E303: Communication Systems

E303: Communication Systems

Professor A. ManikasChair of Communications and Array Processing

Imperial College London

An Overview of Fundamentals of Spread Spectrum:PN-codes and PN-signals

Prof. A. Manikas (Imperial College) EE303: PN-codes & PN-signals v.16c3 1 / 46

Table of Contents1 Introduction 3

Pre—4G Evolution 4Definition of a SSS 7Classification of SSS 11Modelling of b(t) in SSS 12Applications of Spread Spectrum Techniques 13Definition of a Jammer 14Definition of a MAI 15Processing Gain (PG) 16Equivalent EUE 17

2 Principles of PN-sequences 21Comments on PN-sequences Main Properties 22An Important "Trade-off" 26

3 m-sequences 28Shift Registers and Primitive Polynomials 29Implementation of an ‘m-sequence’ 31Auto-Correlation Properties 33Some Important Properties of m-sequences 34Cross-Correlation Properties & Preferred m-sequences 36A Note on m-sequences for CDMA 38

4 Gold Sequences 39Introductory Comments 39Auto-Correlation Properties 41Cross-Correlation Properties 43Balanced Gold Sequences 44

5 Appendices 45A: Properties of a Purely Random Sequence 45B: Auto and Cross Correlation functions of twoPN-sequences 45C: The concept of a ’Primitive Polynomial’in GF(2) 45D: Finite Field - Basic Theory 45E: Table of Irreducible Polynomials over GF(2) 45


Introduction

IntroductionGeneral Block Diagram of a Digital Comm. System (DCS)


Introduction Pre—4G Evolution

Pre-4G Evolution

HSCDS: High Speed Circuit Switched DataGPRS: General Packet Radio Systems (2+)EDGE: Enhanced Data Rate GSM Evolution (2+)UMTS:Universal Mobile Telecommunication Systems (3G)Prof. A. Manikas (Imperial College) EE303: PN-codes & PN-signals v.16c3 4 / 46


Note: CDMA ∈ Spread Spectrum Comms



Industry Transformation and Convergence [from Ericsson 2006, LZT123 6208 R5B]

WCDMA (Wideband CDMA) is a 3G mobile comm system. It is awireless system where the telecommunications, computing and mediaindustry converge and is based on a Layered Architecture design.(Note: CDMA Systems ∈ the class of SSS).


Introduction Definition of a SSS

Definition (Spread Spectrum System (SSS))

When a DCS becomes a Spread Spectrum System (SSS)

Lemma (CS , SSS)

CS , SSS iff

◦ Bss � message bandwidth (i.e. BUE=large)◦ Bss 6= f{message}◦ spread is achieved by means of a code which is 6=f{message}where Bss= transmitted SS signal bandwidth

our AIM: ways of accomplishing LEMMA-1.



N.B.:

PCM, FM, etc spread the signal bandwidth but do not satisfy theconditions to be called SSS

Btransmitted-signal � Bmessage

⇒SSS distributes the transmitted energy over a wide bandwidth

⇒ SNIR at the receiver input is LOW.

Nevertheless, the receiver is capable of operating successfully becausethe transmitted signal has distinct characteristics relative to the noise



(a) SSS: (b) CDMA (K users):



The PN signal b(t) is a function of a PN sequence of ±1’s {α[n]}

I The sequences {α[n]} must agreed upon in advance by Tx and Rx andthey have status of password.

