PERFORMANCE ANALYSIS OF DIRECT SEQUENCE DIFFERENTIAL PHASE SHIFT KEYING (DS-DPSK) WITH SELF-NORMALIZATION AND L-FOLD DIVERSITY IN A FADING CHANNEL

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1 PERFORMANCE ANALYSIS OF DIRECT SEQUENCE DIFFERENTIAL PHASE SHIFT KEYING (DS-DPSK) WITH SELF-NORMALIZATION AND L-FOLD DIVERSITY IN A FADING CHANNEL Charles W. Victory Tri T. Ha Naval Postgraduate School ECE Departmet code EC/HA Moterey, Califoria ABSTRACT The pevormace of a Direct Sequece Differetial Phase Shift Keyig (DS-DPSK) spread spectrum system employig time diversity ad self-ormalizatio over a Riciafiequecy oselective, slowly fadig chael i the presece of pulsed oise iterferece ad additive white Gausia oise (AWGN) is cosidered, A iterferece model which cosiders the possibility that either both, oe or oe of the iformatio bits e~eriece jmig is employed. Worst case pe~orrace uder severe ad moderate fdig coditios are aalyzed. Results idicate that the self-ormalized receiver is effective i mitigatig the eflects of pulse oise jammig for all fadig coditios cosidered. O)C. We also ote that the sigals represetig bit O ad bit 1 are orthogoal to each other over the 2-bit iterval, 2T, The optimum receiver for ocoheret detectio of orthogoal sigals i AWGN accordig to the Bayes criterio is a evelope or square-law detector [1]. We cosider the outputs of a square-law detector with the output VI as the bit O detector ad V z as the bit 1 detector. Assumig a bit O was set, the radom variable VI has a ocetral Chi-squared distributio with 2 degrees of freedom ad the radom variable Vz has a cell Chisquared distributio with 2 degrees of freedom. The probability desity fuctio for VI maybe epressed i the INTRODUCTION followig form We cosider the performace of a Direct Sequece Differetial Phase Shift Keyig (DS-DPSK) spread spectrum system with self-ormalizatio over a Ricia frequecy oselective, slowly fadig chael i the presece of pulsed oise iterferece ad additive white Gaussia oise (AWGN). The system employs L-fold time diversity with i of L chaels, i = 1,2,... L eperiecig iterferece at ay give poit i time. It is assumed that each of the diversity receptios are received i a idepedet fashio. The followig aalysis assumes that the received sigal is perfectly despread. That is we assume bit ad code sychroizatio betwee the trasmitter ad receiver. The geeral form of a DPSK sigal may be writte as fv,(hp) = 1 2(40~ + (s;) ( ep 1 ~ (Vl + 40$2) 2(40: + O) ) 10 U(vl) ()2a& (4fs: + a;) where the average sigal eergy over a l-bit iterval, (2) deoted by Eb, is cx2+ 2cr~ [1], O; is the oise power due s(t) = [ #A[coP r(t) + clp~(t 7 )] coso)ct (1) to AWGN, Io() is the modified Bessel fuctio of the first for O S t S 2T, with Cocl = 1 represetig bit O ad coc~ = -1 represetig bit 1. The fuctio p~(t) represets a rectagular pulse of uit amplitude o the bit iterval O < t < T. The carrier ftequecy is deoted by kid of zero order ad u() is the uit step fuctio. U2 represets tde direct path sigal ecgy while 2~~ represets the diffuse sigal eergy. The probability desity factio for V2 is

