24 1 fourier transform

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Contentsontents Fourier Transforms24.1 The Fourier Transform 2

24.2 Properties of the Fourier Transform 14

24.3 Some Special Fourier Transform Pairs 27

Learning

In this Workbook you will learn about the Fourier transform which has many applications

in science and engineering. You will learn how to find Fourier transforms of some

standard functions and some of the properties of the Fourier transform. You will learn

about the inverse Fourier transform and how to find inverse transforms directly and by

using a table of transforms. Finally, you will learn about some special Fourier transform

pairs.

outcomes
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The Fourier transform

24.1

Introduction

Fourier transforms have for a long time been a basic tool of applied mathematics, particularly forsolving differential equations (especially partial differential equations) and also in conjunction withintegral equations.

There are really three Fourier transforms, the Fourier Sine and Fourier Cosine transforms and acomplex form which is usually referred to as theFourier transform.

The last of these transforms in particular has extensive applications in Science and Engineering, for

example in physical optics, chemistry (e.g. in connection with Nuclear Magnetic Resonance andCrystallography), Electronic Communications Theory and more general Linear Systems Theory.

Prerequisites

Before starting this Section you should. . .

be familiar with basic Fourier series,particularly in the complex form

Learning Outcomes

On completion you should be able to. . .

calculate simple Fourier transforms from thedefinition

state how the Fourier transform of a function

(signal) depends on whether that function iseven or odd or neither

2 HELM (2005):Workbook 24: Fourier transforms


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1. The Fourier transformUnlike Fourier series, which are mainly useful for periodic functions, the Fourier transform permitsalternative representations of mostly non-periodic functions.

We shall firstly derive the Fourier transform from the complex exponential form of the Fourier seriesand then study its various properties.

2. Informal derivation of the Fourier transformRecall that iff(t) is a periodT function, which we will temporarily re-write asfT(t) for emphasis,then we can expand it in a complex Fourier series,

fT(t) =

n=

cnein0t (1)

where0 = 2

T . In words, harmonics of frequencyn0 =n

2

T n = 0, 1,2, . . . are present in

the series and these frequencies are separated by

n0 (n 1)0 =0=2T.

Hence, asTincreases the frequency separation becomes smaller and can be conveniently written as. This suggests that asT , corresponding to a non-periodic function, then 0 andthe frequency representation contains all frequency harmonics.

To see this in a little more detail, we recall ( 23: Fourier series) that the complex Fouriercoefficientscn are given by

cn = 1

T

T2

T

2

fT(t)ein0t dt. (2)

Putting 1

T as

0

2and then substituting (2) in (1) we get

fT(t) =

n=

0

2 T

2

T

2

fT(t)ein0t dt

ein0t.

In view of the discussion above, asT we can put0 as and replace the sum over thediscrete frequenciesn0 by an integral over all frequencies. We replacen0 by a general frequencyvariable. We then obtain the double integral representation

f(t) =

1

2

f(t)eit dt

eit d. (3)

The inner integral (over allt) will give a function dependent only on which we write asF().Then (3) can be written

f(t) = 1

2

F()eit d (4)

where

HELM (2005):Section 24.1: The Fourier transform

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F() =

f(t)eit dt. (5)

The representation (4) off(t) which involves all frequencies can be considered as the equivalentfor a non-periodic function of the complex Fourier series representation (1) of a periodic function.

The expression (5) forF() is analogous to the relation (2) for the Fourier coefficientscn.

The functionF() is called the Fourier transformof the functionf(t). Symbolically we can write

F() = F{f(t)}.Equation (4) enables us, in principle, to writef(t)in terms ofF().f(t)is often called the inverseFourier transform ofF() and we denote this by writing

f(t) = F1{F()}.

Looking at the basic relation (3) it is clear that the position of the factor

1

2 is somewhat arbitraryin (4) and (5). If instead of (5) we define

F() = 1

2

f(t)eit dt.

then (4) must be written

f(t) =

F()eit d.

