Principles of Spread Spectrum Systems

Transcription

1 IMPEIAL COLLEGE of SCIENCE, TECHNOLOGY and MEDICINE, DEPATMENT of ELECTICAL and ELECTONIC ENGINEEING. COMPACT LECTUE NOTES on ADVANCED COMMUNICATION THEOY. Prof. Athanassios Manikas, Autumn 2004 Principles of Spread Spectrum Systems Outline: ì Basic concepts, models and classification and modelling of jammers. ì Principles of PN-Sequences: The significance of auto-correlation for code tracking and for suppressing multipath and ISI. The significance of cross-correlation for code aquisition and for suppressing other users' interference. Trade-off between auto and cross correlation.shift registers. Basic properties of m-sequences. Auto-correlation and cross-correlation functions of m-sequence and their statistics. Gold-sequences. Galois field GFÐ2 Ñ - basic theory. ì PN-Signals: modelling, cross-correlation, auto-correlation functions and power spectral density. Partial correlation properties of PN-signals. Discussion and comments. ì Modelling of BPSK and PSK Direct Sequence SSS (DS-SSS) in a jamming environment. Estimating the SNI at output of the matched-filter receiver. Estimating the message bit error probability. DS-SSS on the (SN/p e,eue,bue)-parameter plane. Comments. ì Frequency Hopping SSS (FH-SSS). PAT-1: Basic Concepts, PN-sequences and PN-signals MSc, MEng

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3 1. Introduction General Block Diagram of a Digital Comm. System (DCS) When a DCS becomes a Spread Spectrum Systems ÐSSS Ñ: H( f) ^ ^ ^ ^ ^ ^ Spread Spectrum (Part-1) 2 Spread Spectrum (Part-1) 3 Ú ì Bss message Bandwidth Ý Ði.e. BUE=largeÑ LEMMA 1: CS SSS iff Û ì B ss=/ f{ message} Ý Ü where B ss =transmitted SS signal bandwidth N.B.: B transmitted-signal B message Ê SSS distributes the transmitted energy over a wide bandwidth Ê SNI at the receiver input is LOW. our AIM: ways of accomplishing LEMMA-1. Nevertheless, the receiver is capable of operating succesfully because the transmitted signal has distinct characteristics relative to the noise Spread Spectrum (Part-1) 4 Spread Spectrum (Part-1) 5

4 BLOCK DIAGAM of a SSS Classification of SSS b(t) interference is averaged over a large time interval transmitted signal is made to avoid the interference a large fraction of time Ê interf œ Æ j(t) jammer of power P j ^ b(t) Dig. Demod ^ ^ e.g. DS/BPSK e.g. FH/FSK DS/PSK Spread Spectrum (Part-1) 6 Spread Spectrum (Part-1) MODELLING of the SIGNAL,Ð>Ñ DS model : bt ÐÑ=!![8Ó.-Ðt nt n FH model : bt ÐÑ=! expš jð# 1kÒ8ÓF1t 9Ò8ÓÑ.cÐt nt Ñ n where Ö! Ò8Ó is a sequence of 1's { kò8ó} is a sequence of integers such that {!Ò8Ó È { kò8ó -ÐtÑ is an energy signal of duration X- and with 9Ò8Ó=random: pdf = 1 9Ò8Ó. rectš Spread Spectrum (Part-1) 8 c Ñ 9Ò8Ó c The PN signal,ð>ñ 3 is a function of a PN sequence of 1's {![ 8]} ìthe sequences Ö! Ò8Ó must agreed upon in advance by Tx and x and they have status of password. ìthis implies that : ˆknowledge of {!Ò8Ó} Ê demodulation=possible ˆwithout knowledge of Ö! Ò8Ó Ê demod.= very difficult ìif Ö! Ò8Ó Ði.e. "password" Ñ is purely random, with no mathematical structure, then ˆ without knowledge of Ö!Ò8Ó Ê demodulation=impossible (See Appendix 4.A for properties of pure random sequences) Spread Spectrum (Part-1) 9

5 ìhowever 1.2.Applications of Spread Spectrum Techniques all practical random sequences have some periodic structure. This means: A. Interference ejection: to achieve interference rejection due to: a Ñ Jamming Ðhostile interferenceñ N.B.: protection against cochannel interference is usually called anti-jamming ÐAJÑ! Ò8Ó œ! Ò8 Ó where œperiod of sequence i.e. pseudo-random sequence (PN-sequence) bñ other users ÐMultiple Access Ñ: Spectrum shared by coordinated " users. cñmultipath: self-jamming by delayed signal Spread Spectrum (Part-1) 10 Spread Spectrum (Part-1) 11 B. Energy Density eduction Ðor Low Probability of Intercept LPI Ñ: LPI' main objectives: añto meet international allocations regulations Jamming source, or, simply Jammer Jammer = intentional (hostile) interference bñto reduce Ðminimize Ñ the detectability of a transmitted signal by someone who uses SPECTAL ANALYSIS cñprivacy in the presence of other listeners C. ange or Time Delay Estimation N.B.: application-a =most important ì the jammer has full knowledge of SSS design except the jammer does not have the key to the PN-sequence generator, i.e. the jammer may have full knowledge of the SSSystem but it does know the PN sequence used. Spread Spectrum (Part-1) 12 Spread Spectrum (Part-1) 13

