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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
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Dirichlet Conditions
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 2 / 44
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.
7/31/2019 1 FourierSeries&Transform
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Fourier Coefficients
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 3 / 44
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.
7/31/2019 1 FourierSeries&Transform
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Fourier Coefficients 2
Paul Lim Fourier Series & Transform 4 / 44
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 Coefficients 3
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 5 / 44
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 Coefficients 4
Fourier Series
yDirichlet Conditions
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
Paul Lim Fourier Series & Transform 6 / 44
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/.
7/31/2019 1 FourierSeries&Transform
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Example 1
Fourier Series
yDirichlet Conditions
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
Paul Lim Fourier Series & Transform 7 / 44
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
yDirichlet Conditions
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
Paul Lim Fourier Series & Transform 8 / 44
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.
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Discontinuous Functions
Fourier Series
yDirichlet Conditions
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
Paul Lim Fourier Series & Transform 9 / 44
At a point of finite discontinuity, xd, the Fourier series converges to
1
2lim!0
f.xd C / C f .xd /:
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Example 2
Fourier Series
yDirichlet Conditions
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
Paul Lim Fourier Series & Transform 10 / 44
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.
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Non-Periodic Functions
Fourier Series
yDirichlet Conditions
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
Paul Lim Fourier Series & Transform 11 / 44
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.
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Example 3
Fourier Series
yDirichlet Conditions
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
Paul Lim Fourier Series & Transform 12 / 44
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
yDirichlet Conditions
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
Paul Lim Fourier Series & Transform 13 / 44
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 :
7/31/2019 1 FourierSeries&Transform
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Example 3 contd
Fourier Series
yDirichlet Conditions
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
Paul Lim Fourier Series & Transform 14 / 44
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)
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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
Paul Lim Fourier Series & Transform 15 / 44
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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
Paul Lim Fourier Series & Transform 16 / 44
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)
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Example 4
Paul Lim Fourier Series & Transform 17 / 44
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
:
7/31/2019 1 FourierSeries&Transform
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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
Paul Lim Fourier Series & Transform 18 / 44
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.
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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
Paul Lim Fourier Series & Transform 19 / 44
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
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Proof of Parsevals Theorem 2
Paul Lim Fourier Series & Transform 20 / 44
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)).
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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
Paul Lim Fourier Series & Transform 21 / 44
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
:
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Fourier Transforms
Fourier Series
Complex Fourier
series
Fourier Transforms
yTransform
yTransform 2
yTransform 3
yExample 6
Dirac -Function
Odd & Even f.x/
Convolution andDeconvolution
Paul Lim Fourier Series & Transform 22 / 44
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Fourier Transforms
Fourier Series
Complex Fourier
series
Fourier Transforms
yTransform
yTransform 2
yTransform 3
yExample 6
Dirac -Function
Odd & Even f.x/
Convolution andDeconvolution
Paul Lim Fourier Series & Transform 23 / 44
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.
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Fourier Transforms 2
Fourier Series
Complex Fourier
series
Fourier Transforms
yTransform
yTransform 2
yTransform 3
yExample 6
Dirac -Function
Odd & Even f.x/
Convolution andDeconvolution
Paul Lim Fourier Series & Transform 24 / 44
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.
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Fourier Transforms 3
Fourier Series
Complex Fourier
series
Fourier Transforms
yTransform
yTransform 2
yTransform 3
yExample 6
Dirac -Function
Odd & Even f.x/
Convolution andDeconvolution
Paul Lim Fourier Series & Transform 25 / 44
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)
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Example 6
Fourier Series
Complex Fourier
series
Fourier Transforms
yTransform
yTransform 2
yTransform 3
yExample 6
Dirac -Function
Odd & Even f.x/
Convolution andDeconvolution
Paul Lim Fourier Series & Transform 26 / 44
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 !/
:
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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
Paul Lim Fourier Series & Transform 27 / 44
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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
Paul Lim Fourier Series & Transform 28 / 44
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.
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Dirac-Function 2
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
Paul Lim Fourier Series & Transform 29 / 44
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)
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Example 7
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
Paul Lim Fourier Series & Transform 30 / 44
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.
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Example 7 contd
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
Paul Lim Fourier Series & Transform 31 / 44
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.
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Relation Of-Function to Fourier Transforms
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
Paul Lim Fourier Series & Transform 32 / 44
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)
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Properties of Fourier Transform
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
Paul Lim Fourier Series & Transform 33 / 44
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.
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Example 8
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
Paul Lim Fourier Series & Transform 34 / 44
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).
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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
Paul Lim Fourier Series & Transform 35 / 44
Odd d E F i
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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
Paul Lim Fourier Series & Transform 36 / 44
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
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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
Paul Lim Fourier Series & Transform 37 / 44
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)
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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
Paul Lim Fourier Series & Transform 38 / 44
C l ti d D l ti
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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
Paul Lim Fourier Series & Transform 39 / 44
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
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Example 9
Paul Lim Fourier Series & Transform 40 / 44
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
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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
Paul Lim Fourier Series & Transform 41 / 44
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
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Convolution Theorem 2
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
Paul Lim Fourier Series & Transform 42 / 44
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
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Fourier Transform in Higher Dimensions
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
Paul Lim Fourier Series & Transform 43 / 44
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
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Fourier Transform in Higher Dimensions 2
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
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)