I This implies that :F knowledge of {α[n]}⇒demodulation=possibleF without knowledge of {α[n]}⇒demod.=very diffi cult

I If {α[n]} (i.e. “password”) is purely random, with no mathematicalstructure, then

F without knowledge of {α[n]}⇒demodulation=impossible

I However all practical random sequences have some periodic structure.This means:

α[n] = α[n+Nc ] (1)

where Nc =period of sequencei.e. pseudo-random sequence (PN-sequence)


Introduction Classification of SSS

Classification of SSS


Introduction Modelling of b(t) in SSS

Modelling of b(t) in SSSDS-SSS (Examples: DS-BPSK, DS-QPSK):

b(t) = ∑n

α[n].c(t − nTc ) (2)

where {α[n]} is a sequence of ±1’s;c(t) is an energy signal of duration Tc =rect

{tTc

}

FH-SSS (Examples: FH-FSK)

b(t) = ∑nexp {j(2πk[n]F1t + φ[n])} .c(t − nTc ) (3)

where {k[n]} is a sequence of integers such that {α[n]} 7→ {k[n]}and {α[n]} is a sequence of ±1’s;c(t) is an energy signal of duration Tcand with φ[n] = random: pdfφ[n] =

12π rect

{ ϕ2π

}Prof. A. Manikas (Imperial College) EE303: PN-codes & PN-signals v.16c3 12 / 46

Introduction Applications of Spread Spectrum Techniques

Applications of Spread Spectrum Techniques

1 Interference Rejection: to achieve interference rejection due to:I Jamming (hostile interference). N.B.: protection against cochannelinterference is usually called anti-jamming (AJ)

I Other users (Multiple Access Interefence - MAI): Spectrum shared by“coordinated “ users.

I Multipath: Self-Jamming by delayed signal

2 Energy Density Reduction (or Low Probability of Intercept LPI). LPI’main objectives:

I to meet international allocations regulationsI to reduce (minimize) the detectability of a transmitted signal bysomeone who uses spectral analysis

I privacy in the presence of other listeners

3 Range or Time Delay Estimation

NB: interference rejection = most important application


Introduction Definition of a Jammer

Jamming source, or, simply Jammer is defined as follows:

Jammer , intentional (hostile) interference

F the jammer has full knowledge of SSS design except the jammer doesnot have the key to the PN-sequence generator,

F i.e. the jammer may have full knowledge of the SSSystem but it doesnot know the PN sequence used.


Introduction Definition of a MAI

Multiple Access Interference (MAI) is defined as follows:

MAI , unintentional interference


Introduction Processing Gain (PG)

PG: is a measure of the interference rejection capabilities

definition:

PG , BssB=1/Tc1/Tcs

=TcsTc

(4)

where B=bandwidth of the conventional system

PG is also known as "spreading factor" (SF)

PG = very important in DS-SSS

PG 6= very important in FH-SSS


Introduction Equivalent EUE

Remember:F Jamming source, or, simply Jammer = intentional interferenceF Interfering source = unintentional interference

F With area-B = area-A we can find NjF Pj = 2× areaA = 2× areaB = NjBj ⇒ Nj =

PjBj



ifBJ = qBss ; 0 < q ≤ 1 (5)

then

EUEJ =EbNJ

=Ps .BJPJ .rb

=Ps .q.BssPJ .B

= PG× SJRin × q (6)

EUEequ =Eb

N0 +NJ(7)

= PG× SJRin × q ×(N0Nj+ 1)−1

(8)

where

SJRin ,PsPJ

(9)



SS Transmission in the presence of a Jammer (or MAI)



SS Reception in the presence of a Jammer (or MAI)


Principles of PN-sequences

PN-codes (or PN-sequences, or spreading codes) are sequences of+1s and -1s (or 1s and 0s) having special correlation properties whichare used to distinguish a number of signals occupying the samebandwidth.

Five Properties of Good PN-sequences:

Property-1 easy to generate

Property-2 randomness

Property-3 long periods

Property-4 impulse-like auto-correlation functions

Property-5 low cross-correlation


Principles of PN-sequences Comments on PN-sequences Main Properties

Comments on PN-sequences Main Properties

Comments on Properties 1, 2 & 3

I Property-1 is easily achieved with the generation of PN sequences bymeans of shift registers,while

I Property-2 & Property-3 are achieved by appropriately selecting thefeedback connections of the shift registers.



Comments on Property-4I to combat multipath, consecutive bits of the code sequences should beuncorrelated.i.e. code sequences should have impulse-like autocorrelation functions.Therefore it is desired that the auto-correlation of a PN-sequence ismade as small as possible.

