Lecturer: Dr. Peter Tsang Room: G6505 Phone: 27887763 E-mail: [email protected]

Lecturer: Dr. Peter Tsang

Room: G6505

Phone: 27887763

E-mail: [email protected]

Website: www.ee.cityu.edu.hk/~csl/sigana/Files: SIG01.ppt, SIG02.ppt, SIG03.ppt, SIG04.ppt

Restrict access to students taking this course.

Suggested reference books

1. M.L. Meade and C.R. Dillon, “Signals and Systems”, Van Nostrand Reinhold (UK).

2. N.Levan, “Systems and Signals”, Optimization Software, Inc.

3. F.R. Connor, “Signals”, Edward Arnold.

4*. A. Oppenheim, “Digital Signal Processing”, Prentice Hall.

Note: Students are encouraged to select reference books in the library.

* Supporting reference

Course outline Week 2-4 : Lecture Week 6 : Test Week 7-10 : Lecture Week 11 : Test

Scores

Tests : 30% (15% for each test)

Exam : 70%

Tutorials Group 01 : Friday

Weeks : 2,3,4,7,8,9 Group 02 : Monday

Weeks : 3,4,5,7,8,9 Group 03 : Thursday

Weeks : 2,3,4,7,8,9

Course outline

1. Time Signal Representation.

2. Continuous signals.

3. Fourier, Laplace and z Transform.

4. Interaction of signals and systems.

5. Sampling Theorem.

6. Digital Signals.

7. Fundamentals of Digital System.

8. Interaction of digital signals and systems.

Coursework Tests on week 6 and 11: 30% of total score.

Notes in Powerpoint Presented during lectures and very useful for studying

the course.

Study Guide A set of questions to build up concepts.

Discussions Strengthen concepts in tutorial sessions.

Reference books Supplementary materials to aid study.

Expectation from students Attend all lectures and tutorials. Study all the notes. Participate in discussions during tutorials. Work out all the questions in the study guide at least

once. Attend the test and take it seriously. Work out the questions in the test for at least one more

time afterwards.

SIGNALSSIGNALS

Information expressed in different forms

Stock Price

Transmit Waveform

$1.00, $1.20, $1.30, $1.30, …

Data File

x(t)

00001010 00001100 00001101

Primary interest of Electronic Engineers

SIGNALS PROCESSING AND ANALYSISSIGNALS PROCESSING AND ANALYSIS

Processing: Methods and system that modify signals

System y(t)x(t)

Analysis:• What information is contained in the input signal x(t)?• What changes do the System imposed on the input?• What is the output signal y(t)?

Input/Stimulus Output/Response

SIGNALS DESCRIPTIONSIGNALS DESCRIPTION

To analyze signals, we must know how to describe or represent them in the first place.

A time signal

-15

-10

-5

0

5

10

15

0 5 10 15 20

t

x(t)

t x(t)

0 0

1 5

2 8

3 10

4 8

5 5

Detail but not informative

TIME SIGNALS DESCRIPTIONTIME SIGNALS DESCRIPTION

1. Mathematical expression: x(t)=Asin(t

2. Continuous (Analogue)

-15

-10

-5

0

5

10

15

0 5 10 15 20

3. Discrete (Digital)

x[n]

n


4. Periodic

-15

-10

-5

0

5

10

15

0 10 20 30 40x(t)= x(t+To)

To

Period = To

5. Aperiodic

-2

0

2

4

6

8

10

12

0 10 20 30 40


6. Even signal txtx

Exercise: Calculate the integral

7. Odd signal txtx

-15

-10

-5

0

5

10

15

-10 -5 0 5 10

-15

-10

-5

0

5

10

15

-10 -5 0 5 10

T

T

tdttv sincos


8. Causality

Analogue signals: x(t) = 0 for t < 0

Digital signals: x[n] = 0 for n < 0


9. Average/Mean/DC value

MTt

tMDC dttx

Tx

1

1

1

Exercise: Calculate the AC & DC values of x(t)=Asin(twith

2

MT

TM

-15

-10

-5

0

5

10

15

0 10 20 30 40

10. AC value

DCAC xtxtx DC: Direct ComponentAC: Alternating Component


11. Energy

dttxE2

Exercise: Calculate the average power of x(t)=Acos(t

12. Instantaneous Power watts

R

txtP

2

13. Average Power

MTt

tMav dttP

TP

1

1

1

Note: For periodic signal, TM is generally taken as To


14. Power Ratio2

11010

P

PlogPR

In Electronic Engineering and Telecommunication power is usually resulted from applying voltage V to a resistive load