6 Multiple Access Interfering (MAI) source (MAI = unintentional interference) Binary Inform. Source 1 TANSMITTES B 1 {a 1[ n]} DS-PSK { α1[ k]} s1( t) T 1 τ 1 CHANNEL ìpsk and FSK modulation techniques will be used (please study BPSK, PSK and BFSK digital modulation schemes). ì N.B.: PSK- is appropriate for applications where phase coherence between Txsignal and x-signal can be maintained over a time interval T phase-coh. which satisfies the following condition: Binary Inform. Source 2 {a 2[ n]} DS-PSK { α2[ k]} s2( t) T 2 τ 2 B ss ECEIVE T^ 1 B^ 1 PSK SSS Demodulator { α1[ k]} where B ss T > B 1 phase-coh. is the transmitted signal Bandwidth. " ss ÐÑ Binary Inform. Source K {a K[ n]} DS-PSK { α [ k]} K sk( t) T K τ K noise FSK- is appropriate for applications where phase coherence between Txsignal and x-signal cannot be maintained over a time interval T phase-coh. which satisfies the condition ÐEquation-1 Ñ due to time varying effects on the communication link. Spread Spectrum (Part-1) 14 Spread Spectrum (Part-1) POCESSING GAIN ÐPGÑ PG: is a measure of the interference rejection capabilities definition: PG F where "ÎX- X F = "ÎX œ X == -= -= - F= bandwidth of the conventional system Spread Spectrum (Part-1) 16 Spread Spectrum (Part-1) 17

7 1.4. EUE and EUE N /;? if FN œ ;F== Ð! ; Ÿ "Ñthen Jamming source, or, simply Jammer = intentional interference Interfering source = unintentional interference I T. F T.q. F N T. < T. F 38, = J = ss EUE œ œ = œ PG SI q N N, N I i.e. EUE/;? œ,! = PG SI38! N ðóóóóóñóóóóóò q Š " 4 œ EUE 4 " Ê dbeeueequf œ dbepgf dbesi38f dbef ; dbš "! 4 Spread Spectrum (Part-1) 18 Spread Spectrum (Part-1) Comments EUE/;? Ðor EUE N Ñ: very important since bit error probabilities are defined as function of EUE Ðor of EUE Ñ /;? N 1.6. CLASSIFICATION OF JAMMES in SSS the smaller the SI38 Ê the better for the signal For a specified performance œ the larger the SI38 Ê the better for the jammer Jammer limits the performance of the communication system i.e. effects of channel noise can be ignored I, I, i.e. Jammer Power Channel Noise Ê EUE/;? œ œ EUEN! N N Å Å N.B. exception to this: 3Ñ non-uniform fading channels 33Ñ multiple access channels b_ number of possible jamming waveforms There is no single jamming waveform that is worst for all SSSs There is no single SSS that is best against all jamming waveforms. Spread Spectrum (Part-1) 20 IMPOTANT NOTE An effective anti-jam CS is one that gives performance close to or better than the BASELINE PEFOMANCE regardless of the type of jammer waveform used. Coding and interleaving can recover most of the performance loss as a result of other jammers and reduce the jammers' effectiveness to that of the BASELINE broadband noise jammer case. Spread Spectrum (Part-1) 21

8 2Þ Principles of PN-sequences ì Five Properties of Good PN-sequences: ì There are two general classes of PN-sequences: aperiodic and periodic. Property-1. easy to generate Property-2. randomness Property-3. long periods Property-4. impulse-like auto-correlation functions Property-5. low cross-correlation Note: PN-sequences are used to distinguish a number of signals occupying the same bandwdith. Spread Spectrum (Part-1) 22 ì Comments on Properties 1, 2 & 3 Property-1 is easily achieved with the generation of PN sequences by means of shift registers, while Property-2 & Property-3 are achieved by appropriately selecting the feedback connections of the shift registers. Spread Spectrum (Part-1) 23 ì Comments on Property-4 to combat multipath, consecutive bits of the code sequences should be uncorrelatedþ i.e. code sequences should have impulse-like autocorrelation functions. Therefore it is desired that the auto-correlation of a PN-sequence is made as small as possible. ì Comments on Property-5 If there are a number of PN-sequences Ö! Ò5Ó ß Ö! Ò5Ó,..., Ö! Ò5Ó 1 # K then if these code sequences are not totally uncorrelated, there is always an interference component at the output of the receiver which is proportional to the cross-correlation between different code sequencesþ Therefore it is desired that this cross-correlation is made as small as possible. Spread Spectrum (Part-1) 24 ì Properties-4 and 5: Trade-off In a CDMA communication environment there are a number of PN-sequences Ö! 1Ò5Ó ß Ö! # Ò5Ó,..., Ö! KÒ5Ó of period which are used to distinguish a number of signals occupying the same bandwdith. Therefore, based on these sequences, we should be able to ˆ combat multipath (which implies that the auto-correlation of a PN-sequence Ö! 3 Ò5Ó should be made as small as possible) ˆ remove interference from other users/signals, ( which implies that the cross-correlation should be made as small as possible). Spread Spectrum (Part-1) 25

