Lecture 2 Review of Fourier Seriessite.uottawa.ca/~sloyka/elg3175/Lec_2_ELG3175.pdf · 2 cos sin ,...

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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

( )H jω( )y t( )

j tx t e

Lecture 2

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

02 /T = π ω

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

jn t Tn t e e n

ωφ = = = ± ±

( ) 0jn tn

x t c e

=−∞

′ = ∑

DC 1st harmonic

Lecture 2

• 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

Tc x t e dt

− π

= ∫ ( ) 0

x t c eT

=−∞

= ω =∑

c x t dtT

Lecture 2

Example of Fourier Series

( ) ( )0 0

1cos( )

j t j tx t t e e

ω − ω

= ω = +

1 1, ,

= ≠ ±

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

. 4 3 2 1 0 1 2 3 40

Lecture 2

Example of Fourier Series

( )1, / 2

0, / 2 / 2

τ < <

sinc ,

sin( )sinc( )

τ τ =

Lecture 2

Q.: How does cn

scale with

the pulse amplitude?

Duration? Period?

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?

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

Gibbs Phenomenon

increasing the number of

terms does not decrease

the ripple maximum!

Lecture 2

Q.: reproduce

these graphs

using a computer

Fourier Series of Real Signals

( ) ( ) ( )

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

2cos sin ,

2 22Re cos , 2 Im sin

n n n nT T

ax t a n t b n 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

( ) ( )

, arg tan /

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

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( ) ( )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

x t c e

=−∞

α = ∑

Lecture 2

Q.: prove these properties

• 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←→∫

dx tjn c

dt←→ ω

( ) ,t F ncx d

jn−∞

τ τ←→

ω∫ 0 0c =

Lecture 2

• 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( )

x t dt cT

=−∞

= ∑∫

Lecture 2

Signal Synthesis via FS

Couch, Digital and Analog Communication Systems, Seventh Edition.

( ) ( )n n

x t a t

=−∞

= ϕ∑

( )x t

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

Lecture 2 Review of Fourier Seriessite.uottawa.ca/~sloyka/elg3175/Lec_2_ELG3175.pdf · 2 cos sin ,...

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