1 fourierseries&transform

7/31/2019 1 FourierSeries&Transform

1/44

Fourier Series

Fourier Series

yDirichlet Conditions

yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic

yExample 3yExample 3 contd

yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution

Paul Lim Fourier Series & Transform 1 / 44


2/44

Dirichlet Conditions

Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic


yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


FIG. 1: An example of a function

that may, without modification, be

represented as a Fourier series.

The particular conditions that a

function f.x/ must fulfil in or-

der that it may be expanded as

a Fourier series are known as

the Dirichlet conditions:

1. f.x/ must be periodic

2. f.x/ is single-valued and

continuous, except possi-

bly at a finite number of fi-

nite discontinuities

3. f.x/ has only a finite

number of maxima & min-ima within one period

4. integral over one period of

jf.x/j must convergeFourier series converge to f.x/

at all points where f.x/ is con-

tinuous.


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Fourier Coefficients

Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic


yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


The Fourier series expansion of the function f.x/ is written as

f.x/ D a02

C1

XrD1

ar cos2rx

L C br sin2rx

L (1)where a0, ar and br are constants called the Fourier coefficients.

For a periodic function f.x/ of period L, the coefficients are

ar D2

L

Zx0CLx0

f.x/ cos

2rx

L

dx (2)

br D2

LZx0CLx0

f.x/ sin2rx

L

dx (3)

where x0 is arbitrary but is often taken as 0 or L=2. The apparentarbitrary factor 1=2 which appears in the a0 term in Eq. (1) is

included so that Eq. (2) may apply for r D 0 as well as r > 0.


4/44

Fourier Coefficients 2


Eqs. (2) and (3) may be derived as follows. Suppose the Fourier seriesexpansion of f.x/ can be written as in Eq. (1),

f.x/ Da

2 C1XrD1

ar cos

2rxL

C br sin2rxL Then multiplying by cos.2px=L/, integrating over one full period in x and

changing the order of summation and integration, we get

Zx0CLx0

f.x/ cos

2px

L

dx D a0

2

Zx0CLx0

cos

2px

L

dx

C1XrD1

arZx0CLx0

cos

2rxL

cos

2px

L

dx

C

1

XrD1

br Zx0CL

x0

sin2rxL

cos2pxL

dx (4)


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Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic


yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


Using the following orthogonality conditions,

Zx0CL

x0

sin2rx

L cos2px

L dx D 0 (5)Zx0CLx0

cos

2rx

L

cos

2px

L

dx D

8

00; r

p

(6)

Zx0CL

x0

sin2rxL

sin2pxL

dx D 8 0

0; r p(7)

we find that when p D 0, Eq. (4) becomes

Zx0CLx0

f.x/ dx D a02

L


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Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic

yExample 3

yExample 3 contd

yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


When p 0 the only non-vanishing term on the RHS of Eq. (4)occurs when r D p, and so

Zx0CLx0 f.x/ cos

2rxL

dx D

ar

2 L:

The other coefficients br may be found by repeating the above

process but multiplying by sin.2px=L/ instead of cos.2px=L/.


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Example 1

Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic

yExample 3

yExample 3 contd

yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


ExampleExpress the square-wavefunction a Fourier series.

FIG. 2: A square-wavefunction.

Solution

The square wave may be represented by

f.t/ D 1 for 1

2

T

t < 0,

C1 for 0 t < 12

T.


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Example 1 contd

Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic

yExample 3

yExample 3 contd

yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


f.t/ is an odd function and the series will contain only sine terms.Using Eq. (3)

br D2

TZT=2T=2 f.t/ sin

2rtT

dt

D 4T

ZT=20

sin

2rt

T

dt

D 2 r

1 .1/r :

The sine coefficients are zero if r is even and equal to 4=r if r is

odd.

) f.t/ D 4

sin !t C sin 3!t

3C sin 5! t

5C

;

where ! D 2=T is called the angular frequency.


9/44

Discontinuous Functions

Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic

yExample 3

yExample 3 contd

yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


At a point of finite discontinuity, xd, the Fourier series converges to

1

2lim!0

f.xd C / C f .xd /:


10/44

Example 2

Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic

yExample 3

yExample 3 contd

yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


ExampleFind the value to which the Fourier series of the

square-wavefunction converges at t D 0.

SolutionThe function is discontinuous at t D 0, and we expect the series toconverge to a value half-way between the upper and lower values;

zero in this case. Considering the Fourier series of this function,

we see that all the terms are zero and hence the Fourier series

converges to zero as expected.


11/44

Non-Periodic Functions

Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic

yExample 3

yExample 3 contd

yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


FIG. 3: Possible periodic exten-

sions of a function.