9 However (see Appendix 4B) # # V auto V -ross a constant which is a function of period 2.1. m-sequences m-seq. : widely used in SSS because of their very good autocorrelation properties. i.e. there is a trade-off between the peak autocorrelation and crosscorrelation parameters. Thus, the autocorrelation and cross-correlation functions cannot be both made small simultaneously. The design of the code sequences should be therefore very careful. PN code generator: is periodic Å i.e the sequence that is produced repeats itself after some period of time Definition of m-sequ. A sequence generated by a linear 7-stages FB shift register is called a maximal length or a maximal sequence if its period is m œ# " Ðwhich is the maximum period for the above shift register generator) The initial contents of the shift register are called initial conditions. Spread Spectrum (Part-1) 26 Spread Spectrum (Part-1) 27 The period depends on the feedback connections (i.e. coefficients -Ñ 3 and 7 œ7+b, i.e. œ# ", when the characteristic polynomial 7 7 " 7 7 "... "!! -ÐHÑ œ - H - H - H - with - = " is a primitive polynomial of degree m. rule: if 0 Ê no connection - 3 =œ " Ê there is connection definition of PIMITIVE polynomial = very important (see Appendix 4C) ì Some Examples of Primitive Polynomials degree-7 polynomial $ $ D+D+ " % % D+D+ " & # & D+D+ " ' ' D+D+ " ( ( D+D+ " ( Appendix 4.E provides some tables of irreducible & primitive polynomial over GF(2)) Implementation ì use a maximal length shift register i.e. in order to construct a shift register generator for sequences of any permissible length, it is only necessary to know the coefficients of the primitive polynomial for the corresponding value of m " f œ œ chip rate œ clock rate - X - Spread Spectrum (Part-1) 28 Spread Spectrum (Part-1) 29

10 1st Implementation: 2nd Implementation: 7 7 " 0 Ê no connection -ÐD Ñœ-7H -7 " H... -H - "! with -! œ" rule: if -3= œ " Ê there is connection either. 8 œ -". 8 " Š -#. 8 # Š ÞÞÞÞ Š where - 3,. 8 œ œ or 0 1 The output sequence {.Ð8Ñ} is a periodic sequence of 0s and 1s which satisfies the following recursive equation:. 8-7 œ. 8 Š -". 8 " Š -#. 8 # Š ÞÞÞÞ Š - 7 ". 8 7 " Note that the sequence of 0's and 1's is transformed to a sequence of by using the following function (o/p œ 1 2 i/p) i.e.![ 8] œ " #. 8 1s Spread Spectrum (Part-1) 30 Spread Spectrum (Part-1) 31 $ ì Example: -ÐHÑ= H H " = primitive Êcoefficient = Ð"ß!ß "ß " Ñ power=m=3 o/p " st # nd $ rd initial condition " " " clock pulse No. " clock pulse No. # clock pulse No. $ clock pulse No. % clock pulse No. & clock pulse No. ' clock pulse No. (! " "!! " "!!! "! "! " " "! " " " Properties of 'm-sequences' In any period there are three main properties/features "Þ e![ 8 ] f ha = a two valued auto-correlation function: V!![ 5] œ "![ 8]![ 8 5] œ œ 5 œ!79. (1) 8œ" " 5Á! œ 7 œ # " i.e. period œ 7. X Spread Spectrum (Part-1) 32 Spread Spectrum (Part-1) 33

11 V!! Ð7 Ñ! _ V!! [ 5].$ Ð7 kt c Ñ k=-_ Ð "Ñ rep e$ Ð7 Ñf rep e$ Ð7Ñf X X - - #Þ There is an approximate balance of " s and 1s i.e. in any one period there are œ œ # 7 " - No. of "= 7 " + œ # 1 No. of "= (2) i.e. Pr Ð + " Ñ Pr Ð " Ñ (3) $Þ half of the runs of consecutive " s or 1s are of length 1 1/4 are of length 2 1/8 are of length 3 />-Þ Intermediate diagram of V Ð7Ñ Final diagram of V Ð7Ñ!!!! $ e.g. H H " " st # nd o/p initial condition " " " clock pulse No. "! " " clock pulse No. #!! " clock pulse No. $ "!! clock pulse No. %! "! clock pulse No. & "! " clock pulse No. ' " "! clock pulse No. ( " " " Spread Spectrum (Part-1) 34 Spread Spectrum (Part-1) 35 7 we have seen that a sequence e![ 8] f of period œ2 ", generated by a linear FB shift register, is called a maximal length sequence. shift-property of 7-sequences: if {![ 8]} is an 7-sequence then {!} +shift Ô{!} Õ=shift Ô{!} Õ (4) In a complete SSS we use more than one different 7-sequences Thus the number of 7-sequs of a given length is an IMPOTANT property Ðbecause in a CDMA system several users communicate over a common channel so that different 7-sequences are necessary to distinguish their signalsñ ÔNumber of 7-sequs of length N Õ m. F Ö N Å where FÖ N œ the No. of +ve integers and relatively prime to " Euler totient function Preferred m-sequences 7-sequences of period are used to distinguish two signals occupying the same bandwidth. A measure of interaction between these signals is their cross-correlation: V!! 3 4 [ 5] œ "! 3[ 8]! 4 [ 8 5] (6) 8œ" However, there exist certain pairs of sequences that have large peaks and noise-like behaviour in their cross-correlation while others exhibit a rather smooth three valued cross-correlation. The latter are called preferred sequences. Spread Spectrum (Part-1) 36 Spread Spectrum (Part-1) 37

