VI. Introduction to the Fourier...

08/24/10 2003rws/jMc-modif SuFY10(MPF)-Section 11 1

[p. 4] Fourier series (FS) review[p. 8] Limiting behavior of FS to non periodic signals[p. 10] Fourier transform (FT) applied to non periodic signals[p. 13] FT applied to periodic signals[p. 23] Frequency response[p. 25] Basic FT properties[p. 28] Uncertainty principle[p. 29] FT examples[p. 34] Convolution property[p. 46] Parseval’s relation[p. 48] using MATLAB to compute the FT[p. 49] AM modulation applications[p. 55] Frequency division multiplexing[p. 57] Quadrature modulation

VI. Introduction to the Fourier Transform


Everything = Sum of Sinusoids

One Square Pulse = Sum of Sinusoids

Non periodic signal can be viewed as the limit of a periodic signal with period = infinityWe are going to take derivations done for Fourier series and extend them …..


Fourier Series: Periodic x(t)

x ( t ) = a k e j ω 0 k t

k = −∞

∞

∑

a k =1T0

x ( t ) e − jω 0 kt dt− T 0 / 2

T0 / 2

∫

0 0 0

0

F u n d a m e n t a l F r e q .2 / 2 ( r d / s )

( H z )T f

fω π π= =

Fourier Synthesis

Fourier Analysis


Recall: Square Wave Signal FS decomposition

ka =

∫−

−=4/

4/0

0

0

0)1(1 T

T

tkjk dte

Ta ω


Spectrum from Fourier Series

⎪⎩

⎪⎨⎧

±±=

±±=≠==

…

…

,4,20

,3,1,00)2/sin(k

k

kkak π

π


What if x(t) is not periodic?

Sum of Sinusoids?Non-harmonically related sinusoids Would not be periodic, but would probably be non-zero for all t.

Fourier transformgives a “sum” (actually an integralintegral) that involves ALLALLfrequenciescan represent signals that are not periodic


Limiting Behavior of FS

0Plot ( )kT a


FS in the LIMIT (long period)( ) ( )0

0 0

2102

12

( )

( ) ( )

j k tT k T

k

j t

x t T a e

x t X j e d

ω ππ

ωπ ω ω

∞

= − ∞

∞

− ∞

=

=

∑

∫

∫∫∞

∞−

−

−

− == dtetxjXdtetxaT tjT

T

tkjTk

ωω ω )()()(2/

2/0

0

0

0

0

0 0

0

0 0

0

2 2l i m l i m

l i m ( )

T T

kT

d kT T

T a X j

π πω ω

ω

→ ∞ → ∞

→ ∞

= =

=


Fourier TransformFor non-periodic signals Fourier Synthesis

Fourier Analysis

{ }( ) ( ) ( ) j tTF x t X j x t e d tωω

∞−

− ∞

= = ∫

∫∞

∞−

= ωω ωπ dejXtx tj)()( 21


Example:x (t ) = e − at u (t )


Example:

T-T

x(t)

t


If x(t) is periodic with period T0 ,0

0 0

0 0

1( ) ( )T

jk t jk tk k

kx t a e a x t e d t

Tω ω

∞−

= − ∞

= =∑ ∫

FT For periodic signals

02

( ) ( )

=

=

j t

jkf t j tk

k

X j x t e dt

a e e dt

ω

π ω

ω∞

−

−∞

∞ ∞−

=−∞−∞

= =

⎛ ⎞⎜ ⎟⎝ ⎠

∫

∑∫


We need to compute

( )jatTF e =

Turns out it is very complicated to compute directlyWe’ll use a shortcut

It turns out:{ } ( ) ( ) 2 ( )jat

Tx t e F x t aπδ ω= ⇔ = −


Example: Compute the FT of the cosine and sine functions


Example: Compute the FT of the delta function δ(t)


If x(t) is periodic with period T0 ,00 0

0 0

1( ) ( )T

jk t jk tk k

kx t a e a x t e d t

Tω ω

∞−

= − ∞

= =∑ ∫0

0W e s h o w e d 2 ( )j k te kω π δ ω ω⇔ −

{ }

{ }

0

0

( )

=

=

jk tT T k

k

jk tT k

k

F x t F a e

F a e

ω

ω

∞

=−∞

∞

=−∞

⎧ ⎫= ⎨ ⎬

⎩ ⎭∑

∑

Summary FT for generic periodic signals


Example: Compute the FT of the square wave signal x(t)

sin( / 2), 0, 1, 3, (i.e., k odd)Recall:

0, k even k

k ka

π = ± ±⎧= ⎨

⎩

( )X jω =


Take advantage of FT linear properties, rewritex(t) in terms of y(t)

2y(t)

y(t)=

How are the FS coefficients of x(t) related to the FS coefficients of y(t)?

