18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 1(14)

Review of Fourier Series

• Why Fourier series? Want to make analysis simple.

• Complex exponents are the eigenfunctions of LTI systems,

• x(t) is presented as linear combination of complex exponents (with different frequencies)

• LTI system response is the same linear combination of individual responses!

( ) ( ) ( ) ( ) ( )y t x t h t h x t d

+∞

−∞

= ∗ = τ − τ τ∫

( ) ( ) ( )j t j tx t e y t H j e

ω ω

= ⇒ = ω

( )H jω( )y t( )

j tx t e

ω=

Lecture 2

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 2(14)

Periodic Signals• Periodic signals are very important class of signals

(widely used), where smallest T is a period,

• Examples: & . Period

• Introduce a set of harmonically-related complex

exponents,

• Construct a periodic signal,

( ) ( ),x t x t T= + for all t

0cos( )tω 0j te

ω

02 /T = π ω

0

2

( ) , 0, 1, 2,...jn t

jn t Tn t e e n

π

ωφ = = = ± ±

( ) 0jn tn

n

x t c e

+∞

ω

=−∞

′ = ∑

DC 1st harmonic

Lecture 2

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 3(14)

• Can be made the same as ?

• Yes, by adjusting cn ,

• {cn} – Fourier series coefficients (or spectral coefficients,

or discrete spectrum of the signal)

• c0 – DC component or average value of x(t),

Fourier Series of Periodic Signal

( )x t′ ( )x t

( )21

nj t

Tn

Tc x t e dt

T

− π

= ∫ ( ) 0

0

2,

jn tn

n

x t c eT

+∞

ω

=−∞

π

= ω =∑

( )0

1

T

c x t dtT

= ∫

Lecture 2

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 4(14)

Example of Fourier Series

( ) ( )0 0

0

1cos( )

2

j t j tx t t e e

ω − ω

= ω = +

1 1

1 1, ,

2 2

0, 1k

c c

c k

−

= =

= ≠ ±

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

1

1

. 4 3 2 1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

.

1T =

f

Lecture 2

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 5(14)

Example of Fourier Series

( )1, / 2

0, / 2 / 2

tx t

t T

< τ=

τ < <

sinc ,

sin( )sinc( )

n

nc

T T

tt

t

τ τ =

π=

π

Lecture 2

Q.: How does cn

scale with

the pulse amplitude?

Duration? Period?

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 6(14)

Example of Fourier SeriesLecture 2

14T T=

18T T=

116T T=

Periodic signal

Its spectrum

A.V. Oppenheim, A.S. Willsky, Signals and Systems, 1997.

Q.: How does cn

scale with

the pulse amplitude?

Duration? Period?

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 7(14)

Convergence of Fourier Series

• Dirichlet conditions:

– x(t) must be absolutely integrable (finite power)

– x(t) must be of bounded variation; that is the number of maxima

and minima during a period is finite

– In any finite interval of time, there are only a finite number of

discontinuities, which are finite.

• Dirichlet conditions are only sufficient, but are not

necessary.

• All physically-reasonable (practical) signals meet these

conditions.

( )Tx t dt < ∞∫

Lecture 2

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 8(14)

Gibbs Phenomenon

increasing the number of

terms does not decrease

the ripple maximum!

Lecture 2

Q.: reproduce

these graphs

using a computer

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 9(14)

Fourier Series of Real Signals

( ) ( ) ( )

{ } ( ) ( ) { } ( ) ( )

0

0 0 0

1

0 0

2cos sin ,

2

2 22Re cos , 2 Im sin

n n

n

n n n nT T

ax t a n t b n t

T

a c x t n t dt b c x t n t dtT T

∞

=

π= + ω + ω ω =

= = ω = − = ω

∑

∫ ∫

( ){ } *Im 0

n nx t c c

−

= ⇒ =• For a real signal,

• Then obtain the trigonometric Fourier series,

• Another form of it is

( ) ( )

( ) ( )

0 0

1

2 2 1

cos

, arg tan /

n n

n

n n n n n n n n

x t x A n t

A c a b c b a

∞

=

−

= + ω + ϕ

= = + ϕ = − = −

∑

Lecture 2

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 10(14)

Properties of Fourier Series

• Linearity:

• Time shifting:

• Time reversal:

• Time scaling:

[ ] [ ] [ ]1 2 1 2( ) ( ) ( ) ( )x t x t x t x tα +β = α +βF F F

0 0

0( ) ( )F F jn t

n nx t c x t t e c− ω

←→ ⇔ − ←→

( ) ( )F F

n nx t c x t c

−←→ ⇔ − ←→

( ) 0( )jn tn

n

x t c e

+∞

αω

=−∞

α = ∑

Lecture 2

Q.: prove these properties

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 11(14)

Properties of Fourier Series

• Multiplication:

• Convolution:

• Differentiation:

• Integration: for

( ) ( )F

k n kkx t y t c c

∞

−=−∞

′ ′′←→∑

( ) ( )F

n nTx y t d Tc c′ ′′τ − τ τ←→∫

0

( ) F

n

dx tjn c

dt←→ ω

0

( ) ,t F ncx d

jn−∞

τ τ←→

ω∫ 0 0c =

Lecture 2

Q.: prove these properties

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 12(14)

Properties of Fourier Series

• Real x(t):

• Real & even x(t):

• Real & odd x(t):

• Parseval’s Theorem:

*

n nc c−

=

{ }, Im 0n n n

c c c−

= =

{ },Re 0n n n

c c c−

= − =

221( )

nT

n

x t dt cT

∞

=−∞

= ∑∫

Lecture 2

Q.: prove these properties

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 13(14)

Signal Synthesis via FS

Couch, Digital and Analog Communication Systems, Seventh Edition.

( ) ( )n n

n

x t a t

+∞

=−∞

= ϕ∑

( )x t

18-Jan-12 Lecture 2, ELG3175 : Introduction to Communication Systems © S. Loyka 14(14)

Summary

• Review of Fourier series

• Periodic signals & complex exponents

• Series expansion of a periodic signal

• Trigonometric form of Fourier series

• Properties of Fourier series

• Reading: the Couch text, Sec. 2.1-2.5; Oppenheim & Willsky text,

Sec. 3.0-3.5. Study carefully all the examples (including end-of-

chapter study-aid examples), make sure you understand and can

solve them with the book closed.

• Do some end-of-chapter problems. Students’ solution manual

provides solutions for many of them.

Lecture 2