A third, more symmetric, alternative is to write

F() = 12

f(t)eit dt

and, consequently:

f(t) = 1

2

F()eit d.

We shall use (4) and (5) throughout this Section but you should be aware of these other possibilitieswhich might be used in other texts.

Engineers often refer toF() (whichever precise definition is used!) as the frequency domainrepresentation of a function or signal andf(t) as the time domain representation. In what followswe shall use this language where appropriate. However, (5) is really a mathematical transformation forobtaining one function from another and (4) is then the inverse transformation for recovering the initialfunction. In some applications of Fourier transforms (which we shall not study) the time/frequencyinterpretations are not relevant. However, in engineering applications, such as communications theory,the frequency representation is often used very literally.

As can be seen above, notationally we will use capital letters to denote Fourier transforms: thus afunctionf(t) has a Fourier transform denoted byF(),g(t) has a Fourier transform writtenG()and so on. The notationF(i),G(i) is used in some texts because occurs in (5) only in thetermeit.



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3. Existence of the Fourier transformWe will discuss this question in a little detail at a later stage when we will also consider briefly therelation between the Fourier transform and the Laplace Transform ( 20). For now we will use(5) to obtain the Fourier transforms of some important functions.

Example 1Find the Fourier transform of the one-sided exponential function

f(t) =

0 t 0

where is a positive constant, shown below:

f(t)

t

Figure 1

Solution

Using (5) then by straightforward integration

F() =

0

eteit dt (sincef(t) = 0 fort 0.

This important Fourier transform is written in the following Key Point:


5


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Key Point 1

F{etu(t)

}=

1

+i, >0.

Note that this real function has a complex Fourier transform.

Note that ifu(t) is used to denote the Heaviside unit step function:

u(t) =

0 t 0

then we can write the function in Example 1 as: f(t) =et

u(t). We shall frequently use thisconcise notation for one-sided functions.

TaskaskWrite down the Fourier transforms of

(a) etu(t) (b) e3tu(t) (c) et

2u(t)

Use Key Point 1:

Your solution

(a)

(b)

(c)

Answer

(a) = 1 soF{etu(t)} = 11 +i

(b) = 3 soF{e3tu(t)} = 13 +i

(c) = 12

soF{e t2u(t)} = 112

+i



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TaskaskObtain, using the integral definition (5), the Fourier transform of the rectangularpulse

p(t) = 1 a < t < a0 otherwise

.

Note that the pulse width is 2aas indicated in the diagram below.

ta a

1

p(t)

First use (5) to write down the integral from which the transform will be calculated:

Your solution

Answer

P() F{p(t)} = aa

(1)eit dt using the definition ofp(t)

Now evaluate this integral and write down the final Fourier transform in trigonometric, rather than

complex exponential form:Your solution

Answer

P() = aa

(1)eit dt=

eit(i)

aa

=eia

e+ia

(i)

= (cos a i sin a) (cos a+i sin a)

(i) =2i sin a

i

i.e.

P() = F{p(t)} =2sin a

(6)

Note that in this case the Fourier transform is wholly real.


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Engineers often call the function sin x

x the sinc function. Consequently if we write, the transform

(6) of the rectangular pulse as

P() = 2asin a

a ,

we can say

P() = 2a sinc(a).

Using the result (6) in (4) we have the Fourier integral representation of the rectangular pulse.

p(t) = 1

2

2sin a

eit d.

As we have already mentioned, this corresponds to a Fourier series representation for a periodicfunction.

Key Point 2

The Fourier transform of a Rectangular Pulse

Ifpa(t) =

1 a < t < a0 otherwise

then:

F{pa(t)} = 2asin aa

= 2a sinc(a)

Clearly, if the rectangular pulse has width 2, corresponding toa= 1 we have:

P1() F{p1(t)} = 2 sin

.

