BER results for a narrowband multiuser receiver based on successive subtraction for M-PSK modulated signals

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1 results for a arrowbad multiuser receiver based o successive subtractio for M-PSK modulated sigals Gerard J.M. Jasse Telecomm. ad Traffic-Cotrol Systems Group Dept. of Iformatio Techology ad Systems Delft Uiversity of Techology Delft, The etherlads g.jasse@its.tudelft.l S. Be Slimae Radio Commuicatio Systems Group Dept. of Sigals, Sesors, ad Systems Royal Istitute of Techology Stocholm, Swede slimae@radio.th.se bstract arrowbad multiuser receiver based o successive sigal detectio ad subtractio, is proposed. The Bit Error Rate () performace for M-PSK modulated sigals is evaluated ad aalytical approximatios for the of the idividual sigals are preseted ad compared to the actual. The effect of iaccurately estimated sigal parameters due to oise is aalyzed. results are preseted for sychroous sigals for both dditive White Gaussia oise (WG) ad Rayleigh fadig chaels. Itroductio importat issue i future mobile commuicatio systems is to achieve a high spectral efficiecy. Icrease of the spectral efficiecy allows for higher data rates or for more users to commuicate simultaeously i the same badwidth. I the third geeratio mobile commuicatio systems, code divisio multiple access (CDM) is applied to achieve this goal. I CDM, the user sigal is coded with a user-sigature which maes it possible to distiguish ad detect users which occupy simultaeously the same chael at the cost of a large icrease of the badwidth required. I this paper, a arrowbad multi-user receiver based o successive sigal detectio ad subtractio, is ivestigated for M-PSK modulated sigals. I this receiver the major sigal is detected ad estimated i the first detector; the estimate is subtracted from the total iput sigal. Subsequetly the ext largest sigal is detected, estimated ad subtracted, ad so o. This priciple is also idicated as "oio peelig" []. By arrowbad we refer to the fact that o spreadig gai or badwidth expadig sigature code is applied to separate the users lie i CDM. The structure of the receiver is show i Figure. The proposed receiver is less efficiet whe the Sigal-to- oise Ratio (SR) at the iput is compared to that required by a QM scheme with the same badwidth efficiecy. However, i a iterferece limited commuicatio system, where a user with a high chael gai ca be received with a high SR at the cost of very low additioal iterferece to the system, this receiver ca be exploited. The sigal state structure of multiple sigals is ivestigated i Sectio. I Sectio 3, the performace for M-PSK modulated sigals, based o successive subtractio, is aalyzed. I Sectio 4, results are preseted for the Gaussia oise chael ad compared to the derived approximatios. lso the performace is show for the Rayleigh fadig chael. Iitially, perfect chael state iformatio for all active sigals is assumed at the receiver. I Sectio 5, the effect of o-perfect parameter estimatio due to the presece of oise is evaluated. I Sectio 6, coclusios are draw.

2 r ( t ) Detector $ d τ + s $ _ Detector $ d - τ + s $ - _ Detector 3 $ d τ + s $ _ Detector $ d Figure : Priciple of the multi-user receiver usig successive detectio ad subtractio. Sigal state structure Let us cosider a multi-user system with active users, all usig the same carrier frequecy f c ad M-PSK modulatio. ssumig a sychroous system, the equivalet lowpass of the received sigal over a give symbol iterval ca be writte i the followig form: r( t) = = = = s ( t) + ( t) e j ϕ + m, π M + ( t), T s t < ( + ) T where T s is the symbol duratio, is the amplitude of the trasmitted sigal, (t) is WG ad = ±, ± 3,..., ( -) is related to the trasmitted symbol of sigal m, ± M s (t). We further assume that these sigals have differet amplitudes with > >... >. I geeral the sigals have uiformly distributed phase ϕ ad idepedet modulated iformatio data d. Thus igorig oise the iterferece term i the above expressio will cause disjoit clouds aroud the major sigal poits. This is show i Figure for four idepedet QPSK modulated sigals where the receiver is loced to the major oe. The tas of the first receiver is to detect the cloud belogig to the correct symbol value of the major sigal. fter subtractio of the major sigal estimate, the remaiig sigal will form clouds aroud the ext major sigal states, ad so o. With sigals preset, the sigal state regios are the area of the clouds which are limited by a ier circle with radius R =,mi i= i ad a outer circle with radius R =,max i= i. For = the sigal states are o a circle with radius R = cetered aroud the state of s (t). s ()

Figure : Sigal cloud aroud the domiat sigal with =, =, = 4, 8, ϕ = / 6 ad radom phases for the other sigals. 4 π 3 4 = 3 Bit Error Rate aalysis Let us first cosider the detectio of sigal (t).

