Course Outline (Tentative)

Course Outline Course Outline (Tentative)(Tentative)

Fundamental Concepts of Signals and Systems Signals Systems

Linear Time-Invariant (LTI) Systems Convolution integral and sum Properties of LTI Systems …

Fourier Series Response to complex exponentials Harmonically related complex exponentials …

Fourier Integral Fourier Transform & Properties … Modulation (An application example)

Discrete-Time Frequency Domain Methods DT Fourier Series DT Fourier Transform Sampling Theorem

Laplace Transform Z Transform

Chapter IVChapter IVFourier Integral

Continuous–Time Fourier Continuous–Time Fourier TransformTransform

So far, we have seen periodic signals and their representation in terms of linear combination of complex exponentials.

Practical meaning: superposition of harmonically related complex

exponentials.

How about aperiodic signals?

Fourier Transform Fourier Transform Representation of Aperiodic Representation of Aperiodic SignalsSignals

(An aperiodic signal is a periodic signal with infinite period.) Let us consider continuous-time periodic square wave

2/ ,0

,1)(

1

1

TtT

Tttx

T2T1T 1T

2T

T

............

)(tx


Recall that the FS coefficients are:

Let us look at it as:

0 10

0

2sin( ) 2, for kk Ta

k T T

0

12sin : samples of an envelope functionk kTTa


as , closer samples, faster rate!as , periodic square wave rectangular pulse (aperiodic

signal) , FS coefficients x T envelope itself

Think of aperiodic signal as the limit of a periodic signal

kTa

0 , T T


Examine the limiting behaviour ofExamine the limiting behaviour of FS FS representation of this signal representation of this signal

)(tx 1 ,0)( Tttx

t

)(tx

1T1T

Consider a signal that is of finite Consider a signal that is of finite duration,duration,


We can construct a periodic signal We can construct a periodic signal out of with period out of with period For For

)(~ tx )(tx T).()(~ , txtxT

1T1T tT2T

2T T

......

)(~ tx

0 0

/ 2

0/ 2

1 2( ) , ( ) , for T

jk t jk tk k

k T

x t a e a x t e dtT T

FS representation of periodic signal


Since 2

for 0 x(t)and 2

for )()(~ TtTttxtx

0 0

/ 2

/ 2

1 1( ) ( )T

jk jk tk

T

a x t e dt x t e dtT T

(Recall and envelope case)kTa

( ) ( ) j tX x t e dt

kTa

00

1( ) ( ) jk t

k

x t X k eT

0 01 ( ) (for )ka X k kT

Define as the envelope of


k

tjkekXtx 000).(

21)(~

,k ,0 ,T 00 As )(~ txx

Hence,

deXtx tj)(21)(

dtetxX tj )()(

: inverse Fourier transform

: Fourier transform (spectrum)

Fourier Transform pair

information needed for describing x(t) as a linear combination of sinusoids

0T2

For

Convergence of Fourier Convergence of Fourier TransformTransform

dttxtx )()(i) x(t) is absolutely integrable,

ii) x(t) has a fiinite number of maxima and minima within any finite interval

iii) x(t) has a finite number of discontinuities within any finite interval. Further more each of these discontinuities must be finite.

deXtx tj)(21)( at any

t except at discontinuities (similar to periodic case).

Dirichlet conditions guarantee that

ExampleExample

0a ),(e x(t) -at tuConsider

0

)( dteeX tjat 0

1 )(

tjaeja

0a ,1

ja

,1)(22

a

wX )(tan)( 1

aX

a1 2

2

ExampleExample

x t ta)

1

1

1,

0,

t Tx t

t T

b)Rectangular pulsesignal

1

1

12 sinTj t

T

TX e dt

X(ω)2T1

1T

ω

-T1 T1

x(t)

t

-

-

( ) 1j tX t e dt

Example (cont’d)Example (cont’d)

1,

0,

Wx t X

W

c) Consider with FT X(ω)

ω-W W

1 sin2

Wj t

W

Wtx t e dt

x(t)

t

W

W

W

NoteNote

sinsinc

important function

1 11

2sin 2 sincT TX T

sin sincWt W Wtx tt

Narrow in Time Domain have Broad FT!Broad in Time Domain Narrow FT!

