Theory of Spread-spectrum Communications - A Tutorial - Pickholtz 1982

8/4/2019 Theory of Spread-spectrum Communications - A Tutorial - Pickholtz 1982

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IEEE TRANSACTIONS ON COMMUNICATIONS, V O L . COM-30,O. 5,MA Y 1982 855

Theory of Spread-Spectrum Communications-A TutorialRAYMOND L. PICKHOLTZ, FELLOW, I ~ E E ,ONALD L. SCHILLING, FELLOW, EEE,

A N D LAURENCE B . MILSTEIN, SENIOR MEMBER, IEEE

AbstracrSpread-spectrum communicat ions , wi th ts nherent ill-terference ttenuation apabi l i ty,has ver he earsbecome nincreasingly popular echnique oruse inmanydifferentsysteml:.Applications range from anti jam systems, to code d ivision mu ltipleaccess systems, o systems designed o combat multipath. It is th eintention of this paper to provide a tutorial treatmentf the theory c%spread-spectrum ommunications,ncluding iscussion on th eappl icationseferredobove, on th eroperties of commomspreading equences,an d on techniques hatca nheused or a( -quisi tion and tracking.

SI. INTRODUCTION

PREAD-spectrum systems have been developedinceabout he mid-1950s. The initial applications have be e1to military antijamming tactical communications, to guidancesystems, to experimental ahtimultipath systems, and t 3other applications [ l ]. A definition of spread spectrurlthat adequately reflects the characteristics of this techniqu:is as follows:Spread spectrum is a means of transmission in whichthe signal occupies a bandwidth in excess of the mini-

mum necessary to send th e information; the band spreadis accom plished by means of a code which is independentof the data, and a synchronized reception w ith the codeat he receiver is used for despreading and subsequentdata recovery.Under this definition, standard modulation schemes such a sFM and PCM wh ich also spread the spec trum of an informa -tion signal do not qualify as spread spectrum.There are many reasons for spreading the spectrum, and ifdone properly, a multiplicity of benefits can accrue simulta-neously: Some of these are0 Antijamming0 Antiinterference0 Low probability of intercept0 Multiple user random access comm unications with selec-tive addressing capabilityi High resolution ranging0 Accurate universal timing.Manuscript received December 22 , 1981; revised February 16, 1982R.L. Pickholtz is with the Department of Electrical Engineering ancComputer Science, George Washington University, Washington, DC20052.D. L. Schilling is with the Department of ,Electricai EngineeringCity College of Ne w York, New York, N Y 10031.L. B. Milstein is with the Depar tmen t of Electrical Engineering antComputer Science, University of California a t Sa n Diego, La JollaCA 92093.

Themeansby which the pectnim is spread is crucial.Several of he techniques are direct-sequence .moduldtionin which a fast pseudorandomly generated sequence causesphase transitions in the carrier containipg data, frequencyhopping, in wh ich the carrier iscaused to shift frequencyin a pseudorandom way, arid time hopping, wherein burstsof signalare initiated at pseudorandom times. Hybrid com-binations of these techniques are frequently used.Although the current applications for spread spectrumcontinue to be primarily for military communications, thereis a growing interest in the use of this technique for mobileradio networks (radio telephony, packet radio, and amateurradio), tiining and positioning systems, somepecializedapplications in satellites, etc. While the use of spread spectrumnaturally means th at each transmission utilizes a large amo untof spectrum, this may be compensated for by the interferencereduction capability inherent in the useof spread-spectrumtechniques, so thata considerable number of users mightshare the same spectral band. There are no easyanswers tothe question of whether spread spectrum is better or worsethan conventional methods for such multiuser channels.However, the one issue that is clear s that spread spectrumaffords an opportunity to give a desiredsignal a power ad-vantage ver many types of interference, including mostintentional interference (i.e., jamniing). In this paper, weconfine ourselves to principles related to the design andanalysis of various important aspects of a spread-spectrumcommunications system. The emphasis will be on direct-sequence techniques aild frequency-hopping techniques.The major systems q uestions associated with the design ofa spread-spectrum system are: How is performance measured?What kind of coded sequences areused and what are theirproperties? How much jamming/interference protection isachievable? What is the perform ance of any user pair in anenvironment where there are many spread spectrum users(code division Multiple accessj? To what exten t does spreadspectrum reduce the effects of multipath? How is the relativetiming of the transmitter-receiver codes established (acquisi-tion) and retained (tracking)?

It is the aim of this tutorial paper to answer some of thesequestions succinctly, and in the process, offer some insightsinto this important communications technique. A gldssafy ofthe sym bols used is provided at the end of the paper.11. SPREADING AND DiMENSIONALITY-PROCESSING GAIN

A fundamental issue in spread spectrum is how thistechniqu e affo rds prote ction against interfering signals with0090-6778/82/0500-0~355$00.750 1982 IEEE


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85 6 IEEE TRANSACTIONS ON COMMUNICATIONS, V O L . COM-30, NO. 5 , MAY 1!282finite power. The underlying principle is that of distributinga relatively low dimensional (defined below) data signal in ahigh dimensional environment so that a jammer with a fixedamount of total power (intent on maximum disruption ofcomm unications) is obliged to either spread that fixed powerover ,all the coordinates, thereby inducing just a little iriter-ference in each coordinate, or else place all of the power intoa small subspace, leaving the remainder of the space inter-ference free.A brief discussion of a classical problem of signal detectionin noise should clarify the emphasis on finite interferencepower. The standard problem of digital transmission ,in thepresence of thermal noise is one where both transmitter andreceiver know the set of M signaling waveforms S i ( t ) ,0 Q t


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PICKHOLTZ e t al . : THEORY OF SPREAD-SPECTRUMOMMUNI2ATIONS 857The received signalr ( t ) = S,(t) +J ( t ) (5 )

is correlated with h e (known) signals so that the output ofthe ith correlator s

Hence,( 7)

k= 1since the second term averages to ze ro. Then, since the signllsare equiprobable,

Similarly, using (1) and (2),var I i ) =x ~ J ~ S S

k .

andE,= - EJn

E,var Ui =- J .nDA measureferformances the signal-to-noise ratiodefined as

This result is independent of th e way that he ammcrdistributes his energy, i.e., regardlessof how J k ischosensubject to the constraint that CkJk = E J , he postprocessirgSN R (11) gives the signalanadvantageof n /D over tk ejammer. This factor n /D is th e processinggain and i t is exactlyequal to the ratio of the dimensionality of the possible sign:dspace and therefore the space in which the amm er mustseek to operate) t o the dimensions needed to actually transm .tthe signals.Using the result tha t he (approximate) dimell-sionality of a signal of duratio n T and of approximate band-width BD is ~ B D T ,e see the processing gain can be wr itte nas

(12)

where Bss is the band wid th in hertz of the (spread-spectrurr)

signals S,(t) and BD is the minimum bandwidth tha t would berequired t o send the information if we did not need to imbedit in the larger bandwidth for protection.A simple illustration of these ideas using random binarysequenceswillbeused to bring out someof these points.Consider the transmission of a single bit +&IT with energyE , of duration T seconds. This signal s one-dimensional. Asshown in Figs. 1 an d 2, the ransmitter multiplies thedatabit @(t) by a binary +1 chipping equence p ( t ) chosenrandomly at rate f, hips/s for a total of f c T chips/bit. Thedimensionalityof the signal d(t)p(t) is then n = f,T. Thereceived signal isr ( t ) = d ( t ) p ( t )+J ( t ) , 0< < T , (13)

ignoring, for the time being, thermal noise.The rece iver, as shown n Fig. 1, performs the correlation

U &3 ( t ) p ( t )dtand makes a decision as to whether ? a f T was sent de-pending upo n U 2 0. The integrand can be expandedas

andhence thedatabit appears in the presenceof a code-modulated jammer.If, for example, J ( t ) is addit ive white Gaussian noise withpower spectral density l ) O J / 2 (two-sided), so is J ( t ) p ( t ) ,an d U is then Gaussian andom variable. Since d ( t ) =+-IT, the conditional mean and variance of U ,assuming

respectively, and the probability of error is [3 ] Q(~-J)where Q(x) b JF ( 1 / 6 ) e - Y 2 / d y . Against white noise ofunlimited pow er, spread spectrum serves no useful purpose,and th e probabil ity of er ror is Q(d-)regardless of themodulationby he code equence.Whitenoiseoccupiesalldimensions with power l ) o J /2 . The situation is different,however, if the amm er power is limited. Then, no t havingaccess to the random sequence p ( t ) , the jamm er with availableenergy EJ (power E J / T ) can dobetter han to apply thisenergy to onedimension. Fo r example, if J ( t ) = &/T,0 < t< T , hen the receiver outp ut is

that *&/T is transm itted, is given by E, an d Eb(r]o~ / 2 ) ,

where the Xis are i.i.d. random variables with P(Xi = +1) =P(Xi = -1) = $.The signal-to-noise atio (SNR) isE2(u>Eb- -var (U) EJ .. (1 7)

Thus, the SNR may be increased by increasing n , he process-1 Independent identically distributed.


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858 IEEE TRANSACTIONS ON COMMUNICATIONS,.VOL. COM-30, NO. 5 , MA Y 1'782

S O U R C E

R A T E = ;-

S p r e a d i n gsequence

D e c i s i o nI N T E G R A T E & v a r i a b l abDUMPU

fP ( t ) Ir a t e = f c JAMMER

S E Q U E N C EG E N E R A T O RE NE RAT OR sequence

RANDOM D e s P r e a d i n g

T R A N S M I T T E RE C E I V E RFig. 1. Direct-sequence spread-spectrum system for transmittingsingle binary digit (baseband).