3 Figure : Sigal cloud aroud the domiat sigal with =, =, = 4, 8, ϕ = / 6 ad radom phases for the other sigals. 4 π 3 4 = 3 Bit Error Rate aalysis Let us first cosider the detectio of sigal (t). ssumig correct decisio for the first s sigals (strogest) ad with coheret detectio, the detector of s (t) should be able to loc o its proper sigal s (t). I that, after phase compesatio the received sample at the iput of this detector (see Figure ) ca be writte as π j m, jθ M i, y, = e + e i +,, =,,..., () i= where θ i, is the total phase of sigal i durig the symbol iterval T s ad, is WG with zero mea ad variace σ. The coditioal symbol error probability i this case ca be writte as follows: π π a = si( π / M ) cosθ d i i i P ( ε )... Q dθ... dθ (3) (π ) σ 0 0 where a d = for BPSK ad a d = for higher modulatio levels. The above expressio ca be solved umerically. However, the computatioal complexity icreases with the umber of sigals. Usig Figure, it is possible to obtai two simplified approximatios to the above symbol error probability. pproximatio : Cosiderig a worst case situatio, the coditioal symbol error probability of sigal ca be approximated as follows: π a si( π / M ) R, max cosθ d P ( ε ) Q dθ (4) π σ 0 This approximatio becomes a exact solutio whe =.

4 pproximatio : For > all sigal states are assumed to be uiformly distributed over the aular regio betwee R, max ad R, mi. The approximated coditioal symbol error probability is ow give by: R π ( ),max a d si( π / M ) r cosθ P ( ε ) Q rdrdθ. (5) π R,max R,mi σ 0 R,mi For large values of, this approximatio is more accurate tha pproximatio as show later i Sectio 4. Example: I the followig, we assume that the sigal amplitudes are geometrically related as:, = α α >, =,,...,. (6) The miimum Euclidea distace D, mi for sigal s is obtaied as α D, mi = α si( π / M ), M (7) α Based o this worst case situatio, the worst case coditioal symbol error probability becomes: ED,mi P ( ) ε adq (8) 0 with a d as earlier defied ad E the average symbol eergy of sigal. It is clear from the above expressio ad (7) that this coditioal error probability is depedet o the parameter α ad the umber of sigals. Let us defie the fuctio g( α, M, ) as follows: α g ( α, M, ) = α si( π / M ). (9) α To be able to detect s, the fuctio g( α, M, ) has to be strictly positive. The fuctio g( α, M, ) has the followig iterestig properties:. for < α < + /si( π / M ), g( α, M, ) decreases with icreasig from the value si( π / M ) for =. This puts a limit o the maximum umber of simultaeous sigals that ca be trasmitted. Table gives the value of α for which the fuctio g( α, M, ) is zero for a give umber of sigals ad modulatio level M.. for α = + /si( π / M ), g( α, M, ) = si( π / M ). 3. for α > + /si( π / M ), g( α, M, ) icreases with icreasig from the value si( π / M ) for =. We otice that the umber of sigals that ca be simultaeously trasmitted is i priciple ot limited if α + /si( π / M ), however, the miimum required value of α = + /si( π / M ) icreases with the modulatio level M.

5 Table : The set of values ( α,, M ) for which the fuctio g( α, ) = 0. M = M = 8 α α (db) α α (db) Taig ito accout the error propagatio from the previous detectors, the average symbol error probability of sigal is obtaied as follows: M Ps, = ( Ps, l ) + P ( ε ) ( Ps, l ), < (0) M l= + l= + ad P = P ( ). s, ε We otice from the above expressio that oly the strogest sigal s (t) does ot experiece error propagatio. M-PSK symbol represets L = log ( M ) bits. error caused by error propagatio results i a error of the correspodig bits with probability 0.5. symbol error i the sigal to be detected results with high probability i oe of the adjacet symbols ad with a appropriate mappig scheme causes oly a sigle bit error. Uder these assumptios the for s ca be approximated by ( M ) P ( ε) Peb, ( Ps, l ) + ( Ps, l ) < () ad M l= + L l= + ad P = P ( ) L. eb, ε / 4 Simulatio results We have compared the approximatios derived i the previous sectios agaist results from simulatios for BPSK modulated sigals. The results are geerated uder the assumptio that amplitude ad phase are accurately ow at the receiver for all sigals. I case of a correct decisio the sigal is completely removed. The simulated is the average over 60,000 radomly geerated sigal costellatios. Figure 3 illustrates the as a fuctio of E / 0 for the case of four sigals ad α =.

6 Simulatio ad pproximatio Simulatio ad pproximatio.e+00.e+00.e-0.e-0.e-04.e-06.e-04.e-06.e-08.e-08.e E pproximatio pproximatio Figure 3: results for the four-sigal case with α = (cotiuous lie represets simulatio, dashed represets approximatio): O = s 4, = s 3, = s, = s, E b 0 = E / 0. It is observed that pproximatio is too pessimistic for the larger sigals because all sigal states are tae at R. pproximatio is very close to reality. For small values, max of (close to ) the calculated symbol error rate becomes slightly optimistic because the desity of the actual sigal states is larger at the outside of the aular regio tha at the iside. I figure 4, the average is show for the Rayleigh fadig chael as a fuctio of the SR ad for two differet values of α. Due to the fadig, the desired amplitude ratios are ot maitaied aymore ad eve the order of sigal powers may chage. 0 0 Rayleigh fadig 0 0 Rayleigh fadig α = (6 db) α =.8 (9 db) Figure 4: results for Rayleigh fadig (dashed lie for a sigle user, cotiuous for the foursigal case): O = s 4, = s 3, X = s, = s, E b 0 = E[ E / 0]. The large sigals ( s 3( t ) ad s 4 ( t) ) clearly show a error floor at high SRs. The small sigal s ( t ) has a worse tha that of the sigle sigal case, however the differece is quite small. 5 Effect of iaccurate parameter estimatio ccurate parameter estimatio is crucial for the operatio of the arrowbad multi-user receiver. I practice parameter estimatio is ot perfect due to the presece of oise ad co-

