Series FOURIER SERIES - … · Table of contents 1. Theory 2. Exercises 3. Answers 4. Integrals 5. Useful trig results 6. Alternative notation 7. Tips on using solutions Full worked

Series

FOURIER SERIES

Graham S McDonald

A self-contained Tutorial Module for learningthe technique of Fourier series analysis

● Table of contents● Begin Tutorial

c© 2004 [email protected]

http://www.cse.salford.ac.uk

mailto:[email protected]

Table of contents

1. Theory

2. Exercises

3. Answers

4. Integrals

5. Useful trig results

6. Alternative notation

7. Tips on using solutions

Full worked solutions

Section 1: Theory 3

1. Theory

● A graph of periodic function f(x) that has period L exhibits thesame pattern every L units along the x-axis, so that f(x + L) = f(x)for every value of x. If we know what the function looks like over onecomplete period, we can thus sketch a graph of the function over awider interval of x (that may contain many periods)

f ( x )

x

P E R I O D = L

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Section 1: Theory 4

● This property of repetition defines a fundamental spatial fre-quency k = 2π

L that can be used to give a first approximation tothe periodic pattern f(x):

f(x) ' c1 sin(kx + α1) = a1 cos(kx) + b1 sin(kx),

where symbols with subscript 1 are constants that determine the am-plitude and phase of this first approximation

● A much better approximation of the periodic pattern f(x) canbe built up by adding an appropriate combination of harmonics tothis fundamental (sine-wave) pattern. For example, adding

c2 sin(2kx + α2) = a2 cos(2kx) + b2 sin(2kx) (the 2nd harmonic)c3 sin(3kx + α3) = a3 cos(3kx) + b3 sin(3kx) (the 3rd harmonic)

Here, symbols with subscripts are constants that determine the am-plitude and phase of each harmonic contribution

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Section 1: Theory 5

One can even approximate a square-wave pattern with a suitable sumthat involves a fundamental sine-wave plus a combination of harmon-ics of this fundamental frequency. This sum is called a Fourier series

F u n d a m e n t a l + 5 h a r m o n i c sF u n d a m e n t a l + 2 0 h a r m o n i c s

x

P E R I O D = L

F u n d a m e n t a l F u n d a m e n t a l + 2 h a r m o n i c s

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Section 1: Theory 6

● In this Tutorial, we consider working out Fourier series for func-tions f(x) with period L = 2π. Their fundamental frequency is thenk = 2π

L = 1, and their Fourier series representations involve terms like

a1 cos x , b1 sinx

a2 cos 2x , b2 sin 2x

a3 cos 3x , b3 sin 3x

We also include a constant term a0/2 in the Fourier series. Thisallows us to represent functions that are, for example, entirely abovethe x−axis. With a sufficient number of harmonics included, our ap-proximate series can exactly represent a given function f(x)

f(x) = a0/2 + a1 cos x + a2 cos 2x + a3 cos 3x + ...

+ b1 sinx + b2 sin 2x + b3 sin 3x + ...

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Section 1: Theory 7

A more compact way of writing the Fourier series of a function f(x),with period 2π, uses the variable subscript n = 1, 2, 3, . . .

f(x) =a0

2+

∞∑n=1

[an cos nx + bn sinnx]

● We need to work out the Fourier coefficients (a0, an and bn) forgiven functions f(x). This process is broken down into three steps

STEP ONEa0 =

1π

∫2π

f(x) dx

STEP TWOan =

1π

∫2π

f(x) cos nx dx

STEP THREEbn =

1π

∫2π

f(x) sinnx dx

where integrations are over a single interval in x of L = 2π

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Section 1: Theory 8

● Finally, specifying a particular value of x = x1 in a Fourier series,gives a series of constants that should equal f(x1). However, if f(x)is discontinuous at this value of x, then the series converges to a valuethat is half-way between the two possible function values

f ( x )

x

F o u r i e r s e r i e s c o n v e r g e s t o h a l f - w a y p o i n t

" V e r t i c a l j u m p " / d i s c o n t i n u i t yi n t h e f u n c t i o n r e p r e s e n t e d

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Section 2: Exercises 9

2. Exercises

Click on Exercise links for full worked solutions (7 exercises in total).Exercise 1.Let f(x) be a function of period 2π such that

f(x) ={

1, −π < x < 00, 0 < x < π .

a) Sketch a graph of f(x) in the interval −2π < x < 2π

b) Show that the Fourier series for f(x) in the interval −π < x < π is

12− 2

π

[sinx +

13

sin 3x +15

sin 5x + ...

]c) By giving an appropriate value to x, show that

π

4= 1− 1

3+

15− 1

7+ . . .

● Theory ● Answers ● Integrals ● Trig ● Notation

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Exercise 2.

Let f(x) be a function of period 2π such that

f(x) ={

0, −π < x < 0x, 0 < x < π .


b) Show that the Fourier series for f(x) in the interval −π < x < π isπ

4− 2

π

[cos x +

132

cos 3x +152

cos 5x + ...

]+

[sinx− 1

2sin 2x +

13

sin 3x− ...

]c) By giving appropriate values to x, show that

(i) π4 = 1− 1

3 + 15 −

17 + . . . and (ii) π2

8 = 1 + 132 + 1

52 + 172 + . . .


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Exercise 3.


f(x) ={

x, 0 < x < ππ, π < x < 2π .


b) Show that the Fourier series for f(x) in the interval 0 < x < 2π is

3π

4− 2

π

[cos x +

132

cos 3x +152

cos 5x + . . .

]−

[sinx +

12

sin 2x +13

sin 3x + . . .

]c) By giving appropriate values to x, show that

(i) π4 = 1− 1

3 + 15 −

17 + . . . and (ii) π2

8 = 1 + 132 + 1

52 + 172 + . . .


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Exercise 4.


f(x) =x

2over the interval 0 < x < 2π.

a) Sketch a graph of f(x) in the interval 0 < x < 4π


π

2−

[sinx +

12

sin 2x +13

sin 3x + . . .


π

4= 1− 1

3+

15− 1

7+

19− . . .


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Exercise 5.


f(x) ={

π − x, 0 < x < π0, π < x < 2π



π

4+

2π

[cos x +

132

cos 3x +152

cos 5x + . . .

