Fourier Analysis S. Awad, Ph.D. M. Corless, M.S.E.E. D. Cinpinski E.C.E. Department University of Michigan-Dearborn Math Review with Matlab: Fourier Series

Fourier Analysis

S. Awad, Ph.D.

M. Corless, M.S.E.E.

D. Cinpinski

E.C.E. Department

University of Michigan-Dearborn

Math Review with Matlab:

Fourier Series

U of M-Dearborn ECE DepartmentMath Review with Matlab

Fourier Analysis: Fourier Series

2

Periodic Signal Definition

Parseval’s Theorem

Fourier Series

Complex Exponential Representation

Magnitude and Phase Spectra of Fourier Series

Fourier Series Representation of Periodic Signals Fourier Series Coefficients Orthogonal Signals

Example: Full Wave Rectifier

Example: Finding Complex Coefficients

Example: Orthogonal Signals



3

For example, the normal U.S. AC from wall outlet has a sine wave with a peak voltage of 170 V (110 Vrms)

The Period of a signal is the amount of time it takes for a given signal to complete one cycle.

What is a Periodic Signal ?

A Periodic Signal is a signal that repeats itself every period



4

General Sinusoid A general cosine wave, v(t), has the form:

)cos()( tMtv

= Phase Shift, angular offset in radians

F = Frequency in Hz

T = Period in seconds (T=1/F)

t = Time in seconds

M = Magnitude, amplitude, maximum value

= Angular Frequency in radians/sec (=2F)



5

General Sinusoid

Plot in Blue:

))60(2sin(5 t

Plot in Red:

)2

)60(2sin(5 t

1 Period = 1/60 sec.

= 16.67 ms.

/2 Phase Shift

Amplitude = 5



6

AC Wall Voltage Sine Wave

1 Period

1 Period



7

Represent Periodic Signals For a general periodic signal x(t) shown to the right:

x(t+nT) = x(t) for all

t

where n is any integer, i.e. n = 0, ± 1, ± 2,…

T

x(t)

-T/2 T/2 t......



8

Frequency of Periodic Signals The frequency of a signal is defined as the

inverse of the period and has the unit “number of cycles/sec.” T

fo1

is the fundamental frequency.of

The frequency of a US standard outlet is 1/T = 60 Hz

T is the period and



9

What is Fourier Series ? Fourier Series is a technique developed by J. Fourier.

This technique (studied by Fourier) allows us to represent periodic signals as a summation of sine functions of different frequency, amplitude, and phase shift.



10

Represent a Square Wave

Represent the Square Wave at the right using Fourier Series

Notice that as more and more terms are summed, the approximation becomes better



11

Fourier Series Representationof Periodic Signals

Any periodic function can be represented in terms of sine and cosine functions:

...2sinsin

...2coscos)(

21

210

tbtb

tataatx

oo

oo

This can also be written as:

1

0 )sincos()(n

onon tnbtnaatx



12

Fourier Series Coefficients

The above a0, an, and bn are known as the Fourier Series

Coefficients. These coefficients are calculated as follows.

1

0 )sincos()(n

onon tnbtnaatx



13

Calculating the a0 Coefficient

ao, the coefficient outside the summation, is known as the average value or the dc component

ao is calculated as follows:

2

2

)(1

T

T

o dttxT

a



14

Calculating the an and bn Coefficients

2

2

,)cos()(2

T

T

on dttntxT

a n = 1, 2,…

2

2

,)sin()(2

T

T

on dttntxT

b n = 1, 2,…

The an and bn coefficients are calculated as follows:



15

Orthogonal Signals

Two periodic signals g1(t) and g2(t) are said to be “Orthogonal” if the the integral of their product over one period is equal to zero.

2/

2/

0)(2)(1T

T

dttgtg



16

Example: Orthogonal Signals

2/

2/

)()( 21

T

T

dttgtg

2/

2/

2/

2/

)cos()sin(22

1

)cos()sin(

T

T

T

T

dttt

dttt

Show that the following signals are orthogonal:

cos(t) (t)g

sin(t) (t)g

2

1



17

Orthogonal Signals

0

)cos()cos(4

1

)2cos(4

1)2sin(

2

1

)cos()sin(22

1

2/

2/

2/

2/

2/

2/

TT

tdtt

dttt

T

T

T

T

T

T



18

Note that the rectified wave has a period equal to one-half of the source wave period.

