed as has
been done previously.
• Note, though, that Fourier transforms can’t
handle initial conditions.
• The transfer function again defined as.
H   
Y  
X  
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Example 10:
Find v0(t) in the circuit shown below for
vi(t)=2e-3tu(t)
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Solution:
2
T heFourier transformof theinput signal is Vi ( ) 
3  j
V ( )
1
T he transferfunctionof thecircuit is H ( )  0

Vi ( ) 1  j 2
Hence,
V0 ( ) 
1
(3  j )(0.5  j )
T akingtheinverseFourier transformgives v0 (t )  0.4(e 0.5t  e 3t )u (t )
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Fourier vs. Laplace
• Let us compare the Fourier transform with
the Laplace transform:
1. The Laplace transform is one-sided in that
the integral is over 0<t< making it only
useful for positive time functions, f (t), t>0
The Fourier transform is applicable to
functions defined for all time.
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Fourier vs. Laplace II
2. For a function that is nonzero for positive time
only, the two transforms are related by:
F    F  s  s  j
–
–
This equation also shows that the Fourier
transform can be regarded as a special case of
the Laplace transform with s= jω
Recall that s=+ jω Therefore, this equation
shows that the Laplace transform is related to
the entire s plane, whereas the Fourier transform
is restricted to the jω axis.
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Fourier vs. Laplace III
3. The Laplace transform is applicable to a
wider range of functions than the Fourier
transform.
– For example, the function tu(t) has a Laplace
transform but no Fourier transform. But Fourier
transforms exist for signals that are not physically
realizable and have no Laplace transforms.
4. The Laplace transform is better suited for
the analysis of transient problems involving
initial conditions, since it permits the
inclusion of the initial conditions, whereas
the Fourier transform does not.
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