Introduction - Basic Concepts 1 CS 2001...Imperial College - Dep. of Electrical & Electronic...

IMPERIAL COLLEGE LONDON,DEPARTMENT of ELECTRICAL and ELECTRONIC ENGINEERING.

COMPACT LECTURE NOTES on COMMUNICATION THEORY.Prof Athanassios Manikas, Autumn 2001

Introduction - Basic Concepts

Outline:

ì Digital Comm. Systems: General Block Structuresì Classification of Signals.ì 0Auto and Cross Correlation Functions and PSD( ).ì FT and Woodwords Notation.ì Additive White Gaussian Noise and its Modelling.ì Tail Function and its Graph.ì Some Useful Appendices.

Imperial College - Dep. of Electrical & Electronic Engineering Compact Lecture Notes

E303/ISE3.2E: Introduction - Basic Concepts 2 A. Manikas

1. GENERAL BLOCK STRUCTURE OF A DIGITAL COMMUNICATION SYSTEM

H( )f

^^^ ^^ ^



• The points may be considered as the input of a Digital Communication System where messages consist of sequences of "symbolA# s" selected from an alphabet e.g. levels of aquantizer or telegraph letters, numbers and punctuations.

• The objective of a Source Encoder (or data compressor) is to represent the message-symbols arriving at point by as few digA# its as possible. Thus, each level (symbol) at point isA#mapped, by the Source Encoder, to a unique codeword of 1s and 0s and, at point , we get a sequence of binary digits.B

• There are two ways to reduce the channel noise/interference effects1. to introduce deliberately some redundancy in the sequence at point A3 and this is what a Discrete Channel Encoder does.

This redundancy aids the receiver in decoding the desired sequence by detecting and many times correcting errors indroduced by the channel;

e.g.repeat each bit of times,

or, a more sophisticated approach, use a mapper: -bits at point A3 -bits at point A4f B 7

Èk n

Note

is the

: measures the amount of redundancy introduced to the data by the

ÚÝÝÝÝÝÛÝÝÝÝÝÜ

k kn ncÀ œ V œrate of code or code-rate"Vc

channel encoder. Note also that BANDWIDTH= by

If limited BANDWIDTH, then there is a need for without

Å "Vc

CLEVER REDUNDANCY need to increase the BANDWIDTH.

2. to increase Transmitter's power - point often very expensive therefore better to trade transmitter's power for channel BA T Ð NDWIDTHÑ

• at point : T waveform.=Ð>Ñ The digital modulator takes at a time at some uniform rate and transmits one of =2 distinct waveforms #cs cs-bits r t ,...Q = Ð Ñ#cs

" .,s t QÐ Ñ Qi.e. we have an -ary communication system.

A new waveform corresponding to a new sequence is transmitted every seconds. If we have one bit at a time # #cs cs cs-bit T =" œ 01

i.e. a binary communication systemÈ =È =

"

#

• at point : The transmitted waveform , affected by the channel, is received at point T noisy waveform . T^ ^<Ð>Ñ œ =Ð>Ñ 8Ð>Ñ =Ð>Ñ

• at point : .B2 a binary sequence^

based on the received signal the digital demodulator has to decide which of the waveforms has been transmitted in a<Ð>Ñ Q = Ð>Ñ3 ny given time interval X-=

• at point : B a binary sequence.^

The channel decoder attempts to reconstruct the sequence at from:B the knowledge of the code used in the channel encoder, andˆ the redundancy contained in the received dataˆ

• at point : A message.^

The source decoder processes the sequence received from the output of the channel decoder and, from the knowledge of the source encoding method used, attempts to reconstruct thesignal of the information source.

message at point A message at point A^ ¶Ð Ñdue to channel decoding errors and distortion introduced by the quantizer