Fig.2. Data bit and chipping sequence.ing gain, and it has the form of (11). As a further indication ofthis parameter, we may c omp ute the probability Pe that thebi t is in error from (16). Assuming that a "&us7' is trans-mitted , we have

P , = P ( U > 0)=P(2" >an)

Eb- > lEJwhere

1 "2 i=lz n 0- (1 +-xi) s a B ernoulli random variable with

n nmean - d a riance -->2 4

an d [x] is defined as the integer portion of X . The partialbinomial sum on the right-hand sideof (18) may be upperbounded [2] by

P e < - 1 (-)ye)Ln2" 1 - a ; ; < a < 1orp e < 2 - " [ l - H ( " ) l ; 1. < a < 12 1 1 )

where H ( a ) & --a log2 a - 1- ) og, (1 - ) s the binaryentropy function. Therefore, for any CY >3 or E , f 0), P,may be made vhishjngly small by increasing n , the processinggain. (The same result is valid even if the jammer u ses a chippattern other than the constant, all-ones used in the exampleabove.) As example, if EJ = 9Eb uammer energy 9. 5 dBlarger th& that of the data), then a = 213 and Pe< 2 -0.0'35nIf y = 20 0 (23 dB processing gain),P , < 7.6 X low6.An approximation to the same result may be obtainedby utilizing a central limit type of argument that says, forlarge n , U in (16) may be treated as if it were Gaussian. Then

P , = P ( U < a ) s Q ( E n )

and, if Eb /EJ = -9.5 dB and n = 200 (23 dB), Pe sQ ( m ) 1.5 X 10- 6. The processing ainan eseento be a multiplier of the "signal-to-jamming" ratio E b / E J .A m ore traditional w ay of describing the .processing gain,which brings in the relative bandwidth of thedata signaland hat of the spread-spectrum mo dulation, is to examinethe power spectrum of an infinite sequence of data, modulatedby the rapidly varying random sequence. The spectrum ofthe random data sequence with rate R = 1/T bitsls is givenby

S , u , = T ( = )in nfT


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PICKHOLTZ er al .: THEORY O F SPREAD-SPECTRUM COMMUNIC kTIONS 859

Fig. 3. Power spectrum of data and of spread signal.and tha t of the spreading sequence [and also that of thr:product d ( t )p ( t )] s given by

Both are sketched in Fig. 3. It isclear tha t if the receive..multiplies the receivedsignal d( t )p ( t ) + J ( t ) by p ( t ) givinl:d ( t ) + J(t)p(t ) , the first term may be extracted virtuallJ.intact with a filter of bandwidth 1/T BD Hz. The seconc.term willbe spreadover at least f, Hzas show n inFig. 3The fraction of power due to the jammer whichcanpas::through the filter is then roughly I/f,T. Thus, the data havea power advantage of n = f,T, the processing gain. As in (12)the processing gain is frequently expressed as the ratio of thcbandwidth of the spread-spectrum waveform to that of thcdata, i.e.,

G Li- f C T = n .ssBD-The n otion of processing gain as expressed in (23) is simplya power improvement factor which a receiver,possessing areplica of the spread ing signal,can achieve by a correlationoperation. It must not be automatically extrapolated toanything else. For exam ple, if we use frequency hopping forspread spectrum employing one of N frequencies every THseconds, the otal bandwidth m ust be approximately N/TH(since keeping the frequencies orthogonal requires frequency

Nowfwe transmit 1 bit/hop, THBD 1 and G p = N , thenumber of frequencies used. If N = 100, G , = 20 dB, whichseems fairly good. But a single spot frequency jammer cancause an average error rate of abou t which is no tacceptable. (A more detailed analysis follows in Section IVbelow.) This effectiveness of partial band jamming can bereduced by the use of coding and interleaving. Codingtypically precludes the possibility of a small number of fre-

Spacing 1 / T H ) .Then, according to (1 2) , G , = (N/TH)/BD.

quenc y slots (e.g., one slot) being jamm ed causingan un-acceptable error rate (i.e.,even f the jammer wipes ou t afew of the code sym bols, depending upon the error-correctioncapability of the ode, hedata may still be ecovered).Interleaving has the effect of randomizing the errors due t othe jammer. Finally, an analogous situation occurs in directsequen ce spreading when a pulse jammer is present.In the design of a practical system, the processing gainG p is not, by itself, a measure ofhow well the system iscapable of performing in a jamming environment. For thispurpose, we usually introduce the jamming margin in decibelsdefined as

This is the residual advantage that the system has against ajammer after we subtractothhe minimum requiredenergy/bit-to-jamming noise power spectral density ratio( E b / q o ) ~ i ~nd implementation and other losses L . Th ejamming margin can be increased by reducing the (Eb/qOJ)minthrough the use of coding gain.We conclude this section by showing that regardless of thetechnique used, spectral spreading provides protection againsta broad-band jammer with a finite power P J . Consider asystem that transmits Ro bits/s designed to operate over abandwidth Bs s Hz in white noise with power density qo W/Hz.For any bit rate R ,

whereP, 2E d ? = signal pow erPN 2 Q ~ B , , noise power.

Then for a specified ( E b / q O ) m i n necessary to achieve mini-


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860 IEEE TRANSACTIONS ON COMMUNICATIONS, V O L . COM-30, NO. 5, M A Y I982

mum acceptable performance,

If a jammer with power PJ now appears, and if we arealready transmitting t he maximu m rate Ro , then ( 2 5 )becomes

or

Thus, if we wish to recover from the effects of the jamm er,the right-hand side of ( 2 7 ) should be not much less than(Eb/qo)min.his clearly requires that we increase Bss, sincefor any finite P J , it is then possible t o ma ke the actorqo/(qo + PJ/B,,) approach unity, and thereby retain the per-formance we had before the jammer appeared.

111.PSEUDORANDOM SEQUENCEGENERATORSIn Section 11, we examined h ow a pu rely random sequencecan be used to spread the signal spectrum. Unfortunately, inorder to despread the signal, the receiver needs a replica ofthe ransmitted sequence (in almost perfect time synchro-nism). In practice, therefore, we generate pseudorandom orpseudonoise (PN) sequences so that the following propertiesare satisfied. They1) areeasy to generate2 ) have random ness properties3) have long periods4) are difficult to reconstructrom hort segment.Linear feedback sh ift register (LFSR) sequences [4] possess

properties 1) an d 3), most of property 2 ) , but not property4). One canonical form of a binary LFSR known as a simpleshift register generator (SSRG) is shown in Fig. 4. The shiftregister consists of binary storage elements (boxes) whichtransfer their contents to the right after each clock pulse (notshown). The contents of the register are linearly combinedwith the binary (0, 1) coefficients ak and are fed back t o thefirst stage. The binary (code) sequence C, then clearly satisfiesthe recursion

The periodic cycle of the tates depends on the initialstate and on the coefficients (feedback taps) ak . For example,the four-stage LFSR generator shown in Fig. 5 has fourpossiblecycles as show n. The all-zeros is always a cycle forany LFSR . For spread.spectrum , we are looking for macimallength cycles, that is, cycles of period 2 -1 (all binary r-tuples.except all-zeros). An example is shown for a four-state registeri n F i g . 6 . T h e s e q u e n c e o u t p u t i s 1 0 0 0 1 1 1 1 0 1 0 1 1 0 0 - ~(period 24 - 1 = 15) if the initial contents of the register(from right to left) are 100 0. It is always possible to ch oosethe feedback coefficients so as to achieve maximal length,as will be discussedbelow.If we do have a maximal length sequence, then this se-quence will have the following pseudorandomn ess properties

1) There is an approximate balance of zeros an d ones( 2 r - 1 ones and 2 1 zeros).2) In any period, half of the runs of consecutive zerosor ones areof length one,one-fourth are of length two ,one-eighth are of length three, etc.3) If we define the k1 sequence C,= 1 - C,, IC =

0, 1, thenheutocorrelationunction R,(r) p 1/LE,& C i C;+ isgiven by -

~41.

r = 0, L , 2L ...where L = 2 - 1. If the code waveform p ( t ) is the square-wave equivalent of the sequences C,, f L % 1, and if we


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PICKHOLTZ et a l . : THEORY OF SPREAD-SPECTRUM COMMUNII:ATIONS 86 1

J l O O Os1100 0001* ?0 10 011u 1011 1101 d o 1 0 0c,& 91010 10010101 0010Fig. 5. Four-stag( LFSR and itsstatecycles.

~ L

e l , c- 43b. -

Fig. 6 . Four-sta ge maximt.1 length LFSR and its state cycles.

define

I 0; otherwisethen

Equation (29), and therefore (30), follow directly from tlte"shift-and-add" property of maximal length (ML) LFSRsequences. This property is that the chip-by-chip sum of anMLLFSR sequence c k and any shift of itself ck+7, f 0is the same sequence (except for some shift). This folloirsdirectly from (28) , sinceL

(cn+ c n + r ) = a k ( Cn - k + c n + r - k ) (mod 2) . (3 :.)k = 1The shift-and-add sequence C,,+ e,,+? isseen to satislythe same recursion as C,,, nd if the coefficients ak yie: dmaximal length, then it must be the same sequence regardlessof the initial (nonzero) state. The autocorrelation properly(29) then follows from the following isomorphism:

Therefore,

0 0 3

O K

11001 0 0 011101111

and if C i is an MLLFSR f 1 sequence, so is C i ck+T' , r f 0.Thus, there are 2'-' 1's and (2r-1 - 1) -1's in th e produc tand (29) follows. The autocorrelation function is shown inFig. 7(a).Property 3) is a most important one for spread spectrumsince the autocorrelation function of the code sequencewaveform p(t ) determines thespectrum . Note that becausep ( t ) is pseudorandom, i t is periodic with period (2'--1)*l / f c , and hence so is R p ( 7 ) . The spectrum shown in Fig. 7(b)is therefore the line spectrum

m#O1

L2 )where

f cf Q = -.2' - 1If L = 2' - 1 isvery large, the sp ectral lines getclosertogether, and for practical purposes, the spectrum may beviewedas being continuous and similar to that of a purelyrandom binary waveform as shown inFig. 3. A different,but comm only used implementation of a linear feedback shiftregister is the modular shift register generator (MSRG) shownin Fig. 8. Additional details on the properties of linear feed.back shift registers are provided in the Appendix.