]+ sinx +

12

sin 2x +13

sin 3x +14

sin 4x + . . .

c) By giving an appropriate value to x, show that

π2

8= 1 +

132

+152

+ . . .


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Exercise 6.


f(x) = x in the range − π < x < π.



2[sinx− 1

2sin 2x +

13

sin 3x− . . .


π

4= 1− 1

3+

15− 1

7+ . . .


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Exercise 7.


f(x) = x2 over the interval − π < x < π.



π2

3− 4

[cos x− 1

22cos 2x +

132

cos 3x− . . .


π2

6= 1 +

122

+132

+142

+ . . .


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Section 3: Answers 16

3. Answers

The sketches asked for in part (a) of each exercise are given withinthe full worked solutions – click on the Exercise links to see thesesolutions

The answers below are suggested values of x to get the series ofconstants quoted in part (c) of each exercise

1. x = π2 ,

2. (i) x = π2 , (ii) x = 0,

3. (i) x = π2 , (ii) x = 0,

4. x = π2 ,

5. x = 0,

6. x = π2 ,

7. x = π.

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Section 4: Integrals 17

4. Integrals

Formula for integration by parts:∫ b

au dv

dxdx = [uv]ba −∫ b

adudxv dx

f (x)∫

f(x)dx f (x)∫

f(x)dx

xn xn+1

n+1 (n 6= −1) [g (x)]n g′ (x) [g(x)]n+1

n+1 (n 6= −1)1x ln |x| g′(x)

g(x) ln |g (x)|ex ex ax ax

ln a (a > 0)sinx − cos x sinhx coshxcos x sinx coshx sinhxtanx − ln |cos x| tanh x ln coshx

cosec x ln∣∣tan x

2

∣∣ cosechx ln∣∣tanh x

2

∣∣sec x ln |sec x + tanx| sech x 2 tan−1 ex

sec2 x tanx sech2 x tanh xcot x ln |sinx| cothx ln |sinhx|sin2 x x

2 −sin 2x

4 sinh2 x sinh 2x4 − x

2

cos2 x x2 + sin 2x

4 cosh2 x sinh 2x4 + x

2

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Section 4: Integrals 18

f (x)∫

f (x) dx f (x)∫

f (x) dx

1a2+x2

1a tan−1 x

a1

a2−x212a ln

∣∣∣a+xa−x

∣∣∣ (0< |x|<a)

(a > 0) 1x2−a2

12a ln

∣∣∣x−ax+a

∣∣∣ (|x| > a>0)

1√a2−x2 sin−1 x

a1√

a2+x2 ln∣∣∣x+

√a2+x2

a

∣∣∣ (a > 0)

(−a < x < a) 1√x2−a2 ln

∣∣∣x+√

x2−a2

a

∣∣∣ (x>a>0)

√a2 − x2 a2

2

[sin−1

(xa

) √a2+x2 a2

2

[sinh−1

(xa

)+ x

√a2+x2

a2

]+x

√a2−x2

a2

] √x2−a2 a2

2

[− cosh−1

(xa

)+ x

√x2−a2

a2

]

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Section 5: Useful trig results 19

5. Useful trig results

When calculating the Fourier coefficients an and bn , for which n =1, 2, 3, . . . , the following trig. results are useful. Each of these results,which are also true for n = 0,−1,−2,−3, . . . , can be deduced fromthe graph of sin x or that of cos x

● sinnπ = 0

−1

0

1

π 2π 3π−3π −2π

s i n ( x )

x−π

● cos nπ = (−1)n

−1

0

1

π 2π 3π−3π −2π

c o s ( x )

x−π

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Section 5: Useful trig results 20

−1

0

1

π 2π 3π−3π −2π

s i n ( x )

x−π

−1

0

1

π 2π 3π−3π −2π

c o s ( x )

x−π

● sinnπ

2=

0 , n even1 , n = 1, 5, 9, ...

−1 , n = 3, 7, 11, ...● cos n

π

2=

0 , n odd1 , n = 0, 4, 8, ...

−1 , n = 2, 6, 10, ...

Areas cancel whenwhen integrating

over whole periods●

∫2π

sinnxdx = 0

●∫2π

cos nxdx = 0

++

−1

0

1

π 2π 3π−3π −2π

s i n ( x )

x−π

+

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Section 6: Alternative notation 21

6. Alternative notation

● For a waveform f(x) with period L = 2πk

f(x) =a0

2+

∞∑n=1

[an cos nkx + bn sinnkx]

The corresponding Fourier coefficients are

STEP ONEa0 =

2L

∫L

f(x) dx

STEP TWOan =

2L

∫L

f(x) cos nkx dx

STEP THREEbn =

2L

∫L

f(x) sinnkx dx

and integrations are over a single interval in x of L

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● For a waveform f(x) with period 2L = 2πk , we have that

k = 2π2L = π

L and nkx = nπxL

f(x) =a0

2+

∞∑n=1

[an cos

nπx

L+ bn sin

nπx

L

]The corresponding Fourier coefficients are

STEP ONEa0 =

1L

∫2L

f(x) dx

STEP TWOan =

1L

∫2L

f(x) cosnπx

Ldx

STEP THREEbn =

1L

∫2L

f(x) sinnπx

Ldx

and integrations are over a single interval in x of 2L

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● For a waveform f(t) with period T = 2πω

f(t) =a0

2+

∞∑n=1

[an cos nωt + bn sinnωt]

The corresponding Fourier coefficients are

STEP ONEa0 =

2T

∫T

f(t) dt

STEP TWOan =

2T

∫T

f(t) cos nωt dt

STEP THREEbn =

2T

∫T

f(t) sinnωt dt

and integrations are over a single interval in t of T

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Section 7: Tips on using solutions 24

7. Tips on using solutions

● When looking at the THEORY, ANSWERS, INTEGRALS, TRIGor NOTATION pages, use the Back button (at the bottom of thepage) to return to the exercises

● Use the solutions intelligently. For example, they can help you getstarted on an exercise, or they can allow you to check whether yourintermediate results are correct

● Try to make less use of the full solutions as you work your waythrough the Tutorial

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Solutions to exercises 25

Full worked solutions

Exercise 1.

f(x) ={

1, −π < x < 00, 0 < x < π, and has period 2π


0

1

π 2π−2π

f ( x )

x−π

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b) Fourier series representation of f(x)

STEP ONE

a0 =1π

∫ π

−π

f(x)dx =1π

∫ 0

−π

f(x)dx +1π

∫ π

0

f(x)dx

=1π

∫ 0

−π

1 · dx +1π

∫ π

0

0 · dx

=1π

∫ 0

−π

dx

=1π

[x]0−π

=1π

(0− (−π))

=1π· (π)

i.e. a0 = 1 .

