Optimization of signal sets for partial-response channels. II.Asymptotic coding gain

For Pt. I see ibid., vol.37, no.5, p.1327-141 (1991). For a linear, time-invariant, discrete-time channel with a given transfer function H(f), and information rate R bits/ T, where T is the symbol interval, an optimal signal set of length K is defined to be a set of 2^RK inputs of length K that maximizes the minimum l₂ distance between pairs of outputs. The author studies the minimum distance between outputs, or equivalently, the coding gain of optimal signal sets as K→∞. He shows how to estimate the coding gain, relative to single-step detection, of an optimal signal set length K when K is large     

Multistage detection in asynchronous code-division multiple-accesscommunications

A multiuser detection strategy for coherent demodulation in an asynchronous code-division multiple-access system is proposed and analyzed. The resulting detectors process the sufficient statistics by means of a multistage algorithm based on a scheme for annihilating successive multiple-access interference. An efficient real-time implementation of the multistage algorithm with a fixed decoding delay is obtained and shown to require a computational complexity per symbol which is linear in the number of users K. Hence, the multistage detector contrasts with the optimum demodulator, which is based on a dynamic programming algorithm, has a variable decoding delay, and has a software complexity per symbol that is exponential in K. An exact expression is obtained and used to compute the probability of error is obtained for the two-stage detector, showing that the two-stage receiver is particularly well suited for near-far situations, approaching performance of single-user communications as the interfering signals become stronger. The near-far problem is therefore alleviated. Significant performance gains over the conventional receiver are obtained even for relatively high-bandwidth-efficiency situations     

Packet error probabilities in frequency-hopped spread-spectrumpacket radio networks-memoryless frequency-hopping patterns considered

The packet error probability induced in a frequency-hopped spread-spectrum packet radio network is computed. The frequency spectrum is divided into q frequency bins. Each packet is exactly one codeword from an (M, L) Reed-Solomon code [M=number of codeword symbols (bytes); L=number of information symbols (bytes)]. Every user in the network sends each of the M bytes of his packet at a frequency chosen among the q frequencies with equal probability and independently of the frequencies chosen for other bytes (i.e., memoryless frequency-hopping patterns). Statistically independent frequency-hopping patterns correspond to different users in the network. Provided that K users have simultaneously transmitted their packets on the channel and a receiver has locked on to one of these K packets, the probability that this packet is not decoded correctly is evaluated. It is also shown that although memoryless frequency-hopping patterns are utilized, the byte errors at the receiver are not statistically independent; instead they exhibit a Markovian structure     

Capacity of frequency-hop spread-spectrum multiple-accesscommunication systems

The information theoretic capacity is considered. In order to account for independent encoding and decoding and private (to the sender and receiver) hopping patterns, an interference channel model is adopted with K sender-receiver pairs with the ith receiver only interested in the message transmitted by the ith sender. Both synchronous and asynchronous hopping patterns are investigated. Although the channel exhibits memory in the latter case, it is possible to compute the capacity region. The asymptotic normalized sum capacity is also computed     

Nonparametric Bayes estimation of a distribution under nominationsampling

Nomination sampling is a sampling process in which every observation is the maximum of a random sample from some distribution. If the data are {(Y_i, K_i), i =1, . . ., n} where K_i is the size of sample i, and Y_i is the maximum of a random sample of size K_i from an unknown Cdf, F; the Bayes estimator of F is derived by discretizing F over a fixed finite partition of the support of F and taking a Dirichlet distribution as the prior for the probabilities of the partitioning intervals. For the flood data of the Nidd River considered by R.A. Boyles and F.J. Samaniego (J. Am. Stat. Assoc., vol.81, p.1039-45, 1986), the plots of the Bayes estimator of F are obtained for several sets of values of the parameters of the Dirichlet distribution     

Interception of frequency-hopped spread-spectrum signals

A frequency-hopped spread-spectrum signal is modeled as a sinusoid that has one of N random frequencies. Coherent and noncoherent interception receiver structures based on Neyman-Pearson detection theory are determined. Under the assumption that there is a single hop per detection period, the optimum receiver structure is shown to consist of a bank of matched filters called the average likelihood (AL) receiver. A suboptimum structure called the maximum likelihood (ML) receiver is also analyzed. It is shown that AL and ML receivers have essentially the same performance. Simple formulas that relate the probability of detection, P_D, to the probability of false alarm, P_F, and the signal-to-noise ratio (SNR) for large N are derived. Receiver structures are also derived and analyzed for the case where the signal hops a number of times in one detection interval. This may correspond to the detection of a multihop signal in one symbol interval or to detection based on integration over a number of symbol intervals. The relationships of P_D to P_F, for both coherent and noncoherent multiple-hop receivers, are examined     

