act 1 2004 - ln - trans - introduction complete 1 2004... · pr pr pr pr pr pr pr pr pr hh h hh h...

IMPERIAL COLLEGE LONDON,DEPARTMENT of ELECTRICAL and ELECTRONIC ENGINEERING.

COMPACT LECTURE NOTES on ADVANCED COMMUNICATION THEORY.Professor Athanassios Manikas, Autumn 2004

Introduction

Aims:

• To define the main performance parameters for DigitalCommunication Systems (DCS) with emphasis given to theChannel Capacity, Energy Utilisation Efficiency and BandwidthUtilisation Efficiency.

• To identify the theoretical limits on the performance of DCS

• To highlight system trade-offs.

Advanced Communication Theory Compact Lecture Notes

Introduction 1 A. Manikas

Glossary

FT Fourier TransformTx TransmitterRx ReceiverADC Analogue-to-Digital ConverterDAC Digital-to-Analogue Converter

SNR Signal-to-Noise power ratioBER Bit-Error-Rate

ESD Energy Spectral DensityPSD Power Spectral Density

ASK Amplitude Shift-Keyed (Digital Modul.)PSK Binary Shift-Keyed (Digital Modul.)FSK Frequency Shift-Keyed (Digital Modul.)

AM Amplitude Modulation (Analogue)PM Phase Modulation (Analogue)FM Frequency Modulation (Analogue)

SSS Spread Spectrum SystemCDMA Code-Division Multiple Access



Notation• time (sec)> œ• frequency (Hz) 0 œ• SNR Signal-to-Noise power ratio at the receiver's input38 œ• SNR Signal-to-Noise power ratio at the receiver's outputout œ (including DAC)• E Energy-per-bit, œ• B Bandwidth of a signal or channel (Hz)œ• EUE Energy Utilisation Efficiencyœ• BUE Bandwidth Utilisation Efficiency (Hz/bits/sec)œ (i.e. BUE = bits/sec/Hz)"

• channel capacity (bits/sec)G œ• probability of a bit in error ( bit-error-rate BER): œ/

• Parameters for a signal :1Ð>Ñ

I œ Energy (J)1

T œ1 Power (W) Ð0Ñ œ ESD Energy Spectral Density (J/Hz)1

Ð0Ñ œ PSD Power Spectral Density (W/Hz)1

V Ð Ñ œ Autocorrelation function11 7

• N.B.: The above are normalised parameters (1 Ohm Resistor)

• other parameters

J œ 1Ð>Ñ max frequ. of (Hz)1

J œ carrier frequ. (freq. of a cosine) (Hz) c œ CR Crest Factor rms (Volts) peak (Volts)

• Additive White Gaussian Noise (AWGN) parameters

œ 8 Ð>Ñ3 channel-noise signal œ 8 Ð>Ñ Î R! single-sided PSD of (W Hz)3



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 B 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 B -bits at point B1f 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



. .

L o

T

L o

T

B U E

D i s c r e t e C h a n n e l

U E C H A N N E L

B , C

D i s c r e t e C h a n n e l

U E C H A N N E L

B , C

M o b i l e C h a n n e lM o b i l e C h a n n e l

DigitalModulator

( )M,Tcs

DigitalDemodul.

( )M,Tcs

H( )f

^ ^

pe EUE corr.= ,f{ }EUEBUE

Comm. Network Mobile Channel

Combined use ofLow-Orbit Satellite &Terrestrial Newtroks

Radio LANS,Wireless ATM, etc.



ìA continuous channel into becomes a discrete channel when ais converted Ð Ñdigital modulator digital demodulator is used to feed the channel and a provides the channel output.

• A digital modulator is described by different channel symbols.QThese channel symbols are ENERGY SIGNALS of duration .X-=

Digital Modulator:

Mapping binary digits to channel symbols

up conversion from baseband to bandpass

Digital Demodulator:

Mapping channel symbols to binary digits

down conversion from bandpass to baseband

Detector with a decision device



• If Binary Digital Modulator Binary Communication SystemQ œ # Ê ÊIf M-ary Digital Modulator M-ary Communication SystemQ # Ê Ê

The probabilistic relationship between input symbols and output symbolsH D is described by the so-called channel transition probability matrix, ,…defined as follows:

, ,

, ,

… œ

Ð l Ñß Ð l Ñß ÞÞÞß Ð l ÑÐ l Ñ Ð l Ñ ÞÞÞß Ð l Ñ

ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞÐ l Ñ Ð l Ñ ÞÞÞ Ð l Ñ

Ô ×Ö ÙÖ ÙÕ Ø

Pr Pr PrPr Pr Pr

Pr Pr Pr

H H HH H H

H H H

" " "

# # #

O O O

L L LL L L

L L L

" # Q

" # Q

" # Q

.