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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-30, NO. 5 , MAY 1'382

I

?CL

11Fig. 7. Autocorrelation function R (7) nd power spectral density ofMLLFSR sequence waveform p($ . (a) Auto correlation function ofp ( t ) . (b) Power spectral density of ( f ) .

fC f

Fig. 8. Implem entation as a modular shift register generator (MSRG).For spread spectrum and other secure communications(cryptography) where one expects an adversary to attempt

to recover the code in order to penetrate he system, prop-erty 4) cited in he beginning of this section is extremelyimportant.Unfortun ately, LFSR sequences donot possessthat property. Indeed, using the recursion (28) or (A8) andobserving only 2 r - 2 consecutive bits n he sequence C,,allows us to solve for the r - middle coefficients and the rinitial b its in the register by linear simu ltaneous equa tions.Thus, even if r = 100 so that the length of the sequence is2' O 0 - 1 1: lo3', we w ould be able to construct the entiresequence from 98 its by solving 198 linear equations

(mod 2), which is neither difficult nor that time consum ing fora large com puter. Moreover, because the sequence C , satisfiesa recursion, a very efficient algorithm is known [7], [8]which solves the equations or which equivalently synthesizesthe shortest LFSR wh ich generates a given sequence.In order to avoid this pitfall, several modifications of theLFSR have been proposed. In Fig. 9(a) the feedback functionis replaced by an arbitrary Boolean function of the contentsof the register. The Boolean fu nction may be implementedby ROM or random logic, and there are an enormous numberof these functio ns (2"). Un fortu nately , very little is know n

[4] in the o pen literature about the properties of such non-


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OUT( 3 )

Fig. 9 . Nonlinear eedbackshiftregisters. a)NonlinearFDBK. Num-ber of Boolean functions = 2!. (b) Linear FSR, nonlinear functionof state, i .e., nonlinear output logic (NOL ).

linear feedback shift registers. Furtherm ore, some nonlinearFSRs ma yhave no cycles or length > 1 (e.g., they ma:,have only a transient that homes towards the all-ones stat:after any initial state). Are there feedback functions hatgenerate only on e cycle of length 2? The answer is yes, anclthere are exactly 2*- - f them [9]. How do we finlithem? Better yet, how do we find a subset of them with allthe good randomness properties? These are, and have been,good esearch problems for quite some time, and unfortu-nately no general theory on this topic currently exists.

A second, more manageable approach is to use an MLLFSI:with nonlinear output logic (NOL) as shown in Fig.9(b:.Some lues abo ut designing the NOL while still retainin:good random ness p roperties are vailable [101-[ 121,and a measure for judging how well condition 4) is fulfilledis to ask : What is th e degree of the shortest LFSR that woulllgenerate the same sequence? A simple example of an LFSRwith NOLhaving three stages is shown in Fig. O(a). Th:shortest LFSR which generates the same sequence (of period7) is shown in Fig. 10(b) and requires si x stages.

When using PN sequences in spread-spectrum systems,several additonal requirements mu st be met.1) The partial correlation of the sequence Cd over awindow w smaller than the full period should be as small aspossib le, i.e., if

n=

should be 4 L = 2 - 1.correlation, i.e.,2) Different code pairs should have uniformly low cross

should be 4 1 for all values of 7.


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864 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-30, NO. 5,MA Y 1982

0 1 0 . . . (period= 7 )

(b)Fig. 10. LFSR with NOL and its shortest l inear equivalent. (a) Three-stage LFSR with NOL. (b) LFSR with f ( x ) = 1 +x + x2 + x3 + x4 +

x5 + x6 which generates the same sequence as that of (a) under theinitial state 1 0 0 0 1 0.

3) Since the code sequences are periodic with period L ,there are two correlation functions (depending on the relativepolarity of one of the sequences in the transition overaninitial point 7 on the other). If we define the finite-cross-correlation function [131 as

then the so-called even and odd cross-correlation functions are,respectively,Rc~cJ"'(.) =fc'c"(7)+fC'C"(L - )

and

and we wantmax I R C ~ C ~ ~ e ( ~ )and max I R C ~ ~ ~ ~ ( 0 ) ( ~ )7 7

to be < 1.The reason for 1) is to keep the "self noise" of the systemas low . as possible since, in practice, the period is very longcompared to the integration time per symbol and there will befluctuation in the sum of any fdtered (w eighted) subseque:nce.This is especially worrisome during acquisition where thesefluctuations cancause alse locking. Bound s on p(w) [[I41and averages over j of p (w ; i,7) re available in the literature.Properties 2 ) an d 3) are both of direct interest whenusing PN sequences for code division multiple access (CDMA)as wi l l be discussed in Section V below. This is to ensureminimal cross interference betw een any pair of users of thecommon spectrum. The most commonly used collection of


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PICKHOLTZ et al.: THEORY OF SPREAD-SPECTRUMOMMUNICATIONS 86 5sequences which exhibit roperty 2) are the Gold codas[15] . These are sequences derived from anMLLFSR,b1.tare not of maximal length. A detailed procedure for theirconstruction is given in the Appendix.Virtually all of the kno wn results abou t he cross-corr:-lation properties of useful PN sequences are summarizedin [16] .As a final comment on the generation ofPN equencc:sfor spread spectrum, it is not at al l necessary that feedbackshift registersbe used. An y technique which can gen 92legood pseudorandom sequences will do. Other techniquesare described in [4], [16] , 17] , for example. Indeed, &egeneration of good pseudorandom sequences i s fundament,Jto other fields, and in particular, to cryptography [18] . . 4good cryptographic system can be used to generate goodPN seq uences, and vice versa. A possible problem is tha t tf especific additional good properties required for an oper-ational spread-spectrum system may not always match tho:erequired for secure cryptographic comm unications.

IV . ANTIJAM C ONSIDERATIONSProbably the single most important application of spreatl-spectrum techniques is that of resistance to intentional inte..-ference or jamm ing. Both direct-sequence (DS) and frequenqr-hopping (FH) systems exhibit this tolerance to jamming,although one might perform better than theother given aspecific type of jammer.The two most common types of jamming signals analyzedare single freq uency sine waves (tones) and broad-ba nd noise.References [19] and [20] provide performance analyses c fDS systems operating in the presence of bo th tone and noiseinterference, and [21] -[26] provide nalogous results fc rFH systems.The simplest case to analyze is that of broad-band noisejamming. If the jamming signal is modeled as a zero-meanwide ense stationary Gaussian oise rocess with a fll tpower spectral density over the bandwidth of interest, the?for a given fixed power PJ available to the jamming signa.,the power spectral density of the jamming signal must bereduced as the bandwidth thathe jammer occupies jsincreased.For a DS system, if we assume that he jamming signzloccupies the total RF band width, typically taken to be twicsthe chip rate, then the despread jammer will occupy an eve.1greater bandwidth and will appear to the final integrateam-dump detection filter as approximately a wh ite noise proces:,.

If, for example, binary PSK is used as the modu lation formal.,then he average probab ility of error will beapproximate!{given by

Fquation (36) is just the classical result for the perform anc:of a coherent b inary com munication system in add itive whit:Gaussian noise. I t differs from the conventional result becaus:an extra erm in thedenom inator of the argument of th:

e(.) function has been added t o account for the jammer. IfP , ,from (36) is plotted versus Eb/l),, fo ra givenvalue ofre P , is the averagesignal pow er, curves such asown in Fig. 1 1 result.sions imilar to (36) are asilyderived forothermodulation formats (e.g.,QPSK), and curves showing theperformance for several different formats are presented, forexample, in [19] .The interesting thing to note about Fig. 1 1is that ora given Q ~ J ,he curve bottomsout as Eb/ l )Ogets larger and larger. That is, the presence of the jamm er willcause an irreducible error rate fo r a given PJ and a given f,.Keeping PJ fixed, the only way to reduce the error rateis to increase f, (i.e., ncrease theamou nt of spreading intho+.system).4, This was also noted at the end of Section 11.

For FH systems, it is not always advantageous for a noisejammer to jam the entire RF bandw idth. That is, for a givenP J , he jammer can often increase its effectiveness by jammingonly a fraction of the total bandwidth. This is termedpartial-band jamming. If it isassumed that he jammer divides itspower uniformly among K slots, where a slot is the region infrequency th at the FH signal occupies on one of its hops, andif there is a total of N slots over which the signal can hop, wehave the following possible s ituations. A ssuming th at heunderlying modulation format is binary FSK(with noncoherentdetection at he receiver), and using the termino logy M A R Kand SPACE to represent the two binary data syv bo ls, on anygiven ho p, if, 1 ) K = 1 , the jammer might jam the M A R K only, jam theSPACE only, or jam neither the M A R K nor the SPACE;2) 1


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l o - *

pe

1 - ~

G = 5 1 1P

1 I I I 1 I I4 6 E ~ J ~ O ( ~ B ) 10 12 1 4 16Fig. 11. Probability of error versus Eb/qo.

power to jammer power per slot at the output of the M A R KBPF. By jammer power per slot, we m ean the total jammerpower divided by henum ber of slots being jammed (i.e.,SJR = ps/(pJ/K)).

The coefficients in front of the exponentials in (37) arethe probabilities of jamming neither the M A R K nor the SPACE,jamming only the M A R K or only the SPACE, or jamming bo ththe M A R K and the SPACE. For example, the probabilityof jamming bo th' the M A R K and the SPACE is iven byK (K - ) / N ( N - 1). In Fig. 1 2, the P, predicted by (37) isplotted versus SNR for K = 1 and K = 100 for a PJ/P, of10 dB. These two 'curves are labeled "uncoded" on the figure.Often, somewh at different model from that usedderivjng'(37) is considered. This latter m odel is used in [ 2 6 ] ,and effectively assumes that' either M A R K and SPACE aresimultaneously jammed or tha t neither of the two is jammed.Fo r t h i s case, a :earameter p , where 0 < p < 1, representingthe fraction of the ban d being jamm ed, is defined. The

resulting average probability of error is then maximized withrespect 't o p (i.e., the worst case p is found), and it is shownin [ 26 ] thate-Pemax>-E b h o

where e is the base of the natural logarithm. It can be seenthat partial band jamming affords the jammer a strategywhereby he candegrade the perform ance significan tly l(i.e.,Pe can b e. forced t o benversely proportion al to E,,/qorather than exponential).For tone jamming, the situation becomes somewhat rnorecomplicated than it is for noise jamming, especially for DSsystems. This isbecause the system performance dependsupon the location of the tone (or tones), and upon whetherthe period of the spreading sequence' is equal to or greaterthan theduration of adata symbol. Oftentimes the effect


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PICKHOLTZ e t al.: THEORY OF SPREAD-SPECTRUM COMMUNIC,%TIONS 8671 0 - 1

10-2

pe

l o - ' \ K = l O Oc 8 l l i 1 2 1 4 1 6

SN R (dB)Fig. 12 . Probabil ty of error versus SNR.

