Non-Binary LDPC Coded DPSK Modulation for Phase Noise Channels

Transcription

1 1 Non-Bnary LDPC Coded DPSK Modulaton for Phase Nose Channels Tudor Nnacs, Balázs Matuz, Ganlug Lva Insttute of Communcaton and Navgaton Deutsches Zentrum für Luft- und Raumfahrt (DLR), Wesslng, Germany e-mal: {tudor.nnacs, balazs.matuz, Gulo Colavolpe Dpartmento d Ingegnera dell Informazone, Unverstà d Parma 43124, Parma, Italy e-mal: gulo@unpr.t Abstract In ths paper, we study dgtal transmsson over an addtve whte Gaussan nose (AWGN) channel wth m- ary dfferental phase-shft keyng (DPSK) modulaton n the presence of phase nose. At the recever sde, non-coherent teratve detecton and decodng s assumed. We present a nonbnary low-densty generator matrx (LDGM) code desgn whch s sutable for both coherent and non-coherent channels. The code constructon s strongly related to the one of non-bnary rregular repeat-accumulate (IRA) low-densty party-check (LDPC) codes. I. INTRODUCTION In the presence of phase nose, detecton may turn out to be a challengng task. Examples are satellte communcatons n Q/V bands, systems wth termnals usng cheap oscllators, and/or systems where short message blocks are employed and the use of plots s lmted. In all these cases non-coherent detecton s a vable opton [1] [4]. A common technque on non-coherent channels s to employ a dfferental modulaton scheme, partcularly n combnaton wth phase-shft keyng (PSK). To ths end, the PSK modulated symbols are further nput to a phase-accumulator before beng sent on the channel. At the recever, non-coherent detectors workng on multple symbols are shown to close the gap w.r.t. deal coherent detecton [2] for suffcently long sequences. Whenever channel codng s also consdered, teratve detecton and decodng at the recever s a common choce [3]. The schemes for teratve detecton/decodng may be dvded nto two groups [4]. For the frst group, standard bnary channel codes are complemented by a further block, for nstance by the aforementoned dfferental modulator, n order to compensate for the phase uncertantes from the channel. Ths results n an teratve demodulaton and decodng scheme, as descrbed n [5] and the references theren, whch de facto corresponds to a seral turbo(-lke) code for whch both component codes exchange soft nformaton. Hgh codng gans have been Ths work has been accepted for publcaton at the IEEE Internatonal Conference on Communcatons 217, IEEE ICC 17: Brdgng People, Communtes, and Cultures. c 217 IEEE. Personal use of ths materal s permtted. Permsson from IEEE must be obtaned for all other uses, n any current or future meda, ncludng reprntng/republshng ths materal for advertsng or promotonal purposes,creatng new collectve works, for resale or redstrbuton to servers or lsts, or reuse of any copyrghted component of ths work n other works. demonstrated wth such schemes, e.g., on non-coherent [3], [5] and fadng [6] channels. For the second group of algorthms, one may modfy the component decoders/structure of the channel code tself to cope wth phase uncertantes. In [7] non-bnary low-densty party-check (LDPC) codes over felds were shown to help resolve the unknown phase on the blockwse non-coherent addtve whte Gaussan nose (AWGN) channel. In [8] LDPC codes were constructed over rngs and used for an AWGN channel wth phase nose. It has been llustrated for quaternary phase-shft keyng (QPSK) modulatons that, gven a certan parttonng of the check nodes n local and global check nodes, moderate phase nose can be handled wth degradatons n performance of roughly.7 db. When PSK modulatons are consdered, t seems to be natural to use channel codes over rngs, preservng the lnearty of the overall scheme. Convolutonal codes over rngs for AWGN channels n absence of phase nose were consdered n [9], [1]. LDPC codes over rngs for the coherent AWGN channel and PSK modulaton were nvestgated n the context of bandwdth-effcent