ì PrÐ l ÑH H −5 5L7 denotes the probability that symbol will appear at theDchannel output, given that was applied to the input.L −7 H

ì The input ensemble , the output ensemble and the matrix Š ‹Hß : Š ‹Dß ; …

fully describe the functional properties of the channel with the followingexpression describing the relationship between and ; :

; œ †… :

ì Note that in a noiseless channel and i.e th matrix is anH œ œD ; : Ð Þ / …

identity matrix).



The joint probabilistic relationship between

input channel symbols and output channel symbols H ,D

is described by the so-called joint-probability matrix,

‰ œ

Ð Ñß Ð Ñß ÞÞÞß Ð ÑÐ Ñ Ð Ñ ÞÞÞß Ð Ñ

ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞÐ Ñ Ð Ñ ÞÞÞß Ð Ñ

Ô ×Ö ÙÖ ÙÕ Ø

Pr Pr PrPr Pr Pr

Pr Pr Pr

L L LL L L

L L L

" " "

# # #

Q Q Q

ß ß ßß ß ß

ß ß ß

H H HH H H

H H H

" # O

" # O

" # O

, ,

, ,

X

‰ is related to the forward transition probabilities of a channel with thefollowing expression compact form of Bayes' Theorem):Ð

‰ œ …Þ Ð Ñdiag : where



1. Measure of Information at the Output of a Channel

In general three measures of information are of main interest:

the Entropy of a Source, and ˆ (bits per source symbol)

. ˆ the Mutual Entropy of a Channel

(bits per channel symbol)

the Discrimination of a Sinkˆ



Mutual Information of a Channel

The mutual information measures the amount of information that the output

of the channel gives about the input to the channel i.e. received message Ð Ñ

Ðtransmitted message .Ñ

That is, when symbols or signals are transmitted over a noisy communication

channel, information is received. isThe amount of information received

given by the information,mutual

H7?> !.



ì Ð ÑFor a discrete memoryless channel in a more compact form ,

H H , 7?> 7?>œ Ð Ñ ´ : … 1 1QX

QŒ ‰ log#Š ‹…Þ:Þ:X

‰ bitssymbol

where , = Hadamard operators ) Ðmultiplication, division =

elements1Q Xðóóóóóñóóóóóòc d"ß "ß ÞÞÞß "

Q



2. Capacity of a Channel

ìThere is a theoretical upper limit to the performance of a specified digitalcommunication system with the upper limit depending on the actual systemspecified. However, in addition to the specific upper limit associated witheach system, there is an overall upper limit to the performance which nodigital communication system, and in fact no communication system at all,can exceed. This bound (limit) is important since it provides the performancelevel against which all other systems can be compared. The closer a systemcomes, performance wise, to the upper limit the better.

ìThe theoretical upper limit was given by Shannon (1948) as an upper boundto the maximum rate at which information can be transmitted over acommunication channel.

This rate is called and is denoted by the symbol .channel capacity G



Shannon's capacity theorem states:

max G œ:

e fH7?>bits

symbol

or, G œ ‚<:-= maxe fH 7?>

bitssec

where denotes the channel-symbol rate (in channel-symbols per sec)<-= with < œ-=

"X-=

i.e. if is maximised with respect to the input probabilitiesH ,7?>Ð Ñ: … : œ : ßáß:Š ‹" M , then it becomes equal to , the channel capacity G (inbits/symbol)



In the case of an additive white Gaussian noisy channel, Shannon's capacitytheorem states:

log 1+SNR bits/symbolsG œ "# # 38a b

or

log SNR bits/secG œ F " # 38a bwhere Bandwidth of the channelF œ

SNR 38TTœ =

8

power of the desired signal at point TT œ=

power of the at point TT œ 893=/ œ R ÞF8 !