Example: Full Wave Rectifier

y(t)=|sin(ot)|

t

y=|x|

x

y

x(t)

tT/2

one period

one period

T

Consider the output of a full-wave rectifier:



19

Function Characteristics The period of y(t) = T/2 and the fundamental

frequency of y(t) is 2o (rad/sec).

1

0 )2cos()(n

on tnaaty

Thus,

Now bn=0 since y(t) is an even function.



20

Finding ao

2/

0

2

0

2

2

)sin(2

)sin(2

1

)(1

T

oo

T

oo

T

T

o

dttT

a

dttT

a

dttyT

a



21

Finding ao

1)2

)2

(cos(

)2

(

2

)0*cos()2

cos(2

)cos(2 2/

0

TT

TT

a

T

Ta

tT

a

o

oo

oo

T

oo

o

* Use o = 2pi/T



22

Finding ao

2

111

1)cos(1

o

o

o

a

a

a



23

Finding an, n = 1, 2, ….

4

0

4

4

)2cos()sin()2(4

)2cos()sin(2

2

T

oo

T

T

oon

dttntT

dttntT

a

2

2

,)cos()(2

T

T

on dttntyT

a



24

Solution for an, n = 1, 2, ….

4

0

4

0

4

0

)12(

)12(cos

)12(

)12(cos4

)12(sin)12(sin2

18

T

o

oT

o

on

T

oon

n

tn

n

tn

Ta

dttntnT

a

)12(

1

)12(

1

2

4

)12(

1

)12(

14

nn

nnT oo



25

Solution for an, n = 1, 2, …. So:

14

4

14

2222

nnan

Thus:

...6cos

35

14cos

15

12cos

3

142)( tttny ooo

Note: We can only obtain an output signal with a nonzero average value by using a nonlinear system with our zero average value input signal



26

Euler’s Identity

)sin()cos( jMMMe j We could also say:

)sin()cos( je j

)sin()cos(

)sin()cos()(

je

jej

j

and ...



27

Representing Sin and Cos with Complex Exponentials

)sin()cos( je j )sin()cos( je j

2)cos(

)cos(2

jj

jj

ee

ee

j

ee

jeejj

jj

2)sin(

)sin(2

Add the equations: Subtract the equations:



28

Complex Exponential Representation

The Sine and Cosine functions can be written in terms of complex exponentials.

tjntjno

oo eetn 2

1cos

tjntjno

oo eej

tn 2

1sin



29

Complex Exponential Fourier Series From previous slides…

10 2

1

2

1)(

n

tjntjnn

tjntjnn

oooo eej

beeaatx

t jn t jn

oo o

e e t n

2

1cos

t jn t jno

o oe e

jt n

2

1sin

Using the Complex Exponential representation of Sine and Cosine, the Fourier series can be written as:

1

0 )sincos()(n

onon tnbtnaatx



30

Fourier Series with Complex Exponentials

10

10

2

1

2

1)(

22

1)(

n

tjnnn

tjnnn

n

tjntjnn

tjntjnn

oo

oooo

ejbaejbaatx

eej

beeaatx

Noting that 1/j = -j, we can write:

10 2

1

2

1)(

n

tjntjnn

tjntjnn

oooo eej

beeaatx



31


110

10

10

)(

)(

2

1

2

1)(

n

tjnn

n

tjnn

n

tjnn

tjnn

n

tjnnn

tjnnn

oo

oo

oo

ececctx

ececctx

ejbaejbaatx

Make the following substitutions:

n

tjnn

oectx )(

)(2

1),(

2

1,00 nnnnnn jbacjbacac



32


The Complex Fourier series can be written as:

n

tjnn

oectx )(

where:

2

2

0)(1

T

T

tjnn dtetxT

c

Complex cn *Complex conjugate Note: if x(t) is real, c-n = cn

*



33

Line Spectra Line Spectra refers the plotting of discrete coefficients

corresponding to their frequencies For a periodic signal x(t), cn, n = 0, ±1, ± 2,… are

uniquely determined from x(t). The set cn uniquely determines x(t)

Because cn appears only at discrete frequencies, n(n = 0, ± 1, ± 2,… the set cn is called the discrete frequency spectrum or line spectrum of x(t).



