48.6

AUTOCORRELATION TECHNIQUES FOR WIDEBAND DETECTION OF FH/DS WAVEFORMSIN RANDOM TONE INTERFERENCE

Andreas Polydoros

and

University of Southern California/Axiomatix Corporation

ABSTRACT

A technique for detecting hybrid frequency-hopping/direct-sequence low-probability-of-inter-cept signals in random tone interference is pre-sented and analyzed. It is assumed that the inter-ference is sufficiently random (arbitrary numberof tones, frequencies, phase, etc.) in each hopinterval that notch filtering is impossible. Thekey element is the use of real-time autocorrelationdevices. It is shown that, compared to the unaccep-tably poor performance of a radiometer, almostperfect performance can be achieved in the absenceof thermal noise. Simulation results are also pre-sented which give excellent support to this theory.

1. INTRODUCTION

In a companion paper [11], a scheme was pro-posed for improving the detection of frequency-hopping (FH) waveforms in wideband additive Gaus-sian noise (AWGN) using samples from the auto-correlation domain. It was shown there that, underfairly general operational assumptions, an appro-ximate gain of T1(X,G)-y2 in decision SNR can beHachieved over the energy discriminator (radio-meter), where YH is the hop SNR. This gain--beingdefined on a per-band basis [1, Section 21--has adirect positive impact on the overall system detec-tion capability. The proposed algorithm, albeitinferior to the optimal likelihood ratio test, hadthe advantage of greatly reduced (hence, manage-able) complexity.

The overall approach has been motivated by therecent implementational feasibility of large time-bandwidth product real-time correlators such as SAWdevices. The purpose of this paper is to showthat, depending on the specific scenario at hand,the use of such devices can bring about impressivegains in detector performance, especially whencompared to simpler alternatives such as the radio-meter. In particular, we shall consider the caseof detecting a frequency-hopping/direct-sequence(FH/DS) hybrid waveform when the dominant compo-nent of the observation "noise"l is random toneinterference, arbitrarily located within the hop-ping bandwidth. In addition, it will be assumedthat the total power of this tone interference ismuch greater than that of the background thermalnoise, so the latter can be ignored. It will thenbe shown that, for observation data with a large

Acknowledgement: This work was supported by theOffice of Naval Research under Contract N00014-82-C-0585.

Jack K. Holmes

-Axiomatix Corporation

time-bandwidth product C (defined in [11), a verysimple algorithm operating on the output of a real-time autocorrelation device can achieve almostperfect performance--in stark contrast with thepoor performance of the radiometer.

Hybrid schemes have become increasingly popu-lar spread-spectrum communication choices due tothe enhanced antijam margin which they offer.Adding DS modulation to FH also improves the anti-intercept capability because the "noiselike"appearance of DS makes detection more difficult[2]. On the other hand, multiple tones constitutea common model of nonwhite interference and canemerge in a number of scenarios, i.e., it can beintentional (jamming of the band) or unintentional(multiple users in a broadcast environment, adjacentradar sources, etc.). It is also conceivable thatthe tone interference has been deliberately inser-ted by the communicator in a pseudorandom manner soas to impede the interceptor's task, while it canbe pseudorandomly avoided by the intended receiver.

In all of the cases previously cited, the toneinterference could be filtered out (notch filtering)and various techniques for doing so effectively haverecently been presented [31. The key provision ofthese techniques is, however, that each tone staysat the notched frequency long enough for the filterto adapt; in the context of LPI detection, then,fast FH or low-duty-cycle pulsed interferencecould present a severe challenge for such systems.It is also clear that, as the number of interferingtones increases, the required complexity (number offilters) would soon surpass acceptable limits.

Here we address exactly those severe caseswherein the number of tones per observation inter-val is both large and arbitrary (random), withunknown frequencies, durations and phases. It willsoon be seen that, in such environments, the radio-meter is doomed to fail as a detector. It there-fore comes as a pleasant surprise to conclude thatthe autocorrelation-domain algorithm proposed herecan provide excellent performance independent ofall those interference parameters. The paper isstructured as follows: Section 2 presents thesystem model and proposed algorithm, while Section3 proceeds with the analysis of its performance;that of the radiometer is also outlined here.Section 4 provides the simulation results andconcludes with a brief discussion,

2. SYSTEM MODEL AND PROPOSED ALGORITHM

The overall system structure is similar to

781CH1909-1/83/0000-0781$1 .00 1 9831EEE

that described in [1, Section 21. Therefore, weshall illuminate only those aspects which are uni-que to the present scenario. Again, our concernis the decision rule performance on a per-bandbasis, as defined below. The autocorrelationdomain will be employed to produce a decision sta-tistic since an optimal rule similar to [1,equation (1)], would, in this case, be hard toestablish and too complicated to implement.

