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Turbo NFSK Iterative Estimation Noncoherent Demodulation and Decoding for Fast Fading Channels Shi Cheng and Matthew C Valenti Don Torrieri West Virginia University U S Army Research Laboratory Morgantown WV Adelphi MD shic mvalenti csee wvu edu dtorrieri arl army mil ABSTRACT This paper considers the problem of communicating over fast fading channels where the channel coherence time is only on the order of a few symbols Since the fading is too fast for coherent reception we employ M ary frequency shift keying with soft noncoherent demodulation Information is encoded by a binary turbo code To improve performance the soft demodulator and decoder work cooperatively through the iterative exchange of extrinsic information During each iteration the receiver estimates the channel state information CSI here defined to be the average received signal energy and noise spectral density for each block of symbols The channel estimator uses the Expectation Maximization EM algorithm and exploits extrinsic information fed from the decoder Simulation results show that for 16 NFSK in block independent Rayleigh fading performance can be within 0 6 dB of the perfect CSI case by using blocks as small as 4 symbols INTRODUCTION Bit interleaved coded modulation BICM 1 combines binary forward error correcting coding with M ary modulation It has become a standard method for signaling over fading channels forming the basis of third generation cellular and 802 11a g wireless networks The performance of BICM can be improved by feeding back a priori information in the form of bit likelihoods from the decoder back to the demodulator Such iterative demodulation and decoding schemes were independently developed by ten Brink 2 Benedetto et al 3 and Li and Ritcey 4 The latter reference terms this technique bit interleaved coded modulation with iterative decoding BICM ID other option is to use orthogonal signaling and noncoherent reception This is more appropriate when either dealing with fast fading or when the oscillators are not stable enough for instance in frequency hopping applications The focus of this paper is on the second option A benefit of orthogonal signaling such as frequency shift keying FSK is that it allows bandwidth efficiency to be traded for energy efficiency If a binary code is combined with nonbinary FSK then BICM ID can improve performance as shown in 6 In 6 the performance of turbo coded FSK using BICM ID was shown under the assumption that the channel fading amplitude was known perfectly at the receiver However in practice this amplitude is not known a priori and therefore must be estimated This paper extends the work of 6 by including the process of channel estimation into the receiver structure To facilitate the development of a pragmatic estimator it is assumed that the channel experiences block fading that is blocks of N consecutive FSK symbols are attenuated by the same channel gain though they could possibly experience different phase shifts Aside from this block fading condition the estimator makes no assumptions regarding the statistics of the channel and in fact estimates each block independently from the other blocks The estimator itself is derived using the expectation maximization EM algorithm 7 which iteratively finds the maximum likelihood ML estimate even though an explicit form is not readily achievable when extrinsic information is fed back to the estimator from the decoder When signaling over a fading channel one of two possible techniques is typically used The first option is to periodically insert pilot symbols into the transmitted signal and then leverage these pilot symbols to perform coherent detection 5 This is effective only if the fading is sufficiently slow and the transmit and receive oscillators relatively stable An Before proceeding further let us stipulate some notational conventions Bold lowercase letters will be used to denote vectors e g x and bold uppercase will be used for matrices e g X All vectors are row vectors but can be transposed into column vectors e g xT Vector elements are plain lowercase letters with subscripts beginning at zero e g x x0 x1 xM 1 Matrices are represented as a row of column vectors e g X xT0 xT1 xTN 1 The function This work was sponsored by the Xenotran Corporation Glen Burnie MD In this paper orthogonal modulation and FSK are used interchangeably u b Encoder b X Modulator Thus the received signal of the th block can be represented as c Y N u z Decoder 1 v z Demodulator Y Figure 1 System model p represents the probability of an event a probability density function or a probability mass function with the context clearly dependent on the argument SYSTEM MODEL The discrete time system model is shown in Fig 1 A vector u 0 1 k of message bits is passed through a binary encoder to produce a codeword b0 0 1 n which is interleaved by a permutation matrix to produce the bit interleaved codeword b b0 The bit interleaved codeword is then passed through a M ary orthogonal modulator to produce the M Nf matrix of symbols S sT0 sTNf 1 where Nf dn log2 M e Each column of S represents one M ary symbol and is represented as an elementary vector em comprised of all zeros except for a one in the mth position Let the set of log2 M code bits that label symbol si i i be represented as b0 b 1 With orthogonal modulation and flat fading that is constant during each symbol period the manner in which the code bits are mapped to symbols is unimportant since the symbols are equidistant and thus a natural mapping suffices In this case si em e0 eM 1 where m 1 X i bk 2k c X N 2 where X and N consist of columns N through 1 N 1 of X and N respectively and N is a M N L noise matrix whose elements are independently and identically distributed i i d complex Gaussian variables whose real and imaginary components each have zero mean and variance N0 2 v In 2 the k i th entry of Y is denoted yki where k 0 1 2 M 1 i 0 1 2 N 1 When the ith symbol is received it is passed through a filter bank with M matched filters one for each FSK tone Since the receiver is noncoherent each matched filter produces a complex quantity The element yki is then the output of the k th matched filter during the ith symbol period The receiver processes the matrix of received symbols Y and produces estimates u of the data bits The receiver is decomposed into a demodulator and a decoder each of which are implemented using the soft input soft output algorithm of 3 and separated by appropriate