The Fourier Transform

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The Fourier Transform

• BEE2113 Signals Systems
• R. M. Taufika R. Ismail
• FKEE, UMP

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Introduction

• Fourier transform is another method to transform
  a signal from time domain to frequency domain
• The basic idea of Fourier transform comes from
  the complex Fourier series
• Practically, many signals are non-periodic
• Fourier transform use the principal of the
  Fourier series, with assumption that the period
  of the non-periodic signal is infinity (T ? 8).

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• Recall the complex form of Fourier series

(2.1)
where
(2.2)

• Substituting (2.2) into (2.1) gives

(2.3)
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• The spacing between adjacent harmonics is

or
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• Then (2.3) becomes

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• Now if we let T ? 8 , the summation becomes
  integration, ?? becomes d?, and the discrete
  harmonic frequency n?0 becomes a continuous
  frequency ?, i.e.

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• Thus, as T ? 8 ,

8

• Therefore

where

• We say that F(?) is the Fourier transform of f
  (t), and f (t) is the inverse Fourier transform
  of F(?)

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• Or in mathematical expression,

and
where F and F -1 is the Fourier transform and
inverse Fourier transform operator respectively
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• Generally, the Fourier transform F(?) exists when
  the Fourier integral converges
• A condition for a function f(t) to have a Fourier
  transform is, f(t) can be completely integrable,
  i.e.
• This condition is sufficient but not necessary

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Comparison between Fourier series and Fourier
transform
Fourier transform
Fourier series

• Support periodic function

• Support non-periodic
• function

• Discrete frequency
• spectrum

• Continuous frequency
• spectrum

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Principle of duality

• The definition of Fourier transform is

(2.4)
and the inverse Fourier transform is
(2.5)

• Note that the function f (t) and its transform
  F(?) can be derived from each other

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• From (2.5),

If we interchange t and ?
And replace ? with -?
or
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• This is an important property to find the Fourier
  transform of certain functions which their
  Fourier integral diverges

• For example, using integration, the Fourier
  transform of 1 is

• At first, we may conclude that 1 has no Fourier
  transform, but in fact, it can be found using the
  principle of duality!

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Note that is neither 0 nor 8 since
where cos 8 and sin 8 do not converge and cos
8 1 and sin 8 1. But
equal to 0 since with condition a is real and
a gt 0.
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Example 1
Using the definition, find the Fourier transform
of d(t). Then deduce the Fourier transform of 1.
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Solution
Recall the sifting property
The Fourier transform of f (t) d(t) is
Then, using the duality principle, the Fourier
transform of 1 is
?
since .
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Example 2
Find the Fourier transform of the
following function (a) (b) (c)
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Solution
(a) Using the definition,
(b) From the result in (a), by replacing a -a,
and using the duality principle,
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(c) From the result in (b),
?
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Example 3
Using the definition, find the Fourier transform
of the following (a) (b) where a ? R and
a gt 0.
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Solution
(a)
?
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(b)
?
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Properties of the Fourier transform

• Linearity
• Time scaling
• Time shifting
• Frequency shifting
• Time differentiation
• Frequency differentiation
• Duality
• Convolution

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If F(?) Ff (t), G(?) Fg (t), a and b are
constants, then
1. Linearity
2. Time scaling
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3. Time shifting
that is, a delay in the time domain corresponds
to a phase shift in the frequency domain.
Phase shift Say . Then

.
4. Frequency shifting
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5. Time differentiation
6. Frequency differentiation
7. Duality
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8. Convolution
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Example 4
Derive the Fourier transform of a single
rectangular pulse of width t and height A,
as shown below
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Solution
Using the definition of the Fourier transform,
?
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Example 5
From the result in Example 4, using the
time shifting theorem, obtain the Fourier
transform of the function g(t)
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Solution
Set A 1 and t 1 of the function f(t) to get
the following waveform
Then write as shown
below


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where and
. Hence,
Therefore, the Fourier transform of g(t) is
time shifting theorem
?
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Method of Differentiation in time domain

• The simplest Fourier transform is on the delta
  function, where Fd(t) 1
• Using this idea, before we transformed a
  function, we differentiate it until its
  derivative is expressed in delta functions form
• The important properties in implementing this
  method are

and
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Example 6
Using time differentiation, find the Fourier
transform of the following function
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Solution
The idea is we differentiate f(t) until its
derivative is expressed in the terms of delta
functions. This required twice differentiation
as shown below
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f t 1
f 1 - t
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Hence,
Taking the Fourier transform for both sides of
eqn,
?
Thus,
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Example 7
Given the Fourier transform of f(t) is
Determine the Fourier transform of the following
(b)
(a)
(c)
(d)
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Solution
(a) Using time scaling property,
?
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(b) This involve the time scaling and time
shifting properties.
?
Note that the time shift is ½, not 1. The
time shift must be determined when
the coefficient of t is 1.
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(c) From the Eulers identity,
Then, using the frequency shifting property,
?
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(d) The frequency shifting property cannot be
apply directly, since there are no term e jat.
However, note that j 2 -1, that is
Then, using the frequency shifting property,
?
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The convolution integral

• A linear system

• In frequency domain

Y(?) X(?)H(?)
X(?)
H(?)

