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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

Course Outline (Tentative)

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Course Outline (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 … - PowerPoint PPT Presentation

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Page 1: 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

Page 2: Course Outline  (Tentative)

Chapter IVChapter IVFourier Integral

Page 3: Course Outline  (Tentative)

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?

Page 4: Course Outline  (Tentative)

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

Page 5: Course Outline  (Tentative)

Fourier Transform Fourier Transform Representation of Aperiodic Representation of Aperiodic SignalsSignals

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

Page 6: Course Outline  (Tentative)

Fourier Transform Fourier Transform Representation of Aperiodic Representation of Aperiodic SignalsSignals

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

Page 7: Course Outline  (Tentative)

Fourier Transform Fourier Transform Representation of Aperiodic Representation of Aperiodic SignalsSignals

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,

Page 8: Course Outline  (Tentative)

Fourier Transform Fourier Transform Representation of Aperiodic Representation of Aperiodic SignalsSignals

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

Page 9: Course Outline  (Tentative)

Fourier Transform Fourier Transform Representation of Aperiodic Representation of Aperiodic SignalsSignals

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

Page 10: Course Outline  (Tentative)

Fourier Transform Fourier Transform Representation of Aperiodic Representation of Aperiodic SignalsSignals

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

Page 11: Course Outline  (Tentative)

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

Page 12: Course Outline  (Tentative)

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

Page 13: Course Outline  (Tentative)

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

Page 14: Course Outline  (Tentative)

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

Page 15: Course Outline  (Tentative)

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

Page 16: Course Outline  (Tentative)

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

Page 17: Course Outline  (Tentative)

Fourier Transform For Periodic Fourier Transform For Periodic SignalsSignals

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)

Page 18: Course Outline  (Tentative)

Fourier Transform For Periodic Fourier Transform For Periodic SignalsSignals

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

Page 19: Course Outline  (Tentative)

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!

Page 20: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransform

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)}

Page 21: Course Outline  (Tentative)

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

Page 22: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransform

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)(

Page 23: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransformExampleExample

Consider

)()()( txtydttdy

Take FT of both sides;

)()( XYj )(1)(

Xj

Y

For )()( ttx 1)( X

j

H

1)(

)()( tueth t

Page 24: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransformExampleExample

Consider (unit step))()( tutx

For ),()( ttg

t

dgtu )()(

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

1( ) ( )Fu tj

Page 25: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransform

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-

Page 26: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransform

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

Page 27: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransform

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)

Page 28: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransform

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

Page 29: Course Outline  (Tentative)

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)

Page 30: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransformConvolution PropertyConvolution Property

)()()()( 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

Page 31: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransformConvolution PropertyConvolution Property

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

Page 32: Course Outline  (Tentative)

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

Page 33: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransformMultiplication PropertyMultiplication Property

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)

Page 34: Course Outline  (Tentative)

Properties of CT Fourier Properties of CT Fourier TransformTransformMultiplication PropertyMultiplication Property

To recover:

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

Page 35: Course Outline  (Tentative)

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).

Page 36: Course Outline  (Tentative)

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

Page 37: Course Outline  (Tentative)

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

Page 38: Course Outline  (Tentative)

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)

Page 39: Course Outline  (Tentative)

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

Page 40: Course Outline  (Tentative)

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

Page 41: Course Outline  (Tentative)

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

Page 42: Course Outline  (Tentative)

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

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

Page 43: Course Outline  (Tentative)

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

)cos( cct

w(t)y(t

) -ωco ωc

o

2

H(ω)

x(t)

lowpass filter

In general,

)cos( cct

y(t)

x(t)

Page 44: Course Outline  (Tentative)

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

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

Page 45: Course Outline  (Tentative)

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

When we apply lowpass filter

)()cos( txcc

output x(t)ccif

2c c output 0

cc must be maintained over time

requires synchronization

Page 46: Course Outline  (Tentative)

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

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

Page 47: Course Outline  (Tentative)

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

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

Page 48: Course Outline  (Tentative)

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

Page 49: Course Outline  (Tentative)

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

Page 50: Course Outline  (Tentative)

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

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

Page 51: Course Outline  (Tentative)

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

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