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STEP TWO

an =1π

∫ π

−π

f(x) cos nxdx =1π

∫ 0

−π

f(x) cos nxdx +1π

∫ π

0

f(x) cos nxdx

=1π

∫ 0

−π

1 · cos nxdx +1π

∫ π

0

0 · cos nxdx

=1π

∫ 0

−π

cos nx dx

=1π

[sinnx

n

]0

−π

=1

nπ[sinnx]0−π

=1

nπ(sin 0− sin(−nπ))

=1

nπ(0 + sinnπ)

i.e. an =1

nπ(0 + 0) = 0.

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STEP THREE

bn =1π

∫ π

−π

f(x) sinnx dx

=1π

∫ 0

−π

f(x) sinnx dx +1π

∫ π

0

f(x) sinnx dx

=1π

∫ 0

−π

1 · sinnx dx +1π

∫ π

0

0 · sinnx dx

i.e. bn =1π

∫ 0

−π

sinnx dx =1π

[− cos nx

n

]0

−π

= − 1nπ

[cos nx]0−π = − 1nπ

(cos 0− cos(−nπ))

= − 1nπ

(1− cos nπ) = − 1nπ

(1− (−1)n) , see Trig

i.e. bn ={

0 , n even− 2

nπ , n odd , since (−1)n ={

1 , n even−1 , n odd

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We now have that

f(x) =a0

2+

∞∑n=1


with the three steps giving

a0 = 1, an = 0 , and bn ={

0 , n even− 2

nπ , n odd

It may be helpful to construct a table of values of bn

n 1 2 3 4 5bn − 2

π 0 − 2π

(13

)0 − 2

π

(15

)Substituting our results now gives the required series

f(x) =12− 2

π

[sinx +

13

sin 3x +15

sin 5x + . . .

]

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c) Pick an appropriate value of x, to show that

π4 = 1− 1

3 + 15 −

17 + . . .

Comparing this series with

f(x) =12− 2

π

[sinx +

13

sin 3x +15

sin 5x + . . .

],

we need to introduce a minus sign in front of the constants 13 , 1

7 ,. . .

So we need sinx = 1, sin 3x = −1, sin 5x = 1, sin 7x = −1, etc

The first condition of sinx = 1 suggests trying x = π2 .

This choice gives sin π2 + 1

3 sin 3π2 + 1

5 sin 5π2 + 1

7 sin 7π2

i.e. 1 − 13 + 1

5 − 17

Looking at the graph of f(x), we also have that f(π2 ) = 0.

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Picking x = π2 thus gives

0 = 12 −

2π

[sin π

2 + 13 sin 3π

2 + 15 sin 5π

2

+ 17 sin 7π

2 + . . .]

i.e. 0 = 12 −

2π

[1 − 1

3 + 15

− 17 + . . .

]A little manipulation then gives a series representation of π

4

2π

[1− 1

3+

15− 1

7+ . . .

]=

12

1− 13

+15− 1

7+ . . . =

π

4.

Return to Exercise 1

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Exercise 2.

f(x) ={

0, −π < x < 0x, 0 < x < π, and has period 2π


π

π 2π 3π−3π −2π

f ( x )

x−π

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STEP ONE

a0 =1π

∫ π

−π

f(x)dx =1π

∫ 0

−π

f(x)dx +1π

∫ π

0

f(x)dx

=1π

∫ 0

−π

0 · dx +1π

∫ π

0

xdx

=1π

[x2

2

]π

0

=1π

(π2

2− 0

)i.e. a0 =

π

2.

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STEP TWO

an =1π

∫ π

−π

f(x) cos nxdx =1π

∫ 0

−π

f(x) cos nxdx +1π

∫ π

0

f(x) cos nxdx

=1π

∫ 0

−π

0 · cos nxdx +1π

∫ π

0

x cos nxdx

i.e. an =1π

∫ π

0

x cos nx dx =1π

{[x

sinnx

n

]π

0

−∫ π

0

sinnx

ndx

}(using integration by parts)

i.e. an =1π

{(π

sinnπ

n− 0

)− 1

n

[−cos nx

n

]π

0

}=

1π

{( 0 − 0) +

1n2

[cos nx]π0

}=

1πn2

{cos nπ − cos 0} =1

πn2{(−1)n − 1}

i.e. an ={

0 , n even− 2

πn2 , n odd , see Trig.

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STEP THREE

bn =1π

∫ π

−π

f(x) sinnxdx =1π

∫ 0

−π

f(x) sinnxdx +1π

∫ π

0

f(x) sinnxdx

=1π

∫ 0

−π

0 · sinnxdx +1π

∫ π

0

x sinnxdx

i.e. bn =1π

∫ π

0

x sinnxdx =1π

{[x

(−cos nx

n

)]π

0−

∫ π

0

(−cos nx

n

)dx

}(using integration by parts)

=1π

{− 1

n[x cos nx]π0 +

1n

∫ π

0

cos nxdx

}=

1π

{− 1

n(π cos nπ − 0) +

1n

[sinnx

n

]π

0

}= − 1

n(−1)n +

1πn2

(0− 0), see Trig

= − 1n

(−1)n

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i.e. bn =

{− 1

n , n even

+ 1n , n odd

We now have

f(x) =a0

2+

∞∑n=1


where a0 =π

2, an =

{0 , n even

− 2πn2 , n odd

, bn =

{− 1

n , n even1n , n odd

Constructing a table of values gives

n 1 2 3 4 5

an − 2π 0 − 2

π ·132 0 − 2

π ·152

bn 1 − 12

13 − 1

415

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This table of coefficients gives

f(x) =12

(π

2

)+

(− 2

π

)cos x + 0 · cos 2x

+(− 2

π· 132

)cos 3x + 0 · cos 4x

+(− 2

π· 152

)cos 5x + ...