communcatons n [11] and t was shown that they may outperform typcal technques lke bnary bt-nterleaved coded modulaton. It was noted that the constructon of such LDPC codes must be taken wth partcular care. Zero dvsors n the party-check matrx may lead to poor mnmum dstances [9]. In ths paper, we consder AWGN channels wth phase nose whch we model as a random-walk process usng the Wener model. We propose a scheme that s a seral concatenaton of two component codes: an outer non-bnary low-densty generator matrx (LDGM) code over a feld of order m and an nner accumulator over the rng of ntegers modulo m. The concatenated scheme resembles an LDPC code wth accumulator structure whose output serves as nput to a m- PSK modulator. At the recever, demodulaton and decodng of the nner component code are done jontly usng a dscretzedphase (DP) detecton algorthm [12], whereas the outer component code s decoded by belef propagaton (BP) on the outer code s bpartte graph. Note that a related concept of jonng detector and check node decoder for bnary repeat-accumulate (RA) codes was ntroduced for multple-nput and multpleoutput (MIMO) communcaton lnks n [13] and adopted to

2 2 u v b x m-psk Phase G modulator accumulator F m F m X X m-dpsk modulator Fg. 1: Transmtter of the proposed scheme. u v c x F G m m-psk accumulator modulator F m F m F m X non-systematc IRA encoder over F m Fg. 3: Structure of a non-systematc IRA encoder wth m-psk. u v a c x G f(v ) Z m m-psk accumulator modulator F m F m Z m Z m X m-dpsk modulator Fg. 2: Equvalent transmtter descrpton. channels wth phase nose n [14]. II. PRELIMINARIES We consder transmsson over an AWGN channel wth phase nose and assume m-ary dfferental phase-shft keyng (DPSK) at the transmtter. Denote the vector of receved samples by r = [r,..., r N ] and the vector of modulated codeword symbols by x = [x,..., x N ]. In the dscretetme baseband model, the relaton between the transmtted and receved symbols can be expressed as r = x e jθ + n (1) = e jφ e jθ + n (2) = e jψ + n where (1) s due to the use of m-dpsk modulaton, for whch x = e jφ, φ {2πl/m}, wth l {,..., m 1}. Further, (2) results from the defnton ψ = [θ + φ ] 2π, havng denoted by [ ] 2π the operaton modulo 2π. We have that n are ndependent complex Gaussan nose samples,.e., n CN (, 2σ 2 ) and the channel phase nose s modeled accordng to the Wener model wth θ = θ 1 + θ, (1) where θ N (, σ 2 ), θ beng unformly dstrbuted n [, 2π). In Secton VI, we also present results for a blockwse noncoherent AWGN channel whch s obtaned by settng σ 2 =. In ths case the phase s constant, but unknown and unformly dstrbuted n [, 2π) over an entre codeword, and we have A. Proposed Scheme r = x e jθ + n. III. TRANSMITTER DESCRIPTION Fgure 1 depcts the transmtter of the proposed scheme. It conssts of the seral concatenaton of a (N, K) lnear block code C over F m and a m-dpsk modulator. Throughout the entre manuscrpt we wll consder m = 2 p, p N, p > 1. Denotng by G the K N generator matrx of C, we have that the nput vector u = [u 1,..., u K ] s encoded nto an (outer) codeword v = [v 1,..., v N ] wth v = ug. The symbols v are then nput to a m-dpsk modulator, whch frst maps each v to a complex m-psk symbol b = e jϕ, wth b = [b 1,..., b N ], and then accumulates the phases, yeldng the modulated codeword symbols x = [x,..., x N ]. By defnng the channel nput alphabet X = { e j2πl/m}, l {,..., m 1} we have that b, x X. Note that the phase accumulator outputs N +1 symbols x, where x = 1. The scheme admts an equvalent representaton whch s provded n Fgure 2. For ths, assume that n an ntermedate step the symbols v F m are mapped by a bjectve functon f : F m Z m to symbols a Z m,.e., to elements of the rng of ntegers modulo m. For the m-dpsk constellaton labels we consder Gray mappng and defne the bjectve mappng f so that the bnary representatons of the F m elements match