1 Parameter Free Iteratve Decodng Metrcs for Non-Coherent Orthogonal Modulaton Albert Gullén Fàbregas and Alex Grant Abstract We study decoder metrcs suted for teratve decodng of non-coherently detected bt-nterleaved coded orthogonal modulaton. We propose metrcs that do not requre the knowledge of the sgnalto-nose rato, and yet stll offer very good performance. I. INTRODUCTION AND SYSTEM MODEL Orthogonal modulaton wth non-coherent (NC) detecton s a practcal choce for stuatons where the receved sgnal phase cannot be relably estmated and/or tracked. Examples nclude mltary communcatons usng fast frequency hoppng, arborne communcatons wth hgh Doppler shfts due to sgnfcant relatve moton of the transmtter and recever, and hgh phase nose scenaros, due to the use of nexpensve or unrelable local oscllators. A common choce for the modulator s frequency shft keyng (FSK). The output alphabet of the FSK modulator s E = {e b : b = 0, 1,..., M 1}, where e b s the canoncal bass vector wth a one at poston b and zeros everywhere else. The channel output y[k] C M at tme k s y[k] = E s h[k]x[k] + n[k], k = 0,..., L 1 (1) where E s s the per-symbol transmt power, h[k] C s the channel gan at tme k, x[k] = (x 0 [k],..., x M 1 [k]) T, the output of the FSK modulator, s all zeros, except for a sngle element x b [k] = 1, correspondng to transmsson on a partcular frequency bn b {0, 1,..., M 1} at tme k, and n[k] s a vector of zero-mean crcularly symmetrc complex Gaussan nose samples, wth varance N 0. In the cases where h[k] = 1, k or h[k] are zero-mean complex The authors are wth the Insttute for Telecommuncatons Research, Unversty of South Australa, Mawson Lakes SA 5095, Australa. e-mal: albert.gullen@unsa.edu.au, alex.grant@unsa.edu.au. Ths work has been supported by the Australan Research Councl (ARC) Grants DP0344856 and DP0558861.
2 Gaussan wth unt varance, we have the addtve whte Gaussan nose (AWGN) and Raylegh fadng channels. The channel transton probabltes wth non-coherent detecton are [2] ) Es p (y[k] x = e b ) = KI 0 (2 h[k] y b [k] N 0 (2) where K s a constant ndependent of the hypothess b and I 0 (.) s the 0-th order modfed Bessel functon of the frst knd. Bt nterleaved coded modulaton wth teratve decodng (BICM-ID) [1] has been recently consdered n [2] for the NC-FSK channel defned by (1) and (2) and a gan s demonstrated by teratng between demodulaton and decodng. In BICM, the codewords c = (c 1,..., c N ) of a bnary code C of length N = ml and rate R are nterleaved and mapped over the sgnal alphabet, chosng frequency bn b = m 1 =0 c π(mk+)2 for transmsson, where m = log 2 M and π(.) denotes the nterleaver permutaton. An mportant consderaton n many applcatons s the amount of channel state nformaton (CSI) avalable at the decoder. Ths may range from full CSI, where the decoder knows the nstantaneous fadng ampltude and the average sgnal-to-nose rato (SNR), to partal CSI, where only the average SNR s known, rght through to no CSI, where not even the SNR s known. The latter case s of nterest for partal band jammng of a fast frequency hopped system, where the resultng SNRs for each of the M frequency bns may vary wth frequency and tme. Valent and Cheng [2] develop decoder metrcs for both the full and partal CSI scenaros, but do not consder the complete absence of CSI. In ths letter we develop low-complexty decoder metrcs sutable for teratve decodng/demodulaton wth no CSI and we llustrate the correspondng effect of loss of CSI on the extrnsc nformaton (EXIT) charts [3] of the demodulator and overall error probablty. II. METRICS FOR ITERATIVE DECODING BICM-ID conssts of exchangng messages between the btwse FSK demodulator and decoder of C n an teratve fashon. The btwse FSK demodulator feeds the decoder of C
3 wth the log-lkelhood ratos p (y[k] e b ) q k, (b) b B 0 L(c π(mk+) ) = log p (y[k] e b ) q k, (b), (3) b B 1 where B a {0,..., M 1} s the set of frequency bns that have the -th bt of the bnary label equal to a (whch mples that B a = M/2) and q k, (b) are the extrnsc probabltes computed by the decoder of C n the prevous teraton (at the frst teraton these are all equal to 0.5). Substtutng (2) nto (3) we obtan the teratve decoder used by [2]. The summatons n (3) may be undesrable from the pont of vew of complexty. To avod these summatons, the log-lkelhood rato (3) may be approxmated n the standard way ) Es L(c[j(k, )]) max log I 0 (2 h[k] y b [k] b B0 N 0 q k, (b) max log I 0 (2 b B1 ) Es h[k] y b [k] q k, (b) N 0 We shall refer to (3) and (4) as the Bessel and Bessel dual-max metrcs respectvely. Note that n order to compute (3) and (4), E s, N 0 or h[k] (or accurate estmates) must be avalable to the recever (full CSI). We wll now develop decoder metrcs that do not depend on E s, N 0 or h[k]. Taylor seres expanson of the Bessel functon I 0 (α) around zero yelds I 0 (α) = 1 + α2 4 + O(α4 ) (5) whch motvates the followng approxmaton of the log-lkelhood ratos (3), L(c[j(k, )]) log E If we further assume that s h[k] 2 N0 2 b B0 (4) M 2 + E s h[k] 2 y N0 2 b [k] 2 q k, (b) b B0 M 2 + E s h[k]. (6) 2 y N0 2 b [k] 2 q k, (b) b B1 y b [k] 2 M/2 we have y b [k] 2 q k, (b) b B 0 L(c[j(k, )]) log y b [k] 2 q k, (b) (7) b B 1