Multiuser signal detection using sequential decoding

The application of sequential decoding to the detection of data transmitted over the additive white Gaussian noise channel by K asynchronous transmitters using direct-sequence spread-spectrum multiple access (DS/SSMA) is considered. A modification of R.M. Fano's (1963) sequential-decoding metric, allowing the messages from a given user to be safely decoded if its E_b/N₀ exceeds -1.6 dB, is presented. Computer simulation is used to evaluate the performance of a sequential decoder that uses this metric in conjunction with the stack algorithm. In many circumstances, the sequential decoder achieves results comparable to those obtained using the much more complicated optimal receiver     

Optimization of signal sets for partial-response channels. I.Numerical techniques

Given a linear, time-invariant, discrete-time channel, the problem of constructing N input signals of finite length K that maximize minimum l₂ distance between pairs of outputs is considered. Two constraints on the input signals are considered: a power constraint on each of the N inputs (hard constraint) and an average power constraint over the entire set of inputs (soft constraint). The hard constraint, problem is equivalent to packing N points in an ellipsoid in min(K,N-1) dimensions to maximize the minimum Euclidean distance between pairs of points. Gradient-based numerical algorithms and a constructive technique based on dense lattices are used to find locally optimal solutions to the preceding signal design problems. Two numerical examples are shown for which the average spectrum of an optimized signal set resembles the water pouring spectrum that achieves Shannon capacity, assuming additive white Gaussian noise     

Quantile estimation based on nomination sampling

Nomination sampling is a sampling process in which every observation is the maximum of a random sample from some distribution. If all samples are taken from a single underlying CDF, F, data can be viewed as consisting of pairs (X_i,K_i) where K_i is the size of sample i and, given K_i=k_i, X_i is distributed according to CDF F^ki. R.A. Boyles and F.J. Samaniego (1986) developed a nonparametric maximum-likelihood estimator of F. In the present work, their approach is extended to obtain estimates of the quantiles of F and to study the limit theory and consistency properties of these estimates. These results generalize the results of T.R. Willemain (1980), who discussed the estimation of the median of F based on nomination samples     

On the Hamming distance properties of group codes

Under certain mild conditions, the minimum Hamming distance D of an (N, K, D) group code C over a non-abelian group G is bounded by D⩽N -2K+2 if K⩽N/2, and is equal to 1 if K>N/2. Consequently, there exists no (N, K, N-K+1) group code C over an non-abelian group G if 1<K<N. Moreover, any normal code C with a non-abelian output space has minimum Hamming distance equal to D=1. These results follow from the fact that non-abelian groups have nontrivial commutator subgroups. Finally, if C is an (N, K, D) group code over an abelian group G that is not elementary abelian, then there exists an (N, K, D) group code over a smaller elementary abelian group G'. Thus, a group code over a general group G cannot have better parameters than a conventional linear code over a field of the same size as G     

Fast K-dimensional tree algorithms for nearest neighborsearch with application to vector quantization encoding

Fast search algorithms are proposed and studied for vector quantization encoding using the K-dimensional (K-d) tree structure. Here, the emphasis is on the optimal design of the K -d tree for efficient nearest neighbor search in multidimensional space under a bucket-Voronoi intersection search framework. Efficient optimization criteria and procedures are proposed for designing the K-d tree, for the case when the test data distribution is available (as in vector quantization application in the form of training data) as well as for the case when the test data distribution is not available and only the Voronoi intersection information is to be used. The criteria and bucket-Voronoi intersection search procedure are studied in the context of vector quantization encoding of speech waveform. They are empirically observed to achieve constant search complexity for O(log N) tree depths and are found to be more efficient in reducing the search complexity. A geometric interpretation is given for the maximum product criterion, explaining reasons for its inefficiency with respect to the optimization criteria     

Asymptotic results for maximum likelihood estimation with an arrayof sensors

In many cases, the maximum likelihood (ML) estimator is consistent and asymptotically normal with covariance equal to the inverse of the Fisher's information matrix. It does not follow, though, that the covariance of the ML estimator approaches the Cramer-Rao lower bound as the sample size increases. However, it is possible to draw such a conclusion for the adaptive array problem in which direction of arrival and signal magnitude are being estimated. Proofs of w-asymptotic efficiency, which comes with a convergence-of-moments condition, and strong consistency (almost-sure convergence) of the ML estimator are given. Strong consistency is also proved for a popular quasi-ML estimator     

On channel estimation and sequence detection of interleaved codedsignals over frequency nonselective Rayleigh fading channels