R, as

The unit is decibel (db)

R

VP

2

Alternative expression for power ratio (same resistive load):

R/V

R/Vlog

P

PlogPR 2

2

21

102

110 1010

2

11020

V

Vlog


15. Orthogonality

Exercise: Prove that sin(tand cos(tare orthogonal for

Two signals are orthogonal over the interval if

021

1

1

dttxtxrMTt

t

MTtt 11,

2

MT


15. Orthogonality: Graphical illustration

x1(t)

x2(t)

x1(t) and x2(t) are correlated.

When one is large, so is the other and vice versa

x1(t)

x2(t)

x1(t) and x2(t) are orthogonal.

Their values are totally unrelated


16. Convolution between two signals

dtxxdtxxtxtxty

122121

Convolution is the resultant corresponding to the interaction between two signals.

1. Dirac delta function (Impulse or Unit Response) (t)

0t

otherwise

tAt

0

0for

where A

Definition: A function that is zero in width and infinite in amplitude with an overall area of unity.

SOME INTERESTING SIGNALSSOME INTERESTING SIGNALS

2. Step function u(t)

0t

otherwise

ttu

0

0for 1

SOME INTERESTING SIGNALSSOME INTERESTING SIGNALS

1

A more vigorous mathematical treatment on signals

Deterministic SignalsDeterministic SignalsDeterministic SignalsDeterministic Signals

A continuous time signal x(t) with finite energy

dttxN

2

Can be represented in the frequency domain

dtetxX tj

Satisfied Parseval’s theorem

dffXdttxN

22

f 2


A discrete time signal x(n) with finite energy

n

N nx2

Can be represented in the frequency domain

n

njenxX

Satisfied Parseval’s theorem

dffXnxn

N

22

1

21

2

deXnx nj

2

1

Note: X is periodic with period = sec/2 rad


Energy Density Spectrum (EDS)

2fXfSxx

Equivalent expression for the (EDS)

where

mj

mxxxx emrfS

n

xx mnxnxmr ** Denotes complex conjugate

Two Elementary Deterministic SignalsTwo Elementary Deterministic SignalsTwo Elementary Deterministic SignalsTwo Elementary Deterministic Signals

Impulse function: zero width and infinite amplitude

1

dtt

Discrete Impulse function

otherwise

nn

0

01

dtxtx

0gdttgt

Given x(t) and x(n), we have

knkxnxk

and

Two Elementary Deterministic SignalsTwo Elementary Deterministic SignalsTwo Elementary Deterministic SignalsTwo Elementary Deterministic Signals

Step function: A step response

Discrete Step function

otherwise

nnu

0

01

otherwise

ttu

0

01

Random SignalsRandom SignalsRandom SignalsRandom Signals

Infinite duration and infinite energy signals

e.g. temperature variations in different places, each have its own waveforms.

Ensemble of time functions (random process): The set of all possible waveforms

Ensemble of all possible sample waveforms of a random process: X(t,S), or simply X(t).t denotes time index and S denotes the set of all possible sample functions

A single waveform in the ensemble: x(t,s), or simply x(t).


x(t,s0)

x(t,s1)

x(t,s2)


Energy Density Spectrum (EDS)

2fXfSxx

Equivalent expression for the (EDS)

where

derfS jxxxx

dttxtxrxx

** Denotes complex conjugate


Each ensemble sample may be different from other.

Not possible to describe properties (e.g. amplitude) at a given time instance.

Only joint probability density function (pdf) can be defined. Given a sequence of time instants

Nttt ,.....,, 21 the samples it tXXi Is represented by:

A random process is known as stationary in the strict sense if

NN tttttt xxxpxxxp ,.....,,,.....,,

2121

Nttt xxxp ,.....,,

21

Properties of Random SignalsProperties of Random SignalsProperties of Random SignalsProperties of Random Signals

is a sample at t=ti itX

The lth moment of X(ti) is given by the expected value

iiii tt

lt

lt dxxpxXE

The lth moment is independent of time for a stationary process.

Measures the statistical properties (e.g. mean) of a single sample.