12 It can be shown that the cross-correlation of preferred sequences takes on values from the set # # " e "ß Vcrossß Vcross # f where Vcross œ 7 # (7) # # " 7 /@/8 7 " A Note on m-sequences for CDMA ì Because of the high cross-correlation between 7-sequences, the interference between different users in a CDMA environment will be large. Therefore, 7-sequences are not suitable for CDMA applications. ì However, in a complete synchronised CDMA system, different offsets of the same 7- sequence can be used by different users. V!! 3 4 [ 5] preferred In this case the excellent autocorrelation properties (rather than the poor cross-correlation) are employed. Unfortunately this approach cannot operate in an asynchronous environment. V!! 3 4 [ 5] non-preferred Spread Spectrum (Part-1) 38 Spread Spectrum (Part-1) '. Partial Auto-Correlation Properties of an m-sequence Ö! Ò8Ó ì Let Ö![ 8 ] be an 7-sequence of " s of length (period) N. Let. Then the partial auto-correlation is given below: V [ 5] œ! 5œ!79.!!![ 8]![ 8 5] œ œ (8) random number 5Á!79. 8œ" Averaging the partial auto-correlation over all starting points yields while its variance is X V!![ 5] œ œ 5 œ!79. (9) 5 Á!79. Z+< V!![ 5] œ œ! 5 œ!79. " (10) Ð" ÑÐ" Ñ 5 Á! 79. ì Note that for the results Equ. ÐÑ 9 and Equ. Ð10 Ñare the same as the results of purely random sequences in Equ. Ð19Ñand Equ. Ð20 Ñ, respectively. SUMMAY of PATIAL AUTO-COELATION POPETIES andom-sequences m-sequences of period Ð Ñ 5 œ! V!! [ 5] œ œ V!! [ 5] œ! 5œ!79.![ 8]![ 8 5] œ random 5Á! œ 8œ" random 5Á!79. V 5 œ 5 œ! 5 œ!79. X [ ] X V [ 5]!! œ!! œ! 5 Á! œ 5 Á!79. 0 if 5œ0 Z+< V [ 5] = Z+<! 5 œ!79. œ V [ 5]!!!! œ if 5 Á 0 œ " Ð" ÑÐ" Ñ 5 Á! 79. Ðsee Appendix 4.A) Spread Spectrum (Part-1) 40 Spread Spectrum (Part-1) 41

13 Partial Autocrrelation expressions for m-seq:! _!!!! k=-_ V Ð7 Ñ V [ 5].$ Ð7 kt c Ñ œ random Xš V!! Ð7 Ñ! _ Xš V!! [ 5].$ Ð7 kt c Ñ k=-_ œ Ð Ñ rep e$ Ð7 Ñf rep e$ Ð7Ñf X X - - œ ðóóóóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóóóóò að "Ñ repx e$ Ð7 Ñf repx e$ Ð7Ñfb - - œ V!! Ð7Ñ œ V!! Ð7Ñ " varš V!! Ð7Ñ œ Ð" ÑÐ" Ñ a rep X e$ Ð7 Ñf+ rep X e$ Ð7Ñf b - - Spread Spectrum (Part-1) 42 Spread Spectrum (Part-1) 43 ì A Note on the Partial Cross-Correlation of two m-sequences Ö! " Ò8Ó +8. Ö! # Ò8Ó The partial cross-correlation between two m-sequences can be evaluated numerically by averaging over all staring points and phase-shifts. The following table presents the results for two 7-sequences of length œ "#(. Partial Cross-Correlation between two 7-sequs of period œ "#( interval length 5 " pure random X V!! " [ Z+< V!![ 5] " m-sequence X V!! [ 5 ] " " " # $ & 10 Z +< V [ 5 ] &Þ!!" "!Þ!!' #!Þ!#& $!Þ!&" %!Þ!** '!Þ##" )!Þ$*"!! " 2 emember that purely random sequences have zero mean and variance. Hence, the above PN-sequences from a 7-stage shift-register Ði.e. œ 127; "#(Ñ can be seen as a good approximation to purely random sequences. 2.2 Þ Gold Sequences ì Although 7-sequences possess excellent randomness Ðand especially autocorrelation Ñ properties, they are not generally used for CDMA purposes as it is difficult to find a set of 7-sequences with low crosscorrelation for all possible pairs of sequences within the set. ì However, by slightly relaxing the conditions on the autocorrelation function, we can obtain a family of code sequences with lower crosscorrelation. ì Such an encoding family can be achieved by Gold sequences or Gold codes which are generated by the modulo-2 sum of two 7-sequences of equal period. Spread Spectrum (Part-1) 44 Spread Spectrum (Part-1) 45

