spread spectrum communications handbook

We will motivate the study of spread-spectrum systems by analyzing a ple game, played on a finite-dimensional signal space by a communicationssystem and a jammer, in which the signal-to-

Part 1

INTRODUCTION TO

SPREAD-SPECTRUM

COMMUNICATION

Chapter 1

A SPREAD-SPECTRUM OVERVIEW

Over thirty years have passed since the terms spread-spectrum (SS) andnoise modulation and correlation (NOMAC) were first used to describe aclass of signaling techniques possessing several desirable attributes for com-munication and navigation applications, especially in an interference envi-ronment What are these techniques? How are they classified? What arethose useful properties? How well do they work? Preliminary answers areforthcoming in this introductory chapter

We will motivate the study of spread-spectrum systems by analyzing a ple game, played on a finite-dimensional signal space by a communicationssystem and a jammer, in which the signal-to-interference energy ratio in thecommunication receiver’s data detection circuitry serves as a payoff func-tion The reader is hereby forewarned that signal-to-interference ratio cal-culations alone cannot illustrate many effects which, in subtle ways, degrademore realistic performance ratios, e.g., bit-error-rate in coded digital SS sys-tems However, the tutorial value of the following simple energy calculationssoon will be evident

sim-1.1 A BASIS FOR A JAMMING GAME

The following abstract scenario will be used to illustrate the need for trum spreading in a jamming environment, to determine fundamentaldesign characteristics, and to quantify one measure of SS system perfor-mance Consider a synchronous digital communication complex in which

spec-the communicator has K transmitters available with which to convey mation to a cooperating communicator who possesses K matching receivers

infor-(see Figure 1.1) Assume for simplicity that the communication signal space

has been “divided equally” among the K transmitters Hence, with a width W ss available for communicating an information symbol in a T ssec-

band-ond interval (0, T s), the resultant transmitted-signal function space of

dimension approximately 2T s W ssis divided so that each transmitter has a

D-dimensional subspace, D ⫽ 2T s W ss /K, in which to synthesize its output

signal Denote an orthonormal basis for the total signal space by ck (t), k⫽

1, 2, , 2T s W ss, i.e.,

(1.1)where the basis functions may be complex valued, and ( )* denotes conju-

gation Then the signal emitted by the k-th transmitter is of the form

(1.2)where

(1.3)

and {a j} is a data-dependent set of coefficients We will refer to the above as

an orthogonal communication system complex of multiplicity K.

Of course, real systems generally radiate real signals The reader may wish

to view m k (t) as the modulation on the radiated signal Re{m k (t) exp (jv c t⫹u)} Without loss of generality, we can dispense with the shift to RF duringthis initial discussion

In a simplified jamming situation, the signal z i (t) observed at the i-th

receiver in the receiving complex might be

(1.4)

where n i (t) represents internally generated noise in the i-th receiver, J(t) is

an externally generated jamming signal, and the K-term sum represents the

total output signal of the transmitter complex One signal processing

cj1t2c*k1t2 dt ⫽ e1, j ⫽ k

0, j ⫽ k

Figure 1.1. The scenario for a game between a jammer and a communication tem complex.

sys-strategy for the i-th receiver is to project the received signal onto the set of basis functions for the i-th transmitter’s signal space, thereby calculating

(1.5)

In the absence of jamming and receiver noise, the properties of the

trans-mitted signals These representations indicate that, in general, the projection

(1.8)

Gaussian, independent, and identically distributed for different values of i and/or j, represents the relatively benign receiver perturbations in the

clas-sified, and from the jammer’s point of view, hopefully are unpredictable bythe receiver

(1.9)

pur-pose, and henceforth, will be assumed zero (In keeping with this tive aspect of communication system design, we also assume that the jammer

sum in (1.9) can be partitioned into K parts, the i-th part representing the

A similar partition holds for the total transmitted signal energy E s, namely

(1.11)

(1.10), (1.11) are a direct result of the orthogonality requirement placed onthe signals produced by the transmitter complex

The above signal representations and calculations have been made underthe assumption that the channel is ideal, causing no attenuation, delay, ordistortion in conveying the composite transmitted signal to the receiver com-plex, and that synchronous clocks are available at the transmitter and

impor-tant considerations have been suppressed in this initial discussion, so that

we may focus on one major issue facing both the communication systemdesigner and the jammer designer, namely their allocations of transmitter

energy and jammer energy over the K orthogonal communication links.

1.2 ENERGY ALLOCATION STRATEGIES

Within the framework of an orthogonal communication system complex of

multiplicity K, let’s consider the communicator and jammer to use the

respec-tively, to the K links.

Communicators’ strategy: Randomly select K S links, K S ⱕ K, for equal

utilized

Jammer’s strategy: Randomly select K Jreceivers for equal doses of

jammed

a single output for the system user The performance measure to beemployed here, in determining the effectiveness of these strategies, will notdepend on specifying a particular diversity combining algorithm

The randomness required of these strategies should be interpreted asmeaning that the corresponding adversary has no logical method for pre-dicting the choice of strategy, and must consider all strategies equally likely.Furthermore, random selection of communication links by the transmittershould not affect communication quality since all available links are assumed

to have equal attributes (Examples of link collections with non-uniformattributes will be considered in Part 2, Chapter 2.)