the labels of the m-dpsk constellaton ponts. E.g. for m = 8, the mappng f s specfed n Table I. Usng arg(b ) = ϕ = 2πa /m and φ = φ 1 + ϕ, x equals x = e jφ = e j l=1 ϕ l = e j2π/m l=1 a l (2) where wthout loss of generalty φ =. Next, the order of the blocks nsde the m-dpsk modulator n Fgure 1 s reversed so that symbols a are frst nput to an accumulator over Z m and afterwards to the m-psk modulator. In partcular, the accumulator computes the output symbol at tme as c = c 1 + a = where the sum s defned over Z m and c =. Now, by mappng c to a m-psk constellaton pont, we obtan x = e jφ l=1 a l = e (j2πc/m) = e j2π/m l=1 a l whch s dentcal to (2) and llustrated n Fgure 2. Fgure 2 allows a dfferent nterpretaton of the encoder. In fact, we can regard the seral concatenaton of the (outer) code C and the (nner) accumulator over Z m as a block code wth nput u defned over F m and output c defned over Z m. Ths s very remnscent, though not equvalent, of the structure of a non-systematc rregular repeat-accumulate (IRA) [15] code over F m as n Fgure 3. The scheme n Fgure 3 s not meant for transmsson over non-coherent channels. Nevertheless, we explot the smlartes to construct the code C for the m-dpsk coded scheme under the assumpton of teratve decodng and detecton. B. Reference Scheme In the sequel a competng state-of-the-art scheme from the lterature [4] wll be also consdered and ts performance presented n Secton VI. It s obtaned by replacng the outer code C n Fgure 2 by a bnary convolutonal code wth generator polynomals (7, 5) n octal notaton, as descrbed

3 3 TABLE I: Mappng between feld elements, ther bnary representaton, rng elements and modulaton symbols wth m = 8. The prmtve polynomal for F 8 s 1 + x + x 3 =. F 8 element Bnary label Z 8 element 8-PSK symbol α 1 1 e jπ/4 α e j3π/4 α e j7π/4 α e jπ/2 α e j3π/2 α e j5π/4 α e jπ n [4]. Further, the scheme s complemented by an nterleaver whch acts on the output bts of the outer code. The nterleaved bts are then mapped to Z m by a mappng functon and passed to the nner encoder,.e., the Z m -accumulator.fnally, the symbols are m-psk modulated. Decodng s performed by teratvely exchangng messages among the nner and outer decoders through an nterleaver. The result s a powerful seral turbo scheme [4], [6], [16] to whch we wll refer as seral turbo code throughout the remanng parts of the paper. IV. CODE DESIGN A. Revew of IRA Codes over F m The IRA encoder n Fgure 3 works as follows. The nformaton vector u s frst multpled by the matrx G yeldng the outer codeword v. Symbols of v are then nput to a tmevaryng accumulator over F m whch produces the codeword symbols through the recurson c = w c 1 + v where (, +) are defned n F m, m = 2 p, p N, p > 1. For the code desgn, we assume that the multplcatve coeffcents used on the feedback branch of the accumulator w wth = 1,..., N 1, are chosen wth unform probablty n F m \{}. Let us ntroduce the N N double-dagonal matrx W whch has form W = 1... w w w N w N 1 1 The generator matrx of the IRA code C IRA s gven by G IRA = G(W 1 ) T. and the codeword c = ug IRA. Consder now a code C SIRA obtaned by appendng to each codeword c of the IRA the nformaton vector u, producng a new codeword [u c]. The code C SIRA s an IRA code wth systematc encoder, havng a generator matrx of the form [ G SIRA = I G(W 1 ) T] and party-check matrx ] H SIRA = [G T W havng denoted by I a K K dentty matrx. By constructon, the IRA code wth non-systematc encodng can be vewed as a punctured verson of the IRA code wth systematc encodng. Hence, the IRA code wth non-systematc encodng can be decoded over the Tanner graph assocated to H SIRA, where the varable nodes assocated wth the frst K columns of H SIRA are not connected to the channel. B. Surrogate Desgn To facltate the code desgn, we follow a surrogate desgn approach. To ths end, we frst desgn the party-check matrx H SIRA of an IRA code