14 The Gold sequence is actually obtained by the modulo-2 sum of two 7-sequences with different phase shifts for 7 the first 7-sequence relative to the second. Since there are œ # " different relative phase shifts, and since we can also have the two 7-sequences alone, the actual number of different Gold-sequences that can be 7 generated by this procedure is # ". These sequences, however, are not maximal length sequences. Therefore, their auto-correlation function is not the two valued one given by Equ. (1). The auto-correlation still has the periodic peaks, but between the peaks the auto-correlation is no longer flat. It can be shown, that the values of the auto-correlation between the peaks are again given by Equ. Ð7 Ñ. m-seq. 127 auto-co lag k lag k Spread Spectrum (Part-1) 46 Spread Spectrum (Part-1) 47 ì Gold-sequences have the same cross-correlation characteristics as preferred 7-sequences, i.e. their crosscorrelation is three valued. ì Gold sequences have higher Vauto and lower Vcross than 7-sequences, and the trade-off (see Equ. 0) between these parameters is thus verified. ì Balanced Gold codes. We should also note that not all Gold codes generated by the above procedure are balanced, i.e. the number of "-1s" in a code period does not always exceed the number of "1s" by one as is the case for 7-sequences. 7 " 7 7 " For example, for 7 odd only # " code sequences of the total # " are balanced, while the rest # code sequences have an excess or a deficiency of -1s. For 7œ(, for instance, only 65 balanced Gold codes can be produced, out of a total possible of " 29. Of these, 63 are non-maximal and two are maximal length sequences. Balanced Gold codes have more desirable spectral characteristics than non-balanced. Balanced Gold codes are generated by appropriately selecting the relative phases of the two original m- sequences. ì SUMMAY: By selecting any preferred pair of primitive polynomials it is easy to construct a very large set of PN-sequences (Gold-sequences). Thus, by assigning to each user one sequence from this set, the interference from other users is minimised. Spread Spectrum (Part-1) 48

15 3. PN-Signal,Ð>Ñ 3.1 Þ Definition of a PN-signal,Ð>Ñ ì A signal,ð>ñ is said to be a PN-Signal if it can be expressed as follows! _ k= _ bt Ð Ñœ![ k. ] -Ðt kx Ñ; with 5X >ŸÐ5 "ÑX (11) or, equivalently, bt Ð Ñ œ! _ -Ðt 8X-Ñ; with 8X- > Ÿ Ð8 "ÑX- 8= _ 3.2 Þ PSD( 0 ) of,ð>ñ if it is a andom Pulse Signal ì For a random pulse signal bt Ð Ñ Ði.e. a sequence of pulses where there is an invariant average time of separation X-= between pulses Ñ with all pulses of the same form but with random amplitudes and statistical independent random time of occurence, then: " PSD b Ð f Ñ = X -=. š ± Fourier Transform of a single pulse ± # where N " - ÐÑœ t!![ k ].-Ðt k XÑœone period; with 5X >ŸÐ5 "ÑX - k= Ðt Ñ is an energy signal which is non-zero in the interval Ð!, X-Ñ and {![ 5]} is an 7-sequence of length NÞ Spread Spectrum (Part-1) 49 Spread Spectrum (Part-1) 50 ì Example: PSD of a random BINAY signal Consider a random binary sequence of 0's and "'s. This binary sequence is transmitted as random signal with "'s and 0's being sent using the pulses shown below. For instance a random binary sequence/waveform could be If "'s and 0's are statistically independent with T <Ð!Ñ œ T <Ð"Ñ œ!þ&, the PSD of the transmitted signal can be estimated as follows: Spread Spectrum (Part-1) 51 Spread Spectrum (Part-1) 52