2 ().fv2( 2p) = +ep -:2 ~( 2) (3) (46: + cr~) with c7~j. The ew desity fuctio for V2 coditioed o the three jammer evets, ~vz ( z O, 1~) may be epressed from equatio 3 by replacig a; with The radom variables VI ad V2 serve as iputs to the self-ormalized receiver. Before describig the selformalized receiver, we first describe the pulse oise iterferece model i the et sectio. PULSE NOISE INTERFERENCE MODEL We cosider a iterferer who jams a fractio p of the iformatio bits (O< p < 1). The fractio of bits ot jammed is equal to (1- p), Pulse oise jammig is eplicitly defied for the case of p <1. We model the iterferig sigal as white Gaussia oise whose PSD is N1/2 p whe the jammer is o ad O whe the jammer is off. The total average PSD is the equal to N1/2. We allow for the possibility that either 2 cosecutive bits, 1 bit or o bits of the DPSK sigal maybe jammed ad defie the followig evet space for these three cases: 11- Evet that either the first bit cotais iterferece ad the secod bit does ot, or that the secod bit cotais iterferece ad the first bit does ot ({I, NI} U {NI, I}). 12- Evet that the first bit ad the secod bit both cotai iterferece {1,1 }. 13- Evet that either the first bit or the secod bit cotai iterferece {NI,NI }. We defie the probabilities of the three evets as Pr(I1) = SELF-NORMALIZED RECEIVER It has bee show that the use of the selformalized receiver show i Figure 1 ca improve the worst case performaces of a frequecy-hopped BFSK sigal uder partial bad jammig iterferece over a Ricia chael [2], I a aalogous way, we seek to improve the performace of a DPSK system employig time diversity uder pulse oise iterferece over a Ricia chael. For the L-fold diversity receiver show, a costat bit rate system is assumed. The iputs to the selformalizig receiver for the # diversity receptio are Vlk ad V2k. The desity fuctios for Vlk ad V2k are the same as derived for VI ad V2 before, with the eceptio that Eb is ow replaced with Eb/L. The radom variables Zlk ad Z2k are defied as v Zlk = lk Vlk + v2k v z2k = 2k lk + 2k (4) (5) Pl~ P@z) = P2 ad Pr(Is) = (1-pl-p2) where o<pl, p2sl. It is assumed that the jammer oise compoet is preset equally i braches VI ad V2. The ad Zlk + Zzk = 1 (6) jammer oise variaces at the receiver braches just prior to the squarig operatio for the three cases are 0; = N1/2p, a;, = N1/p, ad o:, = O. The total 1 coditioal oise variace is defied as o; = O: + a~, for j = 1,2,3. The total coditioal variace for brach 1 o; our receiver is G;,, = (4cf~ + $) ad for brach 2, 0: = O? for j = 1,2,3. The ew desity fuctio for V1? coditioed o the three jammer evets,.fv1 (Vl 0, Ij) may be epressed from equatio 2 by replacig Summig both sides of equatio 6 over L ad rearragig terms yields Z2 = L-Zl (7) From this we ca see that each diversity receptio has a weight of 1 out of L i the decisio process. This prevets ay particular diversity receptio from domiatig the fial decisio process. The probability of bit error as a fuctio of the diversity L is Pb(L, ii, ij = P,(Z1 <2210, Ij;L il, i2) (8) Substitutig equatio 7 ito equatio 8 yields

3 + + v lk Figure 1: Self-Normalized receiver z el lk Z1 k=l pb(l il, i2) = P,( l < L/210, Ij;L il, i2) The desity fuctio for ~k is give by (9) for case 1~j = 1,2,3. The coditioal desity for Z1 may be epressed i the Laplace domai as ~Z1(Zl 10, lj;l, il, i2) = ~-l([~(&,(z~#), IJ)]il X 2((~+l)r)2(l +(l-z1k)) [(7+ 1)r+2(1 zlk)]3 4(7 + l)r(l z~k) + [zyz~k((y+ 1)r)2] ( [(y+ l)r+ 27(1 + Xp - (Y+l)r+z(l-zlk) 2(1 z~k)] ) ( (z~k)) CXZ where y = is the ratio of direct sigal power to 26: diffuse sigal power ad 17 = (?J1+($J%) j is the average bit eergy to thermal oise desity ratio ad () Eb ~ is the average bit eergy to jammer oise power ratio 61 j (lo) [~(&l@klo> 12))] 2 X [@zlk(z@~ 13))1 (L- il - i2) where ~ ad L-l deote the forward ad iverse Laplace trasforms. Sice we model each of the L diversity rece~tios as idepedet evets, the probability. distributio for (L, il, iz) is derived from a rultiomial distributio ad is give as Pr(L, il, i2) = i1!i2!(l il PyP;(l Pl PJ (L- L! i, - i,) ) i2)! (11) (12) We may obtai the desity for Z1 coditioed o bit O beig set by averagig the coditioal desity for 21 i