Example: Take advantage of FT properties


Compute FS coefficients of y(t) and derive those for x(t)

Use the FS Coefficients to derive the FT of x(t)


Square Wave Fourier Transform con’t


Frequency Response

Fourier Transform of h(t) is called the Frequency Response

ωω

jjHtueth t

+=⇔= −

11)()()(

)()( tueth t−=

Magnitude and Phase Plots for H(jω)


ωω

jajH

+=

1)(


Basic Fourier Transform Properties

( )

( ) ( )( )

( ) ( )( ) ( )

0

0

2

0

0 0 0

0 0 0

00

-

0

/ 2

11 2 ( )

2sgn(t)

1/

2

cos( )

sin( ) [ ]

( ), a>0 1/( )

1 sin( ) 0 otherwise

( ) 2 ( )

j t

at

j D

j t

t

t

ju t j

e

t

t j

e u t a j

tt

t D ee

e

ω

ω

ω

δπδ ω

ωπδ ω ω

πδ ω ω

ω πδ ω ω πδ ω ω

ω πδ ω ω πδ ω ω

ω

ω ωωπ

δ

πδ ω ω

−

−

→

→

→

→ +

→ −

→ − + +

→ + − −

→ +

⎧ <→ ⎨

⎩− →

→ −2 / 2 2 e ωπ −→

1 11

2 sin2 sinc T TT ω ωπ ω

⎛ ⎞→ =⎜ ⎟⎝ ⎠

−T1

1

T1

−T1

1

T1

21

1 sinc2

TT ωπ

⎡ ⎤⎛ ⎞→ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦


( ) ( ) ( ) ( )( ) ( )

( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( )

( ) ( )

0

0

1 2 1 1 1 2

0

0

( )* *

1 0

1

2

( ) ( ( ))1( ). ( ) ( )* ( )

2( )* ( ) ( ). ( )

j t

nn

n

t

j t

ax t bx t a X j b X j

x t t X j e

x t X jd x t j X jdtd x t

j X jdt

x d X j X jj

jx at Xa a

X t x j

x t e X j

x t g t X j G j

x t g t X j G j

ω

ω

ω ω

ω

ω

ω ω

ω ω

τ τ ω π δ ωω

ω

π ω

ω ω

ω ωπ

ω ω

−

−∞

+ → +

− →

→ −

→

→

→ +

⎛ ⎞→ ⎜ ⎟⎝ ⎠

→ −

→ −

→

→

∫

Basic Fourier Transform Properties


x(t) X(jω)Real Imaginary Complex Even Odd

real

Real & even

Real & odd

Imaginary

Imaginary & even

Imaginary & odd

Fourier Transform Properties Summary

[from Karris]


Scaling Property

1| |( ) shrinks ( ) expandsa ax at X j ω⇒

)()( 1aa jXatx ω⇔

Consequence Uncertainty Principle

Try to make x(t) shorterThen X(jω) will get widerNarrow pulses have wide bandwidth

Try to make X(jω) narrowerThen x(t) will have longer duration

Cannot simultaneously reduce time duration and bandwidth


Example: x (t ) =1 t < T / 20 t > T / 2

⎧ ⎨ ⎩

( )X jω =Recall:


Example:⎪⎩

⎪⎨⎧

>

<=

b

bjX

ωω

ωωω

0

1)(

( )x t =


Example:( )X jω =

)()( 0tttx −= δ

Example: X( jω) =2πδ(ω −ω0)


( ) ( )k

x t t kTδ+∞

= −∞

= −∑

Example: impulse train


Convolution Property

Convolution in the time-domain corresponds to MULTIPLICATIONMULTIPLICATION in the frequency-domain

y(t) = h(t) ∗ x(t) = h(τ )−∞

∞∫ x(t − τ )dτ

Y ( jω ) = H ( jω ) X ( jω )

System Sx(t) y(t)

h(t)


Proof: see textbook


Convolution: Example 1Convolving two band-limited signals leads to a band-limited signal Input Signal x(t) : “sinc” functionIdeal LPF (Lowpass Filter) h(t) is a “sinc”Output y(t) is bandlimited