The receiving complex, having knowledge of the strategy selected for

receivers remaining in operation However, the amount of jamming energy

denotes a binomial coefficient) Under the equally likely strategy

assump-tion, the probability that the jammer strategy will include exactly N of the

(1.12)

where

(1.13)(1.14)Using (1.12)—(1.14), it is possible to compute the expected total effective jam-

(1.15)

E being the expected value operator Despite the complicated form of

Pr(N), it can be verified that

strat-This idealized situation leads one to conclude, based on (1.17), that thereceiver can minimize the jammer’s effectiveness energy-wise by not using

orthogonal communication system complex should be made as large as

using a single one of the K available communication links, is called a pure spread-spectrum strategy This strategy, with its accompanying threat to use

any of K orthogonal links, increases the total signal-to-jamming ratio from

designated receiver, and therefore qualifies as an anti-jam (AJ) modulation

technique

energy gain EG of the signalling strategy played on the orthogonal

com-munication system complex

(1.18)

Hence, the energy gain for a pure SS strategy is the multiplicity factor of thecomplex In this fundamental form (1.18), the energy gain is the ratio of the

definition of energy gain makes no distinction between diversity and SS

rec-ognize the fact that the quantity called the multiplicity factor, or energy gain

in this chapter, is sometimes referred to as the processing gain of the SS tem This nomenclature is by no means universally accepted, and we will

is the data rate in bits/second It is easily verified from (1.18) that

Two key assumptions were made in showing that the pure SS strategy isbest: (1) The channel is ideal and propagates all signals equally well, and (2)the proper performance measure is the total effective jamming energy Ifeither of the above assumptions is not acceptable, then the jammer’s strat-egy may influence the performance measure, and the optimum diversity fac-

use of bit-error rate (BER) as a performance measure implies that the mum diversity factor can exceed unity

opti-Let’s summarize the requirements characteristic of a digital trum communication system in a jamming environment:

spread-spec-1 The bandwidth (or equivalently the link’s signal-space dimension D)

required to transmit the data in the absence of jamming is much less than

2 The receiver uses inner product operations (or their equivalent) to

con-fine its operation to the link’s D-dimensional signal space, to

demodu-late the signal, and thereby to reject orthogonal jamming waveformcomponents

3 The waveforms used for communication are randomly or domly selected, and equally likely to be anywhere in the available band-

width (or equivalently, anywhere in the system’s 2T s W ssdimensional nal space).

sig-The term pseudorandom is used specifically to mean random in appearance

but reproducible by deterministic means

We will now review a sampling of the wide variety of communication tem designs which possess SS characteristics

sys-1.3 SPREAD-SPECTRUM SYSTEM CONFIGURATIONS

AND COMPONENTS

A pure spread-spectrum strategy, employing only a single link at any time,can be mechanized more efficiently than the system with potential diversity

factor K, shown in Figure 1.1 In an SS system, the K transmitter-receiver

pairs of Figure 1.1 are replaced by a single wideband communication linkhaving the capability to synthetize and detect all of the waveforms poten-tially generated by the orthogonal communication system complex.The pure

SS strategy of randomly selecting a link for communication is replaced with

an equivalent approach, namely, selecting a D-dimensional subspace for

random selection process must be independently repeated each time a bol is transmitted Independent selections are necessary to avoid exposingthe communication link to the threat that the jammer will predict the sig-nal set to be used, will confine his jamming energy to that set, and hence,will reduce the apparent multiplicity and energy gain to unity

sym-Three system configurations are shown in Figure 1.2, which illustratebasic techniques that the designer may use to insure that transmitter andreceiver operate synchronously with the same apparently random set ofsignals The portions of the SS system which are charged with the respon-sibility of maintaining the unpredictable nature of the transmission are

double-boxed in Figure 1.2 The modus operandi of these systems is as

follows:

1 Transmitted reference (TR) systems accomplish SS operation by mitting two versions of a wideband, unpredictable carrier, one (x(t)) modulated by data and the other (r(t)) unmodulated (Figure 1.2(a)).

trans-These signals, being separately recovered by the receiver (e.g., one may

be displaced in frequency from the other), are the inputs to a tion detector which recovers the data modulation The wideband carrier

correla-in a TR-SS system may be a truly random, wideband noise source,unknown by transmitter and receiver until the instant it is generated foruse in communication

2 Stored reference (SR) systems require independent generation at

trans-mitter and receiver of pseudorandom wideband waveforms which areidentical in their essential characteristics (Figure 1.2(b)) The receiver’s

