24 1 fourier trnsfm

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ContentsontentsFouriertransforms

1. The Fourier transform

2. Properties of the Fourier Transform

3. Some Special Fourier Transform Pairs

Learningoutcomes

needs doing

Timeallocation

You are expected to spend approximately thirteen hours of independent study on the

material presented in this workbook. However, depending upon your ability to concentrate

and on your previous experience with certain mathematical topics this time may varyconsiderably.

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The Fourier Transform

24.1Introduction

Prerequisites

Before starting this Section you should. . .

Learning Outcomes

After completing this Section you should beable to. . .


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1. The Fourier TransformThe Fourier Transform is a mathematical technique that has extensive applications in Scienceand Engineering, for example in Physical Optics, Chemistry (e.g. Nuclear Magnetic Resonance),Communications Theory and Linear Systems Theory.

Unlike Fourier series which, as we have seen in the previous two units, is mainly useful forperiodic functions, the Fourier Transform (FT for short) permits alternative representations of,mostly, non-periodic functions.We shall firstly derive the Fourier Transform from the complex exponential form of the FourierSeries and then study various properties of the FT.

2. Informal Derivation of the Fourier TransformRecall that iff(t) is a periodTfunction, 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= 2T

. In words, harmonics of frequencyn0 =n2T

n= 0, 1, 2, . . . are present inthe series and these frequencies are separated by

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

Hence, asT increases the frequency separation becomes smaller and can be conveniently writtenas . This suggests that asT

, corresponding to a non-periodic function then

0

and the frequency representation contains all frequency harmonics.

To see this in a little more detail, we recall (Workbook 23: Fourier Series) that the complexFourier coefficientscn are given by

cn= 1

T

T2

T

2

fT(t)ein0tdt. (2)

Putting 1T

as 02

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

fT(t) =

n=

02 T2

T

2

fT(t)ein0tdt

ein0t.

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

f(t) =

1

2

f(t)eitdt

eitd. (3)

The inner integral (over allt) will give a function dependent only onwhich we write asF().

Then (3) can be writtenf(t) =

1

2

F()eitd. (4)

3 HELM (VERSION 1: March 18, 2004): Workbook Level 224.1: The Fourier Transform


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where

F() =

f(t)eitdt. (5)

The representation (4) off(t) which involves all frequenciescan 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 Transform of the functionf(t). Symbolically we canwrite

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

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

Looking at the basic relation (3) it is clear that the position of the factor 12

is somewhat

arbitrary in (4) and (5). If instead of (5) we define

F() = 1

2

f(t)eitdt.

then (4) must be written

f(t) =

F()eitd.

A third, and more symmetric, alternative is to write

F() = 1

2

f(t)eitdt

and, consequently,f(t) =

12

F()eitd.

We shall use (4) and (5) throughout this section but you should be aware of these other possi-bilities which 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 whatfollows we shall use this language where appropriate. However, (5) is really a mathematicaltransformation for obtaining one function from another and (4) is then the inverse transformationfor recovering the initial function. In some applications of Fourier Transforms (which we shall not

study) the time/frequency interpretations 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: thusa functionf(t) has a Fourier transform denoted byF(),g(t) a Fourier transform writtenG()and so on. The notationF(i),G(i) is used in some texts because occurs in (5) only in thetermeit.

3. Existence of the Fourier TransformWe will discuss this question in a little detail at a later stage when we will also take up brieflythe relation between the Fourier Transform and the Laplace Transform (Workbook 20) whichyou have met earlier.For now we will use (5) to obtain the Fourier Transforms of some important functions.

HELM (VERSION 1: March 18, 2004): Workbook Level 224.1: The Fourier Transform

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Example Find the Fourier Transform of the one-sided exponential function

f(t) =

0 t 0

where is a positive constant.

f(t)

t

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

u(t) =

0 t 0

then we can writef(t) =etu(t).

(We shall frequently use this concise notation for one-sided functions.)

Solution

Using (5) then by straightforward integration

F() =

0

eteitdt (sincef(t) = 0 fort 0.

This important Fourier Transform is written in the Key Point



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Key Point

F{etu(t)

}=

1

+ i, >0.

Note that this real function has a complexFourier Transform.

Write down the Fourier Transforms of

(i) etu(t) (ii) e3tu(t) (iii) et

2u(t)

Your solution

Wehaveusingthegeneralresultjustproved:(i)=1soF{etu(t)}=1

1+i

(ii)=3soF{e3t

u(t)}=13+i

(iii)=12soF{e

t

2u(t)}=11

2+i

Obtain, using the integral definition (5), the Fourier Transform of the rectan-gular pulse

p(t) =

1 a < t < a0 otherwise

.