Figure 3(b) shows the simplestextension to the function shown

in Figure 3(a). However, this ex-

tension has no particular sym-

metry.

Figures 3(c), (d) show exten-

sions as odd and even func-

tions respectively with the ben-

efit that only sine or cosine

terms appear in the resulting

Fourier series.


12/44

Example 3

Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic

yExample 3

yExample 3 contd

yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


ExampleFind the Fourier series of f.x/ D x2 for 0 < x 2.

Solution

We must first make the function periodic. We do this by extendingthe range of interest to 2 < x 2 in such a way thatf.x/ D f .x/ and then letting f .x C 4k/ D f.x/ where k is anyinteger.

FIG. 4: f.x/ D x2, 0 < x 2, with extended range and periodicity.


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Example 3 contd

Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic

yExample 3

yExample 3 contd

yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


Now we have an even function of period 4. Thus, all thecoefficients br will be zero. Now we apply Eqs. (2) and (3) with

L D 4 to determine the remaining coefficients:

ar D 24

Z2

2

x2 cos

2rx4

dx D 4

4

Z2

0

x2 cos

rx2

dx;

Thus,

ar D

2

rx2 sin

rx2

20

4 r

Z20

x sin rx

2

dx

D8

2r2h

x cos rx

2i2

0 8

2r2Z20 cos

rx2

dx

D 162r2

cos r

D16

2r2 .1/r :


14/44

Example 3 contd

Fourier Series


yCoefficients

yCoefficients 2

yCoefficients 3

yCoefficients 4

yExample 1

yExample 1 contd

yDiscontinouos

yExample 2

yNon-Periodic

yExample 3

yExample 3 contd

yExample 3 contd

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


Since the expression for ar has r2 in its denominator, to evaluate

a0 we must return to the original definition,

ar D2

4Z22 f.x/ cos

rx2

dx:

From this we obtain

a0 D 24

Z22

x2 dx D 44

Z20

x2 dx D 83

:

The final expression for f.x/ is then

x2 D 43

C 161XrD1

.1/r2r2

cos rx

2

; for 0 < x 2. (8)


15/44

Complex Fourier series

Fourier Series

Complex Fourier

series

yComplex Series

yExample 4

yParsevals theorem

yProof

yProof 2

yExample 5

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution



16/44

Complex Fourier Series

Fourier Series

Complex Fourier

series

yComplex Series

yExample 4

yParsevals theorem

yProof

yProof 2

yExample 5

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


Using exp.irx/ D cos rx C i sin rx, the complex Fourier series:

f.x/ D1

XrD1

cr exp2irx

L ; (9)where cr D

1

L

Zx0CLx0

f.x/ exp

2irx

L

dx (10)

Eq. (10) can be derived by multiplying Eq. (9) by exp.2ipx=L/before integrating and using the orthogonality relation

Zx0CL

x0

exp2ipx

L exp2irx

L dx D

L; r D p0; r p

We also have the following relations

cr D1

2 .ar i br/; cr D1

2.ar C i br/: (11)


17/44

Example 4


ExampleFind a complex Fourier series for f.x/ D x in the range 2 < x < 2.Solution

cr D1

4

Z22

x exp

i r x

2

dx (Using Eq. (10))

D x2ir

exp i r x

2

22

C Z22

12ir

exp i r x

2

dx

D 1 i r

exp. i r / C exp.ir/ C 1

r22exp

i r x

2 2

2

D 2i r

cos r C 2ir22

sin r D 2i r

.1/r

) xD

1

XrD1

2i.1/rr

exp i r x2

:


18/44

Parsevals theorem

Fourier Series

Complex Fourier

series

yComplex Series

yExample 4

yParsevals theorem

yProof

yProof 2

yExample 5

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


Parsevals theorem gives a useful way of relating the Fouriercoefficients to the function that they describe. It states that

1

LZx0CLx0 jf.x/j

2

dx D1X

rD1 jcr j2

D

1

2a0

2

C 12

1

XrD1.a2r C b2r /:

(12)

This says that the sum of the moduli squared of the complex

Fourier coefficients is equal to the average value of

jf.x/

j2 over

one period.


19/44

Proof of Parsevals Theorem

Fourier Series

Complex Fourier

series

yComplex Series

yExample 4

yParsevals theorem

yProof

yProof 2

yExample 5

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


Consider two functions f.x/ and g.x/, which are (or can be made)periodic with period L, and which have Fourier series:

f.x/ D1X

rD1cr exp

2irxL

;

g.x/ D1

XrD1r exp

2irx

L where cr and r are the complex Fourier coefficients of f.x/ and

g.x/ respectively.