SS waveform generator is adjusted automatically to keep its output inclose synchronism with the arriving SS waveform Data detection, then,

is accomplished by cross-correlation The waveform generators are tialized prior to use by setting certain parameters in the generating algo-rithm, thereby denying the jammer knowledge of the waveform set beingused (even if the jammer has succeeded in determining the generator’sstructure)

ini-3 Matched filter (MF) systems generate a wideband transmitted signal by

pulsing a filter having a long, wideband, pseudorandomly controlledimpulse response (Figure 1.2(c)) Signal detection at the receiveremploys an identically pseudorandom, synchronously controlled,matched filter which performs the correlation computation Matched fil-ter systems differ from SR systems primarily in the manner in which theinner-product detection process is mechanized, and hence, have exter-nally observed properties similar to those of SR systems

Certainly, a pure TR system has several fundamental weaknesses including:(1) The system is easily spoofed since a jammer can in principle transmit apair of waveforms which are accepted by the receiver, (2) relatively poor per-formance occurs at low signal levels because noise and interference are pre-sent on both signals which are cross-correlated in the receiver, (3) the data

is easily determined by any listener who has access to both transmitted nals, and (4) the TR system’s two channels may require extra bandwidth andmay be difficult to match Some of the problems associated with TR systemsmay be mitigated by randomly changing parameters of one of the commu-nication links (e.g., by protecting one of the TR wideband links with an SR-like technique) Historical examples of SR-protected TR systems will begiven in the next chapter

sig-Spread-spectrum waveform generators for SR systems employing the lowing general modulation formats have been built The output of an SS

fol-waveform generator is given the generic name c(t) and is a (possibly

com-plex-valued) baseband representation of the SS waveform

if necessary, extended periodically to give

(1.19)

The utility of this type of signal is limited by the problem of ing recordings to transmitter and receiver so that reuse of a waveform

distribut-is not necessary

2 Frequency hopping (FH): Assuming that p(t) is a basic pulse shape of

c1t2 ⫽ a

n

has the form

(1.20)

In all likelihood, the complex baseband signal c(t) never physically

appears in the transmitter or receiver Instead, the pseudorandomly

synthe-sizer to produce a real-valued IF or RF carrier-modulated version of c(t).

process

3 Time hopping (TH): Assuming that the pulse waveform p(t) has

(1.21)

each interval containing a single pulse pseudorandomly located at one

4 Direct sequence (DS) modulation: Spread-spectrum designers call the

waveform

(1.22)

pseudo-random number generator is linearly modulated onto a sequence of

5 Hybrid modulations: Each of the above techniques possesses certain

advantages and disadvantages, depending on the system design tives (AJ protection is just one facet of the design problem) Potentially,

objec-a blend of modulobjec-ation techniques mobjec-ay provide better performobjec-ance objec-atthe cost of some complexity For example, the choice

(1.23)

may capture the advantages of the individual wideband waveforms

Three schemes seem to be prevalent for combining the data signal d(t) with the SS modulation waveform c(t) to produce the transmitted SS signal x(t).

1 Multiplicative modulation: Used in many modern systems, the

transmit-ted signal for multiplicative modulation is of the form

nal, or multiple frequency-shift-keyed (MFSK) data on a FH signal.Thesemodulation schemes are the ones of primary interest in this book.

2 Delay modulation: Suggested for use in several early systems, and a

nat-ural for mechanization with TH-SS modulation, this technique transmitsthe signal

(1.25)

3 Independent (switching) modulation: Techniques (1) and (2) are

sus-ceptible to a jamming strategy in which the jammer forwards the mitted signal to the receiver with no significant additional delay (a severegeometric constraint on the location of the jammer with respect to thetransmitter and receiver), but with modified modulation This repeaterstrategy, which if implementable, clearly reduces the multiplicity factor

trans-K of the SS system to unity, can be nullified by using a transmitted

sig-nal of the form

(1.26)

Here the data signal, quantized to M levels, determines which of M

key assumption here is that even though the jammer can observe theabove waveform, it cannot reliably produce an alternate waveform

c (j(t)) (t), j(t) ⫽ d(t), acceptable to the receiver as alternate data

modu-lation The cost of independent data modulation is a clearly increased

hardware complexity.

The data demodulation process in a digital SS system must compute innerproducts in the process of demodulation That is, the receiver must mecha-nize calculations of the form

(1.27)

base-band signals usually appears in modulated form as a real IF or RF signal

(1.28)The inner product (1.27) can be recovered from the modulated signal(s) inseveral ways, as illustrated in Figure 1.3 For example, the receiver can first

and then proceed with straightforward correlation or matched filteringoperations using baseband signals On the other hand, as indicated in Figures1.3(c) and 1.3(d), there are alternative ways to compute the inner product,

which do not require that both signals be shifted to baseband first.

In all cases, the heart of the SS receiver is its synchronization circuitry, andthe heartbeats are the clock pulses which control almost all steps in forming

Figure 1.3. Examples of correlation-computing block diagrams.The dashed portions

of the diagrams can be eliminated when the modulations m T (t) and m R (t) are real

and, in addition, the local oscillator is phase-coherent, i.e., fe⬇ 0 Solid line processing

is often called the “in-phase” channel, while the dashed line processing is called the

“quadrature” channel.

