A LOW COMPLEXITY COHERENT CPM RECEIVER WITH MODULATION INDEX ESTIMATION

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1 A LOW COMPLEXITY COHERENT CPM RECEIVER WITH MODULATION INDEX ESTIMATION MESSAI Malek, GUILLOUD Frédéric ad AMIS Karie Istitut Télécom; Télécom Bretage; UMR CNRS 3192 Lab-STICC Techopôle Brest Iroise CS Brest, Frace Uiversité européee de Bretage ABSTRACT I this paper we address the problem of low-complexity coheret detectio of cotiuous phase modulatio (CPM) sigals. We exploit the per-survivor-process techique to build a reduced-state trellis ad apply a Viterbi algorithm with modified metrics. I the case where the modulatio idex ca vary, we propose a maximum-likelihood (ML) estimatio of the modulatio idex ad compare the performace of the resultig structure with a o-coheret receiver structure of the state of the art. Simulatios o a additive white Gaussia oise (AWGN) chael both for biary ad M-ary CPM show the efficiecy of the proposed receiver. Idex Terms CPM, modulatio idex estimatio, persurvivor processig, reduced-complexity, Viterbi decodig, modulatio idex mismatch. 1. INTRODUCTION I applicatios with low-cost trasmitters, the RF frot ed may suffer from o liearities. I this case, cotiuous phase modulatios (CPM) [1], which have a costat amplitude, are a good cadidate to limit the trasmitted sigal distorsio. A CPM is perfectly defied by the symbol alphabet, the frequecy pulse ad the modulatio idex. Whe the modulatio idex varies, either radomly i a cotiuous iterval as for the AIS aalog trasmitter [2] or, itetioally as for the Bluetooth stadard [3], the priority is give to o coheret detectio due to its robustess. Discrimiator detectors [4] are the most used for their low computatio cost, at the expese of severely degraded error rate performace. Efficiet ocoheret receiver structures such as the oe proposed i [5] by Lampe et al. achieve better error rate performace with a reasoable icrease of complexity. The best error rate performace is obtaied through coheret maximum likelihood sequece estimatio (MLSE). Based o the trellis state descriptio of the CPM, the Viterbi algorithm performs optimally, but suffers from a high complexity. Complexity reductio has bee studied i [6], [7], [8], ad [9]. The per survivor processig (PSP) has bee applied i [10] i a specific case (biary full-respose CPFSK) to reduce the trellis state umber ad hece the computatio cost. However, like the Viterbi algorithm, eve a slight modulatio idex variatio yields a severe error rate degradatio. I [11], the PSP has also bee applied to soft-i soft-out (SISO) CPFSK detectio with a irratioal modulatio idex. Furthermore the modulatio idex mismatch betwee the trasmitter ad the receiver is take ito accout i the metric computatio, ehacig the robustess of the resultig modified SISO BCJR algorithm. However a perfect kowledge of the modulatio idex is assumed. I this paper, we geeralize the priciple of the PSP approach for state umber reductio ad metric modificatio for modulatio idex mismatch correctio to ay kid of CPM. We itroduce a maximum-likelihood estimatio of the modulatio idex i the resultig coheret receiver. We compare the proposed coheret receiver with the o-coheret structure of [5] takig ito accout the spectral efficiecy loss ivolved by the ML modulatio idex estimatio. Simulatios carried out o a AWGN chael, for both biary (Bluetooth ad AIS stadards parameters) ad o-biary CPM (Wireless M-Bus stadard parameters), show the efficiecy of the proposed receiver. The paper is orgaized as follows: we first briefly itroduce the system model i Sectio 2. The we defie the geeralized receivers for CPM sigals for ay modulatio idex as well as the modulatio idex estimatio i Sectio 3. The applicatios ad simulatios are give i Sectio 4, followed by a coclusio ad some perspectives i Sectio CPM sigal Model 2. SYSTEM MODEL The complex basebad CPM sigal is defied as: E s(t, α) = T ejφ(t,α), (1) where E is the average symbol eergy, T is the symbol duratio ad φ(t, α) the iformatio-carryig phase give by: φ(t, α) = 2πh tx α i q(t it ). (2)