2. A SIMPLIFIED BLOCK STRUCTURE OF A DIGITAL COMMUNICATION SYSTEM

• A simplified and general block structure of a Digital Communication Systemis shown in the following page and it is common practice its quality to beexpressed in terms of the accuracy with which the binary digits delivered at

the output of the detector/Rx (point ) represent the binary digits that wereBfed into the digital modulator/Tx (point ).B

• It is generally taken that it is the fraction of the binary digits that aredelivered back in error that is a measure of the quality of the communicationsystem. This fraction, or rate, is referred to as the probablity of a bit error ,:/

or, Bit-Error-Rate (point ). BER B



DiscreteInformation

Sink

Discrete Information

SourceTx

Rx

H( )f

T

T

B

B

N.B.: Quality is measured as the Bit Error Rate (BER)



INTERNET

LOCALEXCHANGE( )or a street-cabinet

LOCALEXCHANGE

(or a street-cabinet)

SERVICEPROVIDER

MODEMMODEM

POTS Network(Narrowband Network)


To a POTS line Card

To a POTS line Card

B

TB

T

C H A N N

E

L



3. Digital Transmission of Analogue Signals

ContinuousInformation

Sink

ContinuousInformation

SourceTx

Rx

ADC

DAC

H( )f

A

BA

B^ ^

T

T

N.B.: Quality is measured as the SNR (analogue signals degrade as noise levelincreases)



INTERNET


LOCALEXCHANGE

(or a street-cabinet)

SERVICEPROVIDER

MODEMMODEM

POTS Network(Narrowband Network)


To a POTS line Card

To a POTS line Card

A

T

T CHANNEL

A

A

B

B

A A

Note that like and communications Wireline fiber wireless communicationsare also fully digital.



It is clear from the previous discussion that signals(representing bits) propagate through the networks.

Therefore the following sections are concerned with themain properties and parameters of communicationsignals.



4. Communication Signals

TRANSFORMATION

• Time Domain (TD) Frequency Domain (FD)z-Domains-Domainetc

• Frequency Domain (Spectrum): very important in Communications

TD FDFourier Transform

signal: 1Ð>Ñ KÐ0Ñ system: =impulse response =transfer function2Ð>Ñ LÐ0Ñ



Classification of Signals(according to their )description

Deterministic Signals Random Signals• describable by mathematical function • these are unpredictable; • cannot be expressed as a function • can be expressed probabilistically

3cos(2 1000 )π t

N.B. - Random Signals: very important in Communications



Classification of Signals(according to their )periodicity

Periodic non-periodic Z 96>=

period)X Ð0

J œ!"X!

N.B.:according to Fourier Series Theorem any periodic waveform can berepresented by a sum of sinusoidals having frequencies .F , 2F , 3F , etc! ! !



Classification of Signals(according to their )energy

Energy= signal' _

_#Þ.>

Energy Signals Power Signals ( ) ( )Energy Energy _ œ _

T

N.B.: • Signals of are Signalsfinite duration Energy • Signals are SignalsPeriodic Power



The figures below show the following parameters:• peak (Volts) or peak-to-peak• Energy (J) (or Power (W))• rms (Volts)• Crest Factor CF



Classification of Signals(according to their )spectrum

LOW-PASS BAND-PASS Other types(or baseband) signals signals e.g. all-pass high-pass, etc

- 0J J1 1

F F

Bandwidth= Bandwidth =J F1

where max. frequency of J1 œ 1Ð>Ñ



OPERATIONS


+ +

Š ‹convolutiondenotes



5. IMPORTANT SPECTRUM SHAPES


signal: 1Ð>Ñ KÐ0Ñ |KÐ0Ñl# ESD( ) PSD(0 9< 0Ñ

1. sinewaves/carriers

N.B.: periodic waveform in TD discrete spectrum in FDÊ



2. finite duration (i.e. Energy signals)



1.5 V

t

1.5

3. DC œ f

4. message signals 1Ð>Ñ f - 0 J J1 1

5. transmitted signals ((or, received signals

=Ð>Ñ<Ð>ÑÑ

f

F F



6. An important example in the TD



FD representation of the previous example:



