EE313 Linear Systems and Signals Fall 2010

Initial conversion of content to PowerPointby Dr. Wade C. Schwartzkopf

Prof. Brian L. Evans

Dept. of Electrical and Computer Engineering

The University of Texas at Austin

Fourier Series

11 - 2

Course Outline• Time domain analysis (lectures 1-10)

Signals and systems in continuous and discrete timeConvolution: finding system response in time domain

• Frequency domain analysis (lectures 11-16)

Fourier seriesFourier transformsFrequency responses of systems

• Generalized frequency domain analysis (lectures 17-26)

Laplace and z transforms of signalsTests for system stabilityTransfer functions of linear time-invariant systems

Roberts, ch. 1-3

Roberts, ch. 4-7

Roberts, ch. 9-12

11 - 3

Periodic Signals• For some positive constant T0

f(t) is periodic if f(t) = f(t + T0) for all values of t (-, )

Smallest value of T0 is the period of f(t)

• A periodic signal f(t)Unchanged when time-shifted by one period

May be generated by periodically extending one period

Area under f(t) over any interval of duration equal to the period is same; e.g., integrating from 0 to T0 would give the same value as integrating from –T0/2 to T0 /2

00

0000

12sin

2

22sin)22sin()2sin(

ftf

ftftftf

11 - 4

Sinusoids• Fundamental f1(t) = C1 cos(2 f0 t + )

Fundamental frequency in Hertz is f0

Fundamental frequency in rad/s is = 2 f0

• Harmonic fn(t) = Cn cos(2 n f0 t + n)

Frequency, n f0, is nth harmonic of f0

• Magnitude/phase and Cartesian representations Cn cos(n 0 t + n) =

Cn cos(n) cos(n 0 t) - Cn sin(n) sin(n 0 t) = an cos(n 0 t) + bn sin(n 0 t)

11 - 5

Fourier Series• General representation

of a periodic signal

• Fourier seriescoefficients

• Compact Fourierseries

1

000 sincosn

nn tnbtnaatf

n

nn

nnn

nnn

a

b

bacac

tncctf

1

2200

100

tan

and, , where

cos

0

0

0

0 00

0 00

00

0

sin2

cos2

1

T

n

T

n

T

dttntfT

b

dttntfT

a

dttfT

a

11 - 6

Existence of the Fourier Series• Existence

• Convergence for all t

• Finite number of maxima and minima in one period of f(t)

• What about periodic extensions of

0

0

Tdttf

ttf

1 1-for 1

tt

tg 10for 1

sin

t

tts

11 - 7

Example #1

• Fundamental periodT0 =

• Fundamental frequencyf0 = 1/T0 = 1/ Hz

0 = 2/T0 = rad/s

0

A

f(t)

-A

,15,11,7,38

,13,9,5,18

even is 0

22

22

nn

A

nn

An

bn

2/3

2/1

2/1

2/1

0

10

) sin( ) 22(

) sin( 2

symmetric) odd isit (because 0

plot) theof inspection(by 0

sin) cos(

dttnπtAA

dttnπtAb

a

a

tnπbtnπaatf

n

n

nnn

11 - 8

Example #2

• Fundamental periodT0 =

• Fundamental frequencyf0 = 1/T0 = 1/( Hz

0 = 2/T0 = 1 rad/s

,15,11,7,3

,15,11,7,3 allfor 0

odd

2even 0

2

10

n

n

nn

nC

C

n

n

1

f(t)