16 3.3 Þ Cross Correlation function of two PN-Signals," Ð>Ñ &,# Ð>Ñ The cross correlation between two PN-Signals b"ðt Ñ and b2ðtñ _ b Ðt Ñ=!! [] k.- Ðt kx Ñ " " " - k= b Ðt Ñœ!! []. 6 - Ðt 6X Ñ = _ is given as follows " Ð7Ñ= Ð7Ñ O Ð7Ñ bb X --!! " # - " # " # - (12) where Ú -" Ðt Ñ, -2ÐtÑ: energy signals with 0 Ÿ > Ÿ X- and cross-correlation - " - # Ð7Ñ Û Ü {! },{! }: 7-sequences of length N and cross-correlation [ m] " 2 and Ð7 Ñ= [ m ]. $ Ð7 mt Ñ " # "! #! "! # 3.4 Þ Auto Correlation function of a PN-Signals ì If Ö! = Ö! Ö! and c ÐtÑœc Ð tñ cðt Ñœrect then " 2 " 2 ÐiÑ --Ð 7 Ñ œ --Ð 7 Ñ œ X " 2 - AŠ X 7 - > X-,Ð>Ñ ÐiiÑ!! Ð7 Ñ Ð "Ñ repx e$ Ð7 Ñf repx e$ Ð7Ñf - - Ð iii Ñ," ÐtÑ œ, 2Ð tñ,ðt Ñ Ê the cross-correlation function,, Ð 7 Ñ becomes the " 2 " autocorrelation function,,ð7ñ œ X -- Ð7Ñ O Ð7Ñ which can be expressed as : -!! proof - for you!!!! Ð4 Exam paper 1992) Spread Spectrum (Part-1) 53 Spread Spectrum (Part-1) 54 i.e.,,ð7ñ " œ X -- Ð7Ñ O Ð7Ñ -!! = " " X Ð "Ñ repx X rep -š X-AŠ 7 X X-š X-AŠ 7 X " 7 " 7 X - - i.e Þ,, Ð Ñ=. rep AŠ X Ÿ (13) By using the FT tables the PSD Ð0Ñ of the signal bt Ð Ñ can be found to be: " # " PSD, ÐfÑœFTš b, Ð7 Ñ œ #. comb " sinc Š f. X- Ÿ N. $ ÐfÑ (14)! _!!! m=-c X Þ Partial Auto Correlation function of a PN-Signals,Ð>Ñ ì Let bt ÐÑ be a PN-Signal of period X-. Let M N and bb Ð7Ñ partial autocorr. function of bðtñ " then bbð7ñ= X V--Ð7Ñ O V Ð7Ñ œ random (15) - ðñò!! random " ìxš V,, Ð7Ñ œ X V--Ð7Ñ O Xš V Ð7Ñ -!! " œ X V--Ð7Ñ Oˆ V Ð Ñ -!! 7 " œ X V--Ð7Ñ O V Ð Ñ -!! 7 œv Ð Ñ,, 7 œ " 7 " X. rep X- AŠ - Ÿ 7 rep AŠ X Ÿ (for large ) X - - Spread Spectrum (Part-1) 55 Spread Spectrum (Part-1) 56

17 i.e. Xš V,, Ð7 Ñ rep AŠ 7 X Ÿ (for large ) X - - Var š,, 7 V Ð Ñ œ œ Î " Ð" ÑÐ" Ñ Ð # Ï " rep X- AŠ X - Ÿ 7 # Ñ Ó Ò " Î Ð Ï 7 # Ñ Ó Ò " rep AŠ X Ÿ (for large ) X - - Ú Ý VarÖV!![ 5] # 7 X- Ú #. A š " # 7 X- X-. A š X l7l ŸX- - Var š V,, Ð7Ñ = Û Û 5! Ý V+< š V!! [] k " Ü Ü 7 X- # Spread Spectrum (Part-1) 57 Spread Spectrum (Part-1) 58