Figure 1.3. Continued.

the desired inner product Recovery of Re{(m T , m R )} and Im{(m T , m R)}requires three levels of synchronization.

1 Correlation interval synchronization: Correlators require pulses to

indi-cate when the interval of integration is to begin and when it is to end

In the bandpass correlator of Figure 1.3(d), interval sync not only vides the timing for the sampling operation, but also initializes the nar-rowband filter’s state to zero at the beginning of each correlationinterval Typically, in DS systems these signals correspond to the datasymbol clock pulses In FH systems in which the data symbol timeexceeds the hop time, the interval sync pulses must indicate the dura-tion of a single frequency, since correlation operations spanning randomphase transitions are not generally useful

pro-2 SS generator synchronization: Timing signals are required to control the

epoch of the system’s SS waveform generator’s output and the rate atwhich that output is produced Direct sequence systems employ a clock

3 Carrier synchronization: Ideal reduction of the SS signal to baseband in

the receiver is possible if a local oscillator (or oscillator network) is able whose output is in frequency and phase synchronism with the

carrier sync is often available in DS systems, but usually only frequencysynchronism is attained in FH systems

In some SS systems, the above synchronization signals are derived from asingle clock; in others, the carrier local oscillator is independent of the clocksignals which control its modulation Automatic control circuitry generally

is included to align the receiver’s clocks for proper demodulation of theincoming signal, although some systems have been built in which ultrastableclocks are initially aligned and then are allowed to drift in a free-runningmode until communication is concluded Proper operation of the correlationcomputing circuits generally requires control of the symbol clock epoch to

within a small fraction of the correlation interval’s duration T Similarly, it

is necessary to adjust the SS generator clock’s ticks to within a small tion of the reciprocal of the SS modulation’s short-term bandwidth, i.e., thebandwidth of the energy spectrum of the SS reference waveform within acorrelation interval Section 1.5.3 will indicate that the SS generator clock

respec-tively, to maintain correlator operation at nearly maximum output signal els, as required

lev-Frequency synchronous operation of correlation detectors requires thatthe phase drift between the incoming carrier (excluding SS modulation) andthe receiver’s local oscillator, over a correlation interval, be a fraction of

dur-ing the correlation computation (Phase synchronism of the local oscillator

fre-quency synchronous device requiring that the input to its narrowband filter

be centered in its passband to an error tolerance of a fraction of the rocal of the correlation time

recip-Output threshold crossing techniques, similar to those used in radar tion, are an alternative to MF output sampling in Figure 1.3(b), and may havehigher tolerance to synchronization errors than SR/DS systems However,any realized tolerance to synchronization errors implies a potential weak-ness to repeater jamming

detec-1.4 ENERGY GAIN CALCULATIONS FOR TYPICAL SYSTEMS

The following examples illustrate the energy gain calculation for basic SSsystems In each case, the SS system is viewed (in the terminology of Section1.2) as replacing an orthogonal communication system complex, and its mul-

tiplicity factor K determined, thereby evaluating the energy gain of the

sys-tem via (1.18)

modulation as in (1.22), multiplicative data modulation (1.24), and channel phase-synchronous detection (solid line portion of Figure 1.3(a))

pseudorandom quantities which are known to the receiver, but unknown to

pseudorandom variables and data modulations is given by

(1.29)

冮Ts 0

magnitude p/2 cause orthogonality, the jammer is forced to view his form selection problem as being defined for an orthogonal communication

wave-system complex with multiplicity factor K given by

(1.30)

receiver are then carried out over individual hop times, and the correlator’s

to the receiver but not to the jammer, are th two pseudorandom hopping

possible data symbols Two similar FH waveforms have inner product

(1.31)Orthogonality between two such waveforms is guaranteed, regardless of the

(1.32)

such orthogonal frequency waveforms are used by the communication

during each hop the jammer mmust contemplate combatting a pure SS

Therefore, during each hop time a single link in the orthogonal system plex requires four dedicated orthonormal basis functions (e.g., sines and

hops By the same reasoning the number of basis functions available to the

Example 1.3. One possible hybrid SS communication system employs TH,

FH, and DS modulations to produce the wideband waveform

(1.34)

(1.35)

Two such hybrid SS signals have inner product given by

(1.36)

otherwise Orthogonality of these waveforms can be achieved, regardless of

the following conditions holds:

The signal variables, known a priori to the receiver, but not the jammer, are

observe the signal in a 2M-dimensional space whose basis over the interval

Therefore, the nominal energy gain EG of this system is

(1.40)

indicates that the energy gain for this hybrid SS system in a closely packed

1.5 THE ADVANTAGES OF SPECTRUM SPREADING

We have seen the advantages of making a jammer counteract an ensemble

of orthogonal communication systems The bandwidth increase which mustaccompany this SS strategy has further advantages which we will outlinehere

1.51 Low Probability of Intercept (LPI)

Spectrum spreading complicates the signal detection problem for a veillance receiver in two ways: (1) a larger frequency band must be moni-tored, and (2) the power density of the signal to be detected is lowered inthe spectrum-spreading process The signal may have further desirableattributes based in part on LPI, such as low probability of position fix (LPPF)which includes both intercept and direction finding (DFing) in its evalua-tion, or low probability of signal exploitation (LPSE) which may appendadditional effects, e.g., source identification

sur-For now let’s simply evaluate the power spectral density (PSD) of an SSsignal to determine its general properties Consider a signal of the form

(1.41)

(1.42)

limTSq

Here F2Tis the time-limited Fourier transform

(1.43)

second signal segment Conversion of this energy density to a power

to the relation

(1.44)

for two basic SS modulation designs.