6. Bandwidth of a signal

the range of the significant frequency components in a signal waveform

Examples of message signals (baseband signals) and their bandwidth:

• television signal bandwidth 5.5MHz• speech signal bandwidth 4KHz• audio signal bandwidth 8kHz to 20kHz

Examples of transmitted signals (bandpass signals) and their bandwidth:

• to be discussed latter on• Note that there are various definitions of bandwidth, e.g. 3dB bandwidth, null-to-null bandwidth, Nyquist (minimum) bandwidth



__________ F#

0Hz--// 0 J J J - - -

" "X X

___________________ F"

F œ œ X œ"#Xnull-to-null bandwidth where signal's duration

F œ $# dB bandwidth (Energy Signal)F œ œ$

"XNyquist Bandwidth



7. REDUNDANCY

The degree of in a signal is provided by itsRedundancyautocorrelation function.

For instance the autocorrelation function of a signal 1Ð>Ñis V Ð Ñ11 7

Accumulator

1Ð>Ñ V Ð Ñ11 7

NB:if =fixed then a numberif =variable (i.e. ) then a function of œ 7 77 7 7 7

V Ð Ñ œa V Ð Ñ œ

11

11



8.SIMILARITY

The degree of between two signals is given bySimilaritytheir cross-correlation function.

For instance the cross-correlation function between twosignals and is 1 Ð>Ñ 1 Ð>Ñ V Ð Ñ" # 1 1" #

7

V Ð Ñ1 1" #7

Accumulator1 Ð>Ñ

1 Ð>Ñ

"

2

N.B.:if =fixed then a numberif =variable (i.e. ) then a function of œ 7 77 7 7 7

V Ð Ñ œa V Ð Ñ œ

1 1

1 1

" #

" #



9. SUMMARY

• Parameters for a signal :1Ð>Ñ Energy (J) I œ1

Power (W) T œ1

ESD Energy Spectral Density (J/Hz) Ð0Ñ œ1

PSD Power Spectral Density (W/Hz) Ð0Ñ œ1

Autocorrelation function V Ð Ñ œ11 7 • N.B.: The above are normalised parameters (1 Ohm Resistor)

• other parameters B Bandwidth œ max frequ. of J œ 1Ð>Ñ1

CF Crest Factor œ rms (Volts) peak (Volts)



10. More On Transformations



11. WOODWARD's Notation

• The evaluation of FT, that is

FTe f1Ð>Ñ œ Ð Ñ Ð ÑG f = g t .e dt'_

_ #j ft1

FT"e fKÐ0Ñ œ Ð Ñ Ð0g t G .e dfœ Ñ'_

_ #+j ft1

involves integrating the product of a function and a complex exponential -which can be difficult; so tables of useful transformations are frequently used.

However, the use of tables is greatly simplified by employing Woodward'snotation for certain commonly occurring situations.

• main advantage of using Woodward's notation: allows periodic time/frequencyfunctions to be handled with FT rather than Fourier Series



• 1. rect{ } if

tt

´" ±

! 9>2/<A3=/œ ± "

#

2. sinc Öt× ´ sin ttÐ Ñ11

. tt 0 tt t 0

3 { }

1 if 1 if

A ´Ÿ

"œ Ÿ " Ÿ Ÿ



g t T4. repTÖ × ´ 8g tÐ Ñ !_8 _=

Ð Ñ

where 8=....,-2,-1,0,+1,+2,...

5. combTÖ × ´ 8 8g tÐ Ñ g T . t T!_8 _=

Ð Ñ Ð Ñ$

Å also known assampling function



N.B.:• we can generate any desired "rect" function by scaling and shifting see forÐinstance the following tableÑ

shifting: scaling: shifting+scaling:

rect rect rectg t = t g t = g t =

t

i.e.