I The success of any spread spectrum system relies on certainrequirements for PN-codes. Two of these requirements are:

1 the autocorrelation peak must be sharp and large (maximal) uponsynchronisation (i.e. for time shift equal to zero)

2 the autocorrelation must be minimal (very close to zero) for any timeshift different than zero.

I A code that meets the requirements (1) and (2) above is them-sequence which is ideal for handling multipath channels.



I The figure below shows a shift register of 5 stages together with amodulo-2 adder. By connecting the stages according to the coeffi cientsof the polynomial D5 +D2 + 1 an m-sequence of length 31 isgenerated (output from Q5).The autocorrelation function of this m-sequence signal is shown in theprevious page

1 2 3 4 5

Shift register

Q1 Q2 Q3 Q4 Q5

+Modulo2 adder

i/p

clock

1 2 3 4 5

Shift register

Q3 Q5

+Modulo2 adder

i/p

clock

o/p

1/Tc

(a) (b)



Comments on Property-5I If there are a number of PN-sequences

{α1 [k ]}, {α2 [k ]}, ...., {αK [k ]} (10)

then if these code sequences are not totally uncorrelated, there isalways an interference component at the output of the receiver which isproportional to the cross-correlation between different code sequences.

I Therefore it is desired that this cross-correlation is made as small aspossible.


Principles of PN-sequences An Important "Trade-off"

An Important "Trade-off"

There is a trade-off between Properties-4 and 5.

In a CDMA communication environment there are a number ofPN-sequences

{α1[k ]}, {α2[k ]}, ...., {αK [k ]}of period Nc which are used to distinguish a number of signalsoccupying the same bandwidth.

Therefore, based on these sequences, we should be able toF combat multipath(which implies that the auto-correlation of a PN-sequence{αi [k ]} should be made as small as possible)

F remove interference from other users/signals,(which implies that the cross-correlation should be made as small aspossible).


Principles of PN-sequences An Important "Trade-off"

CorollaryThe following inequality is always valid:

R2auto + R2cross > a constant which is a function of period Nc (11)

i.e. there is a trade-off between the peak autocorrelation andcross-correlation parameters.

Thus, the autocorrelation and cross-correlation functions cannot beboth made small simultaneously.

The design of the code sequences should be therefore very careful.

N.B.:A code with excellent autocorrelation is the m-sequence.

A code that provides a trade-off between auto and cross correlation isthe gold-sequence.


m-sequences

m-sequences

m-seq.: widely used in SSS because of their very good autocorrelationproperties.

PN code generator: is periodicI i.e. the sequence that is produced repeats itself after some period oftime

Definition (m-sequence )A sequence generated by a linear m-stages Feedback shift register is calleda maximal length, a maximal sequence, or simply m-sequence, if its periodis

Nc = 2m − 1 (12)

(which is the maximum period for the above shift register generator)

The initial contents of the shift register are called initial conditions.


m-sequences Shift Registers and Primitive Polynomials

Shift Registers and Primitive Polynomials

The period Nc depends on the feedback connections (i.e. coeffi cientsci ) and Nc = max , i.e. Nc = 2m − 1, when the characteristicpolynomial

c(D) = cmDm + cm−1Dm−1 + ....+ c1D + c0 with c0 = 1 (13)

is a primitive polynomial of degree m.

rule: if ci={0 =⇒ no connection1 =⇒ there is connection

(14)

Definition of PRIMITIVE polynomial = very important(see Appendix C)


m-sequences Shift Registers and Primitive Polynomials

Examples (Some Primitive Polynomials)

degree-m polynomial

3 D3 +D + 1

4 D4 +D + 1

5 D5 +D2 + 1

6 D6 +D + 1

7 D7 +D + 1

Please see Appendix E for some tables of irreducible & primitivepolynomial over GF(2).


m-sequences Implementation of an ‘m-sequence’

