lecture 2 review of fourier seriessite.uottawa.ca/~sloyka/elg3175/lec_2_elg3175.pdf · 2 cos sin ,...
TRANSCRIPT
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