) f.x/g.x/ D1X

rD1

crg.x/ exp

2irx

L


20/44

Proof of Parsevals Theorem 2


Integrating this equation with respect to x over the interval .x0; x0 C L/, anddividing by L, we find

1

LZx0CLx0 f.x/g

.x/ dx D1X

rD1cr

1

LZx0CLx0 g

.x/ exp2irx

L

dx

D1

XrD1 cr"

1

L Zx0CL

x0

g.x/ exp2irx

L dx#

D1X

rD1

cr

r :

Finally, if we let g.x/ D f.x/, we obtain Parsevals theorem (Eq. (12)).


21/44

Example 5

Fourier Series

Complex Fourier

series

yComplex Series

yExample 4

yParsevals theorem

yProof

yProof 2

yExample 5

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution


ExampleUsing Parsevals theorem and the Fourier series for f.x/ D x2,calculate the sum

P1

rD1 r4.

SolutionFind the average value of jf.x/j2 over the interval 2 < x 2,

1

4Z

2

2

x4 dx

D

16

5

:

Now we evaluate the RHS of Eq. (12):

12 a0

2 C 121X1

a2r C

1

2

1X1

b2n D 43

2

C1

2

1XrD1

162

4r4 :

Equating the two expression, we find

1

XrD1

1

r4 D4

90

:


22/44

Fourier Transforms

Fourier Series

Complex Fourier

series

Fourier Transforms

yTransform

yTransform 2

yTransform 3

yExample 6

Dirac -Function

Odd & Even f.x/

Convolution andDeconvolution



23/44

Fourier Transforms

Fourier Series

Complex Fourier

series

Fourier Transforms

yTransform

yTransform 2

yTransform 3

yExample 6

Dirac -Function

Odd & Even f.x/



The Fourier transform provides a representation of functionsdefined over an infinite interval, and having no particular

periodicity, in terms of superposition of sinusoidal functions.

A function of period T may be represented as a complex Fourierseries,

f.t/ D1

XrD1

cre2irt=T D

1

XrD1

crei!r t (13)

where !r D 2r=T. As the period T tends to infinity, thefrequency quantum ! D 2=T becomes vanishingly small andthe spectrum of allowed frequencies !r becomes a continuum.


24/44

Fourier Transforms 2

Fourier Series

Complex Fourier

series

Fourier Transforms

yTransform

yTransform 2

yTransform 3

yExample 6

Dirac -Function

Odd & Even f.x/



The coefficients cr in Eq. (13) are given by

cr D1

T ZT=2

T=2

f . t / e2irt=T dt D !2 Z

T=2

T=2

f . t / ei!r t dt:

(14)Substituting Eq. (14) into (13) gives

f.t/ D

1XrD1

!

2ZT=2T=2 f.u/e

i!ru

d u e

i!r t

: (15)

f.t/ D 12 Z

1

1

d! ei!t Z1

1

du f . u / ei!u: (16)

This result is known as Fouriers inversion theorem.


25/44

Fourier Transforms 3

Fourier Series

Complex Fourier

series

Fourier Transforms

yTransform

yTransform 2

yTransform 3

yExample 6

Dirac -Function

Odd & Even f.x/



From it, we may define Fourier transform of f.t/ by

Qf .!/ D 1p2

Z1

1

f . t / ei!t dt; (17)

and its inverse by

f.t/ D 1p2

Z1

1

Qf .!/ ei!t d!: (18)


26/44

Example 6

Fourier Series

Complex Fourier

series

Fourier Transforms

yTransform

yTransform 2

yTransform 3

yExample 6

Dirac -Function

Odd & Even f.x/



ExampleFind the Fourier transform of the exponential decay function

f.t/ D 0 for t < 0 and f.t/ D Aet for t 0 ( > 0).

SolutionUsing Eq. (17) and separating the integrals into two parts,

Qf .!/

D1

p2 Z0

1

.0/ei!t dtC

A

p2 Z1

0

etei!t dt

D 0 C Ap2

"e

.Ci!/t

C i !

#10

D Ap2. C i !/

:


27/44

Dirac -Function

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function

yDirac -FunctionyDirac -Function 2

yExample 7

yExample 7 contd

yRelation of to FT

yProperties of FT

yExample 8

Odd & Even f.x/

Convolution and

Deconvolution



28/44

Dirac-Function

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function


yExample 7

yExample 7 contd

yRelation of to FT

yProperties of FT

yExample 8

Odd & Even f.x/

Convolution and

Deconvolution


The Dirac -function has the property that

.t/ D 0 for t 0; (19)

Its fundamental defining property isZf.t/.t a/ dt D f.a/ (20)

provided range of integration includes t D a; otherwise 0. Also,

Zb

a

.t/ dt D 1 for all a; b > 0 (21)

and Z.t a/ dt D 1 (22)

provided the range of integration includes t D a.