Implementation of an m-sequenceuse a maximal length shift registeri.e. in order to construct a shift register generator for sequences of anypermissible length, it is only necessary to know the coeffi cients of theprimitive polynomial for the corresponding value of m

fc =1Tc= chip-rate = clock-rate (15)

c(D) = cmDm + cm−1Dm−1 + ....+ c1D + c0 (16)

with c0 = 1 (17)


m-sequences Implementation of an ‘m-sequence’

Example (c(D)= D3+ D + 1 = primitive=⇒power= m = 3)coeffi cients=(1, 0, 1, 1)⇒ Nc = 7 = 2m − 1 i.e.period= 7Tc

o/p1st 2nd 3rd

initial condition 1 1 1

clock pulse No.1 0 1 1







Note that the sequence of 0’s and 1’s is transformed to a sequence of ±1s byusing the following function

o/p = 1− 2× i/p (18)


m-sequences Auto-Correlation Properties

Auto-Correlation Properties

An m-sequ. {α[n]} has a two valued auto-correlation function:

Rαα[k ] =Nc

∑n=1

α[n]α[n+ k ] =

{Nc k = 0 mod Nc−1 k 6= 0mod Nc

(19)

This implies that Rbb(τ) is also a "two-valued"

Rbb(τ):

Remember that a sequence {α[n]} of period Nc = 2m − 1, generatedby a linear FB shift register, is called a maximal length sequence.


m-sequences Some Important Properties of m-sequences

Some Properties of m-sequences

There is an appropriate balance of -1s and +1s

I In any period there are{Nc− = 2m−1 No. of -1sNc+ = 2m−1 − 1 No. of +1s

}i.e.

Pr(+1) ' Pr(−1) (20)

shift-property of m-sequences:I if {α[n]} is an m-sequence then

{α[n]}+ {α[n+m]}︸︷︷︸shift by m

= {α[n+ k ]}︸︷︷︸shift by k 6=m

(21)


m-sequences Some Important Properties of m-sequences

In a complete SSS we use more than one different m-sequencesI Thus the number of m-sequs of a given length is an IMPORTANTproperty

F because in a CDMA system several users communicate over a commonchannel so that different -sequences are necessary to distinguish theirsignals

I Number of m-sequs of length Nc :

No. of m-sequs of length Nc ,1m

Φ {Nc} (22)

where

Φ {Nc} , Euler totient function (23)

= No of (+)ve integers < Nc and relative prime to Nc

I Note: if Nc = p.q where p, q are prime numbers then

Φ {Nc} = (p − 1).(q − 1) (24)


m-sequences Cross-Correlation Properties & Preferred m-sequences

Cross-Correlation Properties and Preferred m-sequences

sequences of period Nc are used to distinguish two signals occupyingthe same bandwidth.

A measure of interaction between these signals is theircross-correlation:

Rαi αj [k ] =Nc

∑n=1

αi [n]αj [n+ k ]

However,I there exist certain pairs of sequences that have large peaks andnoise-like behaviour in their cross-correlation

I while others exhibit a rather smooth three valued cross-correlation.

The latter are called preferred sequences.


m-sequences Cross-Correlation Properties & Preferred m-sequences

It can be shown that the cross-correlation of preferred sequencestakes on values from the set

{−1,−Rcross ,Rcross − 2} (25)

where Rcross =

{2m+12 + 1 m = odd

2m+22 + 1 m = even

(26)

Rbibj (τ) =preferred:

delay τ = kTc

m = 7

Rbibj (τ) = non-preferred:

delay τ = kTc

m = 7


m-sequences A Note on m-sequences for CDMA

A Note on m-sequences for CDMA

Because of the high cross-correlation between m-sequences, theinterference between different users in a CDMA environment will belarge.

I Therefore, m-sequences are not suitable for CDMA applications.

However, in a complete synchronised CDMA system, different offsetsof the same m-sequence can be used by different users.

I In this case the excellent autocorrelation properties (rather than thepoor cross-correlation) are employed.

I Unfortunately this approach cannot operate in an asynchronousenvironment.