over F m for the coherent AWGN channel wthout phase-nose. For ths we rely on standard tools from the lterature. More precsely, our desgn s based on protographs whch are relatvely small bpartte graphs that serve as templates for larger graphs [17]. Sutable protographs can be found, e.g., by densty evoluton [18] whle the progressve edge growth (PEG) algorthm [19] can be used to obtan the Tanner graph of the code. We then adopt the resultng matrx G to descrbe the outer code C n the m-dpsk coded modulaton scheme. More n detal, the code desgn proceeds by optmzng the protograph structure wth respect to the teratve decodng threshold va densty evoluton analyss. Due to the potentally large search space, one may restrct the optmzaton to a specfc famly of IRA protographs. In ths paper we focus on IRA protographs wth base matrx [ ] B SIRA = d 1 1 where the leftmost column of B SIRA s assocated wth the (punctured) systematc part of the codeword, d beng a postve nteger parameter over whch we optmze va Monte Carlo densty evoluton for the coherent AWGN channel. To handle the lack of symmetry n the codebook stemmng from the use of m-psk modulatons wth m > 4, we resort to the use of channel adapters n the analyss [18]. Note that, for a 2 3 protograph, the freedom n desgn s lmted. The upper one entry n the leftmost column s requred for convergence purposes when puncturng. Also, the two rghtmost columns have a fxed structure to realze an accumulator. Resultng teratve decodng thresholds for the coherent AWGN channel for varous d are depcted n Table II. Followng ths, we select a protograph wth d = 2, snce smulaton results suggest lower error floors than for d = 1. Fnally, once H SIRA s avalable, the outer code n Fgure 2 s obtaned by extractng G from H SIRA followng Equaton (3). Due to the low-densty nature of G, the outer code s an LDGM code. Decodng s done va BP on the bpartte graph of the party-check matrx H dec of ts systematc counterpart, where H dec = [ G T I ] s sparse. A. Overvew V. RECEIVER DESCRIPTION A block dagram of the recever s gven n Fgure 4. It conssts of an nner decoder,.e., a m-dpsk detector, as (3)

4 4 TABLE II: Iteratve decodng thresholds (E b /N ) for dfferent values of dfor the coherent AWGN channel. The Shannon lmt correspondng to a rate of 1.5 bts per channel use under 8-PSK modulaton s at (E b /N ) Sh = 1.3 db. r L A,det d (E b /N ) [db] Detector π( ) π 1 ( ) L E,dec Decoder for code C L E,det L A,dec Fg. 4: Structure of the teratve recever. well as an outer decoder for the code C whch s decoded by standard BP [2]. The two decoders teratvely exchange soft nformaton. In detal, the receved word r serves as nput to a maxmum a posteror (MAP) detector whch produces extrnsc soft nformaton L E,det on the symbols a (c.f. Fgure 2). We have that L E,det û / = L APP,det L A,det where the dvson s done element-wse, L APP,det = P (a r) s a m-dmensonal probablty mass functon (p.m.f.) and = P (a ) s ntalzed to [1/m,..., 1/m] pror to startng the teratve process and refned teraton by teraton. The extrnsc nformaton output by the detector s depermuted by the functon π 1 ( ) to obtan the a pror soft L A,det nformaton nput to the decoder L A,dec. The de-permutaton of the p.m.f.s s due to the mappng f 1 ( ) at the encoder. Durng the teraton process, the decoder estmates the MAP probablty P (v L A,dec ) for the outer code symbols v. Only extrnsc / nformaton L E,dec = L APP,dec L A,dec on the symbols v s forwarded, permuted and provded as a pror nformaton L A,det to the detector, where and L A,det ( = π L APP,dec L E,dec P (v L A,dec ) ). Agan the permutaton of the p.m.f.s s due to the mappng f ( ) at the encoder. An exchange of nformaton between detector and decoder s performed for a maxmum number of teratons. 1 Fnally, the teratve scheme provdes an estmate of P (u r), L APP (u ) P (u r) upon whch a decson on the nformaton symbols s made û = arg max u P (u L APP (u )). 