18 4. Appendices Appendix 4.A: Properties of Ö! Ò8Ó if it is a purely random sequence Let the sequence e! [ 8] f be the output of a discrete, memoryless source INFOMATION SOUCE of 1s [ ] 0.5 { [ ]} œ TÐ! 8 œ "Ñ œ Ä! 8 TÐ! [ 8] œ "Ñ œ 0.5 with X e![ 8] f œ! Ð œ " 0.5+(-1)!Þ& œ!ñ (16) Z+< e! [ 8] f œ" # # Ðœ" 0.5+(-1)!Þ&œ"Ñ (17) The auto-correlation of the sequence e! [ 8] f over symbols is defined as follows Ú! # [ ] V [ 5 ]!! 8 œ!" œ 5 œ!!!![ 8]![ 8 5]= Û 8œ" 8œ" 8œ" Ü random 5Á! (18) Therefore the mean and the variance of the autocorrelation function V [ 5] are as follows Ú Ý! # X e! [ 8] f œ!" œ if 5 œ! X [ ]! 8œ" 8œ" V!! 5 œ X e! [ 8]![ 8 5] f œ Û (19) 8œ" Ý! X e! [ 8] fx e! [ 8 5] f œ! if 5 Á! Ü 8œ"!! Z+< V [ 5] œx V [ 5] X V [ 5] œ # #!!!!!! œ " " X e! [ 8]![ 8 5]![ 7]![ 7 5] f X V [ 5]!! œ 8œ" 7œ" Ú # Ý!! # # 2 # X š! [ 8]. X š! [ 7] Xš V!! [0] = œ 0 if 5 œ 0 8 = " 7 = " œ Û # Ý! # # X š! [ 8]. X š! [ 8 5] Xš V!! [ 5] = 0 œ if 5 Á 0 Ü 8 = " One may also define the cross-correlation of two sequences e! "[ 8] fand e! #[ 8] f V [ 5] œ!! [ 8]! [ 8 5] (21)!! " # 8œ" " # # Since e! "[ 8] fand e! #[ 8] f are independent the results are essentially the same as for the auto-correlation of e! "[ 8] f with non-zero lag 5. This shows that completely random sequences have nice auto- and cross-correlation properties. Note that pure random sequences could be used as code sequences, but since the receiver needs a replica of the desired code sequence in order to despread the signal, PN sequences are used instead in practice. (20) Spread Spectrum (Part-1) 59 Spread Spectrum (Part-1) 60 Appendix 4.B: Auto and Cross Correlation functions of two PN-sequences Ö! 3Ò8Ó and Ö! 4Ò8Ó ì Consider the _-sequences of 1s of period : Ö! [ 8 ] œ ÞÞÞÞß! [ "] ß! [ ] ß! ["] ß! [#] ß ÞÞÞÞß! [ "] ß! [ ] ß! ["] ß ÞÞÞÞ Ö! [ 8 ] œ ÞÞÞÞß! [ "] ß! [ ] ß! ["] ß! [#] ß ÞÞÞÞß! [ "] ß! [ ] ß! ["] ß ÞÞÞÞ ì Then, there are three different cross-correlation functions Ú 5!! 3[ 8]! 4[ 8 5] 0 Ÿ5Ÿ " Ý 8œ" ˆaperiodic cross-correlation: G!! [ 5 ] Û !! 3[ 8 5]! 4[ 8 ] " Ÿ5Ÿ! (22) Ý 8œ" Ü! l5 l ˆperiodic cross-correlation: V [ 5 ]!! [ 8]! [ 8 5] (23)!! 3 4 8œ" 3 4 ì Note that: ˆ it is easy to see that V [ 5] œ G [ 5] G [ 5 ] (25)!! 3 4!! 3 4!! 3 4 ˆ the periodic (or even) cross-correlation function has the property V [ 5] œ V [ 5] (26)!! 3 4!! 3 4 ˆ the name of "odd cross-correlation" function follows from the property µ µ V [ 5] œ V [ 5] (27)!! 3 4!! 3 4 ì For a single code sequence, the corresponding autocorrelation functions have similar properties. µ ˆodd cross-correlation function: V [ 5] œ G [ 5] G [ 5 ] (24)!! 3 4!! 3 4!! 3 4 Spread Spectrum (Part-1) 61 Spread Spectrum (Part-1) 62

19 µ ì For best CDMA system performance, all G!! [ 5], V!! [ 5], V!![ 5] should be as small as possible, since they are proportional to the interference from other users. The out-of-phase Ði.e. for lag not equal to zero Ñ autocorrelation functions should also be made as small as possible, since these affect the multipath suppression capabilities and the acquisition and tracking performance of the receivers. We thus define the peak cross-correlation parameters ì Similarly we define the peak autocorrelation parameters Ú V [ ] Ý cross œmaxš ½V!! 5 ½ß að3ß4ß5à 3 4Ñ 3 4 µ µ Û V cross œmaxš ½ V!![ 5] ½ß að3ß4ß5à 3 4Ñ, (28) 3 4 Ý Ü Gcross œmaxš ½G!! [ 5] ½ß að3ß4ß5à 3 4Ñ 3 4 Ú Ý ¼ V [ ] ¼ mod auto œ max V!! 5 ß a3à a5 Á!Ð Ñ, 3 3 µ µ Û V auto œ maxš ½ V!![ 5] ½ß a3à a5 Á!Ðmod Ñ, 3 3 Ý Ü G œ ¼ max G [ 5] ¼ß a3à a5 Á!Ðmod auto!! Ñ 3 3 (29) ì Finally we define Ú Ý Vpeak œ maxevautoßvcrossf µ µ µ Û V peak œ maxš V autoßv cross (30) Ý Ü Gpeak œ maxegauto, Gcrossf µ ì With the above definitions we can see that the smaller the peak correlation parameters Vpeak, V peak and Gpeak, the better the performance of a system. These parameters, however, cannot be made as small as we wish. For example, for a set of O sequences of period, according to the Welch lower bound, O " V É G É peak O " peak O " #O O " Therefore for large values of O and the lower bounds on Vpeak and Gpeak are approximately V È peak G peak É # Moreover, it can show that # # # # auto -ross auto cross V V G G The above shows that not only is there a lower bound on the maximum correlation parameters, but also a trade-off between the peak autocorrelation and cross-correlation parameters. Thus the autocorrelation and cross-correlation functions cannot be both made small simultaneously. The design of the code sequences should be therefore very careful so that all the of above quantities of interest remain as small as possible. # (31) (32) (33) Spread Spectrum (Part-1) 63 Spread Spectrum (Part-1) 64 Appendix 4.C: The concept of a 'Primitive Polynomial' in GF(2) (see Appendix 4E for 'finite field' basic theory) " ì Consider a polynomial fðhñ over the binary field GFÐ2 Ñ: f ÐHÑ= f 8 H f 8 -" H... f " H f! Å Á! The largest power of H with non-zero coef. is called degree of fðhñ over GFÐ2Ñ ì if fðhñ, gðhñ GFÐ2 Ñ then Å Å 7 8 fðhñ gðhñ GFÐ2Ñ œ fðhñ ÞgÐHÑ GFÐ2Ñ ì divisible polynomial: A polynomial gðhñ GFÐ2 Ñis said to divide f ÐHÑ GFÐ2 Ñif b h ÐHÑ: f ÐHÑ=h ÐHÑ.gÐHÑ. Then the polynomial f ÐHÑ is called divisible ì irreducible polynomial: A polynomial fðhñ GFÐ2 Ñ of degree m is called irreducible if f ÐHÑ is not divisible by any polynomial over GFÐ2 Ñ of degree less than m but greater than zero. Ð or equivalently if it cannot factored into polynomials of smaller degree whose coefs are also 0 and " i.e the polynomials belong to GFÐ2ÑÑ f0 Ð ÑÁ0 ì two important properties of irreducible polynomials: if f ÐHÑ=irreducible Ê œ fd Ð Ñ has odd number of terms ì primitive polynomial: Ú fðhñœirreducible Ðof degree mñpolynomial, and if Û fðhñ k ÐH " Ñ k m i.e. f ÐHÑ does not divide H " for any k # " Ü Á! then fðh Ñ primitive polynomial $ # 4 e.g. H H " ; H H " ì only a small number of polynomials are primitive, but am bat least one primitive polynomial. $ # ì examples: 0ÐHÑ= H H " = primitive % # 0ÐHÑ= H H " = irreducible but not primitive Spread Spectrum (Part-1) 65 Spread Spectrum (Part-1) 66