Gold Sequences Introductory Comments

Gold Sequences

Although m-sequences possess excellent randomness (and especiallyautocorrelation) properties, they are not generally used for CDMApurposes as it is diffi cult to find a set of m-sequences with lowcross-correlation for all possible pairs of sequences within the set.

However, by slightly relaxing the conditions on the autocorrelationfunction, we can obtain a family of code sequences with lowercross-correlation.

Such an encoding family can be achieved by Gold sequences or Goldcodes which are generated by the modulo-2 sum of two m-sequencesof equal period.


Gold Sequences Introductory Comments

The Gold sequence is actually obtained by the modulo-2 sum of twom-sequences with different phase shifts for the first m-sequencerelative to the second.

Since there are Nc = 2m − 1 different relative phase shifts, and sincewe can also have the two m-sequences alone, the actual number ofdifferent Gold-sequences that can be generated by this procedure is2m + 1.


Gold Sequences Auto-Correlation Properties

Auto-Correlation Properties

Gold sequences, however, are not maximal length sequences.

Therefore, their auto-correlation function is not the two valued onegiven by Equ. (19), i.e.

{Nc ,−1} (27)

The auto-correlation still has the periodic peaks, but between thepeaks the auto-correlation is no longer flat.


Gold Sequences Auto-Correlation Properties

Example (Gold Sequence of Nc = 127 = 27 − 1)Rbb(τ):

Example (m-sequence of Nc = 127 = 27 − 1)

Rbb(τ):


Gold Sequences Cross-Correlation Properties

Cross-Correlation Properties

Gold-sequences have the same cross-correlation characteristics aspreferred m-sequences,i.e. their cross-correlation is three valued.

Gold sequences have higher Rauto and lower Rcross than m-sequences,and the trade-off (see Equ. 11) between these parameters is thusverified.


Gold Sequences Balanced Gold Sequences

Balanced Gold codes.Balanced Gold Sequence: The number of "-1s" in a code period exceedthe number of "1s" by one as is the case for m-sequences.We should note that not all Gold codes (generated by modulo-2 additionof 2 m-sequences) are balanced, i.e. the number of "-1s" in a code perioddoes not always exceed the number of "1s" by one.For example, for m = odd only 2m−1 + 1 code sequences of the total2m + 1 are balanced, while the rest code 2m−1 − 1 sequences have anexcess or a deficiency of -1s.For m = 7, for instance, only 65 balanced Gold codes can be produced,out of a total possible of 129. Of these, 63 are non-maximal and two aremaximal length sequences.Balanced Gold codes have more desirable spectral characteristics thannon-balanced.Balanced Gold codes are generated by appropriately selecting the relativephases of the two original m-sequences.SUMMARY: By selecting any preferred pair of primitive polynomials it iseasy to construct a very large set of PN-sequences (Gold-sequences).Thus, by assigning to each user one sequence from this set, theinterference from other users is minimised.


Appendices E: Table of Irreducible Polynomials over GF(2)

Appendices1 Appendix A:Properties of a purely random sequence

2 Appendix B:Auto and Cross Correlation functions of two PN-sequences

3 Appendix C:The concept of a ’Primitive Polynomial’in GF(2⊂)

4 Appendix D:Finite Field - Basic Theory

5 Appendix E:Table of Irreducible Polynomials over GF(2)

↘↘↘↘↘Prof. A. Manikas (Imperial College) EE303: PN-codes & PN-signals v.16c3 45 / 46

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Appendices E: Table of Irreducible Polynomials over GF(2)

↘↘↘↘↘


Communication Systems Compact Lecture Notes

Spread Spectrum Systems A. Manikas

Let the sequence [ ] be the output of a discrete, memoryless source α 8

{ [ ]} [ ] 0.5[ ] 0.5

INFORMATION SOURCE of 1s„

TÐ 8 œ "Ñ œTÐ 8 œ "Ñ œ

Ä 8αα

α

with [ ] 0.5+(-1) (2)X α 8 œ ! Ð œ " ‚ ‚ !Þ& œ !Ñ [ ] 0.5+(-1) (3)Z +< 8 œ " Ð œ " ‚ ‚ !Þ& œ "Ñ α # #

The auto-correlation of the sequence [ ] over symbols is defined as follows α 8 Q

[ ] [ ] [ ]= (4)[ ]

randomV 5 ´ 8 8 5

8 œ " œ Q 5 œ !

