lecture 3 review of fourier transform - site.uottawa.ca
TRANSCRIPT
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Review of Fourier Transform
• Fourier series works for periodic signals only. What’s
about aperiodic signals? This is very large & important
class of signals
• Aperiodic signal can be considered as periodic with
• Fourier series changes to Fourier transform, complex
exponents are infinitesimally close in frequency
• Discrete spectrum becomes a continuous one, also
known as spectral density
T →∞
Lecture 3
1(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Fourier Series -> Fourier Transform
14T T=
18T T=
116T T=
Periodic signal
Its spectrum
As T increases, spectral
components are getting closer
and closer, becoming the
continuous spectrum at the limit
A.V. Oppenheim, A.S. Willsky, Signals and Systems, 1997.
Lecture 3
2(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Fourier Transform
• Fourier transform (spectrum):
• Inverse Fourier transform:
• Dirichlet conditions (details on the next page):
– x(t) is absolutely integrable,
– x(t) has a finite number of maxima & minima within any finite
interval,
– x(t) has a finite number of discontinuities within any finite interval.
These discontinuities must be finite.
( ) ( ) 2j ftxS f x t e dt
+∞− π
−∞
= ∫
( ) ( ) 2j ftxx t S f e df
+∞π
−∞
= ∫ ( ) ( )1
2
j txx t S e d
+∞ω
−∞
= ω ω
π∫
( ) ( ) j txS x t e dt
+∞− ω
−∞
ω = ∫
radial frequency
radial frequency
Lecture 3
3(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Convergence of Fourier Transform
• Dirichlet conditions:– x(t) must be absolutely integrable (finite energy)
or
– x(t) has a finite number of maxima & minima within any finite
interval,
– x(t) has a finite number of discontinuities within any finite
interval. These discontinuities must be finite.
• Dirichlet conditions are only sufficient, but are not necessary.
• All physically-reasonable (practical) signals meet these conditions.
( )x t dt
∞
−∞
< ∞∫ ( )2
x t dt
∞
−∞
< ∞∫
Lecture 3
4(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Example: Rectangular Pulse
2
τ
−
2
τ
1
τ
1−
τ
τ
( )x
S f
f
t
signal
spectrum
( )1, / 2
( / )0, / 2
t
x t t
t
< τ= Π τ =
> τ
( )sin
sinc( )x
fS f f
f
π τ
= τ = τ ⋅ τ
π τ
Lecture 3
1
5(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Example: sinc(t)
1
1
2 f−
1
1
2 f
1f− 1f
2
1
2 f−
2
1
2 f
2f− 2f
signal signal
spectrum spectrum
Shortening pulse widens its spectrum!
1( )
xS f
2( )
xS f
Lecture 3
12 f
22 f
6(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Generalized (Singular) Functions
• Unit step function
• Dirac delta function (unit impulse function)
1, 0( )
0, 0
t
u t
t
>=
<
1
t
u(t)
( ) ( ) (0)t x t dt x
∞
−∞
δ =∫
, 0( ) & ( ) 1
0, 0
tt t dt
t
∞
−∞
∞ =δ = δ =
≠∫
t
( )tδ
Lecture 3
7(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Generalized Functions
• Dirac Delta Function as a limit:
• Useful properties of delta function
1/ , / 2( )
0,
t
t
t∆
∆ < ∆Π =
> ∆
t
( )t∆Π
2
∆−
2
∆
1
∆
0
( ) lim ( )t t∆∆→
δ = Π
0 0( ) ( ) ( )t t x t dt x t
∞
−∞
δ − =∫( )
( )du t
tdt
= δ( ) ( )
t
t dt u t
−∞
δ =∫
1( ) ( )at t
a
δ = δ ( ) ( )t tδ − = δ ( ) ( ) (0) ( )x t t x tδ = δ
Fourier transform?
Lecture 3
8(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Fourier Transform of Periodic Signal
• FT of a complex exponent:
• Important property:
• FT of a periodic signal:
• FT of ?