1 In ths work, we fx the number of nternal teratons of the decoder to one and the number of teratons between decoder and detector to 2. B. Dscretzed-Phase Algorthm We now address the operaton of the detector. We am to compute L E,det = P (a r)/p (a ) and thus express the jont p.m.f. p(a, ψ r) as p(a, ψ r) = p(r a, ψ)p(ψ a)p (a) 1 p(r) N p(r ψ ) p(r ψ )p(ψ ψ 1, a )P (a ). =1 Above we have used the fact that due to the Wener model and the dfferental modulaton p(ψ ψ 1,..., ψ, a) = p(ψ ψ 1, a ). Snce the jont p.m.f. p(a, ψ r) s gven as a product of probabltes, the margnal symbol-wse p.m.f. P (a r) can be computed wth the help of a forward/backward recurson on the correspondng factor graph [21] of the functon. From ths we have that P (a r) P (a ) = 2π 2π α(ψ 1 )β(ψ )p(ψ ψ 1, a )dψ dψ 1 (4) where α(ψ ) and β(ψ ) are the forward/backward coeffcents obtaned from the factor graph as [12] α(ψ ) = p(r ψ ) 2π ( ) p(ψ ψ 1, a )P (a ) α(ψ 1 ) dψ 1 a β(ψ ) = p(r ψ ) 2π p(ψ +1, ψ, a +1 )P (a +1 ) β(ψ +1 ) dψ (6) +1, a+1 wth α(ψ ) = p(r ψ ) and β(ψ N ) = p(r N ψ N ). Next we elaborate on the expressons nvolved n the computaton of the forward/backward coeffcents. We have that 1 p(r ψ ) = r 2πσ 2 e e 2σ 2 jψ 2 and from the Wener model n (1) and the dentty φ = φ 1 + ϕ, ψ can be wrtten as ψ = [ψ 1 + ϕ + θ ] 2π. Snce b = e jϕ = e j2πa/m s a determnstc mappng of a, t holds that p(ψ ψ 1, a ) = p(ψ ψ 1, b ) = p (ψ ψ 1 ϕ ) where p ( ) s the probablty densty functon (p.d.f.) of the phase ncrement θ n the Wener process (modulo 2π). Furthermore, we can express where p (ϕ) = + l= (5) g(, σ 2 ; ϕ 2πl) (7) g(µ, σ ; 2 1 ϕ) = e (ϕ µ) 2σ 2. (8) 2πσ 2 2

5 5 For practcal values of σ, the tals of the p.d.f. n (8) tend to zero n almost all ponts except some n the vcnty of µ [14], and we have that and thus (7) smplfes to p (ϕ) g(, σ 2 ; ϕ) p (ψ ψ 1 ϕ ) g(, σ 2 ; ψ ψ 1 ϕ ). (9) Even n ts smplfed form, usng (9) n (4), (5) and (6) nvolves computng ntegrals of contnuous p.d.f.s. A more practcal approach s the use of a DP algorthm [4] whch assumes ψ only takes values n the dscrete set {2πj/L}, j {,..., L 1}, where L s a desgn parameter. Combned wth the fact that for practcal values of σ the p.d.f. n (9) has non-zero values only n the strct vcnty of, the followng approxmaton [5] may be used 1 P, ϕ = P p (ϕ) = 2, ϕ = 2π. L, otherwse In accordance wth [5], a phase dscretzaton factor of L = 8m has been chosen. The parameter P s a desgn parameter, also referred to n lterature as transton probablty of the dscretzed model [14]. By usng ths approxmaton, the ntegrals n (4), (5) and (6) become summatons and the computaton of the forward/backward coeffcents, as well as of the extrnsc probabltes becomes practcal. VI. NUMERICAL RESULTS We target short blocks n the order of a few hundred code bts. We llustrate that, despte the short block-length, the proposed code constructon works well on coherent channels, as well as on blockwse non-coherent AWGN channels and AWGN channels wth (symbol-wse) phase nose, as descrbed n Secton II. A. Coherent AWGN Channel In the followng, we assume 8-DPSK. We desgned an outer LDGM code C over F 8 followng Secton IV. Together wth the nner code,.e., the Z 8 -accumulator we obtan an LDPClke code, as ponted out earler. We focus on short codes wth K = 1 symbols (3 bts) and N = 2 symbols (6 bts). Frst, the performance of the code on a coherent AWGN channel s presented. For both decoder and detector we used the schemes descrbed n Secton V and set the desgn parameter P =.1 whch s complant wth [14]. A consequence of usng the DP-detector s a msmatch to the channel, snce the detector assumes a phase uncertanty whch s not present n the coherent case. As a reference, for the coherent channel we have also smulated the performance of the teratve scheme removng the DP model n the detector. For ths, we run BP on the bpartte graph of the