Examples

1 3 t

1

a(t)=cos(3t) -2<t<2, =0 owy(t)=2/(5+2jt)z(t)=2/πtv(t)=2δ(t+3)-4δ(t-3)

3 42 t

w(t)

−w1

1

w1

ω

X(ω)

b(t)

( ) ( ), ( ) ( ), ( ) ( )* ( )

at btx t e u t h t e u tc t x t h t

− −= ==

w1 =5


Example: Compute the FT for x(t)

( ) ( ) ( )at btx t e u t e u t−= + −


Example: Compute the inverse FT of Y(jω)

sin(2 ) sin( )( )( / 2) ( / 2)

Y j ω ωωω ω

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠


Example: Use FT properties to compute the FT of the signals shown below

2 4

( ) ( ) 2 ( ) ( 2 )sin(400 ( 0.1))( ) 10

( 0.1)( ) 10 ( ) 2 ( 3)t t

x t t t T t Tty t

tz t e u t e u t

δ δ δπ

π− −

= − + − − −−

=−

= − −


Parseval’s Relation for non periodic signals (conservation of energy)

For non periodic signals, the relationship becomes

( ) 2 21x kT

kP x t dt a

T= = ∑∫

( ) ( )2 212xE x t dt X j dω ωπ

+∞ +∞

−∞ −∞= =∫ ∫

Recall Parseval’s relation for periodic signal with period T

Signal Power

Signal Energy


Proof:


Using MATLAB to find Fourier Transforms

MATLAB has the functions fourier.m and ifourier.m

Example: compute the FT of exp(-t2/2)syms t v w xft=exp(-t^2/2);FW=fourier(ft)pretty(FW)

>FW =exp(-1/2*w^2)*2^(1/2)*pi^(1/2)

2 1/2 1/2exp(- 1/2 w ) 2 pi

Compute the Fourier transform of –e-tu(t)+3δ(t)syms t v w xft=sym('(-exp(-t)*Heaviside(t))+3*Dirac(t)');Fw=fourier(ft)


Application of the modulation property to AM modulation

( ) ( ) ( )0cos ?y t s t t Y jω ω= ⋅ → =


• Applications: communication systems; Amplitude modulation (AM) systems.

• Speech exist in the range 300Hz~5KHz

• Atmosphere attenuates signals rapidly in the range 10Hz-->20KHz, and propagates much better at high frequencies

shift speech to higher frequency range where signal propagates better


( ) ( ) ( )cos 4 0 cos 400x t t tπ π= ⋅

Overall period of signal x(t) ?

( ) ( ) ( )

4 4 0 4 4 0 3 6 0 3 6 0

1 c o s 4 4 0 c o s 3 6 0214

j t j t j t j t

x t t t

e e e eπ π π π

π π

− − −

= +⎡ ⎤⎣ ⎦

⎡ ⎤= + + +⎣ ⎦

60 70 80 1009010 20 30 40 500

60 70 80 1009010 20 30 40 500

x(t)

Time t (msec)

envelope

• What does the AM signal look like ?Example:


Spectrum of x(t):

( ) 0jk tk

kx t a e ω⇒ = ∑


Spectrum of x(t) for generic frequencies:

Note:

Change ω2

→

you change where the frequency’s components are for a constant ω1

( ) ( ) ( )

( ) ( )

( ) ( ){( ) ( )

1 2 2 1

1 2 1 2

1 2 1 2

1 2 1 2

co s co s1 co s co s21 ex p ex p4

ex p ex p

y t t t

t t

j t j t

j t j t

ω ω ω ω

ω ω ω ω

ω ω ω ω

ω ω ω ω

=

= + + −⎡ ⎤⎣ ⎦

= + + − +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

+ − + − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

−(ω2 +ω1 ) −(ω2 −ω1 ) ω2 −ω1 ω1 +ω2

ω2ω

ω2

1/4

called carrier frequency


How to recover the original speech signal ?

Demodulate….


Application to Frequency Division Multiplexing

x1 (t)

cos(ωa t)

x2 (t)

cos(ωc t)

x3 (t)

cos(ωc t)

y(t)

Y(ω)


Demodulation


Quadrature Modulator

Compute Y(jω)


Quadrature Demodulator


References[Karris] Signals & Systems, 3rd ed. Orchard Publications, 2007[McClellan] Signal Processing First, J. McClellan, R. Schafer, M. Yoder, 2006, Prentice Hall.

VI. Introduction to the Fourier...

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