Let the FH/DS hybrid signal to be detected berepresented by s(t)=v/'f c(t) coswot, where wo isan unknown frequency within the observed spectralband, c(t) is the DS code of rate RC=Tl and S iscthe signal power. The unknown interference con-sists of M tones (M is a random variable in eachobservation interval, which is equal to the hoptime TH=Rj1), with Ik, wk and fk denoting, respec-tively, the power, radian frequency and phase ofthe kth tone. The total received signal in (O,TH),under hypothesis Hl (signal present), is given by

Mr(t) = VF'i c(t)cosw t+ X v'2kLcos(wkt+k) (1)

Note the absence of thermal noise in (1), as pre-viously discussed. The code c(t) can be modeledas either a random sequence of independent, iden-tically distributed +1's with Pr[c(t)=l]=0.5, or aPN code with a full period equal to TH. As theratio N=TH/TC=RC/RH increases (N denotes the numberof code chips per hop), the performance differencebecomes insignificant, a fact also verified bysimulation.

Let B denote the input observation bandwidth.The presence of the DS code implies that B shouldbe at least equal to Rc or higher, but definitelymuch larger thaR RH. Equivalently, the time-band-width product G=BTH>>1. Furthermore, for simpli-city, we shall assume that all tones have equalpower Ik=I/M, k=l,...,M, where I is the totalinterference power, and they are equi-spaced withinthe bandwidth B. In other words, the frequencyseparation Ifk-fk+ll between adjacent tones equalsB/(M+1), which is much greater than T-1 i.e.,H'Ifk-fk+ll>>TPl. None of the above assumptions iscritical in the forthcoming conclusions; theysimply ease the analytical burden.

The real-time autocorrelation operationproduces the output

Ty(T)= r(t)r(t-T)dt; 0 < T < TH (2)

T

Substituting (1) into (2) and rearranging redefinesy(T) as

I Hy(T) = Sy (T)cosWT+(-)(T -T) I cosW T+

Hf1X nk()+nn(T)] (3)

where yc(T) is the code partial-correlation func-tion (a random variable)

yc(T) = { c(t)c(t-T)dt (4)

and nk(T); k=l,...,M,n (T) are approximately Gaus-sian (via a central limit theorem-type argument),bandpass noise processes (signal x interferenceterms) defined by

najT) =cI(T)coseT--anQ(T)sinwkT

° [kl k(k=O 1XS(T)Jsin CTwhere

aI (u) =

JT

T

ak(T) =fT

T

T)

c(t)cos[(wO-k)t-Pk]dt

c(t)sin[ (w -u -~Id0) k fk]dt

c(t-r)cos[(w -Wk)t-k]dtand T

Sk(T) = c(t-T)sin[(w0-wk)t-PkIdt6kT) J

(5a)

(5b)

(6a)

(6b)

(6c)

(6d)

Before discussing the statistical characteriza-tion of the above noise processes (Section 3), letus examine the noiseless (mean) part of y(r) in(3), as shown in Figure 1. Of particular interesthere are the envelopes of the useful signal (smallshaded triangle) and the interference (large tri-angle), respectively, since the actual components,e.g., the first two terms in (3), are modulated bythe unknown frequencies. The structural differencebetween the two correlations is evident: the DScode superimposed on each hop creates a narrow meanautocorrelation function, since the expected valueof the function yc(T) in (4) is zero for ITI>Tc.Contrary to that, the interfering tones correlatefor the whole interval [O,TH] . Clearly, then, apower (noncoherent) sample at Tl=Tc would measureinterference only1; this sample could be subtractedfrom the corresponding one at TO=O which, under H1,contains the full signal power plus interference.This subtraction would approximately cancel theinterference contribution at TO=O so that, under H1,only the signal would emerge while, under Ho, thestatistic would be almost zero. Thus, the adopteddecision rule is (see Figure 2)

A=y (0) P - Y(T) L A0 (7)

where A0 is a fixed threshold. In the absence ofthermal noise, Ao can be set at a very small(positive) level in order to maximize the detectionprobability. The next section will show that theperformance resulting from (7) is excellent.

3. PERFORMANCE ANALYSIS

We focus here on the statistics of no(T) andn0(T), defined in (5) and (6). Let 9{-}andvar{}indicate the mean and variance, respectively. Itis then easily shown that

ef a CO I=e (-1) }.=- O; j = I,Q

and that

var{a3 (Tr) } = var{ Sk (T)}= {f(Ik() I}k k k

(8)

(9)

In the following, we shall concentrate on offsets

IAny Tl>Tc would also do, especially in the face ofsome uncertainty about Tc.