Note that the pulse width is 2a.

ta a

1

p(t)


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First write down using (5) the integral from which the transform will be calculated.

Your solution

WehaveP()F{p(t)}=

aa

(1)eit

dtusingthedefinitionofp(t)

Now evaluate this integral and write down the final Fourier Transform in trigonometric, ratherthan complex exponential form.

Your solution

Wehave

P()=

aa

(1)eit

dt=

eit(i)

aa

=e

iae

+ia

(i)

=(cosaisina)(cosa+isina)

(i)=

2isina

i

i.e.

P()=F{p(t)}=2sina

(6)

NotethatinthiscasetheFourierTransformiswhollyreal.

Engineers often call the function sinxx

the sinc function. Consequently if we write, the transform(6) of the rectangular pulse as

P() = 2asin a

a ,

we can sayP() = 2asinc(a).

Using the result (6) in (4) we have the Fourier Integral representation of the rectangularpulse.

p(t) = 1

2

2sin a

eitd.

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



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Key Point

The Fourier Transform of a Rectangular Pulse

Ifpa(t) =

1 a < t < a0 otherwise

then:

F{pa(t)} = 2asin aa

= 2asinc(a)

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

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

.

As 0, then 2 sin

2. Also, the function 2 sin

is an even function being the product oftwo odd functions 2 sin and 1

. The graph ofP1() is as follows:

P1()

2

Obtain the Fourier Transform of the two sidedexponential function

f(t) = et t 0

where is a positive constant.

f(t)

t

1


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Your solution

Wemustseparatetherangeoftheintegrandinto[,0]and[0,]sincethefunctionf(t)isdefinedseparatelyinthesetworegions:then

F()=

0

et

eit

dt+

0

et

eit

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.

Note that, as in the case of the rectangular pulse, we have here a real even function oftgivinga Fourier Transform which is wholly real. Also, in both cases, the Fourier Transform is an even(as well as 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

etu(t) =

et t 0

.

4. Properties of the Fourier Transform

1. Real and Imaginary Parts of a Fourier Transform

Using the definition (5) we have,

F() =

f(t)eitdt.

If we writeeit = cos t

isin t

F() =

f(t)cos t dt i

f(t)sin t dt



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where both integrals are real, assuming thatf(t) is real.

Hence the real and imaginary parts of the Fourier transform are:

Re (F()) =

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

f(t)sin tdt.

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

aa

h(t)dt= 2

a0

h(t)dtand

aa

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

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

(ii) f(t) is a real odd function

Your solution

for (i)

Iff(t)isrealandeven

R()ReF()=2

0

f(t)costdt

(becausetheintegrandiseven)

I()ImF()=

f(t)sintdt=0

(becausetheintegrandisodd).

Thus,anyrealevenfunctionf(t)hasawhollyrealFourierTransform.Alsosince

cos(()t)=cos(t)=cost

theFourierTransforminthiscasewillbearealevenfunction.


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Your solution

for (ii)

Now

ReF()=

f(t)costdt

=

(odd)(even)dt=

(odd)dt=0

and

ImF()=

f(t)sintdt

=2

0

f(t)sintdt

(becausetheintegrandis(odd)(odd)=(even)).

Alsosincesin(()t)=sint,theFourierTransforminthiscaseisanoddfunctionof.

These results are summarised in the following Key Point:

Key Point

f(t) F() = F{f(t)}real and even real and evenreal and odd purely imaginary and oddneither even nor odd complex, F() =R() + iI()



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2. Polar Form of a Fourier Transform

Example We have shown that the one-sided exponential function,

f(t) =etu(t)

has Fourier Transform

F() = 1

+ i.

Find the real and imaginary parts ofF() for this case.

Your solution

Wehave

F()=1

+i=

i

2+2

(afterrationalisingi.e.multiplyingnumeratoranddenominatorbyi)

HenceR()=ReF()=

2+2(evenfunctionof)

I()=ImF()=

2+2(oddfunctionof)

We can rewriteF(), like any other complex quantity, in polarform by calculating the magni-tude and the argument (or phase):

|F()| =

R2() +I2()

=

2 +2

(2 +2)2 =

12 +2

and argF() = tan1I()

R()= tan1

.


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|F()|

argF()

/2

/2

1

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, of|F()| and argF() plotted against are often referred to asmagnitudeand phase spectra, respectively.


24 1 fourier trnsfm

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