34

The Cn coefficients are in general complex.

Line Spectra

The standard practice is to make 2 2D plots. Plot 1: Magnitude of Coefficient vs. frequency

The standard practice is to make 2 2D plots. Plot 1: Magnitude of Coefficient vs. frequency Plot 2: Phase of Coefficient vs. frequency



35

Magnitude of Cn

Recall that the magnitude for a complex number a+jb is calculated as follows:

22 bajba



36

Phase of Cn Recall that the phase for a complex number a+jb depends on the quadrant that the angle lies in.

a

bTan 1

Quadrant 1: Quadrant 2:

Quadrant 3: Quadrant 4:

a

bTan 1

a

bTan 1

a

bTan 1

Angle(a+jb) =



37

Amplitude Spectrum of Cn Note: If x(t) is real then |Cn| is

of even symmetry. nn cc

nc

sec)(radoo o2o2



38

Phase Spectrum of Cn

nn cc

Note: If x(t) is real then the Phase of Cn is odd

nc

sec)(radoo o2o2



39

Example: Finding Complex Coefficients

Consider the periodic signal x(t) with period T = 2 sec. Thus:

secsec2

2

sec

2 radradrad

To

x(t)

t-2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.50

1



40

Finding Co(avg)

Co(avg) = 0.5

1

1

5.0

5.0

1

5.0

5.0

1

)( )()()(2

1)(

2

1dttxdttxdttxdttxC avgo

0 0

5.0)5.0(5.02

1

2

1

2

1 5.0

5.0

5.0

5.0

tdt

The area under x(t) from -1 to -.5 and from .5 to 1 is zero.



41

Calculating Cn

0 , 2

sin

nn

nCn

22

0

5.0

5.0

5.0

5.0

5.0

5.0

5.0

5.0

1

1

2/

2/

00

2

1

02

10

2

1

2

1

2

1

)(2

1

)(1

nj

nj

tjn

tjntjntjn

tjn

T

T

tjnn

eenj

dte

dtedtedte

dtetx

dtetxT

C

o

ooo

o

o



42

Now it can be shown that: sin(n/2) = 0 for n = ±2, ±4, … Cn = 0

sin(2/2) = sin() = 0 sin(-4/2) = sin(-2) = 0 etc .

It can be also be shown that: sin(n/2) = -1 for n = 3, 7, 11,… sin(n/2) = 1 for n = 1, 5, 9,…

sin(3/2) = -1 sin(-7/2) = 1 etc .

Factor Evaluation



43

,...11,7,3 , nπ

n)signum(

,...9,5,1n , nπ

signum(n)

etc... 4,2, ,0

nC

C

nC

n

n

n

0 , 2

sin

nn

nCn

Recall:

Factor Evaluation

Co(avg) = 0.5



44

Note: Cn= if Cn is negative Therefore:

0n&evenn ,

0

oddn ,

0

5.||

1

nn

Cn

and

otherwise ,

... 11 7, 3, n ,

0

nC

Summary of Results



45

Plot the Magnitude Response



46

Plot the Phase Response



47

What is Parseval’s Theorem ? Parseval’s Theorem states that the average power of a

periodic signal x(t) is equal to the sum of the squared amplitudes of all the harmonic components of the signal x(t).

This theorem is excellent for determining the power contribution of each harmonic in terms of its coefficients



48

Parseval’s Theorem Average power of x(t) is calculated from the time

or frequency domain by:

)(2

1)(

1

1

2222

2

2

n

nno

T

T

avg baadttxT

P

n n

nonavg cccP1

2222

Time Domain:

Frequency Domain:

Documents

Fourier Analysis S. Awad, Ph.D. M. Corless, M.S.E.E. D. Cinpinski E.C.E. Department University of Michigan-Dearborn Math Review with Matlab: Fourier Series