5 Á !

Q

8œ"

Q

8œ" 8œ"

Q Q#

αα

α αα

Therefore the mean and the variance of the autocorrelation function [ ] are as followsV 5Qαα

[ ] [ ] [ ] (5)[ ] if

[ ] [ ] if X X α α

X α

X α X α

V 5 œ 8 8 5 œ

8 œ " œ Q 5 œ !

8 8 5 œ ! 5 Á !

Q

8œ"

Q8œ" 8œ"

Q Q#

8œ"

Qαα

Appendices

Appendix A: Properties of !!"#$% if it is a purely random sequence



[ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ]

[ ] . [ ]

Z +< V 5 œ V 5 V 5 œ

œ 8 8 5 7 7 5 V 5 œ

œ

8 7

Q Q # Q #

8œ" 7œ"

Q QQ #

Q Q

8 " 7 "

# #

αα αα αα

αα

X X

X α α α α X

X α X α= =

V Q Q œ 5 œ

8 8 5 V 5 Q œ Q 5 Á

X

X α X α X

Q ##

Q

8 "

# # Q#

αα

αα

[0] = 0 if 0

[ ] . [ ] [ ] = 0 if 0

(6)

2

=

One may also define the cross-correlation of two sequences [ ] and [ ] α α" #8 8

[ ] [ ] [ ] (7)V 5 œ 8 8 5Q

8œ"

Q

" #α α" #α α

Since [ ] and [ ] are independent the results are essentially the same as for the auto-correlation of [ ] α α α" # "8 8 8

with non-zero lag . This shows that completely random sequences have nice auto- and cross-correlation properties.5

Note that pure random sequences could be used as code sequences, but since the receiver needs a replica of thedesired code sequence in order to despread the signal, PN sequences are used instead in practice.



ì Consider the -sequences of 1s of period :∞ „ R

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]Ö 8 × œ ÞÞÞÞß R " ß R ß " ß # ß ÞÞÞÞß R " ß R ß " ß ÞÞÞÞα α α α α α α α3 3 3 3 3 3 3 3

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]Ö 8 × œ ÞÞÞÞß R " ß R ß " ß # ß ÞÞÞÞß R " ß R ß " ß ÞÞÞÞα α α α α α α α4 4 4 4 4 4 4 4

ì Then, there are three different cross-correlation functions

aperiodic cross-correlation: [ ] (8)

[ ] [ ] 0

[ ] [ ]ˆ G 5 ´

8 8 5 Ÿ 5 Ÿ R "

8 5 8 " R Ÿ 5 Ÿ !

! 5 R

α α3 4

8œ"

R5

3 4

8œ"

R 5

3 4

α α

α α+

periodic cross-correlation: [ ] [ ] [ ] (9)ˆ V 5 ´ 8 8 5α α3 48œ"

R

3 4α α

odd cross-correlation function: [ ] [ ] [ ] (10)ˆ V 5 œ G 5 G 5 Rµ

α α α α α α3 4 3 4 3 4

Appendix B: Auto and Cross Correlation functions of two PN-sequences !! "#$%& and !! '#$%&



ì Note that:

it is easy to see thatˆ

[ ] [ ] [ ] (11)V 5 œ G 5 G 5 Rα α α α α α3 4 3 4 3 4

the periodic (or even) cross-correlation function has the propertyˆ

[ ] [ ] (12)V 5 œ V R 5α α α α3 4 3 4

the name of "odd cross-correlation" function follows from the propertyˆ

[ ] [ ] (13)V 5 œ V R 5µ µ

α α α α3 4 3 4

ì For a single code sequence, the corresponding autocorrelation functions have similar properties.



