Turbo-NFSK: Iterative Estimation, Noncoherent Demodulation,and Decoding for Fast Fading ChannelsShi Cheng and Matthew C. Valenti Don TorrieriWest Virginia University U.S. Army Research LaboratoryMorgantown, WV Adelphi, MD{shic,mvalenti}@csee.wvu.edu [email protected] paper∗considers the problem of communicating overfast fading channels, where the channel coherence time isonly on the order of a few symbols. Since the fading istoo fast for coherent reception, we employ M-ary frequencyshift keying with soft noncoherent demodulation. Infor-mation is encoded by a binary turbo code. To improveperformance, the soft demodulator and decoder workcooperatively through the iterative exchange of extrinsicinformation. During each iteration, the receiver estimatesthe channel state information (CSI), here defined to bethe average received signal energy and noise spectraldensity for each block of symbols. The channel estima-tor uses the Expectation Maximization (EM) algorithmand exploits extrinsic information fed from the decoder.Simulation results show that for 16-NFSK in block inde-pendent Rayleigh fading, performance can be within 0.6 dBof the perfect CSI case by using blocks as small as 4 symbols.INTRODUCTIONBit interleaved coded modulation (BICM) [1] combines bi-nary forward error correcting coding with M-ary modulation.It has become a standard method for signaling over fadingchannels, forming the basis of third generation cellular and802.11a/g wireless networks. The performance of BICM canbe improved by feeding back a priori information (in theform of bit likelihoods) from the decoder back to the demod-ulator. Such iterative demodulation and decoding schemeswere independently developed by ten Brink [2], Benedettoet al [3], and Li and Ritcey [4]. The latter reference termsthis technique bit interleaved coded modulation with iterativedecoding (BICM-ID).When signaling over a fading channel, one of two possibletechniques is typically used. The first option is to period-ically insert pilot symbols into the transmitted signal, andthen leverage these pilot symbols to perform coherent detec-tion [5]. This is effective only if the fading is sufficiently slowand the transmit and receive oscillators relatively stable. An-∗This work was sponsored by the Xenotran Corporation, GlenBurnie, MD.other option is to use orthogonal signaling and noncoherentreception. This is more appropriate when either dealing withfast fading or when the oscillators are not stable enough, forinstance in frequency hopping applications. The focus of thispaper is on the second option.A benefit of orthogonal signaling, such as frequency shiftkeying (FSK)†is that it allows bandwidth efficiency to betraded for energy efficiency. If a binary code is combined withnonbinary FSK, then BICM-ID can improve performance, asshown in [6].In [6], the performance of turbo coded FSK using BICM-IDwas shown under the assumption that the channel fadingamplitude was known perfectly at the receiver. However, inpractice, this amplitude is not known a priori and thereforemust be estimated. This paper extends the work of [6] byincluding the process of channel estimation into the receiverstructure. To facilitate the development of a pragmaticestimator, it is assumed that the channel experiences blo ckfading, that is, blocks of N consecutive FSK symbols areattenuated by the same channel gain (though they couldpossibly experience different phase shifts). Aside from thisblock fading condition, the estimator makes no assumptionsregarding the statistics of the channel and, in fact, estimateseach block independently from the other blocks. The esti-mator itself is derived using the expectation maximization(EM) algorithm [7], which iteratively finds the maximumlikelihood (ML) estimate, even though an explicit form isnot readily achievable when extrinsic information is fed backto the estimator from the decoder.Before proceeding further, let us stipulate some notationalconventions. Bold lowercase letters will be used to denotevectors, e.g. x, and bold uppercase will be used for matrices,e.g. X. All vectors are row-vectors, but can be transposedinto column vectors, e.g. xT. Vector elements are plainlowercase letters with subscripts beginning at zero, e.g.x = [x0, x1, ..., xM−1]. Matrices are represented as a row ofcolumn vectors, e.g. X = [xT0, xT1, ..., xTN−1]. The function†In this paper, orthogonal modulation and FSK are used inter-changeably.Encoder Modulatorub’ b XDecoder Demodulatoruz’zNcv’ vYΠΠ1−ΠFigure 1: System model.p(·) represents the probability of an event, a probabilitydensity function, or a probability mass function with thecontext clearly dependent on the argument.SYSTEM MODELThe discrete-time system model is shown in Fig. 1. A vectoru ∈ {0, 1}kof message bits is passed through a binary en-coder to produce a codeword b0∈ {0, 1}nwhich is interleavedby a p ermutation matrix Π to pro duce the bit-interleavedcodeword b = b0Π. The bit-interleaved codeword is thenpassed through a M-ary orthogonal modulator to producethe M × Nfmatrix of symbols S = [sT0, ..., sTNf−1] whereNf= dn/ log2Me. Each column of S represents one M-arysymbol and is represented as an elementary vector emcom-prised of all zeros except for a one in the mthposition.Let the set of µ = log2M code bits that label symbol sibe represented as {b(i)0, ..., b(i)µ−1}. With orthogonal mod-ulation and flat fading that is constant during each sym-bol period, the manner in which the code bits are mappedto symbols is unimportant since the symbols are equidis-tant, and thus a natural mapping suffices. In this case,si= em∈ {e0, ..., eM−1} wherem =µ−1Xk=0b(i)k2k. (1)The modulated symbol si, which has unit energy, is thenscaled by a factor of√Es, so that the transmitted symbol isxi=√Essi, and the transmitted matrix of scaled symbols isX = [xT0, ..., xTNf−1].The modulated symbol stream passes through a frequency-nonselective block fading channel, with the fading coefficientconstant within each block, and independent over differentblocks. In the following discussion, we assume Nf= NL,where N is the number of symbols per block, and L is thenumber of blocks per codeword. The `thblock fading co ef-ficient can b e represented as c`= a`exp{θ`√−1}, where a`and θ`are the real-valued amplitude and phase, respectively.Thus, the received signal of the `thblock can be representedasY`= c`X`+ N`, (2)where X`and N`consist of columns `N through