( ) 0
0 02 ( ) ( )
j tx t e f fω
= ↔ πδ ω− ω = δ −
( ) 0
0 02 ( ) ( )
FTjn t
n n n
n n n
x t c e c n c f nf+∞ +∞ +∞
ω
=−∞ =−∞ =−∞
= ↔ π δ ω − ω = δ −∑ ∑ ∑
( ) 2j ftf e dt
+∞± π
−∞
δ = ∫
0cos( )tω
Lecture 3
� prove this property
9(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Properties of Fourier Transform*
• Very similar to those of Fourier series!
• Linearity:
• Time shifting:
• Time reversal:
• Time scaling:
1 21 2( ) ( ) ( ) ( )
F
x xx t x t S f S fα +β ↔α +β
0
0( ) ( ) ( ) ( )j t
x xx t S x t t e S− ω
↔ ω ⇒ − ↔ ω
( ) ( ) ( ) ( )x x
x t S x t S↔ ω ⇒ − ↔ −ω
1( )
xx at S
a a
ω ↔
Lecture 3
*properties are useful for evaluating Fourier transform in a simple way
Prove these
properties.
10(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Properties of Fourier Transform
• Conjugation:
• Differentiation:
• Integration:
• Multiplication:
• Frequency shift (modulation):
*( ) ( ) * ( ) ( )
x xx t S x t S↔ ω ⇒ ↔ −ω
( )( ) ( ) ( )
x x
dx tx t S j S
dt↔ ω ⇒ ↔ ω ω
1( ) ( ) (0) ( )
t
x xx t dt S S
j−∞
↔ ω + π δ ωω
∫
1( ) ( ) ( ) ( )
2
( )* ( )
x y
x y
x t y t S S d
S S
∞
−∞
′ ′ ′↔ ω ω−ω ω =π
= ω ω
∫
0
0( ) ( )j t
x t e Sω
↔ ω−ω
Lecture 3
Pro
ve t
hese p
ropert
ies
11(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Duality of Fourier Transform
( ) ( ) ( ) 2 ( )x x
x t S S t x↔ ω ⇒ ↔ π −ω ( ) ( ) ( ) ( )x x
x t S f S t x f↔ ⇒ ↔ −
A.V. Oppenheim, A.S. Willsky, Signals and Systems, 1997.
Lecture 3
12(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Convolution Property
• This property is of great importance
( ) ( ) ( ) ( ) ( ) ( )x yy t x h t d S H S
∞
−∞
= τ − τ τ↔ ω ω = ω∫
x(t) h(t)
y(t)
( )xS ω ( )H ω ( )yS ω
Cascade connection
of LTI blocks
Lecture 3
A.V. Oppenheim, A.S. Willsky, Signals and Systems, 1997.
• Example: FT of a triangular pulse by convolution
13(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Parseval Theorem• Total energy in time domain is the same as the total
energy in frequency domain:
• - energy spectral density (ESD) of x(t).
Represents the amount of energy per Hz of bandwidth
• Counterpart of Parseval theorem for periodic signals
• Autocorrelation property:
2 22 1( ) ( ) ( )
2x x
E x t dt S f df S d
∞ ∞ ∞
−∞ −∞ −∞
= = = ω ω
π∫ ∫ ∫
2( ) ( )
xE f S f=
( ) ( ) ( )2
( )x x
R x t x t dt S+∞
∗
−∞
τ = − τ ↔ ω∫ ( )0x
R E=
Lecture 3
14(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Parseval Theorem: Example
2sin
?t
dtt
+∞
−∞
=
∫
Lecture 3
15(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Fourier Transform of Real Signal
• if x(t) is real,
• Fourier transform can be presented as
{ } *Im ( ) 0 ( ) ( )
x xx t S S= ⇒ −ω = ω
( )
[ ][ ]
0
1
( ) 2 ( ) cos 2 ( ) ,
Im ( )( ) tan
Re ( )
x
x
x
x t S f f f df
S ff
S f
∞
−
= π + ϕ
ϕ =
∫No negative
frequencies!
Lecture 3
16(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Signal Bandwidth & Negative Frequencies
• What negative frequency means?
• It means that there is term in signal spectrum
• Convenient mathematical tool. Do not exist in practice
(i.e., cannot be measured on spectrum analyzer)
• What is the signal bandwidth? There are many definitions.