1 DS/BPSK modulation: The waveform corresponding to a DS signal

antipodally modulated by binary data is

(1.45)

Hence,

(1.46)

that

(1.47)

where P(f) is the Fourier transform of the chip pulse p(t) Inserting this

transform into (1.46) and ensemble averaging over the data sequence

We consider two possible assumptions regarding the nature of the direct

(1.49)

This PSD is sketched in Figure 1.4 for several possible pulse shapes Note

with some period N, i.e.,

(1.50)

Hence, nothing random remains in (1.48) and the expected value

operator can be dropped Furthermore, the value of the sum on m in

sponding to the smallest value of k such that kN c is a multiple of N.

sequence when the greatest common divisor of N and N c is unity

Summing like terms in (1.53) and using the symmetry of R c (k) gives

2 FH/FSK modulation: The use of M-ary data, frequency-shift-keyed onto

an FH signal at a rate of M Dhops per data symbol, creates the lation

cm⫹kcm⫽ aN⫺1

m⫽0cmNc⫹kcmNc

where p(t) is a hop pulse waveform which is non-zero only in the

random phase variables

(1.57)

the required Fourier transform is given by

(1.58)

where F denotes the ordinary Fourier transform Evaluating the

trans-form in (1.58), substituting in (1.57), and averaging over the randomphase sequence, gives

(1.59)

where P(f) is the transform of the hop pulse Assuming that the data

sequence elements are independent, identically distributed random

(1.60)

We now consider two models for the hopping sequence

a Random Hopping When {f n} is a sequence of identically distributedrandom variables with values selected from a frequency set Ᏺ accord-

ing to a probability distribution function P F(ⴢ), then

Notice that the PSD of the deterministically generated sequence will

match that of the randomly generated sequence with P F (f) ⫽ N f /N Furthermore, the number of hops M Dspanned by a data symbol is not a fac-tor in the spectral density of the resultant signal in either case

A rough measure of the hop pulse transform’s width is 1/T h Since integer

multiples of 1/T hare required for orthogonal randomly phased tones of

1

K⫺1 n⫽ ⫺K

E5 ƒ P1f ⫺ fn⫺ d2 ƒ26

Sm1f2 ⫽ lim

KSq

12KTh aK⫺1 n⫽ ⫺K

E 5 ƒ F2KTh5mT1t26 ƒ26

tion T h , the assumption that the radiated tones d ⫹ f, and , areorthogonal leads to the conclusion that the minimum required bandwidth

The PSDs which we have just calculated correspond to asymptoticallylong-term ensemble-averaged, time-normalized, energy spectra Attempts atinterception of these signals in reality will be made over short time inter-vals, usually much shorter than the periods of the pseudorandom numbergenerators driving the SS waveform generator Hence, signals with compa-rable PSDs may have differing short-term energy density characteristics, andhence, different LPI capabilities

1.5.2 Independent Interference Rejection and Multiple-Access Operation

We have already discussed the energy gain achievable against a jammerwhose radiated signal is generated without knowledge of the key parame-ters used in generating the SS transmitter’s modulation We refer to this type

of interference as independent, the connotation also applying to in-band

interference from other friendly communication systems

The ability of a SS system to reject independent interference is the basisfor the multiple-access capability of SS systems, so called because several SSsystems can operate in the same frequency band, each rejecting the interfer-ence produced by the others by a factor approximately equal to its energy

gain This asynchronous form of spectrum sharing is often called spectrum multiple-access (SSMA) or code-division multiple-access (CDMA).

(1.63)

(1.64)impinging on a receiver which is frequency synchronous with the transmit-ted signal and which computes the inner product of the received signal with

receiver correlator is the sum of two terms, corresponding to the desired

input power, are

xI1t2 ⫽ Re5 2PImI1t ⫺ tI2ej12p1fc⫹¢2t⫹fI26,xT1t2 ⫽ Re5 2PTmT1t2ej12pfct⫹f T26,

MMF>Th, 0 f 0 ⫽ MF, 0 d 0 ⫽ M

d 僆 d

f 僆 f

is the carrier frequency offset between the interference and the transmittedsignal The average signal-to-interference energy at the input to the corre-lator is simply

(1.67)

and the frequency-synchronous correlator’s output signal-to-interferencemeasurement ratio is

(1.68)

The above calculation is based on the supposition that outputs from boththe in-phase and quadrature channels of the correlator are necessary, i.e.,

the average only half of the interference power contributes to the

(1.69)

The key to further analysis is the evaluation of the interference level atthe correlator output

(1.70)

modula-tion relative to the reference modulamodula-tion, this shift being inserted to modelthe fact that the interference is assumed asynchronous with respect to allclocks generating the transmitted signal Stored reference modulations (e.g.,

i.e., they may be made stationary by inserting a time-shift random variable

wide-sense stationary random process This assumption and the further

and jamming identically in average power calculations

1k⫺12T

SIRout1f sync2⫽ 2#SIRout1freq sync2

SIRout 1freq sync2⫽ 冓E5 0 vT1k2 026冔

Averaging (1.70) over t Igives

(1.71)

(1.72)

k-th correlation interval.