(Scaling property)

X(ω)2T1

1T

ω

x(t)

tW

W

W

X(ω)

ω

-W W

-T1 T1

x(t)

t

Fourier Transform For Periodic Fourier Transform For Periodic SignalsSignals

Consider a signal x(t) with Fourier Transform(i.e., a signal impulse of area 2 at ω=ω0)

0( ) 2 . ( )X

Let us find the signal

tjtj edetx 0)( 221)( 0

if X(ω) is a linear combination of impulses equally spaced in frequency, i.e.,

k

k kaX )( 2)( 0

: FS representation of periodic signal

k

tjkkeatx 0)( the

n


Hence, FT of periodic signal is weighted impulse train occuring at integer multiples of ω0

Example:a) Periodic square wave:

kTkak

10sin )(sin2)( 0

10 k

kk

TkX

Recall FS coefficients of periodicsquare wave (from previous chapter)


b) x(t)=sinω0t

o/w ,0

-1k ,21

1 ,21

j

kj

ak )()()( 00 jX

x(t)=cosω0t

)()()( 00 X

c)

k

kTttx )()( impulse train

k T

kX )2( T2)(

X(ω)

-ω0 ω0

Tdtet

Ta

T

T

tjkk

1)(1 2

2

0

FT is again impulse train in frequency domain with period T

2

Properties of CT Fourier Properties of CT Fourier TransformTransform

Notations: ),( )( F Xtx ( ) ( )or X F x t

1. Linearity:

FIf ( ) ( ),x t X ),( )( F Yty

then ( ) ( ) ( ) ( ),Fax t by t aX bY

Prove it as an exercise!


2. Time Shifting:

If ( ) ( ),Fx t X 00then ( ) ( )j tFx t t e X

Proof:

deXtx tj)(21)(

deXttx ttj )(0

0)(21)(

01 ( )

2j t j te X e d

F{x(t-t0)}

Properties of CT Fourier Properties of CT Fourier TransformTransformExampleExample

X(t)

1 2 3 4

11.5

t t

1

X1(t)

21 2

1 t

1

X2(t)

23 2

3

1 21( ) ( 2.5) ( 2.5)2

x t x t x t

, )2sin(2

)(1

X

)23sin(2

)(2 X

52

3sin 2sin2 2( )j

X e

By linearity andtime-shifting propertiesof FT


3. Conjugation / Conjugate Symmetry:( ) ( )Fx t X ( ) ( )Fx t X

)()( XX Opp. pg. 303 for proof !for real x(t)

4. Differentiation & Integration:

deXjdttdx tj)(

21)( ( ) ( )Fdx t j X

dt

* Important in solving linear differential equations:Multiplication in frequency domain

t

F XXj

dx )()0( )(1)(


Consider

)()()( txtydttdy

Take FT of both sides;

)()( XYj )(1)(

Xj

Y

For )()( ttx 1)( X

j

H

1)(

)()( tueth t


Consider (unit step))()( tutx

For ),()( ttg

t

dgtu )()(

1( ) ( ) (0) ( )Fu t G Gj

1( ) ( )Fu tj


5. Time and Frequency Scaling:

If ( ) ( ),Fx t X 1then ( ) ( )Fx at Xa a

(Prove using FT integral)

Remark:

- Inverse relation between time and frequency domains:- A signal varying rapidly will have a transform occupying wider frequency band, and vice versa

)( )( Xtx F-


6. Duality:Observe the FT and inverse FT integrals:

; )()(

dtetxX tj

deXtx tj)(21)(

Recall

1

11 t ,0

t ,1)(

T

Ttx F

1

1sin2)( TX

tWttx

sin)(2 F

W

WX

,0

,1)(2

Symmetry between the FT pairs!

In general;

if ( ) ( ), then ( ) 2 (- )F Fg t f f t g


Example: (t))( tx F 1)X(

By duality;

1)( tx F )( 2 (prove it!)

Dual of the properties:

F

ddX )()(tjtx

F

F)(0 txe tj

)()0( )(1 txtxjt

dX )(

)( 0 X

(frequency differentiation)(frequency shifting)


7. Parseval’s Relation:

if ( ) ( )Fx t X 2 2

-

1then ( ) ( )2

x t dt X d

(Check the proof in Opp. pg.312) total energy in x(t)

energy density spectrum: )( 2X

Properties of CT Fourier Properties of CT Fourier TransformTransformConvolution PropertyConvolution Property

-

)()(y(t) dthx

Take FT.