of a d espread ton e is ap proxim ated as having arisen from a11equivalent am ount of Gaussian noise. In this case, the resultspresented above would be appropriate. However, the Gau ssialapproximation is not always justified, and some conditionsfor its usage are given in [20] and [ 2 7 ] .The situation is simpler in FH systems operating in t h l :presence of partial-band tone jamming, and as shown, forexample, in [24] , the performance of a nonco herent FH-FSE:system in partial-band tone jamming is often virtually thl:sames the performance in partial-band noise jamming.One important consideration in FH systems with eithernoise or ton e jamm ing is the n eed for error-correction coding.This can be seen very simply by assuming that the jam mer i ;much stronger than the desired signal, and th at t choose3to put al l of its power in a single slot (i.e., the jammer jamsone out of N slots). The K = 1 uncoded curveofFig. l:!corresponds to this situation. Then with no error-correctiottcoding, the system will make an error (with high probability)

every time i t hops to a M A R K frequency when the correspond-ing SPACE frequenc y is being jamm ed or vice versa. This willhappen on th e average one ou t of every N hops, so that theprobability of error of the system willbe approximatelyl / N , independent of signal-to-noise ratio. Thiss readilyseen to be the case in Fig. 12 . The useof coding preventsa simple error as caused by aspot jammer from degradingthe system performance. To illustrate this point, an error-correcting code (specifically a Golay code [2]) was sedin conjunction with the system whose uncoded performanceis shown in Fig. 12 , and the performance of the coded systemis lso shown in Fig. 12 . The advantage of using error-correction coding is obvious from co mparing the correspond-ing curves.Finally, there are, of course, many other types of commonjamm ing signals besides broad-b and noise or single frequen cytones. These include swept-frequency jammers, pulse-burstjammers, and repeat jammers. No further discussion of these


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868 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-30, NO. 5, MAY 1982

Fig. 13. DS CDMA system.

TRANSMITTER

+ O I A d t - 7 ) Fit- T ) OS (wot+e IF(t) ZCOSW t0

Fig. 14. DS used to combat mult ipath.jammers w ill be presented in this paper, but references suchas [28] -[30] provide a reasonable description of how thesejammers affect system performance.

V. CODE DIVISION MULTIPLE ACCESS (CDMA)As s well known, the two most comm on multiple accesstechniques are frequency division multiple accessFDMA)

and time division multiple accessTDMA). In FDMA , llusers transmit simultaneously, but use disjoint frequency bands.In TDMA, all users occupy the same RF bandwidth, buttransm it seq uentially in time . When users re allowed totransmit simultaneously in time and occupy the same RFbandw idth as well, some oth er means of separating the signalsat the receiver must beavailable, and CDMA [also termedspread-spectrum multiple accessSSMA)]rovides thisnecessary capab ility.In DS CDMA [31 ] -[33 ], each user is given its own code ,which is ap proximately orthogonal (Le., has low cross correl-ation) with he codes of the othe r users.However,becauseCDMA systems typically are asynchronous (i.e., the transitiontimes of the data symbols of the different users do no t haveto coincide), the design problem is much more complicatedthan hat of having, say, N u spreading sequences with uni-formly low cross correlations such as the Gold codes dis-cussed in Section I11 and in the Appendix. As willbeseenbelow, the key parameters in a DS CDMA system are both thecross-correlation and the partial-correlation functions, and thedesign and optimization of code sets with good partial-correl-ation properties canbe found in many references such as

The system is shown in Fig. 13. The received ignal sW I , P 4 1, and P I .given byr ( t ) = ~ ~ d ~ ( t~ ) p ~ ( tJ co s (mo t + + n,( t> (38)

N U

i= 1

whered i ( t ) = message of ith user an d equals k 1p i ( t ) = spreading sequence waveform of ith userA i = amplitude of ith carrierB i = random phase of it h carrier uniformly distributed

r j = random time delay of ith user uniformly distrib-

T = symbol duration

in [0,27~]uted in [0, TI

n,(t ) = additive'white Gaussian noise.Assuming that the receiver is correctly synchronized to thekth signal, we can set both Tk and 6 k to zero without losingany gene rality. The final test statistic ou t of the inte grate-and-dum p receiver of Fig. 14 is given by

(39)where doub le frequency terms have been ignored.of N u -1 terms of the formConsider the second term on the RHS of (39). It is a sum

Notice that, because the ith signal is no t, in general, in syncwith the kth signal, di(t - ~ ) ill change signs somewh we inthe interval [0, r ] 50 percent of the time. Hence, the a.bove


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PICKHOLTZ e t al .: THEORY OF SPREAD-SPECTRUM COMMUNICATIONS 869integral willbe the sum of two partial correlations of p j ~ : t )an d P k ( t ) , rather than one total cross correlation. Therefore,(39) can be rewritten

i= 1ifkwhere

and

Notice that he coefficients in front of &(Ti) an d $ j k ( i j )can independently have a plus or minus sign due to the datasequen ce of the ith Signal. Also notice that ~ ~ ~ ( 7 ~ )k dik(l i )is the total cross correlation between the ith and kt h spreadi:lgsequences. Finally, the continuous correlation functions&k(7) k Bik(7) can be expresse d in terms o f the discrete evlmand odd cross-correlation functions, respectively, that weredefined in Section 111.While the code design problem in CDMA svery crucialin determining system performance, of potentially greaterimpo rtance in DS CDMA is the so-called near-far problem .Since the N u users re typically geographically separated ,a receiver trying to detect he kt h signal might be mul:hcloser physically to, say, the th ransmitter rather than hekth transmitter. Therefore, if each user transmits with eqtalpower, the signal from he th ransmitter willarrive at hereceiver in question with a larger power than that of the kt hsignal. This particular problem is often so severe that 1)sCDMA cannot be u sed.An alternative to DSCDMA, f course, is FH CDMA[36] -[4 0]. If each user is given a different hopping patteIn ,and if all hopping patterns are orthogon al, the near-far prob-lem will eolved (except for possible spectral spilloverfrom one slot into adjacent slots). However, the hoppillgpatterns arenever truly orthogonal. In particular, any tirlemore than one signaluses the same frequency ata givminstant of time, interference will result. Events of this type aresometimes referred to as hits, and these hits become moreand m ore of a problem as the numb er of users hopping ovel afured bandwid th increases. As is the case when FH is employ cdas an antijam techniqu e, error-correction codin g can be usl:dto significant advantagewhen combined with FHCDM.4.FH CDMA systems have been considered usingone hopper bit, multiple hops per bit (referred to as fast frequen,:yhopping or FFH), and multiple bits per hop (referred to asslow frequency hopping or SFH). Oftentimes the charactc:r-istics of the channel over which the multiple users transnitplay a significant role in influencing which type of hoppilgone employs. An example of this is themultipath channl:l,which is discussed n the next section.

It isclearlyof interest to consid er the relative capacityof a CDMA system compared to FDMA or TDMA. In a per-fectly linear, perfectly synchronous system, the number oforthog onal users fo r all three system s is the sam e, since thisnumber only depends upon the dimensionality of the overallsignal space. In particular, if a given time-band width prod uctG p isdivided up nto , say, G p disjoint time intervals forTDMA, it canalsobe divided into N binary orthogonalcodes (assume that Gp = 2 for somepositive integer m ).m e differences between th e three multiple-accessingtaihniques become apparent whenvarious eal-world con-straints are imposed upo n the ideal situation described above.For exam ple, one attractive feature o f CDMA s tha t it doesnot require the network synchronization that TDMA requires(i.e., f one iswilling to give up som ething in perform ance,CDMA can be (and usually is) operated in an asynchronousmanner). Another advantage ofCDMA is hat it s relatively easyto add additional users to the system. However, probably thedominant reason for considering CDMAs the need, inaddition, or some type of external interference rejectioncapability such as multipath rejection oresistance tointentional jamming.For a n asynchronou s system, even gnoring any near-farproblem effects, the number of users the system can accom-modate is markedly less than Gp. From [31] and [3 5] , arough rule-of-thumb appears to be thata system with pro-cessing ain Gp can support approximately Gp/lO users.Indeed, from [31, eq. (17)] , the peaksignalvoltage to rmsnoise voltage ratio, averaged over all phase shifts, time delay s,and data symbols of the multipleusers,isapproximately givenby- N u - 1 - 12S N R = [3 ~ p-21where the overbar indicates anensembleaverage. From thisequ ation , i t can be seen t ha t, given a value of E&,,, (Nu -l)/Gp should be in the vicinity of 0.1 in order not to have anoticeable effect on system performance.

Finally, oth er factors such as nonline ar receivers influencethe performance of a multiple access system, and, for exam ple,the effect of a hard limiter on a CDMA system is treated in[451-

VI. MULTIPATH CHANNELSConsider a DS binary PSK co mm unication system operatingover a channel which has more than one path linking thetransmitter to the receiver. These different paths might con-sist of several discrete paths, each one w ith a different attenu -ation and time delay relative to the others, or it might con-sist of acontinuu m of paths. The RAKE system describedin [ I ] is an example of a DS system designed to operate ef-fectively in a multipath environment.For simplicity, assume initially there are just wo path s,a direct path and a single multipath. If we assume the timedelay the signal incurs in propagating over the direct path issmaller than that incurred in propagating over the single