In signal processing, often need to measure relation between two or more samples.


are samples at t=t1 and t=t2 21 tXandtX

The statistical correlation between the two samples are given by the joint moment

21212121, tttttttt dxdxxxpxxXXE

This is known as autocorrelation function of the random process, usually denoted by the symbol

2121, ttxx XXEtt

For stationary process, the sampling instance t1 does not affect the correlation, hence

xxttxx XXE21 21 where tt


2

10 txx XEAverage power of a random process

Wide-sense stationary: mean value m(t1) of the process is constant

Autocovariance function:

21212121 ,,21

tmtmtttmXtmXEttc xxttxx

For a wide-sense stationary process, we have

221, xxxxxxx mcttc


22 00 xxxxx mc Variance of a random process

Cross correlation between two random processes:

21212121,, 21 ttttttttxy dydxyxpyxYXEtt

When the processes are jointly and individually stationary,

1111 ttttyxxy YXEYXE


Cross covariance between two random processes:

212121 ,, tmtmttttc yxxyxy

When the processes are jointly and individually stationary,

1111 ttttyxxy YXEYXE

Two processes are uncorrelated if

212121 ,or , ttxyxy YEXEttttc


Power Spectral Density: Wiener-Khinchin theorem

def fjxxxx

2

An inverse relation is also available,

Average power of a random process

dfef fjxxxx

2

00 2

txxxx XEdff


Cross Power Spectral Density: def fjxyxy

2

Average power of a random process

00 2

txxxx XEdff

For complex random process,

fdedefxxxxxxxx

fjfj

22**

*xxxx

For complex random process, ffxyxy*

is a sample at instance n. nXX n or ,

The lth moment of X(n) is given by the expected value

nnln

ln dxxpxXE

Properties of Discrete Random SignalsProperties of Discrete Random SignalsProperties of Discrete Random SignalsProperties of Discrete Random Signals

Autocorrelation

mean theis x

Autocovariance knxxxx XEXEknknc ,,

For stationary process, let knm

2xxxknxxxx mXEXEmmc

knxx XEXEm

The variance of X(n) is given by

22 00 xxxxxc

Properties of Discrete Random SignalsProperties of Discrete Random SignalsProperties of Discrete Random SignalsProperties of Discrete Random Signals

Power Density Spectrum of a discrete random process

fmj

mxxxx emf 2

Inverse relation: 21

21

2 dfefm fmjxxxx

Average power: dffXE xxxxn 21

21

2 0

Mathematical description of signal

kknk

M

kk nanx

cos1

Signal ModellingSignal ModellingSignal ModellingSignal Modelling

are the model parameters. Mkkkkka 1,,,

Harmonic Process model

10or 1 kk

kk

M

kk nanx

cos1

Linear Random signal model

k

knwkhnx

Rational or Pole-Zero model

Signal ModellingSignal ModellingSignal ModellingSignal Modelling

nwnaxnx 1

nwknxanxp

kk

1

Autoregressive (AR) model

q

kk knwbnx

0

Moving Average (MA) model

SYSTEM DESCRIPTIONSYSTEM DESCRIPTION

1. Linearity

System y1(t)x1(t)

System y2(t)x2(t)

IF

System y1(t) + y2(t)x2(t) + x2(t)THEN


2. Homogeneity

System y1(t)x1(t)

System ay1(t)ax1(t)

IF

THEN

Where a is a constant


3. Time-invariance: System does not change with time

System y1(t)x1(t)

System y1(tx1(t

IF

THEN

t

x1(t)

t

y1(t)

t

x1(t

t

y1(t


3. Time-invariance: Discrete signals

System y1 [n]x1[n]

System y1[n - mx1[n - m

IF

THEN

t t

t t

x1[n]

x1[n - m

y1 [n]

y1[n - m

mm


4. Stability

The output of a stable system settles back to the quiescent state (e.g., zero) when the input is removed

The output of an unstable system continues, often with exponential growth, for an indefinite period when the input is removed

5. Causality

Response (output) cannot occur before input is applied, ie.,

y(t) = 0 for t <0

THREE MAJOR PARTSTHREE MAJOR PARTS

Signal Representation and Analysis

System Representation and Implementation

Output Response

Signal Representation and AnalysisSignal Representation and Analysis

An analogy: How to describe people?

(A) Cell by cell description – Detail but not useful and impossible to make comparison

(B) Identify common features of different people and compare them. For example shape and dimension of eyes, nose, ears, face, etc..