-

)()()Y( dtedthx tj

-

)()( ddtethx tj

dexHdHex jj )()()()(-

)()()( XHY

(Additional & very important properties of FT, in terms of LTI systems)


)()()()( XHtxth F

Convolution of two signals in time domain is equivalent to multiplication of their spectrums in frequency domain

:)(H frequency response of the system

Example:a) Consider a CT, LTI system with

)()( 0ttth

Frequency response is

0)( tjeH

)()()()( 0 XeXHY tj )()( 0ttxty

Recall the time-shiftingproperty of FT


b) Frequency-selective filtering: achieved by an LTI system whose frequency response

H(ω) passes desired range of frequencies and stops (attenuates) other frequencies, e.g.,

; ,0

,1)(

c

cH

ttth c

sin)(

1

0-ωc

ωωc

H(ω)

Passband

stopband stopband

ideal lowpass filter

Properties of CT Fourier Properties of CT Fourier TransformTransformMultiplication PropertyMultiplication Property

From convolution property and duality, multiplication in time domain corresponds to convolution in frequency domain.

)()(21)(

PSR )()()( tptstr F

Amplitude Modulation (multiplication of two signals) important in telecommunications !

Example:

Let s(t) has spectrum S(ω)

A

-ω1

ωω1

S(ω)

Baseband signal


p(t)=cosω0t

)()()( 00 P-ω0

ωω0

P(ω)

1( ) ( ) ( )2

R S P

R(ω)

-ω0-ω0-ω1 -ω0+ω1 ω0ω0-ω1 ω0+ω1

2A

- Information (spectral content) in s(t) is preserved but shifted to higher frequencies (more suitable for transmission)


To recover:

-Multiply r(t) with p(t)=cosω0t g(t)=r(t).p(t)-Apply lowpass filter!

Application of Fourier Application of Fourier TheoryTheoryCommunication SystemsCommunication Systems

Modulation: Embedding an info-bearing signal into a second signal.

Definitions:

Demodulation: Extracting the information bearing signal from the second signal.

Info-bearing signal x(t): The signal to be transmitted (modulating signal).

Carrier signal c(t): The signal which carries the info-bearing signal (usually a sinusoidal signal).

Application of Fourier Application of Fourier TheoryTheoryCommunication SystemsCommunication Systems

Modulated signal y(t) is then the product of x(t) and c(t)

y(t)=x(t).c(t)

Objective: To produce a signal whose frequency range is suitable for transmission over communication channel. e.g.:

individual voice signals are in 200Hz-4kHz telephony (long-distance) over microwave or

satellite links in 300MHz-300GHz (microwave), 300MHz-40GHz (satellite)

Information in voice signals must be shifted into these higher ranges of frequency

Application of Fourier Theory Application of Fourier Theory Amplitude Modulation with Complex Exponential Carrier

)()( cctjetc ωc: carrier frequency

Consider θc=0

tj cetxty )()(

From multiplication property

)()(21)(

CXY

)( 2)( cC )()( cXY

Application of Fourier Application of Fourier Theory Theory Amplitude Modulation with Complex Exponential Carrier

-ωm ωm

1 X(ω)

ω *ωc

C(ω)

ω2

ωc+ ωm

1Y(ω)

ωωcωc- ωm

To recover x(t) from y(t):

tj cetytx )()(

(Demodulation)

(Shift the spectrum back)

Application of Fourier Application of Fourier Theory Theory Amplitude Modulation with Sinusoidal Carrier

)cos()( ccttc )cos()( ccttc

y(t)

x(t)

For θc=0 , y(t)=x(t)cosωct , )()()( ccC

)()(21)( cc XXY

-ωc- ωm

Y(ω)

ωc+ ωm

ωωc- ωm ωc-ωc -ωc+ ωm

21

Application of Fourier Application of Fourier Theory Theory Amplitude Modulation with a Sinusoidal Carrier

To recover x(t) from y(t), the condition ωc>ωm must be satisfied! Otherwise, the replicas will overlap.