As 0, then2 sin

2. Also, the function 2 sin

is an even function being the product of two

odd functions2sin and 1

. The graph ofP1() is as follows:

P1()

2

Figure 2



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TaskaskObtain the Fourier transform of the two sided exponential function

f(t) =

et t 0

where is a positive constant.

f(t)

t

1

Your solution

AnswerWe must separate the range of the integrand into [, 0] and [0, ] since the functionf(t) isdefined separately in these two regions: then

F() =

0

eteit dt+

0

eteit dt=

0

e(i)t dt+

0

e(+i)t dt

=

e(i)t

( i)0

+

e(+i)t

(+i)0

=

1

i + 1

+i =

2

2 +2 .


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Note that, as in the case of the rectangular pulse, we have here a real even function oft giving aFourier transform which is wholly real. Also, in both cases, the Fourier transform is an even(as wellas real) function of.

Note also that it follows from the above calculation that

F{etu(t)} = 1+i

(as we have already found)

and

F{etu(t)} = 1 i where e

tu(t) =

et t 0

.

4. Basic properties of the Fourier transform

Real and imaginary parts of a Fourier transformUsing the definition (5) we have,

F() =

f(t)eit dt.

If we writeeit = cos t i sin t, then

F() =

f(t)cos t dt i

f(t)sin t dt

where both integrals are real, assuming thatf(t) is real. Hence the real and imaginary parts of theFourier transform are:

Re (F()) =

f(t)cos tdt Im (F()) =

f(t)sin tdt.

TaskaskRecalling that ifh(t) is even andg(t) is odd then

aa

h(t) dt= 2

a0

h(t) dt and aa

g(t) dt= 0, deduce Re(F()) and Im(F()) if

(a) f(t) is a real even function

(b) f(t) is a real odd function.

Your solution

(a)



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AnswerIff(t) is real and even

R() ReF() = 2

0

f(t)cos tdt (because the integrand is even)

I() ImF() =

f(t)sin tdt= 0 (because the integrand is odd).

Thus, any real even functionf(t) has a wholly real Fourier transform. Also since

cos(()t) = cos(t) = cos tthe Fourier transform in this case will be a real even function.

Your solution

(b)

AnswerNow

ReF() = f(t)cos tdt=

(odd) (even) dt=

(odd) dt= 0

and

ImF() =

f(t)sin tdt= 2

0

f(t)sin tdt

(because the integrand is (odd)(odd)=(even)).Also since sin(()t) = sin t, the Fourier transform in this case is an odd function of.

These results are summarised in the following Key Point:


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Key Point 3

f(t) F() =F{

f(t)}

, F() =R() + iI()

real and evenreal and odd

neither even nor oddpurely imaginary and oddcomplex

real and even

Polar form of a Fourier transform

TaskaskThe one-sided exponential function f(t) = etu(t) has Fourier transform

F() = 1

+i.Find the real and imaginary parts ofF().

Your solution

Answer

F() = 1+i =

i2 +2 .

Hence R() =ReF() =

2 +2 I() =ImF() =

2 +2

We can rewriteF(), like any other complex quantity, in polar form by calculating the magnitudeand the argument (or phase). For the Fourier transform in the last Task

|F()| =

R2() +I2() =

2 +2

(2 +2)2 =

12 +2

and argF() = tan1I()

R()= tan1

.



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|F()|

argF()

/2

/2

1

Figure 3

In general, a Fourier transform whose Cartesian form is F() =R() +iI() has a polar formF() = |F()|ei() where() argF().Graphs, such as those shown in Figure 3, of

|F()

|and argF()plotted against, are often referred

to as magnitude spectra and phase spectra, respectively.

Exercises

1. Obtain the Fourier transform of the rectangular pulses

(a)f(t) =

1 |t| 10 |t| >1 (b)f(t) =

1

4 |t| 3

0 |t| >3

2. Find the Fourier transform of

f(t) =

1 t2

0 t 2

1 +t

2 2 t 0

0 |t| >2Answers

1.(a)F() = 2

sin

(b)F() =sin3

2

2. 1 cos2

2


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24 1 fourier transform

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