Signals can be described by similar concepts: “Decompose into common set of components”

Periodic Signal Representation – Fourier SeriesPeriodic Signal Representation – Fourier Series

Ground Rule: All periodic signals are formed by sum of sinusoidal waveforms

(1)

(2)

11

tnsinbtncosaatx nno

tdtncostxT

a/T

/T

n

2

2

2

(3)

tdtnsintxT

b/T

/T

n

2

2

2

dttxT

a/T

/T

o

2

2

1

Fourier Series – Parseval’s IdentityFourier Series – Parseval’s Identity

Energy is preserved after Fourier Transform

(4)

1

2222

2

2

2

11no

/T

/Tbaadttx

T n

11

tnsinbtncosaatx nno

dttnsintxbtdtncostxadttxa

dttx

/T

/Tn

/T

/Tn

/T

/To

/T

/T

1

2

21

2

2

2

2

2

2

2

Fourier Series – Parseval’s IdentityFourier Series – Parseval’s Identity

dttnsintxbtdtncostxadttxa

dttx

/T

/Tn

/T

/Tn

/T

/To

/T

/T

1

2

21

2

2

2

2

2

2

2

22 11

2 Tb

TaTa nno

22 11

2 Tb

TaTa nno

1

2222

2

2

2

11no

/T

/Tbaadttx

T n


t

x(t)

-t

1

-1T/4-T/4

t x(t)

-T/2 to –T/4 -1

-T/4 to +T/4 +1

+T/4 to +T/2 -1

tdtncostxT

a/T

/T

n

2

2

2

2/

4/

4/

4/

4/

2/

coscoscos2 T

T

T

T

T

T

tdtntdtntdtnT

2/

4/

4/

4/

4/

2/

sinsinsin2T

T

T

T

T

T n

tn

n

tn

n

tn

T

T

2

-T/2 T/2


t

x(t)

-t1

-1T/4-T/4

t x(t)

-T/2 to –T/4 -1

-T/4 to +T/4 +1

+T/4 to +T/2 -1

tdtncostxT

a/T

/T

n

2

2

2

2/

4/

4/

4/

4/

2/

sinsinsin2T

T

T

T

T

T n

tn

n

tn

n

tn

T

2sin

4

4sin

8 Tn

Tn

Tn

Tn

T

2


t

x(t)

-t1

-1T/4-T/4

t x(t)

-T/2 to –T/4 -1

-T/4 to +T/4 +1

+T/4 to +T/2 -1

2

4

4

8 Tnsin

Tn

Tnsin

Tnan

T

2

nsinn

nsin

n

2

2

4

zero for all n

We have, ,ao 0 ,a4

1 ,a 02 ,.......a34

3


t

x(t)

-t1

-1T/4-T/4

t x(t)

-T/2 to –T/4 -1

-T/4 to +T/4 +1

+T/4 to +T/2 -1

T

2

It can be easily shown that bn = 0 for all values of n. Hence,

....tttttx

cos7

7

1cos5

5

1cos3

3

1cos

4

Only odd harmonics are present and the DC value is zero

The transformed space (domain) is discrete, i.e., frequency components are present only at regular spaced slots.


t

x(t)

-t

At x(t)

-/2 to –/2 A

-T/2 to - /2 0

+ /2 to +T/2 0

-/2 /2

-T/2 T/2

T

AdtA

Tdttx

Ta

/

/

/T

/T

o

2

2

2

2

11

2

2

2

2

cos2

cos2

T

T

TtdtnA

Ttdtntx

Tan T

2

2sin

4sin2 2

2

n

Tn

A

n

n

T

A


t

x(t)

-t

At x(t)

-/2 to –/2 A

-T/2 to - /2 0

+ /2 to +T/2 0

-/2 /2

-T/2 T/2

T

2

2sin

4sin2 2

2

n

Tn

A

n

n

T

Aan

It can be easily shown that bn = 0 for all values of n. Hence, we have

1

cos2

2sin2

n/n

/n

T

A

T

Atx


1

cos2

2sin2

n/n

/n

T

A

T

Atx

Note: knyyy 2for 0sin

Hence: ,...,,kn

knk

na 321

2

2for 0

2

4

0

TA

Lecturer: Dr. Peter Tsang Room: G6505 Phone: 27887763 E-mail: [email protected]

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Transcript of Lecturer: Dr. Peter Tsang Room: G6505 Phone: 27887763 E-mail: [email protected]