Example:

-ωm ωm

1 X(ω)

-ωc ωc

1 Y(ω)

ωm+ ωc-ωm- ωc

21

2 m

cFor

Application of Fourier Application of Fourier Theory Theory Demodulation for Sinusoidal Amplitude Modulation

To recover the information-bearing signal x(t) at the To recover the information-bearing signal x(t) at the receiverreceiver

Synchronous Demodulation:

x(t) can be recovered by modulating y(t) with the same sinusoidal carrier and applying a lowpass filter.

ttx c2cos)(

: cos)()( ttxty c

cos)()( ttytw c

ttx c2cos

21

21)(

ttxtxtw c2cos)(21)(

21)(

(need to get rid of the 2nd term in RHS)

Transmitter and receiver are synchronized in phase


ωc

Y(ω)

ωc+ ωm

ωωc- ωm-ωc

21

ωc

C(ω)

ω

-ωc

Apply lowpass filter (H(ω)) with a gain of 2 and cutoff frequency (ωco)ωm<ωco<2ωc-ωm

W(ω)

ω2ωc- ωm

2ωc2ωc

21

41

41

H(ω)

-ωm ωm


)cos( cct

w(t)y(t

) -ωco ωc

o

2

H(ω)

x(t)

lowpass filter

In general,

)cos( cct

y(t)

x(t)


Assume that modulator and demodulator are not synchronized;θc : phase of modulator

φc : phase of demodulator

)cos()cos()()( cccc tttxtw

)2cos(

21)cos(

21)( ccccc ttx

)2cos()(21)()cos(

21)( ccccc ttxtxtw


When we apply lowpass filter

)()cos( txcc

output x(t)ccif

2c c output 0

cc must be maintained over time

requires synchronization


Asynchronous Demodulation: Avoids the need for synchronization between the

modulator and demodulator If the message signal x(t) is positive, and carrier frequency

ωc is much higher than ωm (the highest frequency in the modulating signal), then envelope of y(t) is a very close approximation to x(t)

envelopey(t)

t


Envelope Detector :Envelope Detector :

to assure positivity add DC to message signal, i.e., x(t)to assure positivity add DC to message signal, i.e., x(t)+A > 0+A > 0 x(t) vary slowly compared to x(t) vary slowly compared to ωωcc (to track envelope) (to track envelope)

A

y(t)=(A+x(t)) cosωctx(t)

Tradeoff : simpler demodulator, but requires transmission of redundancy (higher power)

half-wave rectifier!

+

–y(t) C R

+

–w(t)

cos ωct

Application of Fourier Theory Application of Fourier Theory Single-Sideband (SSB) Sinusoidal Amplitude Modulation

For ωm is the highest frequency in x(t), total bandwidth of the original signal 2ωm.

With sinusoidal carrier: spectrum shifted to ωc and -ωc twice bandwidth is required.

Redundancy in modulated signal!

ωm

X(ω)

2ωm

4ωm Y(ω)

ωc-ωc

Solution: Use SSB modulation

Application of Fourier Application of Fourier Theory Theory Single-Sideband (SSB) Sinusoidal Amplitude Modulation

ωm

X(ω)

(DSB)

(SSB)

(SSB)YL(ω)

ωc-ωc

YU(ω)

ωc-ωc

Spectrum with upper sidebands

Spectrum with lower sidebands

Y(ω)

ωc-ωc ωc +ωmlower sideband

upper sideband

upper sideband

lower sideband


For upper sidebands: Apply y(t) to a sharp cutoff bandpass/highpass filter.

-ωc ωc

H(ω)

y(t) yU(t)

Y(ω)

ωc

-ωc

YU(ω)

ωc-ωc


For lower sidebands: Use 90o phase-shift network.

00

)ωj,ωj,

(ωH

x(t)

cos ωct

sin ωct

xp(t)

y1(t)

y2(t)

y(t)

(Trace the operation as exercise!)

AM-DSB/WCAM-DSB/SC

, AM-SSB/WCAM-SSB/SC

: AM, Double (Single) SB, with carrier: AM, Double (Single) SB, suppressed carrier

YL(ω)

ωc-ωc

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Course Outline (Tentative)