(1.73)

Certainly when the correlation time T is not a multiple of the reference

mod-ulation generator’s period (and this must be the case to avoid problems with

Using the fact that the modulation forms are normalized to unit energy,and combining (1.65)—(1.68) and (1.71), demonstrates that the improvement

in signal-to-interference ratio achieved in the correlation calculation is

(1.74)

Again we emphasize that (1.74) applies both to asynchronous access interference and independent jamming

binary antipodal modulation and phase coherent reception over a symbol

c(t) as given in (1.22) Then,

(1.75)

Fourier transform ooperation, and P(f) is the Fourier transform of the chip

pulse p(t) Denoting the period of {c n } by N and time-averaging the squared

(1.76)

averag-ing over the zero-mean independent data symbols, which leads to the result(1.52), breaks up the PSD calculation into a time average of short-termenergy spectra

The SIR improvement ratio for a DS/BPSK system is determined by

cor-relation interval

(1.77)

(1.78)

power normalizations, let’s assume that

(1.79)

(1.80)

This lower bound on signal-to-interference ratio improvement is the energygain indicated in Example 1.1

When the interference is spectrally similar to the transmitted signal, e.g.,

in the SSMA case, and both signals possess the spectra of a purely random

SIRout1f sync2

binary sequence modulated on rectangular chip pulses (1.79), then

could use identical SS waveforms, provided the probability that theyarrive in near synchronism at a receiver (for any reason, natural or jam-motivated) is virtually zero

3 Cross-correlation functions and cross-spectra between asynchronousinterference and desired signal are not a specific factor in the signal-to-interference ratio improvement based on long-term averages

Clearly the use of long-term averages in a SIR-based figure-of-merit has led

to these simple results

Similar results can be achieved for other forms of SS modulation the DSexample was particularly simple because the receiver used only one corre-lator Corresponding analyses of SS systems using higher dimensional sig-nal sets, e.g., FH/FSK, must consider the total signal energy and totalinterference energy collected in a set of correlators

1.5.3 High-Resolution Time-of-Arrival (TOA) Measurements

Not all interference waveforms satisfy the independence and randomlyasynchronous assumptions used in Section 1.5.2 to reaffirm the energy gaincapability of SS systems Here are some examples which are illustrated pic-torially in Figure 1.5

1 Multipath: Additional propagation paths from transmitter to receiver

may produce undesirable interference in a correlator synchronized to asignal arriving via a specified path For example, a single additional pathmay produce an interfering signal of the form (1.64) with

at a fixed delay t Icorresponding to the incremental propagation delaybetween the interference path and the communication signal path.

2 Repeater Jamming: This is a form of artificial multipath, in which the

jam-mer attempts to receive the SS signal, somehow alter the data tion, and then broadcast the result Hence, if the modulationmultiplicatively changed, then the signal retransmitted by the jammermay be of the form (1.64) with

modula-(1.83)

encoun-tered over the propagation path through the surveillance/jamming tem

(1.64) are nearly constant or slowly varying over restricted ranges The

propagation path is used for communication On the other hand, cation via a friendly repeater or an indirect path may result in a negative

The average response of a correlation detector to the above types ofsignal-related interference can be determined by evaluating (1.66) or (1.71),e.g.,

Here it is assumed that the receiver’s correlator reference signal m R (t) is

⌬ are the incremental time and frequency shifts incurred by the interference

signal during the k-th correlation interval At this point, we must depart from

in this interference scenario and cannot be used as an averaging variable

time and freqnecy resolution capability of the SS signal structure, i.e., the

abil-ity of the receiver’s correlation detector to discriminate against versions ofthe transmitted signal which do not arrive in synchronism with the receiver’s

clocks The mean-squared value of the integral in (1.84) is the guity function of the waveforms m I (t) and m R (t) at offsets t Iand⌬, and hence,

cross-ambi-we are embarking on a study of the time- and ensemble-averaged ity function provided by an SS waveform/modulation system The theory of

pro-portional to the rms bandwidth of the signal upon which the correlator acts.Hence, one might expect under certain conditions that SS receivers are espe-cially sensitive to synchronization errors, and possess high time-of-arrival(TOA) resolution capabilities

The evaluation of (1.84) for repeater modulation of the form (1.83),

sta-tionary random process, yields

(1.85)