2 α = {α i } deotes the iformatio sequece. The iformatio symbols α i are assumed to be idepedet ad idetically distributed ad to take values i the M-ary alphabet M = {±1, ±3,..., ±(M 1)}. h tx is the modulatio idex used at the trasmitter. The phase respose q(t) is defied o [0, + ) ad satisfies : q(t) = 1, t LT. (3) 2 Usig the properties of q give i (3), the iformatiocarryig phase durig the -th time iterval, [, ( + 1)T ], N, ca be writte as: φ(t, α) = 2πh tx, = πh tx 2.2. Coheret receiver α i + 2πh tx, = θ tx, + φ tx,(t). (4) From (4) we observe that the modulated sigal over the -th time iterval depeds both o the phase state deoted by θ tx, ad o the L most recet symbols, i.e (α, α 1,..., α +1 ). For a ratioal modulatio idex h tx = ktx p tx, the phase state θ tx, modulo 2π ca take oly p tx or 2p tx differet values for eve ad odd k tx, respectively. Therefore, the phase evolutio ca be described by a fiite-state machie, where each state is represeted by a L-dimesioal vector (θ tx,, α 1, α 2,..., α +1 ) ad where the umber of such states is p tx M L 1. We assume that the sigal is trasmitted over a Gaussia chael. The equivalet basebad received sigal, deoted by r(t), is defied as: r(t) = s(t, α) + (t), (5) where (t) is a realizatio of a zero-mea wide sese statioary complex circularly symmetric Gaussia oise, idepedet of the sigal, ad with double-sided power spectral desity 2N 0 over the badwidth of s(t, α). The MLSEdetector aims at maximizig the scalar product betwee r(t) ad all the realizatios of s(t, α). Assumig N d trasmitted symbols, the MLSE estimatio of the iformatio symbols α 0, α 1,..., α Nd 1 is give by: [ ] Nd T ( α 0, α 1,,..., α Nd 1) = arg max R r(t)s (t, α)dt, α M N d 0 (6) where R(x) deotes the real part of x. As N d T r(t)s (t, α)dt = N d 1 0 =0 r(t)e jφ(t, α) dt, the computatio of all scalar products over [0, N d T ] is equivalet to the sum of all scalar products over a symbol duratio give the previous phase state θ tx,. We itroduce ad deote by BM () i,j the brach metric correspodig to the scalar product associated to the trasitio from state σ i = ( ) θtx,, i α 1, α 2,..., α +1 to state ) σ j = (θ j tx,+1, α, α 1,..., α +2, where α is the symbol related to this trasitio durig the -th time iterval. It is calculated as follows: BM () i,j = R{ 0 r(t)e j(θi tx, +φtx,(t)) dt}. (7) The Viterbi algorithm is based o the recursive computatio of the accumulated metric at state σ i ad time istat ( deoted by CM () i) ad defied as: CM () i = max j CM ( 1) j + BM () i,j. At the ed, the MLSE baseddecisio rule is give by the sequece α yieldig the maximum over i of CM (N) i. We assume that the modulatio idex h tx is ot perfectly kow at the receiver. The coheret detectio requires a estimate of h tx. I that case, N e pilot symbols are iserted at the begiig of the frame. Deotig by H the set of possible values for h tx, the ML estimate of h tx deoted by ĥtx, is computed as: [ ] NeT ĥ tx = arg max R r pilots (t)s i pilots (t)dt, (8) h i H where r pilots (t) is the received sigal ad Spilots i is the modulated traiig sequece with h i. 3. PROPOSED RECEIVERS 3.1. Geeralized No-coheret CPM receiver I this sectio we geeralize to M-ary CPM the o-coheret sequece detectio receiver