20 Appendix 4.D: FINITE FIELD -BASIC THEOY ìconsider a set W = { = " ß = # ß ÞÞÞß = } having elements. A finite field is constructed by defining two binary operations on the set called addition & multiplication such that certain conditions are satisfied. Addition and multiplication of two elements = 3and = 4 are denoted = 3 = 4 and = 3 = 4 respectively. ìthe conditions that must be satisfied for W and the two operations to be a finite field are: 1. The addition or multiplication of any two elements of W must yield an element of W. That is, the set is closed under both addition and multiplication. 2. Both addition and multiplication must be commutative 3. The set W must contain an additive identity element which will always be denoted by 0. = 3! œ = 3 4. The set W must contain an additive inverse element = 3 for every element = 3 = 3 Ð = 3Ñ œ! 5. The set W must contain a multiplicative identity element which will always be denoted by 1. =Þ 3 1 œ= 3 " '. The set W must contain a multiplicative inverse element = 3 for every element = 3 (excluding the additive identity 0) " =Þ= 3 3 œ" 7. Multiplication must be distributive over addition. 8. Both addition and multiplication must be Associative. ìexample It is easy to verify that W œ Ö!ß "ß # with addition and multiplication defined as follows modulo-3! " # modulo-3! " #!! " #!! 0 0 " " #! " # #! " # 0 2 " is a field of 3 elements e.g. additive inverse 0 œ 0 " multiplicative inverse 1 œ " "œ# " # œ# #œ" ìexample It is easy also to verify that W œ Ö!ß ", with addition and multiplication defined as follows: modulo-2! " modulo-2! "!! "!! 0 " " 0 " 0 1 is a field of 2 elements e.g. additive inverse 0 œ 0 " multiplicative inverse 1 œ 1 "œ1 ìnote that W œ Ö!ß " field above is the binary number field. Furthermore that addition can be performed electronically using EXCLUSIVE-O gate and multiplication can be performed using an AND-gate. Spread Spectrum (Part-1) 67 Spread Spectrum (Part-1) 68 ìan Important esult (presented without proof): The set of integers Wœ{0, 1, 2,..., " }, is prime and whereœ ß + ddition and multiplication are carried out modulo- is a field. These fields are called prime fields. Appendix 4.E: Table of Irreducible Polynomials over GF(2) (from "Error-Correcting Codes" by Peterson & Weldom MIT Press, 1972) ìsubtraction and Division: The operations of subtraction and division are also easily defined for any field using the addition and multiplication tables, 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 the multiplicative inverse. For example for the field WœÖ0,1,2} subtraction is defined by 1 + ( 2) = = 2. " Similarly, 1 ƒ 2 œ 1 Þ (2 ) œ 1Þ2 œ 2. ìnote that nonprime fields do not necessarily employ modulo- arithmetic. 7 7 ìfields can be constructed having any prime number of elements : or :. A field having : elements is called an 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. ìthe remainder of this discussion will be concerned exclusively with the binary number field GF(2) and its 7 extensions GF(2 ). The reason for this is that the electronics used to implement the code generators is binary, and 7 some of the shift register generators will be shown to generate the elements of GF(2 ) Spread Spectrum (Part-1) 69 Spread Spectrum (Part-1) 70

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