• In time domain

y(t) x(t)h(t)
x(t)
h(t)
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• Convolution

• Which means the integral is evaluated in domain ?

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• The steps of convolution

1. Create a dummy variable (i.e. ?) and make
each waveform a function of ?.
2. Time-invert one of the waveforms (doesnt
matter which).
3. Shift the inverted waveform by t. this allow
the function to travel along ?-axis.
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4. Find the product of the intersection
5. For a given time t, integrate the product for
the intersection range a lt ? lt b.
6. Combine all the integrals after the
travelling waveform moved for -8 lt t lt 8.
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Example 8
Find the convolution of the two signals
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Solution
1. Write f and g as functions of ?.
g(?)
f(?)
g(?) ?
?
?
2. Choose one of the functions as the
travelling wave. Suppose that we choose
f. Reflect f(?) about the vertical axis to get
f(-?).
f(-?)
g(?)
g(?) ?
?
?
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3. Write f(-?) as f(t-?) . Since t varies from -8
to 8, it means we bring f to -8 and shift
it to 8 direction.
f(t-?)
g(?)
?
t
t - 1
0
1
45. Along the journey of f(t - ?), observe
the intersections between f and g.
Then integrate along the intersection ranges
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(i) -8 lt t lt 0 No overlap. f g 0
(ii) 0 lt t lt 1
f(t-?)
g(?)
?
t
t - 1
0
1
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(iii) 1 lt t lt 2
f(t-?)
g(?)
?
t - 1
t
1
0
(iv) 2 lt t lt 8 No overlap. f g 0
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6. Combine all the integral for -8 lt t lt 8 to
obtain f g
f g (t)
½
t
0
2
1
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Example 9
Find . Given
and
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Solution
Sketch the graph x and h
h(t)
x(t)
1
1
t
t
0
-1
1
0
1
-1
-1
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Suppose that we choose h to be the
travelling wave. After express x and h as
functions of ?, then flip h and bring it to -8
x(?)
1
?
0
-1
1
-1
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(i) -8 lt t lt -2 No overlap. x h 0
(ii) -2 lt t lt -1
x(?)
1
?
0
-1
1
-1
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(iii) -1 lt t lt 0
x(?)
1
?
0
1
-1
-1
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(iv) 0 lt t lt 1
x(?)
1
?
0
1
-1
-1
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(v) 1 lt t lt 2
x(?)
1
?
0
1
-1
-1
(vi) 2 lt t lt 8 No overlap. x h 0
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Combine all the integral for -8 lt t lt 8 to obtain
x h
x h (t)
1
t
2
-1
-2
1
-1
0
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Circuit application

• Can solve circuit problem with various kind of
  voltage and current source sinusoid,
  exponential, impulse, step function, etc.
• Similar with phasor analysis

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Example 10
Using the Fourier transform technique, determine
the output voltage of the following
circuit.
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Solution
Transform the cct. into frequency domain
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Using voltage divider rule,
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Example 11 (P.P.18.8, pg.831)

• Find the current in the following
  circuit. Given that .

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Solution
Transform the cct. into frequency domain
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Using current divider rule,
The inverse Fourier of cannot be found
directly from the table. Therefore we use
the inverse Fourier formula.
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Example 12

• Determine in the following circuit.

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Solution
Transform the cct. into frequency domain
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Using nodal analysis,
where
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Using partial fraction,
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Pb.18.43, pg.845
Find vo(t) in the circuit of Fig. 18.45, where
is 5e- t u(t) A.
Figure 18.45
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Pb.18.44, pg.845
If the rectangular pulse in Fig. 18.46(a) is
applied to the circuit in Fig. 18.46(b), find
vo at t 1 s.
Figure 18.46
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Parsevals theorem

• Parsevals theorem demonstrates one practical use
  of the Fourier transform.
• It relates the energy carried by a signal to the
  Fourier transform of the signal.
• If p(t) is the power associated with the signal,
  the energy carried by the signal is

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• It is convenient to use a 1 W resistor as the
  base for energy calculation.
• For a 1 W resistor,
• where f(t) stands for either voltage or current.
• The energy delivered to the 1 W resistor is

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• Parsevals theorem states that this same energy
  can be calculated in the frequency domain as
• since F(w)2 is an even function.
• We may also calculate the energy in any frequency
  band w1 lt w lt w2 as

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Example 13 (P.P.18.9, pg.834)

• (a) Calculate the total energy absorbed by a 1 W
  resistor with i(t) 10e-2 t A in the time
  domain.
• (b) Repeat (a) in the frequency domain.

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Solution
(a)
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(b)
From the table,
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Example 14 (P.P.18.10, pg.835)

• A 1 W resistor has i(t) e-tu(t) A. What
  percentage of the total energy is in the
  frequency band -4 lt w lt 4 rad/s?

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Solution
From the table,
Hence,
and
The total energy dissipated by the 2 W resistor
is
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The energy dissipated in the frequency -4 lt w lt
4 rad/s is
Therefore, its percentage of the total energy is