ì For best CDMA system performance, all [ ], [ ], [ ] should be as small as possible, since they areG 5 V 5 V 5µ

α α α α α α3 4 3 4 3 4

proportional to the interference from other users.

The out-of-phase i.e. for lag not equal to zero autocorrelation functions should also be made as small as possible,Ð Ñsince these affect the multipath suppression capabilities and the acquisition and tracking performance of thereceivers.

We thus define the peak cross-correlation parameters

[ ] , (1

[ ]

[ ]

V œ V 5 ß aÐ3ß 4ß 5à 3 4Ñ

V œ V 5 ß aÐ3ß 4ß 5à 3 4Ñµ µ

G œ G 5 ß aÐ3ß 4ß 5 à 3 4Ñ

cross

cross

cross

max

max

max

α α

α α

α α

3 4

3 4

3 4

4)

ì Similarly we define the peak autocorrelation parameters

[ ] mod ,

[ ] mod ,

[ ] mod

V œ V 5 ß a3à a5 Á !Ð RÑ

V œ V 5 ß a3à a5 Á !Ð RÑµ µ

G œ G 5 ß a3à a5 Á !Ð R

auto

auto

auto

max

max

max

R

R

R

α α

α α

α α

3 3

3 3

3 3Ñ

(15)



ì Finally we define

(16) ,

V œ V ßV

V œ V ßVµ µ µ

G œ G G

peak auto cross

peak auto cross

peak auto cross

max

maxmax

ì With the above definitions we can see that the smaller the peak correlation parameters , and , theV V Gµ

peak peak peakbetter the performance of a system. These parameters, however, cannot be made as small as we wish. For example,for a set of sequences of period , according to the Welch lower bound ,O R

(17)V R G Rpeak peak O" O"RO" #ROO"

Therefore for large values of and the lower bounds on and are approximatelyO R V Gpeak peak

(18)V R G peak peak R

#

Moreover, it can show that

(19)V V R G G # # # #-

R#auto ross auto cross

The above shows that not only is there a lower bound on the maximum correlation parameters, but also a trade-offbetween the peak autocorrelation and cross-correlation parameters. Thus the autocorrelation and cross-correlationfunctions cannot be both made small simultaneously. The design of the code sequences should be therefore verycareful so that all the of above quantities of interest remain as small as possible.



ì Consider a polynomial over the binary field GF 2 : f f = f f .... f fÐH Ð ÐH H H H ÅÁ !

Ñ Ñ Ñ 8 8 " " !8 8 "

--

The largest power of with non-zero coef. is called of over GF 2H Ñ Ñdegree fÐH Ð

ì if , GF 2 then GF 2 GF 2f g

f gf g

ÐH ÐH ÐÅ Å7 8

ÐH ÐH − ÐÐH ÐH − Ð

Ñ Ñ − ÑÑ Ñ ÑÑ Þ Ñ Ñ

ì divisible polynomial:A polynomial GF 2 is said to divide GF 2 if .g f h : f =h .gÐH Ð ÐH Ð b ÐH ÐH ÐH ÐHÑ − Ñ Ñ − Ñ Ñ Ñ Ñ ÑThen the polynomial is called divisiblef ÐHÑ

ì irreducible polynomial:A polynomial GF 2 of degree is called irreducible if is not divisible by any polynomial overf m f ÐH Ð ÐHÑ − Ñ ÑGF 2 of degree less than but greater than zero.Ð Ñ mÐ "or equivalently if it cannot factored into polynomials of smaller degree whose coefs are also 0 and i.e thepolynomials belong to GF 2Ð ÑÑ

Appendix C: The concept of a 'Primitive Polynomial' in GF(2) (see Appendix 4E for 'finite field' basic theory).