Ð Ñ Ð Ð Ñ Ð Ñ Ñ

Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ

Ÿ

7

7

7

š › ›t tT T

t t-T T

š

" " " " " "# # # # # #

"#

7

7

t + i.e. t i.e. t +Ÿ Ÿ Ÿ Ÿ Ÿ7 7 7"# # # # #

T T T T

• effects of temporal scaling:

as FT becomes narrower and amplitude rises -function atT Ä _ Ê Ê $0 frequency when T Ä _



12. Additive White Gaussian Noise (AWGN)

Í

Types of Channel Signals• bandpass• bandpass• allpass • bandpass

=Ð>Ñ œ<Ð>Ñ œ8 Ð>Ñ œ8Ð>Ñ œ3

SNRin at point T

at pointœ T

T

desired

noise T



Comments on 8 Ð>Ñ3



Comments on 8Ð>Ñ





13. Q-function or Tail(T)-function• Consider a random signal with an amplitude probability density functionBÐ>Ñpdf (x). Then the probability that the amplitude of is greater than 3B BÐ>ÑVolts (say) is given as follows:

PrÐBÐ>Ñ $Z Ñ œ Þ.$Z

_

Bpdf (x) x

• If pdf (x) = Gaussian of mean and standard deviation (notation used:B B B. 5pdf (x) N( , )), then the above area is defined as the Q-function (orB B Bœ . 5Tail-function)

i.e.: pdf (x) x$Z

_

B Þ. œ? Tš ›l$ l.5

B

B

Note: if =0 and =1 then pdf (x) x 3. 5 ?B B B

$Z

_

Þ. œ Te f



14. Tail Function GraphThe graph below shows the Tail function which represents the area from to of theTe fB B _

Gaussian probability density function N(0,1), i.e.

.Te fB œ .C'B

_"

#

C#È 1

expŠ ‹#

pB

T Te f e fB B

pB Note that if then may be approximated by B 'Þ& ¸ Þ T Te f e fB B " B

# Þ B #È 1expš ›#



15. AppendicesAppendix-1: Expectation, Moments and StationarityEXPECTATION

• Consider a r.s. characterized by its amplitude pdf i.e. pdf The or value of is defined as follows:Z Ð Ðt . expected mean V tÑ ÑV

pdfX .š ›V t v. v .dvÐ ÐÑ ´ Ñ ´'_

_"V V

• V t V t V t . expected mean V tConsider a function of the r.s. i.e. and the pdf of i.e. pdf The or value of isÐ Ð Ð ÐÑ Ñ Ñ Ñf{ } f{ }V

defined as follows:

pdfXš ›f{ } f{ }.V t v v .dvÐ ÐÑ Ñ´ '_

_V

PROPERTIES OF EXPECTATION

• if then V 0 V 0 Xš ›• c.V =c. VX Xš › š ›• V V VX X Xš › š › š ›" " #+V = +#

• 1 =1; X Xš › š ›c =c

• =statistical independent If f { }.f { } f { } f { }— X — X X —Ð Ð Ð Ð Ð Ðt ,V t V t t V t tÑ Ñ Ñ Ñ Ñ Ñthen š › š › š ›" # " #= .

MOMENTS of a RANDOM SIGNALConsider a r.s. V t :Ð Ñ• definitions:

is called momentof of

X . . X .š › š ›V t = V tV t V t

Ð ÐÐ Ð

Ñ ´ Ñ ´Ñ Ñ

"V V V1 moment 1 centralst st

V t V tV t V t

X . X .š › š ‹ ›Ð ÐÐ Ð

Ñ ´ Ñ ´Ñ Ñ

##

#

V Vis called momentof of

2 moment

st Š 2 centralnd

..... .....