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87 0 IEEE TRANSACTIONS ON COMMUNICATIONS, V O L . COM-30,O. 5 , MAY 1982multip ath, and if it is assumed that the receiver s synchro-nized to the time delay and RF phase associated with thedirect path, then the ystem is as shown in Fig. 14 . The receivedsignal is given by

r ( t ) = A d ( t ) p ( t ) COS &lot + d d ( t - ) p ( t- )CO S (&lot+ e ) +n,(t) (41)where T is the differential time delay associated with the twopaths and is assumed to be in the interval 0 < T < T , 6 is arandom phase uniformly distributed in [0, 2n], and a is therelative attenuation of themultipath relative to the directpath. The output of the integrate-and-dump detection filteris given by

g ( T ) = A + f * d p ( ~ ) a Ai (7 ) ] cos e (4 2)where p ( ~ ) nd ;(T) are partial correlation functions of thespreading sequence p ( t ) and are given by

l TP ( 7) A F P ( t )P ( t - ) t (43)and

l Tb(7) 2 F P ( t )P ( t - ) t . (44)Notice that the sign in front of the second term on theRHS of (42) canbeplus or minus with equal probabilitybecause this term arises from he pulse preceding the pulse

of interest (Le., if the ith pulse is being detec ted, this term arisesfrom the i - 1 h pulse), and this latter pulse will be f the samepolarity as thecurrent pulse only 50 percent of the time.If the signs of these two pulses happen to be the same, andif T > T, where T, is the chip duration, hen p ( 7 ) + { (T )equals the autocorrelation function of p ( t ) (assuming that afull period of p ( t ) is contained n each T second symbol),and this latter quantity equals -(l/L) , where L is the periodof p ( t ) . In other words, the power in the undesired compon entof the received signal has been attenuated by a factor of L 2 .If the sign of the preceding pulse is opposite to that of thecurrent pulse, the attenuation of the undesired signal will beless than L 2 , and typically can be much less than. L 2 . This isanalogous, of course, to the partial correlation problem inCDMA discussed in the previous section.The case of more than two discrete paths (or a continuumof paths) results in qu alitatively the same effects in hatsignals delayed by amoun ts outside of+T c seconds about acorrelation peak in the autocorrelation function of p ( t ) areattenuated by an amou nt determined by the processing gainof the system.If FH is employ ed instead of DS spreading, improve mentin system perform ance is again possible, bu t through a differ-ent mechanism. As was seen in the two previous sections, FHsystems achieve their processing gain through interferenceavoidance, not interference attenuation (as in DS systems).

This same qualitative difference is true again if the interfere:nceis multipath. As long as the signal s hopp ing fast enoughrelative to the differential time delay between the desiredsignal and themu ltipath signal (or signals), all (or most)of the multip ath energy will fall in slots that are orthogona lto the slot that the desired signal currently occupies.Finally, the problems treated in this and the previous twosections are often all present in a given system, and so the useof an appropriate spectrum -spreading technique can alleviateall three problems at once. In [41] and [ 4 2 ] , th eointproblem of multipath and CDMA is treated, and in [43] and[4 4 ] , the joint problem of multipath and intentional nter-ference is analyzed. As indicated in Section V , if only multipleaccessing capability is needed, there are systems other hanCDMA that can be used (e.g., TDMA). How ever, when m ulti-path is also a problem, the choice of CDMAas the multipleaccessing technique isespecially appropriate since the samesignaldesignallows bo th many simultaneous users and m-proved performance of each user individually relative to themultipath channel.In the case of signals transmitted over channels degradedby both multipath and intentional interference, either factorby itselfsuggests the consideration of a spectrum -spreadingtechnique (in particular, of course, thententionalnter-ference), and when all three sources of degradation are presentsimultaneously, spread spectrum is a virtual necessity.

VII. ACQUISITIONAswe have een in the previous sections, pseudon oisemodulation employing direct sequence, frequency hopping,and/or time hopping is used in spread-spectrum system:s toachieve bandwidth spreading which is large compared t o the

bandwidth required by the information signal. These PN modu-lation techniques are typically characterized by their very lowrepetition-rate-to-bandwidth ratio and, as a result, synchroni-zation of a receiver to a specified modulation constitutesa major problem in the design and operation of sprsead-spectrum comm unications systems [46] -[50].It is possible, in principle, for spread-spectrum receiversto use matched filter or correlator structures to synchronize tothe incoming waveform. Consider, for example, a direct-sequence amplitude modulation synchronization system asshow n in Fig. 15(a). In this figure, the locally generated codep ( t ) is available with delays spaced one-half of a chip (TJ2)apart to ensure correlation. If the region of uncertainty ofthe code phase s N , chips, 2N, correlators are employed.If no infor matio n is available regarding the chip unce rtaintyand the PN sequence repeats every, say, 2047chips, then 4094correlators are employed. Each correlator is seen to exanlineh chips, after which the correlator outputs V,, V I , .-,V 2 ~ , - - 1 re compared and the largest output is chosen.As h increases, the probability of m aking an error n syn-chronization decreases; however, the acquisition time in-creases. Thus, h is usually chosen as a compromise between theprobability of a synchronization error and the time to acquirePN phase.A second example, in which FH synchronization is em-ployed, is show n in Fig. 15(b ). Here the spread- spectrum signal


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PICKHOLTZ et a l . : THEORY OF SPREAD-SPECTRUM COMMUN IC.\TIONS

IncomingDS signal

I

()--Tk*andpass filter square device law m elay hops

square Jawfilterevice 1 ho p

code start Thresho ld

V.

87


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872 IEEE TRANSACTIONSN COMMUNICATIONS, V O L . COM-30, No. 5, MAY 1982hops over, for example, m = 500 distinct frequencies. Assumethat the frequency-hopping sequence is fl , 2, ...,f, and thenrepeats. The correlator then consists of m = 500 mixers,each followed by a bandpass filter and square law detector.The delaysare inserted so that when the correct sequenceappears, the voltages V I , V 2 , . e , V , will occur at he sameinstant of time at he adder and will, therefore, with highprobability, exceed the threshold level indicating synchron-ization of the receiver to the signallWhile the above techniquFof using a bank of correlatorsor matche d filters provides a means for rapid acquisition, aconsiderable reduction in complexjty, size, and receiver costcan be a chieved by using a single correlator or a single matchedfilter and repeating the procedure for each possible sequenceshift, However, these reductions are paid for by the increasedacquisition time needed when performing a serial rather than aparallel operation. One obvious question of interest is there-fore the determination of the .tradeoff between the nu mberof parallel correlators (or matched filters) used and the costand time to acquire. It is interesting to note tha t this tradeoffmay become a mo ot poin t in several years as a result of therapidly advancing VLSI technology.Nomatterwhat synchronization technique is employed,the time to acquire depends on the length of the correlator.For .example, in the system depicted in Fig. 15 (a), the inte-gration is performed over h chips where h depends on thedesired. probability of making a synchronization error (i.e.,of deciding that a given sequence phase is correct when indeedit is not), the signal-to-the rmal noisepower ratio, and thesignal-to-jammer power ratio. In addition, in the presence offading, the fading characteristics affect the number of chipsand hence the acquisition time.The importance that one should attribute to acquisitiontime, comp lexity, and size depends upon the intended appli-cation. In tactical military communications systems, whereusers are mobile and push-to-talk radios are employed, rapidacquisition is needed. However, in applications where syn-chronization occurs once, ay, each day, he time to syn-chronize is no t a critical parameter. In e ither case, nceacquisition has been achieved and the communication hasbegun, it is extremely importit not to lose synchronization.Thus , while the acquisition process involves a search throughthe region of time-frequency uncertainty and a determinationthat the locally generated code and the incoming code aresufficiently aligned, the next step, called tracking, is neededto ensure t h a t the close alignment is maintained. Fig. 16 showsthe basic synchronization.system. In this system, the incomingsignal is first locked in to th e loca l PN signal genera tor usingthe acquisition circuit, and then kept in synchronism using thetracking circuit. Finally, the data are demodulated.

One popular method of acquisition iscalled the slidingcorrelator and is shown in Fig. 17. In this system, a singlecorrelator i s used rather han L correlators. Initially, theoutput phase k of the local PN generator is set to k = 0 and apartial correlation is performed by examining h chips. Ifthe integrator output falls below the threshold and thereforeis deemed too small, k is set to k = 1 and the procedure isrepeated. Th e determination that acquisition has taken place

, TRACKING 1 >. I;DATADEMODCIRCUITS --4A hv

LOCAL PN--L SIGNAL t SYNC -GENERATOR CONTROL

A

r v vACQUISITION

CIRCUITS-ig . 16. Funct ional d iag ram of synchronization subsystem.is made when the integrator outp ut VI exceeds the thresholdvoltage V T ( ~ ) .It should be clear that in the worst case, we may have toset k = 0 , 1, 2, -, an d UV,-l before finding thecorrectvalue of k . If, during each correlation, X chips are examined,the worst case acquisition time (neglecting false-alarm anddetection probabilities) is

In he 2N,-correlator system, Ta cq ,rn ax= TJ, and s o wesee that there is a time-complexity tradeoff.Another technique, proposed by Ward [46] , called rapidacquisition by sequential estimation, is illustrated in Fig. 18.When switch S is in position 2, the shift register forms a PNgenerator and generates the same sequence as the inpu t signal.Initially, in order to synchronize the PN generator to theincoming signal, switch S is thrown to position 1. The firstN chips received at he nput are loaded into the register.When the register is fully loaded, switch S is thrown toposition 2. Since the PN seque nce generator generates; thesame sequence as the incoming waveform, the sequencesat positions 1 and 2 mus t be identical. Th at such is the case isreadily see-n from Fig. 19 which show s how the code p ( t - Tc )is initially generated. Comparing this code generator to thelocal generator shown in Fig. 18; we see that with the switchin position 1, once the register s filled, the outpu ts of bo thmod 2 adders are identical. Hence, the bit stream at positions1 and 2 are the same and switch S can be thrown to position2 . Once switch S is thrown to position 2 , correlation is begunbetween the incoming code p ( t - T,) in white noise and thelocally generated PN sequence. This correlation is performedby first multiplying the wo waveforms and then examiningh chips in the integrator.When no noise is present, the N chips are correctly loadedinto he shift register, and therefore he acquisition time isTacq = NT,. However,when noise is present, one or :morechips may be incorrectly loaded into the register. The resultingwaveform a t 2 will the n not be of the sam e phase as the se-quence generated at 1. If the correlator output after h7, ex-


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PICKHOLTZ et ai.: THEORY O F SPREAD-SPECTRUM COM MUNIC/,TIONS$I

873Ic+GENERATOR AND CLOCK PULSESFig. 17. The .slidingcorrelator.

Fig. 18 . Shift regi iter acqu isition circuit.

3 mo d2 :-adder

shift register

~ s p ~ t - j T C )Fig. 19. Th eequivalent ransmitter SRSG.