4 equatio 11 over the probability distributio i equatio 12. The epressio is give as 10 \ L L iz fz,(zlp = ~ ~ fz, (zip $;L> il> Q i2=oi1=0 E@o.16 db L.{ w wusi.. p=, ~ o~.ol Pr(L, il, i2) (14) Fially the epressio for probability of bit error follows o! s 0,i~,ydm) from equatio 9 ad is Figure 2: Performace of self-ormalized receiver b = ( 2f#l lo)~zl (15) for pulse jammig fractios p = 1,0,25,0.1,0.01 ad worst case for diversity order L =1, E@. = 15 db ad y=o. This last epressio must be evaluated umerically. 10. E~N,-16 d. L_4 w NUMERICAL RESULTS,..,- I the followig aalysis, we will be iterested i ~. determiig receiver worst case performace. Worst case performace represets a composite performace by,,-c, obtaiig the value of the jammig fractio, p that produced the highest probability of bit error as a fuctio of the bit eergy to iterferece oise desity ratio, E@I $6 40 &ldbj Worst case performace was determied by umerical Figure 3: Performace of self-ormalized receiver search sice o aalytical solutio could be produced. for pulse jammig fractios p = 1,0.25,0.1,0.01 Performace curves of the self-ormalized receiver for ad worst case for diversity order L =4, EbfNo = 15 diversity orders L = 1 (o diversity) ad L = 4 ad jammer db ad y = O. fractios p = 0.01,0.1,0.25,1, worst case withejno = 15 db ad y = O (Rayleigh fadig) are show i Figures 2 Figure 5 shows the optimum value of p as a fuctio of Y ad 3 respectively. It is see that the self-ormalized with diversity order as a parameter ad with EJNO = 15 db receiver has completely egated the effects of pulse oise ad E@l = 20 db. We observe the overall tred that the jammig sice the worst case performace curve coicides optimum value of p decreases for icreasig y. From these with the cotiuous jammig curve (p = 1). plots, we ca see that there appears to be a lower limit as to &e effectiveess of pulse oise jammig. For eample, for Aother idicator of the effectiveess of the selfa diversity order of L = 4, pulse oise jammig is o loger ormalized receiver is that for icreasig diversity order, effective below Y = 3. For y greater tha 3, higher order the optimum value of p icreases forcig the iterferer to a diversities would be required to reder the pulse oise more cotiuous jammig strategy. This result is see i jammer ieffective. Figure 4 where the optimum value of p is plotted as a fuctio of the diversity order with E/N. = 15 db, y = 5 ad E@J1 as a parameter. We also observe that the optimum value of p (kcreases as ~@I icreases.

5 ,0LZ J dlversliy L - Eb/Nl.OdB - EbN,=5 db Eb/N,.10 db _ EbiN,.15 db EbN~20 dll - Ebfl,.25 db E@I~o db EJ?4,u35dB ~ Ebi?+40 db Figure 4: Optimum value of p as a fuctio of diversity order witheb/no = 15 db, y = 5 ad E~N1 as a parameter. was determied that the self-ormalizig receiver was effective i mitigatig the effects of pulse oise j ammig for all fadig coditios cosidered. It was oticed that higher diversity orders tha those cosidered would be required to reder the pulse oise jammer ieffective for less severe fadig coditios. This was observed for values of Y betwee O ad 10. REFERENCES 1. Proakis, Joh G., Digital Comwzicatios, 3rd cd., McGraw-Hill, Ic., New York, NY, Robertso, R. Clark ad Ha, Tri T., Error Probabilities of Fast Frequecy-Hopped FSK with Self-Normalizatio Combiig i a Fadig Chael with Partial-Bad Iterferece; IEEE Joural o Selected Areas i Commuicatios, vol. 10, o. 4, May L=l 1.2 L=3 L.4 L+ _ L=6 10 : I i Figure 5: Optimum value of p as a fuctio of y with diversity order as a parameter for E@. = 15 db ad E@T1 = 20 db. CONCLUSIONS We have cosidered the performace of a Direct Sequece Differetial Phase Shift Keyig (DS-DPSK) spread. spectrum system over a Ricia frequecy oselective, slowly fadig chael i the presece of pulsed oise iterferece ad additive white Gaussia oise (AWGN). The receiver employs L-fold time diversity with self-ormalizatio ad soft decisio equaj gai combiig. The performace of the self-ormalizig receiver uder coditios of severe ad moderate fadig was aalyzed. It