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PE R FO R M A N C E O F A NONCOHERENT RAKE RECEIVER AND C ONVOLUTIONAL CODING WITHRICEAN FADING AND PULSE-NOISE JAMM INGKyle Kowa lske and R. Clark RobertsonDepartment of Electrical and Computer Engineering, Code E C R cNaval Postgraduate SchoolMonterey, C alifom i, 93943-5121email: crobertson@nps.navy.milfax: (831) 656-2760

Abstract-The effect of pulse-noise jammingnoncoherent RAKE receiver with convolutional codingin the presence of Ricean fading is analyzed. The effectof additive white Gaussian noise (AWGN) is alsoincluded in the analysis. The maximum-likelihoodRAKE receiver for the combination of pulse noise andAWGN is derived. Pulse jam min g usually has asignificant effect when soft decision decoding is used,however, we show tha t the m aximum-likelihood RAK Ereceiver effectively mitigates pulse noise jamming.Ha rd decision decoding redu ces the effect of pulse noisejamm ing, however, the maximum-likelihood RAK Ewith soft decision decoding performs better th an har ddecision decoding in the presence of pulse-noisejamming.

I. INTRODUCTIONIn [l] it was shown that the performance of aconvolu tional code with soft decision decodin g can besignificantly degraded by pulse noise jamming. The

performa nce was even worse when the constraint length ofthe convolutional code was increased. The authors alsoshowe d a noise-n ormalize d receiver, which m ultiplies thereceived signal by the inverse of the variance, improvesthe probability of bit error. This paper extends the a nalysisfrom [ l ] by consid ering a RA KE receive r with a frequenc yselective fading channel. The maximumlikelihoodreceiver for a combination of AWGN and pulse-noisejamm ing is derived and the derivation shows that the noisenormalized receiver in [I ] is an optimal maximum-likelihood receiver. The performance of the maximum-likelihood RAKE receiver with a constraint length 9, rateK convolu tional code is analyze d. Both hard and softdecision decoding will be considered. A block diagram ofthe system to be analyzed is shown in figure 1. Th eperformanceof the maximum -likelihood receiver will alsobe compared with the performance of a RAKE receiveroptimized for signals received with additive whiteGaussian noise (AWG N). This comparison is of interestbecause com mercially available RAK E receivers currently

in use are optimized for AWGN channels, and resistanceto jammin g is important for military communication.

RALEECmmFigure 1: BFSK Comm unication System With a RAKEReceiver, rate lnConvolutional Coding and Pulse-Noise Jamm ing.11.MAXIMUM-LIKELIHOOD RAKE RECEIVERFOR PULSE NOISEAND GAUSSIAN NOISE

We will now consider a BFSK signal that is receivedwith both AWGN and a noise like pulse jamming signal.It will be assumed that the jamming signal is either on oroff for a full b t period and jams an integer number of bits.The interleaver randomizes the bits that are jammed, so itwill be assumed that the jammed bits and the unjammedbits occur independently at the inp ut to the decoder.The maximum-likelihood receiver ma ximizes the ratio ofthe joint probability density functions for soft decisionreceiver outputs when bit 1 is sent and the soft decisionreceiver outuuts when bit 0 is sent

- . .where y is the received sequence of soft decision receiveroutputs. If the ratio of the joint density func tions is greate rthan one, the receiver assumes bit 1 was sent. If the ratiois less than one, the receiver assumes bit 0 was sent.Since soft decision decoding uses a sequence of softdecision receiver outputs to make a bit decision, we willassume that the joint density functions are a combination

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of d soft decision receiver outputs. The probability densityfunction of the random variable y at the output of aquad ratic detector, given that a bit 0 as transmitted is

where 0 U, for AWGN only, 0 U: +0: hen thejamming signal is present and &U is the received signalamplitude. The Gaussian noise variance is

(3) b

whe re c is the code rate and 7 the du ration of a bit. Whenthe pulse-noise jamm ing signal is on the variance is

where r is the duty cycle of the jamm er. Since it isassumed that the bits arrive independently at the RAKEreceiver, the joint density function is the product of themarginal density functions. Hence, the joint densityfunction for the soft decision receiver output assuming abinary 0 was sent and that i of them are jammed and d-ihave only AW GN can is

Since it was assumed that each RA& finger has flatfading, the received signal power 2 a 2 on the yl ( m )branch will be the same as 2 a 2 on the yo(m) ranch.Hence, the ln[ I , (e)] adds the same value on both sidesof (4), and we can further simplify the maximum-likelihood receiver to

(4)

The joint density function for the soft decision receiveroutput assuming a binary 1 was sent can be written bychanging yo ( m ) o yl ( m ) in (3). Substituting these jointdensity fun ctions into ( I ) taking the natural log of bothsides and rearranging terms we obtain

which shows that the optimal receiver weights the receivedsignal with the inverse of the variance, 1/u: for bits withonly AWGN and I/(ui+0:) hen the jamming signalis present. This weightin g by th e inverse of the variancewas fnst evaluated in [3,4] for norrcoherent frequencyhopped signals with soft decision convolutional coding.The analysis in this section shows that weighting each bitby the inverse of the variance is the optimal maximum-likelihood weighting for a noncoherent RAKE receiverwith convolutional coding and soft decision decoding.111. PERFORMANCE ANALYSIS FORNONCOHERENT RAKE RECEIVERA diagram of a non coherent BFSK RAKE receiver isshown in Figure 2. The transmitted signal for a bit 0