V t V tV t V t

X . X .š › š ‹ ›Ð Ð ´Ð Ð

Ñ ´ Ñ Ñ Ñ

kkV V

kis called moment

of of k moment k central

th thŠ

• the moment is called of the r.s. 2 central variancend V tÐ Ñ

i.e. = V t = v . v .dv5 X . .V V V# # #š › 'Ð Ð Ð ÐÑ Ñ Ñ ÑpdfV

• An important property of the of the r.s. variance V t :Ð Ñ

= proof for you5 . .V V2V

## ÑÐ

STATIONARITY:• a r.s. is said to be stationary in the STRICT-SENSE iff all its averages are time invariant. i.e. and + have theZ Ð Z Ðt tÑ Ñ7

same statistics .a7• a r.s. is said to be stationary in the WIDE-SENSE iff the mean and the autocorrelation function are time invariant

i.e.

X .š ›Z ÑÐt = "V

.X 7 7š ›Z Ñ Ñ ÑÐ Z Ð Ðt t+ =R VV i.e a function only of the time difference 7

This is a less demanding form of stationarityClearly, a STRICT-SENSE stationary process is also WIDE-SENSE stationary but the converse is not guaranteed.



Appendix-2: pdf's with Extensive Applications in Communications:

UNIFORM DISTRIBUTION• If the r.v. is to take on value within a given range (e.g. )• equal likely any a to +a

BINOMIAL DISTRIBUTION

• Consider an experiment having only A and B(e.g. 0 and 1), mutually exclusive.2 possible outcomes That means that a r.v. takes only 2 possible values, the first (A, say) with probability and the second (B) with• p probability " :.

• Repeat the experiment and take a r.v. .n-times —• The Probability that the event A, say, will happen exactly in will be given by:B 8-times -trials

pdf—ÐBÑ Ð" = p . p)Š ‹8B

B 8B

where Š ‹8B Bx Ð8BÑx

8x= =binomial coef

B-times

8-trials

N.B.: B-timesmean œ 8:variance œ 8:Ð" :Ñ

POISSON DISTRIBUTION• In the Binomial Distribution if while then the Binomial is approximated by8 :Ä _ Ä, 0,

the Poisson Distribution given by

pdf where —ÐBÑ= e . - -B

Bx - œ 8:

• In practice the approximation is used if 50 while 5N.B.: n8 8:

GEOMETRIC DISTRIBUTION• Consider an experiment having only 2 possible outcomes (A, B), say, mutually exclusive.

That means that the random variable takes only 2 values and• Pr( =A)=• p Pr( =B)=1• p

• Repeat the experiment up to the time you obtain an A (say), and take a r.v. .—• The probability that the event A will obtained after B'sB "

is given by:

pdf—ÐBÑ œ Ð" Ñp .pB"

-timesB



GAUSSIAN DISTRIBUTION

pdf—ÐBÑ œ "

#

Ð Ñ#É 15

.5

—

—

—#

#

#.expŠ ‹ x

• pdf of thermal noise= Gaussian• Central Limit Theorem: The pdf of a linear combination of statistically independent andn

identically distributed rvs (with finite mean and variance) tends to a Gaussian pdf as8 Ä _.

i.e. . = if where =rvs & =coefficients!3

3 3=1

n! • —3 3 • !

then as or a large number) pdf =Gaussiann (Ä _ Ê —

• N.B.: if pdf =Gaussian for then pdf =Gaussian for any • —3 0 i n nŸ Ÿ

CHI-SQUARE DISTRIBUTION• Consider that you have a r.v. with pdf =gaussian N(0, )• 5• •

• if is another r.v. such that = then pdf— — •#—ÐBÑ œ

" B

# B #É 1 5 5•

•##. 0expŠ ‹ B

• N.B.:if = + +.. and pdf =gaussian pdf =chi-square (see a mathematical handbook)— • • •"

# # ## n • —3

Ê

RAYLEIGH DISTRIBUTION• Consider that you have two r.v's , with pdf =gaussian N(0, ) (i.e these 2 r.v's are• • 5" # • •3

Gaussian r.v.'s)• If is another r.v. such that = + then pdf =Rayleigh— — • •È

"# #

# —

i.e. pdf—ÐBÑ œ B B#5 5• •

# #

#

. expŠ ‹ B !