4 whch s ndependent of E s, N 0 and the fadng ampltudes h[k]. The nterpretaton of (7) s nterestng. The recever frst measures the receved energes at every frequency bn and computes the emprcal probablty at every bn as the fracton of the total receved energy present n a gven bn. Obvously, the normalzaton factor (the total energy M 1 =0 y [k] 2 ) cancels n (7). We can further approxmate (7) usng the dual-max method as follows, L(c[j(k, )]) max b B 0 log( y b [k] 2 q k, (b)) max log( y b [k] 2 q k, (b)) (8) b B1 whch yelds the correspondng parameter free dual-max metrcs. We now present some numercal examples whch demonstrate the utlty of the parameter free metrcs. Snce we are nterested n applcaton of the metrcs to teratve decodng, t s of nterest to compare the correspondng EXIT charts [3]. Fgure 1 shows EXIT charts for soft demodulaton usng the Bessel metrcs (3) (sold), dualmax Bessel (4) (dashed), and the parameter free metrcs (7) (dashed-dotted) and (8) (dotted) for M = 4, 16, 64 n the AWGN channel. The curves exhbt an almost-lnear behavor, wth Bessel metrcs and parameter free metrcs resultng n smlar slopes. Ths mples that at hgher SNR, the parameter metrcs wll have the same EXIT chart, whch wll help n assessng the performance degradaton due to the lack of CSI. Further, we observe that the parameter free metrcs are nformaton lossy, namely, when the nput mutual nformaton s I n = 1, the output mutual nformaton s lower than that obtaned wth Bessel metrcs. Fnally, and perhaps most surprsng, the parameter free dual-max metrc (8) s sgnfcantly better than (7) at low I n, despte the reducton n computatonal complexty. Applcaton of the dual-max approxmaton followng the Taylor approxmaton seems to regan some of the loss from the deal Bessel metrcs. Smlar charts are obtaned for the Raylegh fadng channel. From now, we concentrate n comparng the metrcs (3) and (8). Fgure 2 show the EXIT charts and smulated trajectores for metrcs (3) (left) and (8) (rght) wth the (25, 27, 33, 37) 8 convolutonal code and 64-FSK n the AWGN channel. Whle the
5 EXIT analyss predcts the threshold behavor qute accurately for (3), the EXIT chart analyss s slghtly pessmstc n the case of metrcs (8). Ths s due to the fact that the Gaussan approxmaton nherent n the EXIT analyss s not accurate. Recall that metrcs (8) are a result of three consecutve approxmatons to (3) and therefore, some loss n the Gaussanty of the teratve process s expected. The predcted EXIT chart thresholds are E b /N 0 = 2.5091 db for metrcs (3) and E b /N 0 = 3.8391 db for (8). Smulatons show that bt-error rate of 10 5 s acheved at 4 db and 4.6 db respectvely, thus mplyng that the EXIT chart analyss s slghtly optmstc. The penalty for not knowng the channel s 0.65 db only. Table I summarzes the smulated bt-error rate for BICM wth an outer rate R = 1/4 repeat-accumulate code and 4, 16 and 64-ary NC-FSK n the AWGN and Raylegh fadng channels. The smulatons were performed usng 10, 000 nformaton bts per codeword and 20 decodng teratons (one teraton of the RA decoder per demodulaton teraton). The results n the table hghlght the small loss for not knowng E s, N 0 or the fadng ampltude. III. CONCLUSIONS We present a low complexty method of computng metrcs suted for teratve demodulaton/decodng of M-ary non-coherent orthogonal modulaton that does not requre any knowledge of the sgnal-to-nose rato or fadng coeffcents at the recever. The method s based on the frst-order Taylor seres expanson of the Bessel functon. The proposed method performs very close of the deal metrcs and enables the use of methods such as BICM over non-coherent channels wthout sde nformaton. REFERENCES [1] X. L and J. A. Rtcey, Trells-coded modulaton wth bt nterleavng and teratve decodng, IEEE J. Select. Areas Commun., pp. 715 724, Aprl 1999. [2] M. C. Valent and S. Cheng, Iteratve demodulaton and decodng of turbo-coded M-ary noncoherent orthogonal modulaton, IEEE J. Select. Areas Commun. (Specal Issue on Dfferental and Noncoherent Wreless Communcatons), vol. 23, no. 9, pp. 1739 1747, Sept. 2005. [3] S. ten Brnk, Convergence of teratve decodng, Electroncs Letters, 35(10):806-808, May 1999.
6 I out 1 0.8 Bessel (3) Bessel DM(4) Approx (7) Approx DM (8) 4 16 64 0.6 4 16 64 0.4 0.2 0 0.2 0.4 0.6 0.8 1 I n Fg. 1. EXIT charts of the Bessel and parameter free metrcs for M = 4, 16, 64 on the AWGN channel wth SNR= 6 db. I out 1 I out 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.2 0.4 0.6 0.8 1 I n 0 0.2 0.4 0.6 0.8 1 I n Fg. 2. Trajectores for M = 64 n the AWGN channel wth a (25, 27, 33, 37) 8 convolutonal code usng Bessel (3) at E b /N 0 = 3 db (left) and parameter free metrcs (8) at E b /N 0 = 4 db (rght). TABLE I E b /N 0 AT 10 5 FOR M = 4, 16, 64 AND RA CODE OF R = 1/4. AWGN Raylegh Fadng M Metrcs (3) Metrcs (8) Metrcs (3) Metrcs (8) 4 5.5 db 5.9 db 6.1 db 8.1 db 16 3.9 db 4.3 db 4.3 db 5.8 db 64 3.5 db 4.1 db 3.7 db 5.1 db