Optimum threshold diversity reception of binarynoncoherent frequency shift keying

A.M. Maras, MSc, PhDH.D. Davidson, BScA.G.J. Holt, DSc

Indexing terms: Noise and interference, Telecommunication

Abstract: The optimum diversity rule for recep-tion of binary noncoherent frequency-shift keyingin slow and nonselective Rayleigh fading(multipath) and additive non-Gaussian noise isderived for the critically important threshold case(of small nonvanishing signals and large samplesize). The relevant expression for the probability oferror illustrates the significant improvementachieved, even with a small number of diversitychannels, i.e. 20-30 dB gain from two to five chan-nels for 10"5 error probability.

1 Introduction

Many telecommunication systems are increasinglyaffected by non-Gaussian noise, i.e. noise of a highlystructured form. Underwater acoustic communications isone example, where noise due to reverberation is highlynon-Gaussian, and it is found to fit Middleton's class Anoise [1] very closely, which is also an appropriate modelof interference caused by collections of intentionally radi-ated signals, as in the crowded HF band. Another equallyimportant example is the transmission of binary, or evenM-ary, suitably modulated signals through the atmo-sphere, where the main source of interference is atmo-spheric noise, which can be adequately modelled by theimpulsive Middleton's class B noise model. The develop-ment of these statistical/physical noise models, which areanalytically tractable and experimentally verifiable, hasresulted in the extension and correction [2] of the theoryof optimum signal detection, so that the canonical detec-tion algorithms (in signal and noise statistics) are locallyoptimum Bayes and they are also, with the inclusion ofthe correct bias term, asymptotically optimum. These twofeatures guarantee minimum possible probability oferror expressions, for both coherent and incoherentmodes, which may become equal to zero as the samplesize becomes infinitely large. However, the above analyti-cal results can only be obtained in the critically impor-tant threshold regime, i.e. for small but nonvanishingsignals, and large, independent noise samples. Moreover,the above theory allows comparison to be made withsuboptimum receivers, which are optimum in Gaussianor other types of noise. Thus the performance of these

Paper 6347F (E8), first received 29th June 1987 and in revised form20th May 1988The authors are with the Department of Electrical and ElectronicEngineering, University of Newcastle upon Tyne, Newcastle upon TyneNE1 7RU, United Kingdom

canonical optimum threshold receivers can be measuredand evaluated against the performance of other, morewell known receivers. Indeed, it has been found thatthe performance of the optimum threshold receivers ismuch better than that of those receivers which are opti-mum in white Gaussian noise by 20-30 dB in effectivesignal-to-noise ratio (SNR). Furthermore, the per-formance of the suboptimum receivers is also known tobe further degraded with Rayleigh fading, as in the caseof a quadrature receiver in active sonar when the multi-path returns overlap [3] (the quadrature receiver isoptimum for the noncoherent detection of binary FSKor, more generally, orthogonal signals in white Gaussiannoise). The purpose of this paper is twofold: first, it aimsto show the severe degradation in the performance of theoptimum threshold noncoherent receiver for frequency-shift keyed (FSK) signals when Rayleigh fading is present,and secondly, it aims to demonstrate the significantimprovement in performance achieved with the newlyderived optimum noncoherent threshold detector, usingan expression for its error probability. Even thoughdiversity reception of noncoherently received binary FSKsignals in Gaussian noise is a well known and understoodtechnique (see Reference 4 for a recent survey), similartechniques in non-Gaussian noise and interferenceenvironments have not yet been established.

2 Optimum threshold FSK detection withRayleigh fading

The incoherently received FSK signals are

Sj(t) = <xJ(2S) cos (cojt + <f>),

0<t<T, ; = l, 2 (1)

where a is the signal amplitude, momentarily assumed tobe constant; S is the signal power; cOj (j = 1, 2) are thecarrier angular frequencies; <$> is the carrier phase, a uni-formly distributed random variable, and T is the inverseinformation rate. It is easily shown from the results inReference 5 that the threshold error probability (for aconstant amplitude signal) is

P(e)/a = \ exp {SL(2) (2)

where N is the sample size (N > 1). Zl2) is given by

ti2) ^ J" [ £ In Pz(x)Jpz(x) dx560 IEE PROCEEDINGS, Vol. 135, Pt. F, No. 6, DECEMBER 1988

where pz(.) is the noise probability density function(PDF). When a varies according to Rayleigh's distribu-tion law, i.e. its PDF is given by

p(a) = exp [ - —; (3)

with 2a2 being the total mean intensity of a; the expres-sion for the error probability is found to be

P(e) =1

(4)

where /? = 2cr2S, the total average signal power. Plots ofthe threshold P(e) against 5 (which is also the SNRbecause the received signals are normalised with respectto the total mean intensity of the noise), from eqns. 2 and4 with N = 10, 100, 1000 and Zi2) = 1340, are given inFig. 1 when a2 = 2<r2 = 1. It is evident from Fig. 1 that

-60 -55 -50 -45 -40 -35 -30 -25

5 (SNR).dB

Fig. 1 Threshold performance of binary noncoherent FSK with Ray-leigh fading and no fading with N = 10, 100, 1000 andIi2) = 1340((x2 = 2a2 = 1)

the threshold performance of FSK is heavily degradedwhen fading is present, but improves as N increases.