7 chael sigals. I [] it is show that by usig traiig sequeces with low correlatio, sigals ca be separated almost completely by applyig the bootstrappig techique. So eve i a multi-user eviromet it is feasible to estimate the parameters of each sigal with early the same accuracy as for the Gaussia oise chael. The amplitude ad phase errors caused by oise have a statistical ature. I [3] Garder showed that for small phase errors, δ the variace of the phase error is iversely proportioal to the SR i the estimatio θ circuit. For the variace of the ormalized amplitude error expressio ca be derived: BL 0 σ δθ = σ δ = = SR P Here L L s SR is the SR withi the equivalet oise badwidth δ = δ a idetical / () B L of the estimatio circuit, ad P s = / is the sigal power. Sice δ ad δ θ are idepedet stochastic variables, the residual power P after subtractio is foud by usig eq. () as P = E [ Ps ( δ + δθ )] = Ps ( σ δ + σ δ ) = BL 0 θ Thus iaccurate sigal suppressio i the multi-user detector by phase ad amplitude errors ca be modelled by addig the oise power B L 0 i the badwidth of the parameter estimatio circuit after every subtractio. Sice the traiig sequeces are early ucorrelated, the estimatio errors due to oise are idepedet for each of the sigals. The coditioal symbol error probability taes the form of (3) with σ replaced by σ + ( ) BL 0. The degradatio the icreases with the umber of sigals ad is largest for the mior sigal. If the badwidth of the estimatio circuit is much smaller tha the matched filter badwidth, B MF, of the detector the this degradatio ca be eglected for << L = B MF / B L. L represets is the legth of the used traiig sequece. I practice, this assumptio holds true otherwise the sigals caot be separated for parameter estimatio. Figure 5 shows the effect of estimatio error o the for traiig sequeces of legth L = 5 ad 3 i a four-sigal eviromet with α = ad BPSK modulatio. It is observed that the degradatio for the mior sigal (worst case) is relatively small. (3) Simulatio for L = 5.E+00.E-0.E-0.E-03.E-04.E-05.E-06.E-07.E-08.E-09.E Simulatio for L = 3.E+00.E-0.E-0.E-03.E-04.E-05.E-06.E-07.E-08.E-09.E Case Case B Figure 5: results for the four-sigal case with traiig sequece legth:. L = 5, B. L = 3 (cotiuous lie represets ideal case ad dashed with estimatio errors): O = s 4, = s 3, = s, = s, α = (6 db).

8 5 Coclusios The performace of a subtractive multi-user receiver for arrowbad M-PSK modulated sigals is aalyzed i this paper. For a multi-sigal eviromet two approximatios for the were derived ad compared to simulated results. It has bee show that for sigals with geometrically related amplitudes, two importat cases occur. If the amplitude relatio betwee successive sigals (ordered i amplitude) α + /si( π / M ), i priciple a arbitrary umber of sigals ca be staced. I practice, however, where sigal parameter estimatio is iaccurate, this will ot be feasible. For α < + / si( π / M ), iheret errors occur if too may sigals are staced; the maximum umber umber of sigals decreases with decreasig α ad with icreasig M. It has also bee show that to compesate for iaccurate sigal parameter estimatio, a extra margi should be added. This margi ca be reduced cosiderably if the legth of the traiig sequece L is chose such that << L. The performace i the Rayleigh fadig chael shows a error-floor for the error probability of the differet sigals. This is due to the fact that fadig maes the iterferece more Gaussia. Refereces [] T.M. Cover, J.. Thomas, Elemets of Iformatio Theory, Joh Wiley & Sos Ic., ew Yor, 99. [] M. Moretti, E. ostrato, S. Piageri, G.J.M. Jasse, " Frequecy Estimatio Scheme for a Two-Sigal Eviromet", IEEE Proc. VTC'99, pp , msterdam, The etherlads, September 999. d [3] F.M. Garder, "Phaseloc Techiques", ed., Joh Wiley & Sos Ic., ew Yor, 979.

Orthogonal Gaussian Filters for Signal Processing

Orthogonal Gaussian Filters for Signal Processing Orthogoal Gaussia Filters for Sigal Processig Mark Mackezie ad Kiet Tieu Mechaical Egieerig Uiversity of Wollogog.S.W. Australia Abstract A Gaussia filter usig the Hermite orthoormal series of fuctios