• Defined for positive frequencies only.
• Determines the frequency band over which a substantial
part of the signal power/energy is concentrated.
• For band-limited signals
2j fte− π
max min max min, , 0f f f f f∆ = − ≥
Lecture 3
17(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Power and Energy
• Power Px& energy E
xof signal x(t) are:
• Energy-type signals:
• Power-type signals:
• Signal cannot be both energy & power type!
• Signal energy: if x(t) is voltage or current, Exis the
energy dissipated in 1 Ohm resistor
• Signal power: if x(t) is voltage or current, Pxis the power
dissipated in 1 Ohm resistor.
2( )
xE x t dt
∞
−∞
= ∫21
lim ( )2
T
xT
T
P x t dtT→∞
−
= ∫
xE < ∞
0xP< < ∞
Lecture 3
18(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Energy-Type Signals (summary)
• Signal energy in time & frequency domains:
• Energy spectral density (energy per Hz of bandwidth):
• ESD is FT of autocorrelation function:
2 22 1( ) ( ) ( )
2x x x
E x t dt S f df S d
∞ ∞ ∞
−∞ −∞ −∞
= = = ω ω
π∫ ∫ ∫
2( ) ( )
x xE f S f=
( ) ( ) ( ) ( )x x
R x t x t dt E f+∞
∗
−∞
τ = − τ ↔∫ ( )0x x
R E=
Lecture 3
19(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Power-Type Signals: PSD
• Definition of the power spectral density (PSD) (power per
Hz of bandwidth):
• where is the truncated signal (to [-T/2,T/2]),
• and is its spectrum (FT),
( ) ( )2
( )lim
T
x x xT
S fP f P P f df
T
∞
−∞→∞
= ⇒ = < ∞∫
Lecture 3
( ), / 2 / 2 ( ) ( )
0, otherwiseT
x t T t Ttx t x t
T
− ≤ ≤ = Π =
( )Tx t
{ }( ) ( )T TS f FT x t=
( )TS f
20(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Power-Type Signals
• Time-average autocorrelation function:
• Power of the signal:
• Wiener-Khintchine theorem :
( ) ( ) ( )1
lim2
T
xTT
R x t x t dtT
∗
−→∞
τ = − τ∫
( ) ( ) { }( )x x x xP P f df P f FT R
∞
−∞
= ⇒ = τ∫
( ) ( )21
lim 02
T
x xTT
P x t dt RT −→∞
= =∫
Lecture 3
21(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Periodic Signals
• Power of a periodic signal:
• Autocorrelation function:
• Power spectral density (PSD):
221( ) (0)
x n xT
n
P x t dt c RT
∞
=−∞
= = =∑∫
( ) ( ) ( ) 021 jn
x nTn
R x t x t dt c eT
∞
ω τ∗
=−∞
τ = − τ = ∑∫
( ) { }2
( )x x n
n
nP f FT R c f
T
∞
=−∞
= τ = δ −
∑
Lecture 3
( ) 0jn t
n
n
x t c e
+∞
ω
=−∞
← = ∑
Prove these
properties!
22(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Relation Between Fourier Transform &
Series
• consider a periodic signal x(t)=x(t+T)
• truncate it (one period only):
• find FT of the truncated signal xT(t):
• Fourier series of the original periodic signal x(t) is
• Continuous spectrum is the envelope of discrete
spectrum (see slide 2)!
( ), / 2 / 2( )
0, otherwiseT
x t T t Tx t
− < ≤=
( ) ( )T
T xx t S↔ ω
0
1( )T
n xc S n
T= ω
Lecture 3
� prove this
23(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Fourier Transform of Single Pulse ~ Envelope of
Fourier Series of Pulse Train
A.V. Oppenheim, A.S. Willsky, Signals and Systems, 1997.
Lecture 3
FT/S
24(25)
20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka
Summary
• Definition of Fourier Transform
• Properties of Fourier Transform
• Signal bandwidth
• Signal power & energy. Energy and power-type signals
• Fourier transform of periodic signals
• Relation between Fourier series & transform
• Reading: Couch, 2.1-2.6; Oppenheim & Willsky, Ch. 1, 3 & 4. 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 3
25(25)