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874 IEEE TRANSACT IONS ON COMMUNICATIONS, V O L . COM-30, NO. 5 , MAY 1982

ceeds the threshold voltage, we assume that synchronization hasoccurre d. If, howevel, the outp ut is iess than the thresholdvoltage, switch k is thrown to position 1, the register s re-loaded, and the procedure is repeated.Note hat in bo th Figs. 17 and 18, correlation occurs fora time AT, before predicting whether or not synchronismhas occurred. If, however, the correlator output is examinedafter a time nT , and a decision made at each n < X as towhether 1) synchronism has occurred, 2) synchronism hasnot occurred, or 3) a decision c annot be m ade with sufficientconfidence and therefore an additional chip should be ex-amined, then he average acquisition time canbe reduceds u b s t ~ t i a l l y .

One can approximately calculate, he mean acquisition timeof a parallel search acquisition system, such as the systemshown in Fig. 1 5, by noting that after integrating over X chips,a correct decision wiil be made with probability PD where PDis called the probability of detection . If, however, an incorrectoutput is chosen, we will, after examining an additionalh chips, again make a determination of the correct output.Thus , on the average, the acquisition time is

Tats = ~ T , P D + ~ ~ T , P D ( ~ - P D ) + ~ ~ T ~ , ( ~ - ~ D ) ~*.XTC- (46)PD--

where it is assumed that we co ntinue searching every X chipseven after a threshold has been exceeded. This is not, ingeneral, the wayan actual system would operate,but doesallow a simple approximation t o the rue acquisition time.Calculation of the mean acquisition time when singthe sliding correlator shown in Fig. 17 can be accomplishedin a similar manner (again making the approximation that wenever stop searching) by noting hat we are initially offsetby a random number of chips A as shown in Fig. 20(a).After the correlator of Fig. 17 finally slides by these Achips, acquisition can be achieved with probability P o . (Notethat this P D differs from he PD of (4 6), since the atterPD accounts for false synchronizations due to a correlatormatch ed to an incor rect phase having a larger outpu t voltagethan does the correlator matched to thecorrect phase.) If,due to an incorrect decision, synchronization is not achieved

at that time, L additional chips must then be examined beforeacquistion canechievedagain with probability P D ) .

We first c alculate the average time needed t o slide by the Achips. To see how this time can change, refer to Fig. ;!O(b)which indicates the time required if we are not synchronized.X chips are integrated, and if the integrator output VI < V ,(the threshold voltage), a 4 chip delay s generated, and wethen process an additional i\ chips, etc. We n ote that in orderto slide A chips in 3 chip intervals, this process must occur2A times. Since each repetition takes a time (X i-)T,., thetotal elapsed time is 2A@ + i ) T c .Fig. 20(b) assumes tha t at theend of each examinationinterval, VI< V ,. However, if a false alarm occurs and VI >V,, no slide of T c / 2 will occur until after an additional h chipsare searche d. This is shown in Fig. 20(c). In this case, the totalelapsed time is 2A(h + i ) T , + AT,. Fig. 20jd) shows the: casewherealselarms occurred twice. Clearly, neither theseparation between these false alarms nor where they occuris relevant. The total elapsed time is now 2A(h + 4)7, +2hT,.In general, the average elapsed time to reach the correctsynchronization phase is

- n= 1= 2A(h + i ) T , + h T c P ~(i - F) (47)

where PF is the false d ar m probability. Equa tion (47) is fora givenvalue of A . Since A is a random variable which isequally likely to take on any integer value from 0 to L-1,F s / ~ust be averaged over all A . Therefore,

Equation (48) is the average time needed to slide throughA chips. If, after sliding through A chips, we do no t detectthe correct phase, we must now slide through an additionalL chips. The mean time to do this is given by (47) , with Areplaced by L . We sha ll call thi s time TsIL(49)


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PICKHOLTZ et a l . : THEORY OF SPREAD-SPECTRUM COMMUNIC kTIONS 875

. to da ta demod, .

l o c a l EN sequence ..p ( t + T ) 3

.;&;1,3 ED1

, '.iJ'i*

clockoopl o c a lg en era to rPN + J CO f i b t e r

Tp ( t - + + T I

band:>assd e t e c t o r'f i l e renvelope

EDFig. 21 . Delay-locked loop f o ~racking direct-sequence PN signals.

The mean time to acquire a signalcan now be written a:;

or

VIII. TRACKINGOnce acquisition, or coarse synchronization, has beeraccomplished, tracking, or fine synchronization, takes place.Specifically, this must include chip synchronization and, fo1coherent systems, carrier phase locking. In many practicalsystems, no data are transmitted for a specified time, suffi.ciently long to ensure that acquisition has occurred. Duringtracking, data are transmitted and detected. Typical referencesfor tracking loops are [51] -[54].The basic tracking loopor a direct-sequence spread-spectrum system using PSK data trans mission is show n in Fig.2 1 . The incoming carrier at frequency fo is amplitude modu-

lated by the product of the data d ( t ) and the PN sequencep ( t ) . The tracking loop contains a local PN generator whichis offset in phase from the incoming sequence p ( t ) by a time Twhich is ess than one-half the chip tim e. To provide "fine"synchronization, the local PN generator generates two se-quences, delayed from each other byone chip. The twobandpass filters are designed to have a tw o-sided bandwidthB equal to twice the data bit rate, i.e.,

In this way the data are passed, but the product of the twoPN sequences p ( t ) and p ( t T T c / 2 + T) is averaged. Th eenvelope detector eliminates the ata since Id(t) = 1 .As a result, the outpu t of each envelope detector is approx-imately given by

where R p ( x ) is the au tocorrelation function of the PN wave-form as show n in Fig. 7(a). [See Section I11 for a discussion ofthe characteristics of R p ( x ) . ]The output of the adder Y(t) s shown in Fig. 2 2 . We seefrom this figure that, when T is positive, a positive voltage,proportional to Y , nstructs the VCO t o increase its frequency,thereby forcing T to decrease, whilewhen T is negative, a


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876 IEEE TRANSACTIONS ON COMMUNICATIONS, V O L. COM-30, NO. 5 , MAY 1982Y L

/ /

Fig. 22. Variat ion of Y with 7.negativevoltage instructs he VCO to reduce its frequency,thereby forcing 7 to increase toward 0.When the tracking error r is made equal to zero, an outputof the local PN generator p(r + 7) = p ( r ) is correlated with theinput signal p ( r ) .d ( t ) cos (a,,+ e ) to formp2(r )d( r ) COS (oat + e) = d( r ) CO S (oat + e) .

This despread PSK signal is inputted to the data dem odulatorwhere the data are detected .

An alternate technique for synchronization of a DS systemis to use a tau-dither (TD) loop. T his tracking loop is a delay-locked lo op with only a single arm, as show n in Fig. 23(a).The control (or gating) waveforms g( r ) , &r) , and g(r) areshown in Fig. 23(b), and are used to generate bo th arms ofthe DLL even though only on e arm is present. The TD loop isoften used in lieu of the DLL because of its simplicity.The operation of the loop is explained by observing thatthe control waveforms generate the signalV,(t) =g(r)p(r + 7 - T c / 2 )+ gr)p(r + 7 + Tc/2). (53)

Note that either one or the other, but not both, f these wave-forms occursat each instant of time. The voltage V,(r) thenmultiplies the incoming signald( r )p ( r ) CO S (oat + e) .

The ou tput of the bandpass filter is thereforeEf(f>= d(t)g(t) p(t)p(t + 7 + Tc/2) I

+ d(t)g(r) Ip(t)pO + 7 - Tc/2) I (54)where, as before, he average occurs because the bandpass

filter isdesigned to pass thedata and control signals, bu tcuts offwellbelow the chip rate. The data are eliminatedby the envelope detector, and (54) then yieldsE&) =g(t) lRp(r Tc/2)l + g ( t ) l R p ( 7 - T c / 2 ) 1 . (55)The input Y(r) o the loop filter isY ( r )= Ed(r )g r )

=g( f ) IR p (7 -T c /2 ) I -~ (~ ) IR p ( .-~ c / 2 ) I (5 )where the - sign was introd uced by th e inversion causetl by

The narrow-band loop filter now averages Y(r).Sinceeach term is zero half of the time, the voltage into the VCOclock is, as befo re,

g ( 9 .

Vc ( 0= lRp(r- Tc/2) I - Rp(r+ T c / 2 ) . (57)A typical tracking system for an FSK/FH spread-spectrumsystem is show n in Fig. 24. Waveforms are shown in Fig,. 25.

Once again, we have assumed that, although acquisition hasoccurred, there is still an error of 7 seconds between transi-tions of the incoming signals frequencies and the locallygenerated frequenc ies. The bandpass filter BPF is made :suffi-ciently wide to pass the prod uct signal V,(r) when V,(t )andV2( r ) are at he same frequency fi, but sufficiently na.rrowto reject V p ( r ) when Vl(r) and V2(r) are atdifferent fre-quencies fi and fi+ . Thus, the output of the envelope d.etec-to r V,(r), shown inFig. 24, is unity when V,(r ) and .V2(r)are at the same frequency and is zero when V,(t ) and .V2(r)are a t different frequencies. From Fig. 25, we see that V,(t) =V,(t) Vc(t)and is a three-level signal. This three-level signalis filtered t o form a dc voltage which, in this case, presentsa negative voltage to the VCO.


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PICKHOLTZ et al. : THEORY O F SPREAD-SPECTRUM COMM UNICP.TIONSg: 877

T


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878 IEEE TRANSACTIONS ON COMMUNICATIONS,VOL. COM-30, O. 5 , MAY 1982

envelopedetectorvg(t)

3 BPF

v2 (t) wavefomlocal FH V C ( t ) LPFb-ynthesizer V,(t)frequency e PN code clocke VcO-*generator4 -r

fHFig. 24 . Tracking loop for FH signals.

incoming k T H l/fH 4signalVI (t) f O f a1l o c a l FHsignalV,(t) 0 f I f I f

V f (t; h tFig. 2 5. Waveforms for tracking an FH signal.