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T=mTc; m=0,1,2,.. . ,N. In fact, only T=0 and T=Tcare of interest (see (7)). Then,lengthy manipula-tions can establish the following facts:

(a) The variance in (9) is given by

2 (N-m)T2 26n(onI(MT )) } = 2 Sa [7v(f-f )T ] (10)k c 2 k c

where Sa(x)=sinx/x is the sampling function.

(b) The components ul(mTc) and jk(mTc); j=I,Q; Z=I,Q; are mutually uncorrelated which, by theGaussian assumption, renders them independent; thenthe noises nk(mTc) and n6(mTc) are also independent

(c) Each noise process has mutually indepen-dent in-phase and quadrature components.

(d) Noises corresponding to different fre-quencies (e.g., no(T) and no (T) with k#Q) areapproximately independent.

(e) Processes nk('r) and no(T), however, havedifferent properties, i.e., nk(T) is a highlycgrrelated process (as a function of T), whileno(T) generally is not; in fact, samples of n6(T)taken Tc apart could be uncorrelated.

Of the above conclusions, (e) is probably themost interesting from a performance viewpoint: Ifthe noise nO(T) had been as highly correlated asna(T), the decision rule (7) performance would havebeen perfect since the same random noise samplewould be obtained at '=O and T=Tc; thus, they wouldcancel out. This not being true, a slight degra-dation in performance is expected, as was alsoobserved in the simulation. Nonetheless, since themean part of the interference (second term in (3))does cancel out, the performance of this scheme isfar superior to that of the radiometer, which isoblivious to that term. In fact, the radiometeroutput is merely the value y(0O), which is domina-ted by the power of the random interference(Figure 1). Without further assistance, it isimpossible to determine if there is any signal inthe total observed power since the interferencecontribution is random and, hence, unknown.

As mentioned above, the high correlation ofak(T) implies that j(O)znjT

c); j=I,Q. On the

other hand, the degree of correlation betweenSk(O) and Sk(Tc) varies with the frequency diff-erence Afk=fu-fk. For the special case whereinone of the interfering frequencies fk coincideswith fo, it is easily seen that SSTO(OS-jTO(Tc);J=I,Q. Those facts are used in the subsequentanalysis.

We can now turn to the decision rule (7):upon squaring, taking the difference, lowpassfiltering (e.g., rejecting double-frequency terms),assuming that fk0=fo for one frequency and usingthe above conclusions, it follows that

222 Si 2 2IS(T_y (T ) )+(S) (n (O)-n (TC)) |LP

k +0) kF 0): TH_ otk(Tc)+k Pk(Tc)}yc(Tc)j

HA[.7 k ) S(C)+STH(- (H- (TC))2 (11)Same within a totally insignificant change.

where

k X (c )+n (r ) 12)

is the tstalBP equivalent noise. Let us note that,at least for the case of a full PN code period perhop, yc(Tc) is approximately zero, which consider-ably simplifies (11). Furthermore, even for therandom-code model, a comparison between analysisand simulation for the one-tone case (see nextsection) has indicated that the impact of the rvyc(TC) is unnoticeable; hence, setting yc(Tc)=Oseems to be a reasonable approximation for thegeneral case. Still, a full analysis of (11) with-out further simplifying assumptions is extremelycomplicated. In order to gain some insight here,we shall focus on the one-tone random interference(M=1), with the reasonable conjecture that themulti-tone case should provide analogous conclu-sions.

I Since, in this case, S1(Tc)zSI(O)=aI(OY)al(Tc)=al and n2(0)=n2(TC), it follows from (11)that

2(STH) 2 1= 121+2yI (1+2,/y a, )l (13)

where we have defined the signal-to-interferenceratio yI4S/I and the normalized random variable

A a.1 -l Ma =-=T cos4b I c (14)l,norm TH H 1 k=l n

In (14), cn; n=l,...,N are the code chips and flis the random phase of the interference. Clearly,under Ho, A=0 with probability 1. Under H1, itcan be shown that

(ST )2{e|AH1l-= 2 (1+2yI ) (15a)

and

var{AIH } = 2(STH)4 (Ny1) -l (15b)

Based on (15) and the Gaussian assumption about A,the performance of this decision scheme is pre-dicted by the detection probability

p =1lQ e(-+Y A-)I (16)

where AO=AO/(STH) is a normalized threshold and

A -1/2 = z2Q(x) = (21T) exp(- )dz (17)

x

is the Gaussian integral function. In the absenceof thermal noise, At can be set arbitrarily closeto zero; thus, it always yields the zero false-alarm rate PFA=O. In practice, At would be setaccording to the thermal noise level and thedegree of uncertainty about the power S of thedetected signal.3

The performance predicted by (16) is indeedexcellent. We note that increasing the inter-ference power (hence, increasing y-1) actuallyhelps detection instead of deterring it, while ithas no effect on false alarm. For instance,it canbe shown that, if N>60, then pcorr>99% indepen-

D

3Alternatively, this can be expressed in terms ofthe uncertainty regarding the transmitter/inter-cept receiver's true distance.