for Bluetooth systems referred to as adaptive NSD (No-coheret sequece detectio) ad developed i [5]. No-coheret receivers are desiged to solve the problem of robustess to modulatio idex variatio whe a perfect estimatio caot be guarateed. I [5] a trellis is defied with a reduced umber of states. The states correspodig to [, ( + 1)T ] deped o (α,..., α +1 ) oly. The phase state θ tx, is determied by employig the per-survivor processig approach (PSP) [12]. I order to be robust to phase variatios, a adaptive phase referece recursively updated q ref () is added. So, the ew expressio of the brach metric becomes: BM () i,j [ ] =R{ q (i) ref () r(t). e j(θi tx, +2πhtx k=+1 α kq(t kt )) dt}, where the phase referece is updated as follows: q (j) ref ( + 1) = βq(i) ref () + (1 β) r(t). e j(θi tx, +2πhtx k=+1 α kq(t kt )) dt, (9) (10)

3 where q (k) ref is the phase referece related to the kth state ad i is the previous state of the state j. The parameter β, 0 β < 1 acts as a forgettig factor. However, the estimatio of modulatio idex h tx i the NSD receiver is doe with the help of the first N e received symbols to adaptively choose the modulatio idex yieldig the maximum cumulative metric (9): { ( ) ˆα 1, ˆα 2,..., ˆα Ne, ĥtx = arg max ˆα 1,ˆα 2,...,ˆα Ne,h i H { N et R r(t) ( Sseq(t) i ) } (11) } q ref (t)dt h i 0 where S i (t) correspods to the modulatio of (ˆα 1,..., ˆα Ne ) with h tx = h i. After the estimatio period ĥtx is used for detectig the remaiig sequece. This receiver does ot require the isertio of pilots Geeralized coheret CPM receiver We first assume a perfect kowledge of the modulatio idex h tx. The proposed receiver relies o the decompositio of h tx i the form h tx = h rx + h with h rx beig a ratioal umber. The key idea is to use the Viterbi algorithm with modified brach ad state metrics o a trellis desiged from h rx. It takes ito accout a phase differece proportioal to h ad computed o a per survivor processig basis [12]. The PSP applicatio for state umber reductio purpose was itroduced i [10], [11] i the case of biary full respose CPFSK. I [11], the brach metric computatio takes ito accout h, ad is applied i a SISO algorithm amely the BCJR [13]. I this sectio we geeralize the PSP brach metric computatio of [13] to ay kid of CPM modulatio, whatever the modulatio idices h tx ad h rx, the value of L or the frequecy pulse g(t) are. Goig back to the iformatio-carryig phase expressio give i (4), we express it as a fuctio of h rx ad h : φ(t, α) = πh tx = πh rx + 2πh rx + 2π h α i + 2πh tx α i + π h α i = θ rx, + φ rx,(t) + λ i + 2π h (12) with θ rx, = πh rx α i, φ rx, (t) = 2πh rx ad λ i = π h α i. The first two terms i (12) are tracked by the Viterbi algorithm ad the third term is the resultig phase differece which is built up at every symbol. This accumulatio is calculated usig the PSP techique by associatig to each state a additioal parameter λ i. The last term of (12) is calculated at the output of the matched filter. We ca write the followig relatio betwee the corrected ad the referece phase states. θ i tx = λ i + i krx p rx π, i = {0, 1, 2,..., 2p rx 1} = λ i + θ i rx. (13) The oly additioal task that eeds to be performed is the PSP update of the phase differece for each state, λ i. The update equatio is give by: λ i +1 = λ j + α (j,i)π h (14) where j is selected from the M previous states of σ i, as the idex of the maximum cumulative metric ad α (j,i) is the correspodig data symbol ivolved i the trasitio from state σ j to σ i i the receiver trellis structure. We defie a ew expressio of the brach metric σ j σ i used i the proposed algorithm: ( ) θrx(λ j j ), α 1, α 2,..., α +1 α ( ) θ i rx (λ i +1), α, α 1, α 2,..., α +2. The brach metric is the calculated as follows: BM () j,i =R{e jλj r(t). e j(θj rx +2πhrx k=+1 α kq(t it )) e j2π h k=+1 α kq(t kt ) dt} (15) I the simulatios, h tx is estimated from N e pilot symbols by applyig Equatio (8). The brach metrics are computed by replacig h by h i (15). 