+ sinx− 12

sin 2x +13

sin 3x− ...

i.e. f(x) =π

4− 2

π

[cos x +

132

cos 3x +152

cos 5x + ...

]+

[sinx− 1

2sin 2x +

13

sin 3x− ...

]and we have found the required series!

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(i) π4 = 1− 1

3 + 15 −

17 + ...

Comparing this series with

f(x) =π

4− 2

π

[cos x +

132

cos 3x +152

cos 5x + ...

]+

[sinx− 1

2sin 2x +

13

sin 3x− ...

],

the required series of constants does not involve terms like 132 , 1

52 , 172 , ....

So we need to pick a value of x that sets the cos nx terms to zero.The Trig section shows that cos nπ

2 = 0 when n is odd, and note alsothat cos nx terms in the Fourier series all have odd n

i.e. cos x = cos 3x = cos 5x = ... = 0 when x = π2 ,

i.e. cos π2 = cos 3π

2 = cos 5π2 = ... = 0

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Setting x = π2 in the series for f(x) gives

f(π

2

)=

π

4− 2

π

[cos

π

2+

132

cos3π

2+

152

cos5π

2+ ...

]+

[sin

π

2− 1

2sin

2π

2+

13

sin3π

2− 1

4sin

4π

2+

15

sin5π

2− ...

]=

π

4− 2

π[0 + 0 + 0 + ...]

+

1− 12

sinπ︸︷︷︸=0

+13· (−1)− 1

4sin 2π︸︷︷︸

=0

+15· (1)− ...

The graph of f(x) shows that f

(π2

)= π

2 , so that

π

2=

π

4+ 1− 1

3+

15− 1

7+ ...

i.e.π

4= 1− 1

3+

15− 1

7+ ...

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Pick an appropriate value of x, to show that

(ii) π2

8 = 1 + 132 + 1

52 + 172 + ...

Compare this series with

f(x) =π

4− 2

π

[cos x +

132

cos 3x +152

cos 5x + ...

]+

[sinx− 1

2sin 2x +

13

sin 3x− ...

].

This time, we want to use the coefficients of the cos nx terms, andthe same choice of x needs to set the sinnx terms to zero

Picking x = 0 givessinx = sin 2x = sin 3x = 0 and cos x = cos 3x = cos 5x = 1

Note also that the graph of f(x) gives f(x) = 0 when x = 0

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So, picking x = 0 gives

0 =π

4− 2

π

[cos 0 +

132

cos 0 +152

cos 0 +172

cos 0 + ...

]+sin 0− sin 0

2+

sin 03

− ...

i.e. 0 =π

4− 2

π

[1 +

132

+152

+172

+ ...

]+ 0− 0 + 0− ...

We then find that

2π

[1 +

132

+152

+172

+ ...

]=

π

4

and 1 +132

+152

+172

+ ... =π2

8.


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Exercise 3.

f(x) ={

x, 0 < x < ππ, π < x < 2π, and has period 2π


π

0 π 2π−2π

f ( x )

x−π

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STEP ONE

a0 =1π

∫ 2π

0

f(x)dx =1π

∫ π

0

f(x)dx +1π

∫ 2π

π

f(x)dx

=1π

∫ π

0

xdx +1π

∫ 2π

π

π · dx

=1π

[x2

2

]π

0

+π

π

[x

]2π

π

=1π

(π2

2− 0

)+

(2π − π

)=

π

2+ π

i.e. a0 =3π

2.

Toc JJ II J I Back


STEP TWO

an =1π

∫ 2π

0

f(x) cos nxdx

=1π

∫ π

0

x cos nxdx +1π

∫ 2π

π

π · cos nxdx

=1π

[ [x

sinnx

n

]π

0

−∫ π

0

sinnx

ndx

]︸︷︷︸

using integration by parts

+π

π

[sinnx

n

]2π

π

=1π

[1n

(π sinnπ − 0 · sinn0

)−

[− cos nx

n2

]π

0

]

+1n

(sinn2π − sinnπ)

Toc JJ II J I Back


i.e. an =1π

[1n

(0− 0

)+

(cos nπ

n2− cos 0

n2

)]+

1n

(0− 0

)

=1

n2π(cos nπ − 1), see Trig

=1

n2π

((−1)n − 1

),

i.e. an =

{ − 2n2π , n odd

0 , n even.

Toc JJ II J I Back


STEP THREE

bn =1π

∫ 2π

0

f(x) sinnxdx

=1π

∫ π

0

x sinnxdx +1π

∫ 2π

π

π · sinnxdx

=1π

[ [x

(−cos nx

n

)]π

0−

∫ π

0

(− cos nx

n

)dx

]︸︷︷︸


+π

π

[− cos nx

n

]2π

π

=1π

[ (−π cos nπ

n+ 0

)+

[sinnx

n2

]π

0

]− 1

n(cos 2nπ − cos nπ)

=1π

[−π(−1)n

n+

(sinnπ − sin 0

n2

) ]− 1

n

(1− (−1)n

)= − 1

n(−1)n + 0 − 1

n

(1− (−1)n

)Toc JJ II J I Back


i.e. bn = − 1n

(−1)n − 1n

+1n

(−1)n

i.e. bn = − 1n

.

We now have

f(x) =a0

2+

∞∑n=1


where a0 = 3π2 , an =

{0 , n even

− 2n2π , n odd , bn = − 1

n

Constructing a table of values gives

n 1 2 3 4 5an − 2

π 0 − 2π

(132

)0 − 2

π

(152

)bn −1 − 1

2 − 13 − 1

4 − 15

Toc JJ II J I Back


This table of coefficients gives

f(x) = 12

(3π2

)+

(− 2

π

) [cos x + 0 · cos 2x + 1

32 cos 3x + . . .]

+(− 1

) [sinx + 1

2 sin 2x + 13 sin 3x + . . .

]

i.e. f(x) = 3π4 − 2

π

[cos x + 1

32 cos 3x + 152 cos 5x + . . .

]−

[sinx + 1

2 sin 2x + 13 sin 3x + . . .

]

and we have found the required series.

Toc JJ II J I Back



(i) π4 = 1− 1

3 + 15 −

17 + . . .

Compare this series with

f(x) =3π

4− 2

π

[cos x +

132

cos 3x +152

cos 5x + . . .