(1.83) and multipath (1.84) respectively, and

(1.86)

Henceforth, for simplicity we set

(1.87)Equation (1.85) indicates that the effect of the jammer’s added modula-

We will now evaluate the multipath interference measure for two basic

SS waveform designs

simply the SS code c(t) in (1.22) Breaking up the integral over the data

(1.88)

The integral in (1.88) can be simplified by using the fact that the pulse

(1.89)Then

n⫽ 1k⫺12Nc a

q m⫽⫺q

d:m>Nc;cmcn

We will carry out the ensemble-averaging process under the

moments

(1.93)

(1.94)and the fact that

(1.95)

Expanding the squared sum in (1.92) as a double sum and simplifying, gives

(1.96)

With the aid of (1.89) and (1.97), the average ambiguity surface (1.96)

interfer-ence’s incremental delay exceeds a chip time This is a firm basis for ing that the TOA resolution of a high energy gain DS/BPSK system isapproximately one chip time

stat-2 FH/FSK modulation: When FH signalling of the form (1.56) is employed,

signal with the reference

(1.98)

Figure 1.6. Normalized RMS detector output as a function of normalized time

mis-match t I /T c for several values of normalized frequency mismatch nT c Random DS

modulation is assumed, with an energy gain of N c⫽ 100 and square chip pulses.

k-th correlator output (1.86) is

(1.99)

(1.101), produces a variety of expressions depending on the values of

N and the data-induced frequency shifts For simplicity, we assume that

all the involved data variables are equal Ensemble-averaging (1.101)produces

capa-Before proceeding, we will develop a useful upper bound on (1.103)

0 Xp1t, n; Th2 02⫽¢ a

f–苸fPF1f¿2PF1f–2 0 Xp1t, f¿ ⫹ f– ⫹ n; Th2 02

⫹eju¿Xp1tI⫺ Th, fNI⫹k⫹ d:1NI⫹k2>MD;

0 vM1k2 02⫽ 0 ejuXp1tI, fNI⫹k⫺1⫹ d:1NI⫹k⫺12>MD;⫺ fk⫺1⫺ dR⫹ n; Th2

⫺tI⫽ NITh⫹ tI, 0ⱕ tI 6 Th,

2

0 vM1k2 02⫽ ` 冮kTh

m

equally likely and uniformly spaced 1/T hHz apart (the minimum ing required for orthogonality over a hop time) For this calculation’spurposes, this assumption is conservative in the sense that the hop fre-quencies in an operational system may be further apart to support FSKdata and maintain the orthogonality of all possible waveforms.Assumingthat the pulse shape is rectangular, the corresponding ambiguity func-tion (1.97) can be overbounded by

This is the basis for stating that a high energy gain FH/FSK communication

frequency-synchronous correlation detectors should be able to reject thedesired FH/FSK signal, if it arrives out of time synchronism by more than

1.6 DESIGN ISSUES

Based on the analyses presented in this chapter, spread-spectrum techniquespromise an attractive approach to the design of communication systemswhich must operate in an interference environment However, major designissues have been obscured thus far by the apparent simplicity of the concept

1 How does the receiver acquire and retain synchronization with thereceived signal’s clocks, especially in the presence of interference?

2 Can the appearance of randomness in modulation selection be achieveddeterministically by a stored-reference SS system?

3 In what ways does the communication vs jamming game change whenthe payoff function is bit-error rate?

4 What are the effects of imperfect system operation/modelling (e.g., chronization errors, non-uniform channel characteristics across the com-munication band) on performance estimates?

syn-5 How should data detectors be designed when the nature of the

inter-ference is not known a priori?

These are a sample of the questions which must be answered before a istic system design can be achieved

real-Several truths have been demonstrated by this introductory treatment.First, excess bandwidth is required to employ randomized signalling strate-

gies against interference That is, spreading the spectrum is necessary in thisgaming approach to interference rejection Secondly, if either side, commu-nicator or jammer, fails to completely randomize its signalling strategy, theopponent in principle may observe this fact, adapt to this failing, and takeadvantage of the situation Barring this event, the energy gain of the SS sys-tem on the average will be achieved by the communicator unless the jam-mer can avoid complying with a rule of the game.

1.7 REFERENCES

In most cases the results of this chapter and the issues raised herein will

be fully discussed in greater detail in later chapters The references listedhere are general references to background material, tutorials, and papercollections

1.7.1 Books on Communication Theory

[1] W B Davenport, Jr., and W L Root, Random Signals and Noise New York:

[4] J M Wozencraft and I M Jacobs, Principles of Communication Engineering,

New York: John Wiley, 1965.

[5] M Schwartz, W R Bennett, and S Stein, Communication Systems and

Techniques New York: McGraw-Hill, 1966.

[6] A J Viterbi, Principles of Coherent Communication New York: McGraw-Hill,

[12] W C Lindsey, Synchronization Systems in Communication and Control.

Englewood Cliffs, NJ: Prentice-Hall, 1972.

[13] W C Lindsey and M K Simon, Telecommunication Systems Engineering.