e.g. pdf = Rayleighenvelope of Gaussian noise

Appendix-3:Four Useful Combinatorial-Analysis Properties

1. Factorial: with 08x " ‚ # ‚ $ ‚ ÞÞÞÞ ‚ Ð8 #Ñ ‚ Ð8 "Ñ ‚ 8 x ´ "´

N.B.: if =large then (Stirling's approximation)8 8x œ # 8 8 /È 1 . .8 8

2. FUNDAMENTAL CONCEPT:If an even can happen in any one of ways and if, when this has occurred, another event 8"

can happen in any one of ways, then the No. of ways in which both events can happen in8#

the specified order is 8" #‚ n .

3. PERMUTATIONS:A permutation of different objects taken objects at a time is an of out of8 3 3 arrangment8 objects given of the arrangment.with attention to the order

No. of permutations of objects taken at a time =8 3 8xÐ83Ñx

4. COMBINATIONS:A combination of different objects taken objects at a time is an of out of 8 3 3 8 selectionobjects given of the arrangment.with no attention to the order

No. of different permutations of objects taken at a time =8 3 8x3x Ð83Ñx

Appendix-4:General Block Structure of an Analogue Communication System

FOURIER TRANSFORMS - TABLES

DESCRIPTION FUNCTION TRANSFORM

1 Definition

2 Scaling

3 Time shift

4 F

g t G f = g t .e dt

g |T| . G fT

g t T G f . e

Ð Ñ Ð Ñ Ð Ñ

Ð Ñ Ð Ñ

Ð Ñ Ð Ñ

'_

_ #

#

j ft

tT

j fT

1

1

requency shift

5 Complex conjugate

6 Temporal derivative

7 Spectral deriva

g t . e G f F

g t G f

. g t j f . G f

Ð Ñ Ð Ñ

Ð Ñ Ð Ñ

Ð Ñ Ð Ñ Ð Ñ

j Ft

* *

ddt

#

8

1

#8

8 1

tive

8 Reciprocity

9 Linearity

1

Ð Ñ Ð Ñ Ð Ñ

Ð Ñ Ð Ñ

Ð Ñ Ð Ñ Ð Ñ Ð Ñ

j2 t .g t . G f

G t g f

A . g t B . h t A . G f B . H f

1 8 ddf

8

8

0 Multiplication

11 Convolution

12 Delta function

13 Constant

g t . h t G f * H f

g t * h t G f . H f

t

f



Ð Ñ "

" Ð Ñ

$

$

DESCRIPTION FUNCTION TRANSFORM

rect sinc

sinc rect

14 Rectangular function { }if

15 Sinc function

t f = t

t

´ Ð Ñ" ±

! 9>2/<A3=/

Ð Ñ

œ ± "# sin f

f1

1

Ð Ñ

Ð Ñ Ð Ñ

Ð Ñ

Ð

f

u t = f

t =

16 Unit step function

17 Signum function

18 Decaying exponentialt

œœ "ß > !!ß > !

"ß > ! "ß > !

"#$

jf#1

sgn jf1

wo-sided

19 Decaying exponentialone-sided

20 Gaussian function

21 Lambda func

Ñ

Ð ÑÐ Ñ

e

e .u t

e e

0

|t|

|t|

t

#" #

" #" #

+ f

j f+ f

Ð Ñ

Ð Ñ

1

11

#

#

1 1# #

tion { } 1 if 11 if

22 Repeated function

23 Sampled

A t ft 0 tt t 0

g t = g t * t | |. G f

´ Ð ÑŸ


œ Ÿ " Ÿ Ÿ

sinc

rep rep comb

#

"T T Te f e f e f$ "

T

function comb rep repT T Te f e f e fg t = g t . t | |. G f Ð Ñ Ð Ñ Ð Ñ Ð Ñ$ "

"T

Introduction - Basic Concepts 1 CS 2001...Imperial College - Dep. of Electrical & Electronic...

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