]−

[sinx +

12

sin 2x +13

sin 3x + . . .

]Here, we want to set the cos nx terms to zero (since their coefficientsare 1, 1

32 , 152 , . . .). Since cos nπ

2 = 0 when n is odd, we will try settingx = π

2 in the series. Note also that f(π2 ) = π

2

This givesπ2 = 3π

4 − 2π

[cos π

2 + 132 cos 3π

2 + 152 cos 5π

2 + . . .]

−[sin π

2 + 12 sin 2π

2 + 13 sin 3π

2 + 14 sin 4π

2 + 15 sin 5π

2 + . . .]

Toc JJ II J I Back


and

π2 = 3π

4 − 2π [0 + 0 + 0 + . . .]

−[(1) + 1

2 · (0) + 13 · (−1) + 1

4 · (0) + 15 · (1) + . . .

]then

π2 = 3π

4 −(1− 1

3 + 15 −

17 + . . .

)1− 1

3 + 15 −

17 + . . . = 3π

4 − π2

1− 13 + 1

5 −17 + . . . = π

4 , as required.

To show that

(ii) π2

8 = 1 + 132 + 1

52 + 172 + . . . ,

We want zero sinnx terms and to use the coefficients of cos nx

Toc JJ II J I Back


Setting x = 0 eliminates the sin nx terms from the series, and alsogives

cos x+132

cos 3x+152

cos 5x+172

cos 7x+ . . . = 1+132

+152

+172

+ . . .

(i.e. the desired series).The graph of f(x) shows a discontinuity (a “vertical jump”) at x = 0

The Fourier series converges to a value that is half-way between thetwo values of f(x) around this discontinuity. That is the series willconverge to π

2 at x = 0

i.e.π

2=

3π

4− 2

π

[cos 0 +

132

cos 0 +152

cos 0 +172

cos 0 + . . .

]−

[sin 0 +

12

sin 0 +13

sin 0 + . . .

]

andπ

2=

3π

4− 2

π

[1 +

132

+152

+172

+ . . .

]− [0 + 0 + 0 + . . .]

Toc JJ II J I Back


Finally, this gives

−π

4= − 2

π

(1 +

132

+152

+172

+ . . .

)and

π2

8= 1 +

132

+152

+172

+ . . .


Toc JJ II J I Back


Exercise 4.

f(x) = x2 , over the interval 0 < x < 2π and has period 2π

a) Sketch a graph of f(x) in the interval 0 < x < 4π

π

3π 4π0 2π

f ( x )

xπ

Toc JJ II J I Back



STEP ONE

a0 =1π

∫ 2π

0

f(x) dx

=1π

∫ 2π

0

x

2dx

=1π

[x2

4

]2π

0

=1π

[(2π)2

4− 0

]i.e. a0 = π.

Toc JJ II J I Back


STEP TWO

an =1π

∫ 2π

0

f(x) cos nxdx

=1π

∫ 2π

0

x

2cos nxdx

=12π

{[x

sinnx

n

]2π

0

− 1n

∫ 2π

0

sinnxdx

}︸︷︷︸


=12π

{(2π

sinn2π

n− 0 · sinn · 0

n

)− 1

n· 0

}

=12π

{(0− 0)− 1

n· 0

}, see Trig

i.e. an = 0.

Toc JJ II J I Back


STEP THREE

bn =1π

∫ 2π

0

f(x) sinnxdx =1π

∫ 2π

0

(x

2

)sinnxdx

=12π

∫ 2π

0

x sinnxdx

=12π

{[x

(− cos nx

n

)]2π

0

−∫ 2π

0

(− cos nx

n

)dx

}︸︷︷︸


=12π

{1n

(−2π cos n2π + 0) +1n· 0

}, see Trig

=−2π

2πncos(n2π)

= − 1n

cos(2nπ)

i.e. bn = − 1n

, since 2n is even (see Trig)

Toc JJ II J I Back


We now have

f(x) =a0

2+

∞∑n=1


where a0 = π, an = 0, bn = − 1n

These Fourier coefficients give

f(x) =π

2+

∞∑n=1

(0− 1

nsinnx

)

i.e. f(x) =π

2−

{sinx +

12

sin 2x +13

sin 3x + . . .

}.

Toc JJ II J I Back



π4 = 1− 1

3 + 15 −

17 + 1

9 − . . .

Setting x = π2 gives f(x) = π

4 and

π

4=

π

2−

[1 + 0− 1

3+ 0 +

15

+ 0− . . .

]π

4=

π

2−

[1− 1

3+

15− 1

7+

19− . . .

][1− 1

3+

15− 1

7+

19− . . .

]=

π

4

i.e. 1− 13

+15− 1

7+

19− . . . =

π

4.


Toc JJ II J I Back


Exercise 5.

f(x) ={

π − x , 0 < x < π0 , π < x < 2π, and has period 2π


π

0 π 2π−2π

f ( x )

x−π

Toc JJ II J I Back



STEP ONE

a0 =1π

∫ 2π

0

f(x) dx

=1π

∫ π

0

(π − x) dx +1π

∫ 2π

π

0 · dx

=1π

[πx− 1

2x2

]π

0

+ 0

=1π

[π2 − π2

2− 0

]i.e. a0 =

π

2.

Toc JJ II J I Back


STEP TWO

an =1π

∫ 2π

0

f(x) cos nxdx

=1π

∫ π

0

(π − x) cos nxdx +1π

∫ 2π

π

0 · dx

i.e. an =1π

{[(π − x)

sinnx

n

]π

0

−∫ π

0

(−1) · sinnx

ndx

}︸︷︷︸


+0

=1π

{(0− 0) +

∫ π

0

sinnx

ndx

}, see Trig

=1

πn

[− cos nx

n

]π

0

= − 1πn2

(cos nπ − cos 0)

i.e. an = − 1πn2

((−1)n − 1) , see Trig

Toc JJ II J I Back


i.e. an =

0 , n even

2πn2 , n odd

STEP THREE

bn =1π

∫ 2π

0

f(x) sinnxdx

=1π

∫ π

0

(π − x) sinnxdx +∫ 2π

π

0 · dx

=1π

{[(π − x)

(−cos nx

n

)]π

0−

∫ π

0

(−1) ·(−cos nx

n

)dx

}+ 0

=1π

{(0−

(−π

n

))− 1

n· 0

}, see Trig

i.e. bn =1n

.