outer code C and of the nner accumulator. Ths procedure would not work on the phase nose channel and s only shown here to dscuss the sub-optmalty of the DP algorthm for the selected block length. CER Seral turbo, BP+DP (msm.), bl. non-coh. Seral turbo, BP+DP (msm.), coh. Seral turbo, BP (w/o DP), coh. LDPC, BP+DP (msm.), bl. non-coh. LDPC, BP+DP (msm.), coh. LDPC, BP (w/o DP), coh. (6, 3) RCB, coh E b /N [db] Fg. 5: CER vs. E b /N of LDPC and seral turbo code on a coherent AWGN channel under BP w/o DP and BP+DP decodng. Also the performance on the blockwse non-coherent AWGN channel under BP+DP decodng s gven. Smulaton results n terms of CER versus E b /N are presented n Fgure 5 together wth the random codng bound (RCB) [22] as a benchmark. We observe that the proposed scheme s wthn 1 db from the correspondng RCB. Further, for the proposed block length, owng to the msmatched DPdetector, a slght loss of.2 db w.r.t. matched detecton s vsble. Fnally, we also smulated the performance of the seral turbo code from Secton III-B. Also n ths case we assumed both a detector w/o the DP algorthm for the coherent case, as well as a msmatched DP-detector. The turbo code exhbts a further loss of more than 1 db compared to the proposed constructon. B. Blockwse Non-coherent AWGN Channel In Fgure 5 we have also depcted the performance of the LDPC code on a blockwse non-coherent AWGN channel, for whch the phase s unknown, but constant over an entre codeword duraton. We assume agan the DP-detector n place, yeldng a msmatch w.r.t. the channel, snce phase transtons among modulaton symbols wthn a codeword are not present. Compared to the coherent settng wth perfect phase knowledge one may observe from Fgure 5 that the proposed scheme wth the DP-detector only loses.2 db, thus yeldng approxmately the same performance as on the coherent channel. Smlarly the turbo code s smulated and a loss n the order of 1 db w.r.t. the LDPC scheme s vsble.

6 6 CER Seral turbo, BP+DP (msm.), σ = 2 Seral turbo, BP+DP (msm.), coh. LDPC, BP+DP (msm.), σ = 2 LDPC, BP+DP (msm.), coh E b /N [db] Fg. 6: CER vs. E b /N of LDPC and seral turbo code on a coherent, as well as non-coherent AWGN channel wth σ = 2. C. Non-coherent AWGN Channel wth Phase Nose We apply the proposed algorthm on the phase nose channel as descrbed n Secton II. We ntroduce a phase nose wth σ = 2. Note that typcal Dgtal Vdeo Broadcastng Satellte 2 (DVB-S2) consumer grade equpment experences values smaller than ths, e.g. as can be observed by comparng the phase nose mask of DVB-S2 [23] to the power spectral densty (PSD) of the Wener process havng σ = 2. In Fgure 6 CER performances are shown, recappng also the results for a coherent AWGN channel wth msmatched detecton. Observe a loss w.r.t. the coherent case of merely.3 db underlnng the robustness of the proposed scheme. Also the performance of the seral turbo code s depcted for the same settng. Also here we experence smlar degradaton w.r.t. the phase nose free case. Stll a gap of around 1 db compared to the LDPC code remans. VII. CONCLUSIONS We proposed non-systematc LDGM codes over fnte felds F m concatenated wth m-dpsk modulaton, where the feld order s matched to the modulaton order. To obtan the LDGM code we rely on a surrogate desgn whch s nspred by F m IRA LDPC codes. Thanks to ts accumulator-based structure, the resultng code concatenaton performs well not only on coherent AWGN channels, but also on blockwse non-coherent AWGN channels, as well as on AWGN channels wth phase nose. A notceable gan s experenced w.r.t. standard seral turbo codes. REFERENCES [1] J. Proaks and M. Saleh, Dgtal Communcatons. New York, NY, USA: McGraw-Hll, 28. [2] D. Dvsalar and M. K. Smon, Multple-symbol dfferental detecton of MPSK, IEEE Trans. Commun., vol. 38, no. 3, pp. 3 38, Mar [3] G. Colavolpe, A. Barber, and G. 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