4. APPLICATIONS AND SIMULATIONS The proposed algorithms ca be used i all cases where the origial Viterbi algorithm fails to achieve the best trade-off betwee performace ad complexity. I all simulatios we take a oversamplig of 8 samples per symbol iterval ad we deote by N d the data legth. For the coheret detectio, we take ito accout the spectral efficiecy loss i the computatio of E b N Bluetooth The first example correspods to the Bluetooth stadard which employs the biary Gaussia frequecy-shift keyig (GFSK) modulatio with a frequecy pulse legth L = 3 ad a frame legth N d = The modulatio idex is assumed to vary i a fiite alphabet H kow at the receiver [5], [14]. For the modulatio idex estimatio, the best traiig sequece has all symbols equal to 1, sice it allows to have

4 the maximum decorrelatio betwee the referece sigals Sseq. i I Figure 1 we compare the coheret receiver to the adaptive NSD. We fix h tx = 0.35 ad h rx = 2 3. h rx = 2 3 esures the best state umber reductio. We fix the forgettig factor β = 0.9 ad N e = 50 i the case of NSD ad let N e vary i the case of the coheret receiver. We observe that the coheret receiver requires at least N e = 15 to outperform the NSD. For N e = 20 its performace coicides with the optimum MLSE oe ad exhibits a gai of 1 db compared to NSD for a = MLSE 3GMSK N SD N e = 50 µ h = α = 0.9 N e = 40, µ h = N e = 20, µ h = N e = 40, µ h = N e = 30, µ h = N e = 5 N e = 10 N e = 15 N e = 20 MLSE h tx = 0.35 = 7 20 NSD α = Fig. 1. Bluetooth detectio for a trasmissio with h tx = 0.35 ad a coheret receptio with h rx = 2 3 with N d = Automatic Idetificatio System (AIS) The secod applicatio case is the radom variatio of the modulatio idex aroud its omial value as i AIS systems due to a aalog low-cost trasmitter [2]. The AIS stadard uses a biary GMSK modulatio with BT = 0.4. At the trasmitter, we geerate radomly the trasmissio modulatio idex h tx i the iterval [ ]. At the receptio we discretize this iterval with a step µ h to estimate h tx i the same maer as i (8). For the NSD receiver we fix the forgettig factor β = 0.9, µ h = ad N e = 50 ad let µ h ad N e vary for the coheret receiver. The MLSE performace is plotted as a lower boud. The coheret receiver is built with h rx = 2 3. From Figure 2, we see that for µ h = ad N e = 40 the coheret receiver outperforms the NSD receiver with a gai of 0.3 db ad a loss of oly 0.2 db compared to the lower boud for a = Wireless M-Bus The third applicatio case is the demodulatio of quaterary GFSK sigals. This applicatio is kow as Wireless M- Fig. 2. AIS detectio for a trasmissio with radom modulatio idex h tx ad a coheret receptio with h rx = 2 3 with N d = 256 Bus which is a Europea Stadard for wireless Meterig BUS [15]. To lower the cost ad the power cosumptio, a aalog trasmitter ca be implemeted o the meters, yieldig idex modulatio variatio as for the AIS case. The best traiig sequece has all symbols equal to 3. I the quaterary modulatio case, we foud from several simulatios that the value of h rx = 2 5 is the best choice for a compromise betwee performace ad complexity. The performace of Figure 3 are obtaied by averagig over several values of h tx take radomly from the iterval [ ]. For µ = ad N e = 50 we obtai ear optimal performace with a egligible loss of 0.3 db compared to the optimum MLSE ad a gai of 0.2 db compared to the NSD receiver oe for a =. 