ì two important properties of : if has odd number of termsirreducible polynomials f =irreduciblef 0 0 f D

ÐHÐÐ

Ñ ÊÑ ÁÑ

ì primitive polynomial:

if irreducible of degree polynomial, and

i.e. does not divide for any

f m

f f k

ÐH Ð

ÐH ÐH " ÐH " # "Á !

Ñ œ Ñ

Ñ Ñ Ñ Hk k m

then f primitive polynomialÐH ´Ñ

e.g. ; H H " H H "$ # 4

ì only a small number of polynomials are , at least one polynomial.primitive m primitivebut a b

ì examples: 0 ÑÐH H H "= = primitive$ #

0 ÑÐH H H "= = irreducible but not primitive% #



ìConsider a set = { } having elements.W = ß = ß ÞÞÞß = Q" # Q

A finite field is constructed by defining two binary operations on the set called addition & multiplication such thatcertain conditions are satisfied. Addition and multiplication of two elements and are denoted and= = = =3 4 3 4

= =3 4 respectively.

ìThe conditions that must be satisfied for and the two operations to be a finite field are:W1. The addition or multiplication of any two elements of must yield an element of . W W That is, the set is closed under both addition and multiplication.

2. Both addition and multiplication must be commutative

3. The set must contain an element which will always be denoted by 0.W additive identity = ! œ =3 3

4. The set must contain an element for every element W = =additive inverse 3 3

= Ð = Ñ œ !3 3

5. The set must contain a element which will always beW multiplicative identity denoted by 1. 1= Þ œ =3 3

' W = =. The set must contain a element for every element multiplicative inverse "3 3

(excluding the additive identity 0) = Þ= œ "3

"3

7. Multiplication must be over addition.distributive8. Both addition and multiplication must be .Associative

Appendix D: FINITE FIELD -BASIC THEORY



ìEXAMPLEIt is easy to verify that with addition and multiplication defined as followsW œ Ö!ß "ß #× modulo-3 modulo-3

0 00 1 20 2

! " # ‚ ! " #! ! " # ! !" " # ! "# # ! " # "

is a field of 3 elementse.g. additive inverse 0 0 multiplicative inverse 1 œ œ ""

" œ # # œ #"

# œ "ìEXAMPLE

It is easy also to verify that , with addition and multiplication defined as follows:W œ Ö!ß "× modulo-2 modulo-2

0 0 10

! " ‚ ! "! ! " ! !" " "

is a field of 2 elementse.g. additive inverse 0 0 multiplicative inverse 1 1 œ œ"

1 " œ

ìNote that field above is the binary number field. Furthermore that addition can be performedW œ Ö!ß "×electronically using EXCLUSIVE-OR gate and multiplication can be performed using an AND-gate.



ìAn Important Result (presented without proof):The set of integers {0, 1, 2, . . . , },W œ Q "

where is prime and ddition and multiplication are carried out modulo-Q ß

+ Q

is a field. These fields are called .prime fields

ìSubtraction and Division:The operations of subtraction and division are also easily defined for any field using the addition and multiplicationtables, just as is done with the real-number field.Subtraction is defined as the addition of the additive inverse and division is defined as multiplication by themultiplicative inverse.For example for the field 0,1,2} subtraction is defined by 1 + ( 2) = 1 + 1 = 2.W œ Ö Similarly, 1 2 1 (2 ) 1 2 2.ƒ œ Þ œ Þ œ"

ìNote that nonprime fields do not necessarily employ modulo- arithmetic.Q

ìFields can be constructed having any prime number of elements or . A field having elements is called an: : :7 7

extension field of the field having elements.:

ìFinite fields are often referred to as Galois fields, using the notation GF( ) for the field having elements.Q Q

ìThe remainder of this discussion will be concerned exclusively with the binary number field GF(2) and itsextensions GF(2 ). The reason for this is that the electronics used to implement the code generators is binary, and7

some of the shift register generators will be shown to generate the elements of GF(2 )7



(from "Error-Correcting Codes" by Peterson & Weldom MIT Press, 1972)Appendix E: Table of Irreducible Polynomials over GF(2)

E303: Communication Systems

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