3 Optimum threshold diversity reception

The simple alternative hypothesis test for each of the Lchannels is

wn = <xJ(2S) cos (coxt

c o s (°>21

) + zJLt) (5a)

') + *#) (5b)

where 0 ^ t ^ T, n = 1, . . . , L.The amplitudes <xn obey eqn. 3; </> are uniformly dis-

tributed, and {zj are mutually independent, identicallydistributed, stationary noise processes. Owing to theassumptions of large and independent noise samples andof slow, nonselective Rayleigh fading, the optimumthreshold decision rule is

(6)

where

RZj = \ E IK,) COS ft); t,

r N 12+ Y,Kwnt) sin cojtA , 7 = 1,2,...

and

= J- In pz(wn)ax

Eqn. 6 shows that the optimum threshold diversity recei-ver is a linear combination of n copies of the binary FSKthreshold detector employed in the single channel case.From eqn. 24 of Reference 6, the PDFs of the normaliseddecision variables xn and yn are found to be

= *n expx2 + 2a2SNZ<2)]

2 Jx J0(xnN/2a2SJVZ<2>) (7a)

and

-yl/2) (7 b)respectively. To find p(xn), we average the envelope dis-tribution in eqn. la over the density of an as given in eqn.3, to obtain

SJVZ<2> 2(1

Let now un = x2/2 and vn = yl/2. Then the PDFs of thenew variables, from eqns. 1b and 8, are given by

and

= exp(-yn) (9b)

Having obtained the statistics of un and vn (n = 1, . . . , L),which are related to Rn2 and Rnl of eqn. 6, we may nowobtain the threshold performance, i.e. an expression forP(e), of the optimum diversity receiver.

4 Probability of error

Define next the following random variables

U = £ un and V = £ vnn = l n = l

These variables are equal to the sum of random variableswhich are asymptotically normal and uncorrelated, andthus independent, and so their respective PDFs are easilyshown to be given by

P(U) = uL - l

<T2SNti2))L(L-\)\

x exp -[-i + . w ] (1Oa)

and

Hi

(106)

Assume now that hypothesis Hx of eqn. 5 is true. Thenan error occurs when U > V, since both U and V aredirectly proportional to the sum of the squares, as seen

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from the optimum decision rule, eqn. 6. Therefore, bysymmetry, the error probability of binary FSK signals forthe optimum threshold detection situation described byeqn. 6 is given by

P(e; L) = p(U) p{V)

n - 1 ) !

(11)

with 0 < p < 1, N > 1 and Zl2) ^ 1.The expression for P(e; L) against ft, the average SNR,

is plotted in Fig. 2 for L = 1, 2, 3, 4, 5, 6 and fixed values

-60 -50 -40 -30 -20 -10£ (average SNR).dB

Fig. 2 Optimum diversity threshold performance of binary noncoherentFSK with number of channels L = 1, 2, 3, 4, 5, 6 with U2) = 1340 andN = 100

of Zi2) (=1340) and JV(= 100). It is obvious that a signifi-cant improvement in performance is achieved with diver-sity, i.e. a gain of almost 30 dB with five diversitychannels for a desirably low probability of error of 10 ~4.It is also to be noted that when L = 1 in eqn. 11 itreduces to eqn. 4 as expected.

5 Conclusions

In this paper we have derived the error probability ofbinary FSK signals for optimum threshold noncoherentdiversity reception in arbitrary non-Gaussian noise, andslow, nonselective Rayleigh fading. Since our detector islocally optimum Bayes and also asymptotically optimum,P(e; L) of eqn. 11 is the minimum possible value that canbe obtained, and also P(e; L) -> 0 as N -*• oo. Further-more, the chosen modulation scheme is by no meansaccidental, for it can be shown to be the optimum choicein noncoherent optimum threshold multichannel commu-nications with Rayleigh fading. In addition, we have beenmotivated by active underwater acoustic communicationswhere noncoherent FSK is employed [3], and the noise/interference environment is highly non-Gaussian. Wehave thus demonstrated the importance of utilising diver-sity and the significant gains achieved, e.g. approximately35 dB for five channels when P(e; L) = 10 ~4. Finally, webelieve that our approach can be applied to more generalfading channels, such as the slow, selective Rayleighfading channels.

6 Acknowledgments

The authors wish to express their sincere gratitude to theSERC Marine Technology Program for its financialsupport.

7 References

1 WILSON, G.R., POWELL, D.R., and FRASER, M.E.: 'Statisticalcharacterisation of underwater acoustic signals'. Oceans '82, IEEEConference Record, Washington DC, 1982, pp. 241-249

2 MIDDLETON, D., and SPAULDING, A.D.: 'Optimum reception innon-Gaussian electromagnetic interference environments. Pt. II —Optimum and suboptimum threshold signal detection in class A andB noise'. NTIA Tech. Rep. 83-120, US Dept. of Commerce, Washing-ton DC, 1983

3 HIGGINS, R.C.: 'Performance degradation in a quadrature receiverfor CW signals corrupted by multipath', J. Acoust. Soc. Am., 1981, 69,pp. 728-731

4 STEIN, S.: 'Fading channel issues in systems engineering', IEEE J.,1987, SAC-5, pp. 68-89

5 SPAULDING, A.D., and MIDDLETON, D.: 'Optimum reception inan impulsive interference environment. Pt. II — Incoherent recep-tion', IEEE Trans., 1977, COM-16, pp. 924-934

6 MARAS, A.M., DAVIDSON, H.D., and HOLT, A.G.J.: 'Resolutionof binary signals for threshold detection in narrowband non-Gaussian noise', IEE Proc. F, Commun., Radar & Signal Process.,1985,132, (3), pp. 187-192

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