It is readily seen that when V 2 ( t ) as frequency transitionswhich precede those of the incoming waveform Vl ( t ) , hevoltage into he VC O willbe negative, thereby delaying thetransition, while f the local waveform frequency transitionsoccur after he incoming signal frequency transitions, thevoltage into the VC O will be positive, thereby speeding up thetransition.The role of the tracking circuit is to keep the offset time

T sma ll. However, even a relatively small T can have a majorimpacton he probability of error of the received data.Referring to the DS system of Fig. 21, we see that if T is notzero, the nput to thedata demodulator is p ( t ) p ( t + ~ ) d ( t )cos (wo t + e ) rather than p 2 ( t ) d ( t ) cos (mot + 0 ) = d ( t )cos (wo t + e). The datadem odulator removes the carrierand then averages the remaining signal, which in this case s

p ( t ) p ( t + ~ ) d ( t ) . he result is p( t )p( t + T)d(t). Thus, theamplitude of the data has been reduced by p ( t ) p ( t +3R p ( 7 ) < 1. For example, if T = TJ10, hat -data amplitudeis reduced to 90 percent of its value, and the power is reducedto 0.81. Thus, the probability of error in correctly detectingthe data is reduced from.=?(e)t o


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and at an Eb/qO of 9.6 dB, Pe is increased from lo-' t3

IX. CONCLUSIONSThis tutorial paper looked t some of the theoreticalissues nvolved in the design of a spread-spe ctrum comm uni-cation system. The topics discussed included the characte1-istics of PN sequ ences, the resulting processing gain when usin::either direct-sequence or frequency-hopping antijam consideI-ations, multiple access when using spread spectrum, multipatheffects, and acquisition and tracking systems.No attem pt was made to present other than fundamentalconcepts; indeed, to adequately cover the spread-spectrunlsystem completely is the task for an entire ext [55], [ 5 6 ] .Furthermore, to keep this paper reasonably concise, t h t !authors chose to ignore both practical system consideration:;such as those encountered when operating at, say, HF, VHF,or UHF, 2nd technology considerations, such as the role o:..

surface acoustic wave devices and charge-coupledevice::in the design of spread-spec trum systems.Spread spectrum has for far too long been considered :.technique with very limited applicability. Such is not thccase. In addition to military applications, spread spectrunis being co nsidered for co mm ercial applications such as mobilctelephone and m icrowave comm unications in congested areasThe authorshop e hat this tutorial will result in moraengineers and educators becoming aware of the potential ofspread spectrum, the dissemination of this information in theclassroom, and the use of spread spectrum (where appropriate)in the design of communication systems.

APPENDIXALGEBRAIC PROPERTIES OF LINEAR FEEDBACK

SHIFT REG ISTER SEQUENCESIn order to fully appre ciate the tudy of shift registersequences, it is desirable to introduce he polynomial repre-sentation (or generating function) of a sequence

If thesequen ce is periodic with period L , .e.,

L - 1C(x)(l - - x L ) = C , x i Q R ( x ) (A21i= 0

with R ( x ) the (finite) polynomial representation of on eperiod.Thus, for any periodic sequence of period L ,R (x)

1 - x L 'C(x) = -. deg R ( x )


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880 IEEE TRA NSACTIONS ON COMMUNICATIONS, V O L . COM-30, NO. 5 , MA Y 1982

initialcontents bo b1

Fig. 26 . Binary modularhift register generato rwitholynomialfn/r(x)= 1+ a l x +a2$ + - . + a r - l x r - l + x r .It is easy t o see (by multiplying and equating coefficients oflike powers) that if

1 1- -f ( x ) 1+ a ,x + a2x2 + *'.+arxr= c +ClX + c2x2+ ...= C(X)

thenr

cn = x k c n - k ,k= 1

so tha t (except for initial conditions) f lx ) completely describesthe maximal length sequence. Now what properties must f ( x )possess t o ensure tha t the sequence is maximal length? Asidefrom the fact that f(x) must divide 1 + x L , i t isnecessary(but not sufficient) that f ( x ) be irreducible, i.e., f(x) # f, x)- f 2 ( x ) . uppose that flx) = fl x) f2 (x ) with fl x) of degreer1, 2(x) of degree r 2 , and r1 + r2 = r . Then we can write, bypartial fractions,1 4 x 1 P(x) deg a ( x )< I- = - +-.

f(x> fl (x) f2@> ' deg P(x)< 2The maximum period of the expansion of the first term is2"-1 and that of the second term is 2r2-1. Hence, theperiod of l/f(x) < east common multiple of (2"--1, 2"-1)< 2'-3. This is a contradiction, since if f (x) were maximallength, the period of l / f lx ) would be 2'-1. Thus, a necessarycondition tha t he LFSR is maximal length is that f ( x ) isirreducible.A sufficient condition is that f(x) is primitive. A primitivepoiynomial of degree r over GF(2) is simply one for w hich theperiod of the coefficients of l/f(x) is2'-1.However, addi-tional insight canbe had by examining the oots of f ( x ) .Since f(x) is irreducible over G F ( 2 ) ,we must imagine that the

roots are elements of some arger (extension) field. Suppos ethat 01 is such an element and that f la) = 0 = 01+ t r y - 1cyr-' + + a, 01 + 1 or

a ' = a r - l a r - l + - a * +a l a+ 1 . (A5)We see that al l powers of a can be expressed in terms of alinear combination of a'-' , -, a, since any powerslarger than r - maybe reduced using A5).Specifically,suppose we have some power of a hat we represent as

/ 3 b b o + b l a + b 2 a 2* * . + b r - l a ' - l . (A61Then if we multiply this p by a ~d use (A5), we obtain

Pa = b r - 1 + (b o + b r - 1al)a + (6 1 + br- 1a2)a2+ ...+ ( b r - 2 + b r - l a r - ])a'-' (A7)

The observations abovemaybe expresse d in anoth er, :morephysical way with the ntroduction of an LFSR in modularform [called a modu lar shift register generator (MSRG)]shown in Fig. 26. The feedback, modulo 2 , is between thedelay elements. The binary contents of the register at any timeare shown as b o , b , .-, b r u 1 . This vector canbe iden-tifiedwith /3 as

~ = b o + b ~ ~ + ~ - + b r ~ ~ a ' - ' + - + [ b o , b l , - ~ b rthe contents of the first stagebeing identified with th e co-efficient of (YO, hose of the second stage with the coefficientof a', etc. A fter one clock pulse, it is seen that the re,@stercontents correspond to

= br- 1 + (b o + b r - 1). + *..+ (br- 2 + br - la,- 1)CY'-- b r - 1, '- 3 br-2 + br - la,- 11.


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PICKHOLTZ et a l . : THEORY OF SPREAD-SPECTRUMOMMUNICATIONS 881Thus, he MSRG s an a-multiplier. Now f ao a , a2,a3 ..,L = 2'- 1 are all distinct, we call a a primitive elemer tof GF(2'). The register in Fig. 26 cycles through all statcs(starting in any non zero state), a nd hence generates a maxim:dlength equence. Thu s,another wayofdescribing that t h epolynomial ~ M ( x )s primitive (or maximal length) is that : thas a primitive element in GF(2') as a root.Theresn intimate relationship between the MSR(;shown in Fig. 26 and the SSR G shown in Fig. 4. From Fig. 25it is asily een that the ou tpu t sequence C satisfies therecursion

aL- l

Multiplying both sides by x n and summ ing yields

n=--m k= O n=Or - 1 - 1

r- 1

k=O n=Oor

"_ 1C(X) = g & J ( x )x ' c kx-kC(X) . (A101

k=O

gM(x) is the first term on the right-hand side of (A9) and ira polynomialofdegree < r whichdepends on the initialstate. Then we have

(A1 11where

(recall that in GF(2) , minus is the same as plus) is the char.acteristic (or connection) polynomial of the MSRG. Since thtsequence C [of coefficients of C (x) ] when f M ( x ) s primitivedepends onZy on f M ( x ) (discounting phase), the relationshillbetween the SSRG and the MSRG which generates the samtsequence is

(A1 211&-(x) is called the reciprocal polynom ial of f ( x ) and is ob.tained from f ( x ) by eversing the orderof the coefficientsThere are several good tables of irredu cible and primitivepolynom ials available [ 2 ] [ 5 ] , [ 6 ] and although he tables

TABLE ITHE NUMBER OF MAXIMAL LENGTH LINEAR SRG SEQUENCESOF DEGREE r = X(r) = @ ( 2 r- ) / r

", P- 2- 11 12 3 13 74 15 2

6

_-( r )

5 I 31.. '

910111213141516171819202122

6312 725 551 1

1 , 0 2 32,0474,0958,191

16,38332,76765,535

131,071262,143524,287

1,048,5752,097,1514,194,303

618164860

17 614 4630756

1,8002,0487,7108 , 064

27,59424,00087,672

120,032

do not list all the prim itive polynom ials, algorithms exist [ 7 ]whichallowone to generateall primitive polynom ials of agivendegree if oneof them is know n. Thenumber h(r) ofprimitive polynomials of degree r is [ 4 ]N 2 r - 1)

h(r) = rwhere $ ( m ) is thenum ber of integers less than m whichare relatively prime to m (Euler totient fu nction). The growthof this num ber with r is shown in TableI.The lgebra f L F S R s is useful in onstructing odeswithuniformly low cross correlation know n asGoldcodes.The nderlying principle of these codess ased n thefollowing theorem [151.

If fl ( x ) is the mininial polynomial of the primitive elementa E GF(2') an d f t (X) is the minimal polynomial of a', whereboth fl ( x ) an d f t ( X ) are of degreer and

r+l2 2 + 1 ,2 2 + 1 ,

t = -+ 2 r oddr even,then heproduct f ( x ) 4 l ( x ) f r ( x ) determines nLFSRwhichgenerates 2 + 1 different sequences correspondingto the 2 + 1 states in distinct cycles) of period 2 - 1, an dsuch that for any pair C' nd C",


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88 2 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-30, NO. 5, M A Y 1982

(b)Fig. 27 . Tw o implementations of LFSR which generate Gold codes oflength 2 5 - 1 = 3 i with maximum cross correlation t = 9 . (a ) LFSRwith f(x) = 1 + x + x3 + x9 + x10. (b) LFSR which generates se-: ( 1 + ~ 2 x 5 ) . ( 1 + ~ 2 + ~ 4 + ~ 5 ) =ienc es corresponding to f(x) =1+ x + $ +x9 +x10.