Toc JJ II J I Back


In summary, a0 = π2 and a table of other Fourier cofficients is

n 1 2 3 4 5

an = 2πn2 (when n is odd) 2

π 0 2π

132 0 2

π152

bn = 1n 1 1

213

14

15

∴ f(x) =a0

2+

∞∑n=1


=π

4+

2π

cos x +2π

132

cos 3x +2π

152

cos 5x + . . .

+ sinx +12

sin 2x +13

sin 3x +14

sin 4x + . . .

i.e. f(x) =π

4+

2π

[cos x +

132

cos 3x +152

cos 5x + . . .

]+ sinx +

12

sin 2x +13

sin 3x +14

sin 4x + . . .

Toc JJ II J I Back


c) To show that π2

8 = 1 + 132 + 1

52 + . . . ,

note that, as x → 0 , the series converges to the half-way value of π2 ,

and thenπ

2=

π

4+

2π

(cos 0 +

132

cos 0 +152

cos 0 + . . .

)+ sin 0 +

12

sin 0 +13

sin 0 + . . .

π

2=

π

4+

2π

(1 +

132

+152

+ . . .

)+ 0

π

4=

2π

(1 +

132

+152

+ . . .

)giving

π2

8= 1 +

132

+152

+ . . .


Toc JJ II J I Back


Exercise 6.

f(x) = x, over the interval −π < x < π and has period 2π


−π

0

π

π 2π 3π−3π −2π

f ( x )

x−π

Toc JJ II J I Back



STEP ONE

a0 =1π

∫ π

−π

f(x) dx

=1π

∫ π

−π

xdx

=1π

[x2

2

]π

−π

=1π

(π2

2− π2

2

)i.e. a0 = 0.

Toc JJ II J I Back


STEP TWO

an =1π

∫ π

−π

f(x) cos nxdx

=1π

∫ π

−π

x cos nxdx

=1π

{[x

sinnx

n

]π

−π

−∫ π

−π

(sinnx

n

)dx

}︸︷︷︸


i.e. an =1π

{1n

(π sinnπ − (−π) sin(−nπ))− 1n

∫ π

−π

sinnxdx

}=

1π

{1n

(0− 0)− 1n· 0

},

since sinnπ = 0 and∫

2π

sinnxdx = 0,

i.e. an = 0.

Toc JJ II J I Back


STEP THREE

bn =1π

∫ π

−π

f(x) sinnxdx

=1π

∫ π

−π

x sinnxdx

=1π

{[−x cos nx

n

]π

−π

−∫ π

−π

(− cos nx

n

)dx

}

=1π

{− 1

n[x cos nx]π−π +

1n

∫ π

−π

cos nxdx

}=

1π

{− 1

n(π cos nπ − (−π) cos(−nπ)) +

1n· 0

}= − π

nπ(cos nπ + cos nπ)

= − 1n

(2 cos nπ)

i.e. bn = − 2n

(−1)n.

Toc JJ II J I Back


We thus have

f(x) =a0

2+

∞∑n=1

[an cos nx + bn sinnx

]with a0 = 0, an = 0, bn = − 2

n (−1)n

and

n 1 2 3

bn 2 −1 23

Therefore

f(x) = b1 sinx + b2 sin 2x + b3 sin 3x + . . .

i.e. f(x) = 2[sinx− 1

2sin 2x +

13

sin 3x− . . .

]and we have found the required Fourier series.

Toc JJ II J I Back



π4 = 1− 1

3 + 15 −

17 + . . .

Setting x = π2 gives f(x) = π

2 and

π

2= 2

[sin

π

2− 1

2sin

2π

2+

13

sin3π

2− 1

4sin

4π

2+

15

sin5π

2− . . .

]This gives

π

2= 2

[1 + 0 +

13· (−1)− 0 +

15· (1)− 0 +

17· (−1) + . . .

]π

2= 2

[1− 1

3+

15− 1

7+ . . .

]i.e.

π

4= 1− 1

3+

15− 1

7+ . . .

Return to Exercise 6Toc JJ II J I Back


Exercise 7.

f(x) = x2, over the interval −π < x < π and has period 2π


0

π2

π 2π 3π−3π −2π

f ( x )

x−π

Toc JJ II J I Back



STEP ONE

a0 =1π

∫ π

−π

f(x)dx =1π

∫ π

−π

x2 dx

=1π

[x3

3

]π

−π

=1π

(π3

3−

(−π3

3

))

=1π

(2π3

3

)

i.e. a0 =2π2

3.

Toc JJ II J I Back


STEP TWO

an =1π

∫ π

−π

f(x) cos nxdx

=1π

∫ π

−π

x2 cos nxdx

=1π

{[x2 sinnx

n

]π

−π

−∫ π

−π

2x

(sinnx

n

)dx

}︸︷︷︸


=1π

{1n

(π2 sinnπ − π2 sin(−nπ)

)− 2

n

∫ π

−π

x sinnxdx

}

=1π

{1n

(0− 0)− 2n

∫ π

−π

x sinnxdx

}, see Trig

=−2nπ

∫ π

−π

x sinnxdx

Toc JJ II J I Back


i.e. an =−2nπ

{[x

(− cos nx

n

)]π

−π

−∫ π

−π

(− cos nx

n

)dx

}︸︷︷︸

using integration by parts again

=−2nπ

{− 1

n[x cos nx]π−π +

1n

∫ π

−π

cos nxdx

}

=−2nπ

{− 1

n

(π cos nπ − (−π) cos(−nπ)

)+

1n· 0

}

=−2nπ

{− 1

n

(π(−1)n + π(−1)n

)}

=−2nπ

{−2π

n(−1)n

}

Toc JJ II J I Back


i.e. an =−2nπ

{− 2π

n(−1)n

}

=+4π

πn2(−1)n

=4n2

(−1)n

i.e. an =

{ 4n2 , n even

−4n2 , n odd.