Englewood Cliffs, NJ: Prentice-Hall, 1973.

[14] R E Ziemer and W H Tranter, Principles of Communications Boston, MA:

Houghton Mifflin, 1976.

[15] J J Spilker, Jr., Digital Communications by Satellite Englewood Cliffs, NJ:

Prentice-Hall, 1977.

[16] S Haykin, Communication Systems New York: John Wiley, 1978.

[17] A J Viterbi and J K Omura, Principles of Digital Communication and Coding.

New York: McGraw-Hill, 1979.

[18] J G Proakis, Digital Communication New York: McGraw-Hill, 1983.

1.7.2 Books on Resolution and Ambiguity Functions

[19] C E Cook and M Bernfeld, Radar Signals New York: Academic Press, 1967 [20] A W Rihaczek, Principles of High Resolution Radar New York: McGraw-Hill,

[22] Proceedings of the 1973 Symposium on Spread Spectrum Communications.

Naval Electronics Laboratory Center, San Diego, CA, March 13—16, 1973.

[23] Spread Spectrum Communications Lecture Series No 58, Advisory Group for

Aerospace Research and Development, North Atlantic Treaty Organization, July 1973 (AD 766914).

[24] R C Dixon, ed., Spread Spectrum Techniques New York: IEEE Press, 1976 [25] R C Dixon, Spread Spectrum Systems New York: John Wiley, 1976.

[26] L A Gerhardt and R C Dixon, eds., “Special Issue on Spread Spectrum

Communications,” IEEE Trans Commun., COM-25, August 1977.

[27] D J Torrieri, Principles of Military Communication Systems Dedham, MA:

Artech House, 1981.

[28] C E Cook, F W Ellersick, L B Milstein, and D L Schilling, eds., “Special Issue

on Spread Spectrum Communications,” IEEE Trans Commun., COM-30,

May 1982.

[29] J K Holmes, Coherent Spread Spectrum Systems New York: John Wiley, 1982 [30] R H Pettit, ECM and ECCM Techniques for Digital Communication Systems.

Belmont, CA: Lifelong Learning Publications, 1982.

[31] MILCOM Conference Record, 1982 IEEE Military Communications

Conference, Boston, MA, October 17—20, 1982.

[32] Proceedings of the 1983 Spread Spectrum Symposium, Long Island, NY.

Sponsored by the Long Island Chapter of the IEEE Commun Soc., 807 Grundy Ave., Holbrook, NY.

1.7.4 Spread-Spectrum Tutorials and General Interest Papers

[33] R A Scholtz, “The spread spectrum concept,” IEEE Trans Commun.,

COM-25, pp 748—755, August 1977.

[34] M P Ristenbatt and J L Daws, Jr., “Performance criteria for spread spectrum

communications,” IEEE Trans Commun., COM-25, pp 756—763, August 1977.

[35] C L Cuccia, “Spread spectrum techniques are revolutionizing

communica-tions,” MSN, pp 37—49, Sept 1977.

[36] J Fawcette, “Mystic links revealed,” MSN, pp 81—94, Sept 1977.

[37] W F Utlaut, “Spread spectrum—principles and possible application to

spec-trum utilization and allocation,” ITU Telecommunication J., vol 45, pp 20—32, Jan 1978 Also see IEEE Commun Mag., Sept 1978.

[38] “Spread Spectrum: An Annoted Bibliography,” National Telecommunications and Information Administration, Boulder, CO, May 1978 (PB 283964).

[39] J J Spilker,“GPS signal structure and performance characteristics,” Navigation,

vol 25, pp 121—146, Summer 1978.

[40] W M Holmes, “NASA’s tracking and data relay satellite system,” IEEE

Commun Mag., vol 16, pp 13—20, Sept 1978.

[41] R E Kahn, S A Gronemeyer, J Burchfield, and R C Kunzelman, “Advances

in packet radio technology,” Proc IEEE, vol 66, pp 1468—1496, Nov 1978 [42] A J Viterbi, “Spread spectrum communications—myths and realities,” IEEE

Commun Mag., vol 17, pp 11—18, May 1979.

[43] P W Baier and M Pandit, “Spread spectrum communication systems,”

Advances in Electronics and Electron Phnysics, vol 53, pp 209—267 Sept 1980.

[44] R L Pickholtz, D L Schilling, and L B Milstein, “Theory of spread spectrum

communications—a tutorial,” IEEE Trans Commun., COM-30, pp 855—884,

May 1982.

[45] N Krasner, “Optimal detection of digitally modulated signals,” IEEE Trans.

Commun., COM-30, pp 885—895, May 1982.

[46] C E Cook and H S Marsh, “An introduction to spread spectrum,” IEEE

Commun Mag., vol 21, pp 8—16, March 1983.

[47] M Spellman, “A comparison between frequency hopping and direct sequence

PN as antijam techniques,” IEEE Commun Mag., vol 21, pp 26—33, July 1983 [48] A B Glenn, “Low probability of intercept,” IEEE Commun Mag., vol 21, pp.

26—33, July 1983.