3. Bandwidth and Channel Symbol Rate



• we have seen that a digital modulator is described by different channel symbols which are ENERGY SIGNALSQ of duration .X-=

pe= f{EUE, corr.}pe= f{EUE, corr.}

EUEBUE

DigitalModulator

(M,Tcs)

DigitalDemodul.

(M,Tcs)

k

H(f)

Discrete Channel

AN

AL

OQ

UE

CH

AN

NE

L

B,C

+ noise

DigitalModulator

(M,Tcs)

DigitalDemodul.

(M,Tcs)

k

H(f)

Discrete Channel

AN

AL

OQ

UE

CH

AN

NE

L

B,C

+ noise



4.ENERGY UTILIZATION EFFICIENCY (EUE)ìThe parameter is a measure of how efficiently the system utilises theEUE

available energy in order to transmit information in the presence ofadditive white Gaussian noise of double-sided power spectral density N /! #(i.e. ) and it is defined as follows:PSDn3

(f)=N /! #

EUE œ IR

,

!

Note that . It willEUE is directly related to the received signal powerbe appreciated of course that this is, in turn, directly related to thetransmitted power by the attenuation factor introduced by the channel.

ìClearly, a question of major importance is how large EUE needs to be inorder to achieve communication at some specific bit error probability:/. Obviously the smaller EUE to achieve a specified error probability thebetter.



5. BANDWIDTH UTILIZATION EFFICIENCY (BUE)ì FThe measures how efficiently the system utilises the bandwidth, ,BUE

available to send information and it is defined as follows:

BUE œ F<,

where denotes the bit rate.<,

ìSpecifically, the BUE indicates how much bandwidth is being used pertransmitted information bit and hence, for a given level of performance,the smaller BUE the better since this means that less bandwidth is beingused to achieve a given rate of data transmission.

ìN.B.: is known as rB, signalling speed



6.VISUAL COMPARISON• By using and the can be expressed as followsEUE BUE SNRin

SNR =.....=inP

N Bœ s!

EUEBUE

• By determining the and , that system canEUE BUE of any particular systembe represented as .a point in the plane (EUE,BUE)

it is desirable for this point to be as close to the origin as possible

log bits/sec

/ = log bits/sec/Hz

ÚÝÝÛÝÝÜŠ ‹Š ‹

G œ F "

G F "

#

#

EUEBUE

EUEBUE



• N.B.:ˆa line from origin represents those points (systems) in the plane for

which the SNRin=constantˆBy comparing points representing one system with those representing

another Ê VISUAL COMPARISON !



7.THEORETICAL LIMITS on the PERFORMANCE of Digital Communication Systems

We have seen that the capacity of a white Gaussian channel of bandwidth Bis bits/secG œ Ð" Ñ œ Ð" ÑB B SNRlog log# #

TT

=

8in

Do not forget that the above Equation refers to bandlimited white-noisechannel with a constraint on the average transmitted power.

• Question: if then ? and particularly if B= then C =?B C= Å _ _

Answer: From the capacity-equation it can be seen that B CÅ Ê Å C = B but, when tends to then _ _ "Þ%% P

Ns!

C

B

C_



• LIMIT-1: limit on bit rateˆwhen binary information is transmitted in the channel, should ber,

limited as follows: r C, Ÿ

ideal case: r =C,

• LIMIT-2: limit on EUEˆthe best Energy Efficiency is EUE=0.693. This is the ultimate limit

below which no physical channel can transmit without error

i.e EUE 0.693



• LIMIT-3: threshold channel capacity curve

This is the curve for a bit rate equal to itsEUE=f BUEš › r,

maximum value,

i.e. r = C , Ê EUE= # BUE-

-

"

"1

BUE

EUE

BUE

Shannon ’s T hreshold Ca a i y vp c t Cur e0.693

No physical realizable CS could occupy a point in the plane(EUE,BUE) lying .below this theoretical channel capacity curve



Important Note: Information and data bitsAdvanced Communication Theory Compact Lecture Notes




8. Overview Comparison of ACSs and DCSs.ì EUE= BUE for various known Communication Systems required tof( )

produce SNR =40dBout

EUE

BUE

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