5. CONCLUSIONS AND PERSPECTIVES I this paper, we have exteded the PSP applicatio ad brach metric modificatio of trellis-based coheret receivers, iitially proposed for biary CPFSK, to detect M-ary CPM sigals with ay modulatio idex ad phase respose. We have applied this geeralizatio to cases where the modulatio idex is ukow at the receiver ad has to be estimated. We have simulated the error rate performace i three differet applicatios, amely Bluetooth, Automatic Idetificatio System (AIS) ad the Wireless Meterig Bus Europea stadard. The error rate performaces have bee compared to the performace of a adaptive o-coheret receiver proposed for Bluetooth detectio, which is, to the best of the authors kowledge, oe of the most promisig receiver for the Bluetooth stadard, where robustess agaist ukow modulatio idex is a cosiderable advatage. For this com-

5 pariso, we have exteded the adaptive o-coheret receiver to o-biary CPM. The compariso shows that, despite the ukow modulatio idex, the coheret receiver outperforms the o-coheret receiver. LMSE N SD N e = 50 µ = α = 0.9 N e = 50 µ = N e = 50 µ = Fig. 3. Wireless M-Bus detectio for a trasmissio with radom modulatio idex h tx ad a coheret receptio with h rx = 2 5 with N d=1000 quaterary symbols REFERENCES [1] J.B. Aderso, T. Auli, ad C.E. Sudberg, Digital Phase Modulatio, Spriger, [2] D. Boacci, J.P. Millerioux, R. Prevost, J. Lemaitre, M. Coulo, ad J. Toureret, Advaced cocepts for satellite receptio of AIS messages (Toulouse Space Show, Toulouse, 25/06/ /06/2012), [3] Specificatio of the Bluetooth System, December 1999, Bluetooth Special Iterest Group documet. [4] M.K. Simo ad C. Wag, Differetial versus limiter discrimiator detectio of arrow-bad fm, Commuicatios, IEEE Trasactios o, vol. 31, o. 11, pp , Nov [5] L. Lampe, R. Schober, ad M. Jai, Nocoheret sequece detectio receiver for bluetooth systems, Selected Areas i Commuicatios, IEEE Joural o, vol. 23, o. 9, pp , Sept [6] A. Svesso, Carl-Erik Sudberg, ad Tor Auli, A class of reduced-complexity viterbi detectors for partial respose cotiuous phase modulatio, Commuicatios, IEEE Trasactios o, vol. 32, o. 10, pp , [7] S.J. Simmos ad P.H. Wittke, Low complexity decoders for costat evelope digital modulatios, Commuicatios, IEEE Trasactios o, vol. 31, o. 12, pp , [8] A. Svesso, Reduced state sequece detectio of partial respose cotiuous phase modulatio, Commuicatios, Speech ad Visio, IEE Proceedigs I, vol. 138, o. 4, pp , [9] G.K. Kaleh, Simple coheret receivers for partial respose cotiuous phase modulatio, Selected Areas i Commuicatios, IEEE Joural o, vol. 7, o. 9, pp , [10] M.J. Miller, Detectio of CPFSK sigals usig per survivor processig, i Military Commuicatios Coferece, MILCOM 98. Proceedigs., IEEE, 1998, vol. 2, pp vol.2. [11] S. Zarei, W. Gerstacker, G. Kilia, ad W. Koch, A iterative detectio algorithm for coded cpfsk sigals with irratioal modulatio idex, i Sigal Processig Coferece (EUSIPCO), 2012 Proceedigs of the 20th Europea, Aug 2012, pp [12] R. Raheli, A. Polydoros, ad Chig-Kae Tzou, Persurvivor processig: a geeral approach to MLSE i ucertai eviromets, Commuicatios, IEEE Trasactios o, vol. 43, o. 234, pp , [13] L. Bahl, J. Cocke, F. Jeliek, ad J. Raviv, Optimal decodig of liear codes for miimizig symbol error rate (corresp.), Iformatio Theory, IEEE Trasactios o, vol. 20, o. 2, pp , Mar [14] N.. Ibrahim, L. Lampe, ad R. Schober, Bluetooth receiver desig based o lauret s decompositio, Vehicular Techology, IEEE Trasactios o, vol. 56, o. 4, pp , July [15] Specificatio of Wireless M-bus System, , Wireless M-bus EN Stadard.