Toc JJ II J I Back


STEP THREE

bn =1π

∫ π

−π

f(x) sinnxdx =1π

∫ π

−π

x2 sinnxdx

=1π

{[x2

(− cos nx

n

)]π

−π

−∫ π

−π

2x ·(− cos nx

n

)dx

}︸︷︷︸


=1π

{− 1

n

[x2 cos nx

]π

−π+

2n

∫ π

−π

x cos nxdx

}

=1π

{− 1

n

(π2 cos nπ − π2 cos(−nπ)

)+

2n

∫ π

−π

x cos nxdx

}

=1π

{− 1

n

(π2 cos nπ − π2 cos(nπ)

)︸︷︷︸=0

+2n

∫ π

−π

x cos nxdx

}

=2

πn

∫ π

−π

x cos nxdx

Toc JJ II J I Back


i.e. bn =2

πn

{[x

sinnx

n

]π

−π

−∫ π

−π

sinnx

ndx

}︸︷︷︸


=2

πn

{1n

(π sinnπ − (−π) sin(−nπ))− 1n

∫ π

−π

sinnxdx

}

=2

πn

{1n

(0 + 0)− 1n

∫ π

−π

sinnxdx

}

=−2πn2

∫ π

−π

sinnxdx

i.e. bn = 0.

Toc JJ II J I Back


∴ f(x) =a0

2+

∞∑n=1


where a0 = 2π2

3 , an =

{4

n2 , n even−4n2 , n odd , bn = 0

n 1 2 3 4

an −4(1) 4(

122

)−4

(132

)4

(142

)i.e. f(x) =

12

(2π2

3

)− 4

[cos x− 1

22cos 2x +

132

cos 3x− 142

cos 4x . . .

]+ [0 + 0 + 0 + . . .]

i.e. f(x) =π2

3− 4

[cos x− 1

22cos 2x +

132

cos 3x− 142

cos 4x + . . .

].

Toc JJ II J I Back


c) To show that π2

6 = 1 + 122 + 1

32 + 142 + . . . ,

use the fact that cos nπ =

{1 , n even

−1 , n odd

i.e. cos x − 122 cos 2x + 1

32 cos 3x − 142 cos 4x + . . . with x = π

gives cos π − 122 cos 2π + 1

32 cos 3π − 142 cos 4π + . . .

i.e. (−1) − 122 · (1) + 1

32 · (−1) − 142 · (1) + . . .

i.e. −1 − 122 − 1

32 − 142 + . . .

= −1 ·(

1 +122

+132

+142

+ . . .

)︸︷︷︸

(the desired series)

Toc JJ II J I Back


The graph of f(x) gives that f(π) = π2 and the series converges tothis value.

Setting x = π in the Fourier series thus gives

π2 =π2

3− 4

(cos π − 1

22cos 2π +

132

cos 3π − 142

cos 4π + . . .

)π2 =

π2

3− 4

(−1− 1

22− 1

32− 1

42− . . .

)π2 =

π2

3+ 4

(1 +

122

+132

+142

+ . . .

)2π2

3= 4

(1 +

122

+132

+142

+ . . .

)i.e.

π2

6= 1 +

122

+132

+142

+ . . .


Toc JJ II J I Back

EE 102 spring 2001-2002 Handout #23

Lecture 11The Fourier transform

• definition

• examples

• the Fourier transform of a unit step

• the Fourier transform of a periodic signal

• properties

• the inverse Fourier transform

11–1

The Fourier transform

we’ll be interested in signals defined for all t

the Fourier transform of a signal f is the function

F (ω) =∫ ∞

−∞f(t)e−jωtdt

• F is a function of a real variable ω; the function value F (ω) is (ingeneral) a complex number

F (ω) =∫ ∞

−∞f(t) cosωt dt − j

∫ ∞

−∞f(t) sinωt dt

• |F (ω)| is called the amplitude spectrum of f ; � F (ω) is the phasespectrum of f

• notation: F = F(f) means F is the Fourier transform of f ; as forLaplace transforms we usually use uppercase letters for the transforms(e.g., x(t) and X(ω), h(t) and H(ω), etc.)

The Fourier transform 11–2

Fourier transform and Laplace transform

Laplace transform of f

F (s) =∫ ∞

0

f(t)e−st dt

Fourier transform of f

G(ω) =∫ ∞

−∞f(t)e−jωt dt

very similar definitions, with two differences:

• Laplace transform integral is over 0 ≤ t < ∞; Fourier transform integralis over −∞ < t < ∞

• Laplace transform: s can be any complex number in the region ofconvergence (ROC); Fourier transform: jω lies on the imaginary axis


therefore,

• if f(t) = 0 for t < 0,

– if the imaginary axis lies in the ROC of L(f), then

G(ω) = F (jω),

i.e., the Fourier transform is the Laplace transform evaluated on theimaginary axis

– if the imaginary axis is not in the ROC of L(f), then the Fouriertransform doesn’t exist, but the Laplace transform does (at least, forall s in the ROC)

• if f(t) �= 0 for t < 0, then the Fourier and Laplace transforms can bevery different


examples

• one-sided decaying exponential

f(t) ={

0 t < 0e−t t ≥ 0

Laplace transform: F (s) = 1/(s + 1) with ROC {s | �s > −1}Fourier transform is∫ ∞

−∞f(t)e−jωt dt =

1jω + 1

= F (jω)

• one-sided growing exponential

f(t) ={

0 t < 0et t ≥ 0

Laplace transform: F (s) = 1/(s − 1) with ROC {s | �s > 1}f doesn’t have a Fourier transform


Examples

double-sided exponential: f(t) = e−a|t| (with a > 0)

F (ω) =∫ ∞

−∞e−a|t|e−jωt dt =

∫ 0

−∞eate−jωt dt +

∫ ∞

0

e−ate−jωt dt

=1

a − jω+

1a + jω

=2a

a2 + ω2

−1/a 1/a0

1

t

f(t

)

−a a0

2/a

ω

F(ω

)


rectangular pulse: f(t) ={

1 −T ≤ t ≤ T0 |t| > T

F (ω) =∫ T

−T

e−jωt dt =−1jω

(e−jωT − ejωT

)=

2 sinωT

ω

−T T0

1

t

f(t

)

−π/T π/T

0

2T

ωF

(ω)

unit impulse: f(t) = δ(t)

F (ω) =∫ ∞

−∞δ(t)e−jωt dt = 1


shifted rectangular pulse: f(t) ={

1 1 − T ≤ t ≤ 1 + T0 t < 1 − T or t > 1 + T

F (ω) =∫ 1+T

1−T

e−jωt dt =−1jω

(e−jω(1+T ) − e−jω(1−T )