(8)and the transmitted signal for a bit 1

(9)

is f i A c c ( t ) co s (mot+@)is &&c(t) co s (W , t + 6 )

I 1 4 4

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SummerU SummerUr - lecision LogicMigure 2: N oncoheren t FSK RAKE Receiver.

where &A is the unfaded comprised of a carrieramplitude, @ and 6 are andom phase offsets, and c ( t ) isa chipping sequence that satisfiesl N- ~ c ( t - r n ~ , ) x c ( t - k ~ , )6 (m k )Nwhich is a good approximation for long chippingsequences. When this signal is transmitted over afrequency selective fading channel the received signal is

(10)

Ir ( 1 ) = c, i 4 a i c ( t -iT, )cos[ w, ( t- iT, ] (1 1)where each of the 1 components is a flat fading signal andcould have a different Ricean fading coefficient a,.Th eRicean fading is assumed to vary slowly compared to theduration of a bit so it can be treated as a constant ov er a bitduration. In order to demodulate the received signal r( t )the RAKE receiver correlates against two signals

(12)andS =& ( t ) cos (W ] t) (13)If a bit 0was transmitted then the output of the m ixer onthe j * finger of the RAKE receiver will be

,=I

So=&(r) co s (coot)

when r(t) is mixed with So and

for r(t)mixed with SI.The integrator then correlates thesesignals, which will eliminate any delayed components ofthe received signal that do not have the same delay as thelocally generated chipping sequence c ( t ) Thetransmitted signals are orthogonal soIoros (woI)cos wll)dt=0 (16)In the absence of noise the output of the integrator on thesignal branch will be A p j , and the non-signal branchintegrator output will be zero. Tbe integrator outputs aresquared and com bined. The results of correlating with SIare then subtracted from the correlation with So, and ifthe result is greater than zero the receiver decides that a bit 0 was transmitted. Otherwise, it decides a bit I wastransmitted.A. Ricean Fading Analysis with AWGN and Pulse-Noise JammingWe will now derive the performance for a noncoherentRAKE receiver with AWGN and pulse noise jamming.The conditional density of the random variable XI, at theoutput of a quadratic detector, given a signal amplitude&a, is

where Zo (e) represents the modified Bessel function ofthe first kind and order zero. Th e average received poweron the k* finger is a : , the subscript l k is used toidentify the k h finger of the signal branch and the fadingchannel is modeled by assuming a, to be a Riceanrandom variable. The probability demity function of theRicean random variable is

-

(18)Where ai is the average power of the direct componentand 252: is the average power of the diffuse component.The total received power on the k finger is ai + 252:.

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Ihe conditioning of XI , on a , can now be removed byintegrating over the pdf for a,

a

f X , , ( k = I X l k (It I k If4 ('itSubstituting (17 ) and (18) into (19), we obtain

(19)0

The' pdf at &e output of the summer is then theconvolution of pdf from all the fingers of the RAKEreceiver. In order to avoid doing these convolutionsdirectly, the Laplace transform of the pdf from each fingeris multiplied together and the inverse transform is then thepdf at the output of the summer. The Laplace transform ofthe pdf for XI , is obtained from-F X , , ('1 = I X, , ( exp -'%k )&k (21)

0Substituting (20) into (21) and integrating we get

1+23(m:tu;)sff' Jexp -The Laplace transform of the pdf at the output of thesummer is

LF x > ( s ) = n F x , , ( s ) (23)The pdf for the n oise only branch can be found from (20)by setting a: = 52: =0

k= l

Where the subscript 2k identities the k h finger on thenoise only branch, U: =0,' hen only AWGN is presentan d 0: =U: +U, when pulse-noise jamming is present.The Laplace transform of (22) is

2

and the Laplace transform of the pdf at the output of thesummer for the noise only branch is the product of theLapla ce transforms of all the noise only RAKE fingers.

B. Soft Decision DecodingThe bit error rate for a system with soR decisiondecoding is upper bounded bye

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where a is a number greater than the real part of anysingularity of Q (s) . The trapezoidal rule is then used tointegrate the numerically inverted data points to producethe probability of bit error. In figure 3 we show theprobability of bit error for a three finger RAKE with softdecision decoding when EbI N , = 1 5 d B , the ratio ofdirect to diffuse power is lOdB and the jamming signal hasa one percent duty cycle (r=O.Ol) , ten percent duty cycle( 1 4 . 1 ) and a jamming signal that was on all the time(Fl) .