29/44

Dirac-Function 2

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function


yExample 7

yExample 7 contd

yRelation of to FT

yProperties of FT

yExample 8

Odd & Even f.x/

Convolution and

Deconvolution


Eq. (20) can be used to derive further useful properties of the Dirac-function:

.t/

D.

t / (23)

.at/ D 1jaj.t/ (24)t.t/ D 0: (25)


30/44

Example 7

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function


yExample 7

yExample 7 contd

yRelation of to FT

yProperties of FT

yExample 8

Odd & Even f.x/

Convolution and

Deconvolution


ExampleProve that .bt/ D .t/=jbj.

Solution

Let us consider the case where b > 0. It follows thatZ1

1

f.t/.bt/ dt DZ1

1

f

t 0

b

.t 0/

dt 0

b

D 1b

f.0/ D 1b

Z11

f.t/.t/ dt;

where we have made the substitution t 0 D bt . But f.t/ is arbitraryand therefore .bt/ D .t/=b D .t/=jbj for b > 0.


31/44

Example 7 contd

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function


yExample 7

yExample 7 contd

yRelation of to FT

yProperties of FT

yExample 8

Odd & Even f.x/

Convolution and

Deconvolution


Now consider the case where b D c < 0. It follows thatZ1

1

f.t/.bt/ dt DZ1

1

f

t 0

c

.t 0/

dt 0

c

DZ11

1c

f

t0

c

.t 0/ dt 0

D 1c

f.0/ D 1

jbj

f.0/

D 1jbjZ1

1

f.t/.t/ dt

where we have made the substitution t 0

Dbt

D ct . But f.t/ is

arbitrary and so

.bt/ D 1jbj.t/;

for all b.


32/44

Relation Of-Function to Fourier Transforms

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function


yExample 7

yExample 7 contd

yRelation of to FT

yProperties of FT

yExample 8

Odd & Even f.x/

Convolution and

Deconvolution


Referring to Eq. (16), we have

f.t/ D 12

Z1

1

d! ei!tZ1

1

du f . u / ei!u

DZ11

duf.u/

12

Z11

ei!.tu/ d!

Comparison of this with Eq. (20) shows that we may write the

-function as

.t u/ D 12

Z1

1

ei!.tu/ d!: (26)

The Fourier transform of a -function is

Q.!/ D 1p2

Z1

1

.t/ei!t dt D 1p2

: (27)


33/44

Properties of Fourier Transform

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function


yExample 7

yExample 7 contd

yRelation of to FT

yProperties of FT

yExample 8

Odd & Even f.x/

Convolution and

Deconvolution


Fourier transform of f.t/ is de-noted by Qf .!/ or F.

Differentiation:

F

f0.t / D i ! Qf .!/

(28)

Ff00.t / D i !Ff0.t/

D !2 Qf .!/;

Scaling:

Ff.at/ D 1a

Qf!

a

:

(29)

Translation:

Ff.tCa/ D eia! Qf .!/:(30)

Exponential multiplica-

tion:

F

et

f.t/ D Qf .!Ci/;

(31)

where may be real,

imaginary or complex.


34/44

Example 8

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function


yExample 7

yExample 7 contd

yRelation of to FT

yProperties of FT

yExample 8

Odd & Even f.x/

Convolution and

Deconvolution


ExampleProve relation F f0.t/ D i ! Qf .!/.

Solution

Calculating the Fourier transform of f0.t / directly, we obtain

F

f0.t /

D 1p

2

Z1

1

f0.t/ei!t dt

D 1p2

hei!tf.t/

i1

1

C 1p2

Z1

1

i !ei!tf.t/ dt

D i ! Qf .!/;

if f.t/ ! 0 at t D 1 (as it must since R11

jf.t/j dt is finite).


35/44

Odd & Even f.x/

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

yOdd & Even f

yOdd & Even f 2

Convolution and

Deconvolution


Odd d E F i


36/44

Odd and Even Functions

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

yOdd & Even f

yOdd & Even f 2

Convolution and

Deconvolution


Consider an odd function f.t/ D f .t /, whose Fouriertransform is given by

Qf .!/

D1

p2 Z1

1

f.t/ei!t dt

D 1p2

Z1

1

f.t/.cos !t i sin !t / dt

D 2i

p2Z10

f.t/ sin !t dt;

since f.t/ and sin !t are odd, whereas cos !t is even.