)

=−e−jω

jω

(e−jωT − ejωT

)=

2 sin ωT

ωe−jω

−π/T π/T

0

2T

ω

|F(ω

)|

−π/T π/T−π

−π

0

(phase assuming T = 1)

ω

�F

(ω)


Step functions and constant signals

by allowing impulses in F(f) we can define the Fourier transform of a stepfunction or a constant signal

unit step

what is the Fourier transform of

f(t) ={

0 t < 01 t ≥ 0 ?

the Laplace transform is 1/s, but the imaginary axis is not in the ROC,and therefore the Fourier transform is not 1/jω

in fact, the integral∫ ∞

−∞f(t)e−jωt dt =

∫ ∞

0

e−jωt dt =∫ ∞

0

cosωt dt − j

∫ ∞

0

sin ωt dt

is not defined


however, we can interpret f as the limit for α → 0 of a one-sided decayingexponential

gα(t) ={

e−αt t ≥ 00 t < 0,

(α > 0), which has Fourier transform

Gα(ω) =1

a + jω=

a − jω

a2 + ω2=

a

a2 + ω2− jω

a2 + ω2

as α → 0,a

a2 + ω2→ πδ(ω), − jω

a2 + ω2→ 1

jω

let’s therefore define the Fourier transform of the unit step as

F (ω) =∫ ∞

0

e−jωt dt = πδ(ω) +1jω


negative time unit step

f(t) ={

1 t ≤ 00 t > 0

F (ω) =∫ 0

−∞e−jωt dt =

∫ ∞

0

ejωt dt = πδ(ω) − 1jω

constant signals: f(t) = 1

f is the sum of a unit step and a negative time unit step:

F (ω) =∫ ∞

−∞e−jωt dt =

∫ 0

−∞e−jωt dt +

∫ ∞

0

e−jωt dt = 2πδ(ω)


Fourier transform of periodic signals

similarly, by allowing impulses in F(f), we can define the Fourier transformof a periodic signal

sinusoidal signals: Fourier transform of f(t) = cosω0t

F (ω) =12

∫ ∞

−∞

(ejω0t + e−jω0t

)e−jωt dt

=12

∫ ∞

−∞e−j(ω−ω0)tdt +

12

∫ ∞

−∞e−j(ω+ω0)tdt

= πδ(ω − ω0) + πδ(ω + ω0)

ω

F (ω)

ω0−ω0

ππ


Fourier transform of f(t) = sinω0t

F (ω) =12j

∫ ∞

−∞

(ejω0t − e−jω0t

)e−jωt dt

=12j

∫ ∞

−∞e−j(ω−ω0)tdt + − 1

2j

∫ ∞

−∞e−j(ω0+ω)tdt

= −jπδ(ω − ω0) + jπδ(ω + ω0)

ω

F (ω)

ω0

−ω0

−jπ

jπ


periodic signal f(t) with fundamental frequency ω0

express f as Fourier series

f(t) =∞∑

k=−∞ake

jkω0t

F (ω) =∞∑

k=−∞ak

∫ ∞

−∞ej(kω0−ω)t dt = 2π

∞∑k=−∞

akδ(ω − kω0)

ω

F (ω)

ω0 2ω0 3ω0−ω0−2ω0−3ω0

2πa0

2πa1

2πa2 2πa3

2πa−1

2πa−22πa−3


Properties of the Fourier transform

linearity af(t) + bg(t) aF (ω) + bG(ω)

time scaling f(at) 1|a|F (ω

a)

time shift f(t − T ) e−jωTF (ω)

differentiation df(t)dt jωF (ω)

dkf(t)

dtk (jω)kF (ω)

integration∫ t

−∞ f(τ)dτ F (ω)jω + πF (0)δ(ω)

multiplication with t tkf(t) jk dkF (ω)

dωk

convolution∫ ∞−∞ f(τ)g(t − τ) dτ F (ω)G(ω)

multiplication f(t)g(t) 12π

∫ ∞−∞ F (ω̃)G(ω − ω̃) dω̃


Examples

sign function: f(t) ={

1 t ≥ 0−1 t < 0

write f as f(t) = −1 + 2g(t), where g is a unit step at t = 0, and applylinearity

F (ω) = −2πδ(ω) + 2πδ(ω) +2jω

=2jω

sinusoidal signal: f(t) = cos(ω0t + φ)

write f asf(t) = cos(ω0(t + φ/ω0))

and apply time shift property:

F (ω) = πejωφ/ω0 (δ(ω − ω0) + δ(ω + ω0))


pulsed cosine: f(t) ={

0 |t| > 10cos t −10 ≤ t ≤ 10

−10 0 10−1

−0.5

0

0.5

1

t

f(t

)

write f as a product f(t) = g(t) cos t where g is a rectangular pulse ofwidth 20 (see page 12-7)

F(cos t) = πδ(ω − 1) + πδ(ω + 1), F(g(t)) =2 sin 10ω

ω


now apply multiplication property

F (jω) =∫ ∞

−∞

sin 10ω̃ω̃

(δ(ω − ω̃ − 1) + δ(ω − ω̃ + 1)) dω̃

=sin(10(ω − 1))

ω − 1+

sin(10(ω + 1))ω + 1

−1 0 1−4

−2

0

2

4

6

8

10

12

ω

F(ω

)


The inverse Fourier transform

if F (ω) is the Fourier transform of f(t), i.e.,

F (ω) =∫ ∞

−∞f(t)e−jωt dt

then

f(t) =12π

∫ ∞

−∞F (ω)ejωt dω

let’s check

12π

∫ ∞

ω=−∞F (ω)ejωt dω =

12π

∫ ∞

ω=−∞

(∫ ∞

τ=−∞f(τ)e−jωτ

)ejωtdω

=12π

∫ ∞

τ=−∞f(τ)

(∫ ∞

ω=−∞e−jω(τ−t)dω

)dτ

=∫ ∞

−∞f(τ)δ(τ − t)dτ

= f(t)


Documents

Series FOURIER SERIES - … · Table of contents 1. Theory 2. Exercises 3. Answers 4. Integrals 5. Useful trig results 6. Alternative notation 7. Tips on using solutions Full worked