Figure 3: Performance of NoncoherentRAKE ReceiverWith Soft Decision Decoding and Pulse-NoiseJammingWhen the jammer duty cycle r is decreased, the probabilityof bit error increases. lhis is due to the fact that thejamming power increases as the duty cycle decreasesuj = N j / r c , T , where N j is the jamming noise powerspectral density, c, is the code rate, and Tb is the bitduration. Soft decision decoding uses a sequence of softdecision receiver outputs to make a bit decision. If any ofthese soft decision receiver outputs have a large variance,then the probability of bit error will significantly increase.The jamm ed b it will also affect future bit decisions, whichwill cause a burst of decoding errors.

2

C. Hard Decision Decod ingFor hard decision Viterbi decoding, Pd in (26) is

Lwhen d is odd, and

when d is even. In (32) and (331, the channel transitionprobability p isp = r (P, 1o2 o,' of)+1- r ) 4 u2=u i) (34)where r is the duty cycle of the jammer. In figure 4 weshow the probability of bit error for a three finger RAKEwith hard decision decoding when Eb / N o =15dB, heratio of direct to diffuse power is lOdB and the jammingsignal has a one percent duty cycle (F0.0l). ten percentduty cycle (F0.1) and a jamming signal that was on all thetime ( P I ) .

EblNiFigure 4 Performance of Noncoherent RAKE ReceiverWith Hard Decision Decoding and Pulse-NoiseJamming.Hard decision decoding m akes bit decisions on a bit by bitbasis, hence, hard decision decoding limits the effects ofpulse-noise jamming to a single bit. As a result, harddecision decoding can perform better the soft decisiondecoding w hich is susceptible to decoding a burst of errorswhen the pulse-noise jammer is present. Hence, harddecision decoding can provide bit error probabilitks of10" or lower for a pulse-noise jamming duty cycle is onepercent, which is a significant improvement over the softdecision decoding that required about 30 dB Eb / N j toachieve this bit error probability.D. Maximu m-Likelihood Noncoherent RAKE Receiver

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From (7) we see that the maximumlikelihoodnoncoherent RAKE receiver scales the received signal bythe inverse of the variance. We will begin with the effectof this scaling on the signal branch by definingZ,, -X,, , which is the random variable on the k mringer of the signal branch. n e df for Z,,s

1U2

where a, is assumed to be a Ricean random variabledefined in (16), 0; 0, hen only AW GN is present andU, =Uo+U , when pulse-noise jamming is present.Substituting (35) and (18) into (19)we get

2 2 2

,Th e L aplace transform of .(36) is

an d the Laplace transform at the output of the summer isthe product of all the signal branch fingers as shown in(23). The pdf for the noise only branch is

,an d the Laplace transform of (38) is(39)

The Laplace transform of the output of the noise onlysummer is the product of all the noise only RAKE fingerLaplace transforms. We use a three finger RAKE withEb / N o = 1 5 d B . use a lOdB ratio of direct-tediffusepower to calculate the performance of the maximum-likelihood noncoherent RAKE receiver against a jammingsignal with o ne p ercent duty cycle (r=O.Ol), ten percentduty cycle ( ~ 0 . 1 )nd a jamm ing signal that was on all thetime ( ~ 1 ) . he results are presented in figure 5 . Themaximumlikelihood noncoherent RAKE receiverimproves the performance when pulse-noise jamming ispresent. In fact, the worst case jamm ing at low SM valueswa s achieved when the jammer was on all the time. Thisshows that pulse-noise jamming is not effective against amaximumlikelihood receiver designed for both AWGNand pulse-noise jamming.

Figure 5: Performanceof Maximum-Likelihood RAKEReceiver With Soft Decision Decoding and Pulse-NoiseJamming.IV. Conclusions

In this paper, the performance of a noncoherent RAK Ereceiver with pulse-noise jamming, and frequencyselective fading was examined. The maximum likelihoodreceiver for a noncoherent system with both AWGN andpulse jamming was derived. We found that the maximum-likelihood RAKE receiver effectively mitigates theeffects of pulse noise jamm ing when a constraint length 9rate y2 soft decision convolutional coding is used. We a lsoshowed that the noise normalized receiver in [ l] is amaximaLlikelihood receiver for AWGN and pulse-noisejamming. The conventional RAKE receiver with softdecision decoding is very susceptible to pulse-noisejamm ing. Decreasing the duty cycle of the pulse jamm ersignificantly increases the bit emor rate. n e maximum-likelihood RAKE receiver limits the effect of pulsejamming by weighting each bit by the inverse of thevariance. This requires the variance to be measured on abit by bit basis, which significantly complicates thereceiver. However, it also greatly improves theperformance against pulse noise jamming.

REFERENCESTedesso, Thomas W. an d Robertson, R. Clark,Performance Analysis of a SFWNCBFSK CommunicationSystem with Rate YI Convolutional Coding in the Presenceof PaniakBand Noise lamming, MILCOM 1998, October1998.Simon, Richard M ., Stroot,Michael T. , and Weiss, GeorgeH., Numerical Inversion of Laplace Transforms withApplication to Percentage Labled Mitoses Experiments,Computers an d Biomedical Research. Vo l 5, pp. 596-607,1972.

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