Note that Qf .!/ D Qf .!/, i.e. Qf .!/ is an odd function of !.

Odd d E F ti 2


37/44

Odd and Even Functions 2

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

yOdd & Even f

yOdd & Even f 2

Convolution and

Deconvolution


Hence

f.t/ D 1p2

Z1

1

Qf .!/ei!t d!

D 2ip2

Z10

Qf .!/ sin !t d!

D 2 Z

1

0

d! sin !t Z1

0

f.u/ sin !u du :Thus we may define the Fourier sine transform pair for odd

functions:

Qfs

.!/D r

2

Z10

f.t/ sin !t dt; (32)

f.t/ Dr

2

Z1

0

Qfs.!/ sin !t d!: (33)


38/44

Convolution and Deconvolution

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution

yConvolution and

Deconvolution

yExample 9

yConvolution Thm

yConvolution Thm 2

yFT in 3D

yFT in 3D contd


C l ti d D l ti


39/44

Convolution and Deconvolution

Fourier Series

Complex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution

yConvolution and

Deconvolution

yExample 9

yConvolution Thm

yConvolution Thm 2

yFT in 3D

yFT in 3D contd


The convolution of the functions f and g is defined as

h.z/ DZ1

1

f.x/g.z x/ dx (34)

and is often written as f g.

The convolution is commutative (f g D g f), associative and

distributive.

E l 9


40/44

Example 9


ExampleFind the convolution of the function f.x/ D .x C a/ C .x a/ with thefunction g.y/ plotted in the Figure below.

FIG. 5: The convolution of two functions f.x/ andg.y/.

Solution

Using the convolution integral Eq. (34),

h.z/ DZ1

1

f.x/g.z x/ dx

D Z1

1.x C a/ C .x a/g.z x/ dx D g.z C a/ C g.z a/:

C l ti Th


41/44

Convolution Theorem

Fourier SeriesComplex Fourier

series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution

yConvolution and

Deconvolution

yExample 9

yConvolution Thm

yConvolution Thm 2

yFT in 3D

yFT in 3D contd


Consider the Fourier transform of the convolution [Eq. (34)],

Qh.k/ D 1p2

Z1

1

dz eikzZ

1

1

f.x/g.z x/ dx

D 1p2

Z11

dx f .x/Z11

g.z x/ eikz dz

If we let u D z x in the second integral we have

Qh.k/ D 1p2

Z1

1

dx f .x/

Z1

1

g.u/eik.uCx/ du

D1

p2 Z1

1

f.x/eikx dx Z11

g.u/ eiku du

D 1p2

p

2 Qf .k/ p

2 Qg.k/

Dp

2 Qf .k/ Qg.k/: (35)

Convolution Theorem 2


42/44

Convolution Theorem 2


series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution

yConvolution and

Deconvolution

yExample 9

yConvolution Thm

yConvolution Thm 2

yFT in 3D

yFT in 3D contd


Hence the Fourier transform of a convolution is equal to theproduct of the separate Fourier transforms multiplied by

p2 ; this

is called the convolution theorem.

The converse is also true, namely, that the Fourier transform of theproduct f .x/g.x/ is given by

Ff .x/g.x/

D1

p2 Qf .k/

Qg.k/: (36)

Fourier Transform in Higher Dimensions


43/44

Fourier Transform in Higher Dimensions


series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution

yConvolution and

Deconvolution

yExample 9

yConvolution Thm

yConvolution Thm 2

yFT in 3D

yFT in 3D contd


Fourier transform of f.x;y;z/ is

Qf .kx; ky; kz/ D1

.2/3=2

f.x;y;z/eikxxeikyyeikzzdxdydz

(37)

and its inverse by

f.x;y;z/ D1

.2/3=2 Qf .kx; ky; kz/eikxxeikyyeikzzdkxdkydkz

(38)

Fourier Transform in Higher Dimensions 2


44/44

Fourier Transform in Higher Dimensions 2


series

Fourier Transforms

Dirac -Function

Odd & Even f.x/

Convolution and

Deconvolution

yConvolution and

Deconvolution

yExample 9

yConvolution Thm

yConvolution Thm 2

yFT in 3D

yFT in 3D contd

i

Denoting the vector with components kx , ky , kz by k and that withcomponents x, y, z by r , we can write the Fourier transform pair

Eqs. (37), (38) as

Qf .k/ D 1.2/3=2

Zf .r/ eikr d3r (39)

f .r/D

1

.2/3=2Z Qf .k/ e

ikr d3k (40)

We may deduce that the three-dimensional Dirac -function can be

written as

.r/ D 1.2/3

Zeikr d3k: (41)

1 fourierseries&transform

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