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dently of yi, as long as A <0.4, i.e., one cantolerate 40%, uncertainty about the signal powerand still expect excellent detection capabilities,regardless of the interference power. The aboveconclusion is rather insensitive to N in that, forN=10, the corresponding minimum pcorr is 90%.D

The radiometer performance (depicted in [1,Figure 2]) is easier to analyze for an arbitraryM and is based on the fact that its output Yrad=y(0)can be written as

Y ST±+ITH+24M( ak(O)) (17)rad H k ()

Let Mmax be the maximum number of tones which canbe expected in any hop. Here, for simplicity,we consider only the case Mmax=l, where Prob [oneinterfering tone]=Pr[no interfering tone]=1/2 foreach hop. Assuming that the threshold yo is setat yo=IlTH, so that4 pFA=O, it can then be shownthat

grad l--~QL12i if 1 (18)

pD (rr18j)e Q 1i if by<l2 24 I-(18

Thus, the obtainable performance decreases withdecreasing signal-to-interference ratio yI and inthe limit

li rad 1 ( 1i-*0 D 4 max (19)

YI+O corrwhich is certainly poor compared to PDC . It canalso be shown that the radiometer performance is adecreasing function of Mmax so that the aboveresults constitute an upper bound for the generalcase. Finally, we note that the question of thres-hold setting is much more crucial for the radio-meter than the correla-tor since the performance ofthe latter is effectively independent of the jam-ming power.

4. SIMULATION RESULTS AND DISCUSSION

The correlator with one tone interferencewas simulated by computer; the results shown inFigure 3 are based on 10,000 independent trials.Also shown in this Figure are the analytical pre-

dictions (dotted lines) whose agreement with thesimulation is quite striking, even for such lowvalues of N as N=3 and N=10. Since detectionprobability is monotonically increasing with N(see (16)), those values represent worst-casedesigns which nonetheless yield excellent perfor-mance results. It was somewhat surprising to findthat the Gaussian model provides such an accurateanalytical prediction, even for N=3; furthermore,setting yc(Tc) equal to zero proved to be a well-justified simplification. Note that no false alarmwas observed (PFA=O) and that performance is prac-tically insensitive to the amount of interferenceinserted. In contrast, the radiometer performance(as evidenced by (18)) deteriorates rapidly withdecreasing S/I, as expected. Finally, let usmention that the performance shown in Figure 3 isfor a nonoptimized (arbitrarily chosen) thresholdH=S2T2 /N. In the absence of thermal noise, further

improvement can be attained for the correlator bydecreasing A* to a very small (but positive) value.

Although analysis and simulation are not yetavailable for the multitone case, it is anticipatedthat the gap between the radiometer and the corre-lator performances will increase as the interfe-rence-to-noise ratio increases, independently ofMmax. That again is due to the relative insensi-tivity of the correlator to the interferencenuisance parameters.

Comments pertaining to the power measurementof (7) are the same as those In l]. Furthermore,we note that, if the carrier frequency f0 of thesignal sought is known, narrow bandpass filteringof the output y(T) in (3) prior to the power measu-rement will further enhance performance.

REFERENCES

[1] A. Polydoros and J.K. Holmes, "AutocorrelationTechniques for Wideband Detection of FH Waveformsin White Noise," MILCOM '83 Proceedings.

[2] A. Polydoros, "LPI Signal Detection Investiga-tion: The Direct-Seqence Case," Axiomatix FinalReport No. R8308-1, September 1983.

[3] Special Issue on Spread Spectrum Communica-tions, IEEE Trans. on Comm., May 1982.

4ThisHis selected so as to match the zero PFA ofthe correlator; another choice of Yo would lead toPFA=0. 5.

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Envelope of Spread Signal

Figure 1. Mean of Output y(T): The Signal and Interference Additive Components

Dec i sion

Figure 2. Block Diagram of the Decision Rule in the Autocorrelation Domainy Truncation at 0.9999

0.

0.999

0.99

0.98

0.95

0.90

0.80

0.70

Figure 3. PD Versus I/S for

I -Interference Power (dB)S Signal Power

the Real-Time Autocorrelator (Nonoptimized Threshold)

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