Futhermore, RCIC"(T)s only a three-valued function orby iriteger T.A minimal polynomial of a is simply the smallest degreemonic3 polynomial for which a s a root. With the help of atable of primitive polynomials, we cai~dentify m him al poly-nomials of powers of a and easily construct Gold codes: Forexarinple, if r = 5 and. t = 23 + 1 = 9 ,using [2] we find thatfl(x) = 1 -k x2 + x5' an d f9(x) = 1 + x + x4 + x5.Thenf (x ) = 1 + x + x 3 + x9 + x1 . The two ways to representthis LFS R (in MSRG\ form) are show n Fig. 27. Fig.27(a)shows one long nonmaximal length register of degree 10whichgenerates sequences of period 25 .- 1 = 31. Since there are2 l 0 - 1 possible nonzero initial states, the number of initialstates that result in distinct cyClesi~(2~ )/(z5 - ) = 2' +1 = 33. Each of these initial states specifies a different G oldcode of length 31. Fig. 27(b) shows how the same result canbe obtiiined by adding the outpu ts of the wo MLFSR's ofdegree 5 together modulo two. T his follows simply from theobservation that the sequence(s) generated by f i x ) are just thecoefkcients in the expansion of l/f(x) = l/f1(x).f9(x). Byusing partial fractions, one cansee tha t the resulting coeffi-cients are the (modulo two) sum of the coefficients of likepowers in the expansion of l/fl(x) and l/f9(x). Naturally, thesequence resulting will depend on the relative phases of thetwo degree-5 registers. As before, there are (2 l - 1)/(25 - ) =

3 A monicpolynomial is on ewhose oefficient of it shighestpower is unity.

25 + 1 = 33 relativephases which result in33differentsequences satisfying the cross-correlation bound given by

GLOSSARY OF SYMBOLS{0: 1) feedback taps for LFSR.One-sided bandwidth (Hz) for data signal(s).One-sided bandwidth (Hz) of basebandspread-spectrum signal.Generating function of C,; C(x) =( 0 , l ) LFSR sequence.{ 1 , -1) LFSR sequen ce.Dimensionality of underlying signal space.Data sequence. waveform.Initial offset, in chips, of incoming signal andlocally generated code.Direct sequence.Energy/iilformation bit.Jammer energy over the correlation interval.Ener@/symbol.Characteristic (connection) polynomial of anLFSR, f i x ) = 1 + a l X + + ~ i - 1 '--'X'.Chip rate; T , = l/f,.Frequency hopping.Processing gain.Jammer signal waveform.

cnxn


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Number of frequencies jammed by partia.- [ M .Wozencraftnd 1. M .acobs, Principles of Communicationband jammer.Period of PN sequence. W .Golomb, Shift RegisterSequences . Sa nFrancisco, CA:Implementation losses. [5] R. W. Marsh, Table of Irreducibleolynomialsver GF( 2)Number of chips examined during each searcn ThroughDegree 19 . Washington, DC: NSA,957.[6 ]W. tahnke, Primitive inary polynom ials, M ath .Compur . ,inhe processf acquisition. vol.7,p. 977-980, Oct. 1973.Number of binary maximal length PN%%das [7 ] E. R.Berlekamp, AlgebraicCoding heory. New York: Mc-of d eg ee r (length L = 2 - ). Graw-Hill , 1968.Signal alphabe t size. [8] J . L.Mass ey, Shift-register synthesis and BCH decoding , IEEETrans . Inform. T heor y , vol. IT-1 5, pp . 122-1127, Ja n. 19 69.Jamming m argin. [9 ] N . G . deBrui jn,A combinatorial problem , in KoninklijkeNumber of ch ips/bit or number of dimensi0r.s Neder landsAkademieVanWetenschappenProc.. 1946, pp. 758-of spread signal space. , .,:764.[10&E. J. Groth , Generation of binary equenceswithcontrollableNumber of frequencies in FH . comolexitv. IEEE Trans . Inform. Theory, vol.T-17,p. 288-

ftgineering. New York: Wiley, 1965.Holden Day, 1967.

Number of chips in ncertainty region ; I tstart of acquisition.Num ber of users in CDMA system.One-sided white noise power spectral densityP J /2 f , = power density of jammer.Additive wh ite G aussian noise (AWGN).Spreading sequence waveform.Probability of detection.Probability o f error.Probability of false alarm.Jammer power.Noise pow er.Signal powe r.Pseudonoise sequence.Num ber of stages of shift register.Received waveform.Data rate (bits/s).Autocorrelation function.Cross-correlation fu nction of two (periodic:)? 1 sequences en,$(1/T) j i p ( t ) p ( t - ) d t (partial correlationfunction).Transmitted signal waveform.Power spectral density of spreading sequencewaveform [also denoted Sss(f)] .Signal-to-noisepower ratio.Signal-to-jammerpower ratio.Signal or symbol d uration.Chip duration.Time to h op one frequency; l/TH = hoppingrate.Correlator output voltage.

@/Hz).

( U T ) S?P(t) P ( t - w .

ACKNOWLEDGMENTThe authors wish to thank the anonymous reviewers fc .rtheir constructive suggestions in the final preparation of t h s [271paper.

[281REFERENCES

[ I ] R .A .Scholtz,Theorigins of pread-spectrumcommunications, [29i[2 ]W .W. Peterson nd E. J. Weldon, Jr . , Er r orCor r ec t ingCode: , , [301this issue, pp. 822-854.2n ded .Camb ridge, MA: M.I.T. Press, 1972.

296,=Ma{1971.E. L.Key, An analysis fhe tructure and complexity ofnonlinear binaryequence generators, IEEE Trans. Inform.T heor y , vol. IT-22, pp. 732-736, Nov. 1976.H .Beke r, Multiplexed hift egister equences,presented atCR YP T0 81 Workshop, Santa Barbara, CA, 1981.J. L. Mass ey and J. J. Uhran, Sub-baud coding, in Proc. 13thAnnu.Allerton Conf. Circui tan dSys t .Theory, Monticello, IL,J. H. Lindholm, An analysis of the pseudo randomness propertiesof the subsequences of long m-sequences, IEEE Trans. Inform.T heor y , vol. IT-14, 1968.R .Gold ,Op timal binary equences or pread pectrum mul-tiplexing, IEEE T r ans . Inform. T heor y , vol. IT-13, pp. 619-621,1967.D. V . Sarwate and M . B . Pursley, Cross correlation properties ofpseudo-random and related sequences, Proc. IEEE, vol. 68 , pp.598-619, May 1980.A .Lempel ,M .Cohn, and W. L. Eastman, A new class ofbalanced inaryequenceswith ptimal utocorrelation rop-erties, IBM Res. Rep. RC 5632, Sept. 1975.A . G . Konheim, Cryptography,APrimer . New York:Wiley,1981.D. L. Schi l l ing, L. B.Milstein, R. L. Pickholtz, and R. Brown,Optimization of the processing gain of an M-ary direct sequencespread spectrum communication system, IEEE Trans. Commun.,vol. COM -28, pp. 1389-1398, Aug. 1980.L. B. Milstein, S . Davidovici, and D. L. Schilling, The effect ofmultiple-tonenterferingignals on a directequencepreadCOM-30, pp . 43 64 46 , Mar . 1982.spectrum communication system, IEEE Trans. Co mmun. , vol.S . W. Houston, Tone and noise jamming performanceof a spreadspectrum M-ary FSK and 2, 4-ary DPSK waveforms, in P r oc .Nut . Aer os p . Elec t r on . Co nf . , June 1975, pp. 51-58.G . K .Huth , Optimization of coded pread pectrum ystemsperformance, IEEE Trans . Commun. , vol.COM-25,pp. 763-770, Aug.1977.R. H. Pettit, A susceptibility analysis of frequency hopped M-aryNCPSK-Partial-band noise on CW tone jamming , presented atthe Symp. Syst . Theory, May 1979.L. B. Milstein, R. L. Pickholtz, D. L. Schilling, Optimization ofm un. , vol . COM-28, pp. 1062-1079, July 1980.the processinggain ofan FSK-FH system, IEEE Trans. Com-M. K. S imon and A. Polydoros, Coherent detection of frequency-hopped quadrature modulations in the presence of jamming-Part I:QPSK and QASK; Part 11: QPR class I modulation, IEEE Trans.Commun . , vol. COM-29, pp. 1644-1668, Nov. 1981.A. J.Viterbi and I. M .acobs,Advan ces in coding andmodulation for noncoherentchannelsaffected by fading,partialband, andmultiple ccess interference, in Advances in Com-municat ionSys tems, vol. 4. New York:Academic,1975.J. M. Aein and R. D. Turner, Effect of co-channel interference onCPSK carriers. IEEE Trans . Commun. . vol . COM-21. DD. 783-

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790, July 1973.R. H. Pettit . Error orobabilitv for NCFSK withinear FM. ..

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[31]M . B . Pursley,Performanceevaluation or phase-coded spreadspectrummultiple-access communication-Part I: Systemanaly-sis, IEEE Trans.C o m m u n . , vol.COM-25 ,pp. 795-799, Aug.1977 .[32] K . Yao,Errorrobab ility of asynchron ous spread-spectrummultiple access communication systems, IEEE Trans. Commun. ,[33] C. L. Weber , G . K. Huth, and B.H.Batson, Performanceconsiderationsof codedivisionmultipleaccess systems, IEEE

T r a n s . V e h . T e c h n o l . , vol. VT-30, pp. 3-10, Feb. 1981.[34]M . B . Pursleyan dD .V.Sarwate,Performanceevaluation orphase-coded pread pectrummultiple-access communication-Part 11. Code equenceanalysis, IEEETrans. C o m m u n . , vol.[35]M . B . Pursley ndH .F.A.Roefs, Numerical valuation ofcorrelation parameters for optimal phases of binary shift-registersequences, IEEE Trans.C o m m u n . , vol.COM -25, pp. 1597-1604, Aug. 1977.[36] G . Solomon,Optimal requencyhoppingsequence s or multipleaccess, in Pro c . 1973 Symp. Spread Spec t rum Commun., vol. 1,[37]D . V. Sarwate ndM . B . Pursley, Hoppingpatterns or fre-quency-hoppedmultiple-accesscommunication, in Proc.1978IEEE In t . Conf . Commun. , vol. 1, pp. 7 .4 .1-7.4.3.[38]P . S . Henry,Spectrumefficiency of a frequency-hopped-DPSKspread spectrum mobile radio system,EEE Trans. Veh. Te chno/ . ,

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Theory of Spread-spectrum Communications - A Tutorial - Pickholtz 1982

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