Due to the receiver complexity introduced by interleaving, the implementation of maximum likelihood (IML) decoding of interleaved coded signals transmitted over frequency nonselective Rayleigh fading channels has been shown to be practically impossible. As an alternative, a two-stage receiver structure has been proposed, where the channel estimation and sequence decoding are done separately. The channel estimation issue in a two-stage receiver is examined in detail in this paper. It is shown that although an optimum (maximum a posteriori (MAP)) channel estimation is not possible in practice, it can be approached asymptotically by joint MAP estimation of the channel and the coded data sequence. The implementation of the joint MAP estimation is shown to be an ML sequence estimator followed by an minimum mean-square error (MMSE) channel estimator. Approximate fill sequence estimation using pilot symbol interpolation is also studied, and through computer simulations, its performance is compared to receivers using hit sequence estimation. The effect of decision delay (DD), prediction order, and pilot insertion rate (PIR) on the reduced complexity ML sequence estimation is investigated as well. Finally, a practical receiver is proposed that makes the best compromise among the error performance, receiver complexity, DD, and power (or bandwidth) expansion due to pilot insertion     

A convolutional single-parity-check concatenated coding scheme forhigh-data-rate applications

The coding scheme uses a set of n convolutional codes multiplexed into an inner code and a (n,n-1) single-parity-check code serving as the outer code. Each of the inner convolutional codes is decoded independently, with maximum-likelihood decoding being achieved using n parallel implementations of the Viterbi algorithm. The Viterbi decoding is followed by additional outer soft-decision single-parity-check decoding. Considering n=12 and the set of short constraint length K=3, rate 1/2 convolutional codes, it is shown that the performance of the concatenated scheme is comparable to the performance of the constraint length K=7, rate 1/2 convolutional code with standard soft-decision Viterbi decoding. Simulation results are presented for the K=3, rate 1/2 as well as for the punctured K=3, rate 2/3 and rate 3/4 inner convolutional codes. The performance of the proposed concatenated scheme using a set of K=7, rate 1/2 inner convolutional codes is given     

Asymptotically optimal classification for multiple tests withempirically observed statistics

The decision problem of testing M hypotheses when the source is Kth-order Markov and there are M (or fewer) training sequences of length N and a single test sequence of length n is considered. K, M, n, N are all given. It is shown what the requirements are on M , n, N to achieve vanishing (exponential) error probabilities and how to determine or bound the exponent. A likelihood ratio test that is allowed to produce a no-match decision is shown to provide asymptotically optimal error probabilities and minimum no-match decisions. As an important serial case, the binary hypotheses problem without rejection is discussed. It is shown that, for this configuration, only one training sequence is needed to achieve an asymptotically optimal test     

Genetic algorithm assisted joint multiuser symbol detection andfading channel estimation for synchronous CDMA systems

A novel multiuser code division multiple access (CDMA) receiver based on genetic algorithms is considered, which jointly estimates the transmitted symbols and fading channel coefficients of all the users. Using exhaustive search, the maximum likelihood (ML) receiver in synchronous CDMA systems has a computational complexity that is exponentially increasing with the number of users and, hence, is not a viable detection solution. Genetic algorithms (GAs) are well known for their robustness in solving complex optimization problems. Based on the ML rule, GAs are developed in order to jointly estimate the users' channel impulse response coefficients as well as the differentially encoded transmitted bit sequences on the basis of the statistics provided by a bank of matched filters at the receiver. Using computer simulations, we showed that the proposed receiver can achieve a near-optimum bit-error-rate (BER) performance upon assuming perfect channel estimation at a significantly lower computational complexity than that required by the ML optimum multiuser detector. Furthermore, channel estimation can be performed jointly with symbol detection without incurring any additional computational complexity and without requiring training symbols. Hence, our proposed joint channel estimator and symbol detector is capable of offering a higher throughput and a shorter detection delay than that of explicitly trained CDMA multiuser detectors     

New results in the theory of identification via channels

The identification capacity is the maximal iterated logarithm of the number of messages divided by the blocklength that can be reliably transmitted when the receiver is only interested in deciding whether a specific message was transmitted or not. The identification coding theorem of R. Ahlswede and G. Dueck (1989) for single-user discrete memoryless channels states that the identification capacity is equal to the Shannon capacity. A novel method to prove the converse to the identification coding theorem is shown to achieve the strong version of the result. Identification plus transmission (IT) coding, a variant of the original problem of identification via channels, is proposed in the context of a common problem in point-to-multipoint communication, where a central station wishes to transmit information reliably to one of N terminals, whose identity is not predetermined. The authors show that as long as log log N is smaller than the number of bits to be transmitted, IT codes allow information transmission at channel capacity     

Balancing sets of vectors

For n>0, d⩾0, n≡d (mod 2), let K(n, d) denote the minimal cardinality of a family V of ±1 vectors of dimension n, such that for any ±1 vector w of dimension n there is a v∈V such that |v- w|⩽d, where v-w is the usual scalar product of v and w. A generalization of a simple construction due to D.E. Knuth (1986) shows that K(n , d)⩽[n/(d+1)]. A linear algebra proof is given here that this construction is optimal, so that K(n, d)-[n/(d+1)] for all n≡d (mod 2). This construction and its extensions have applications to communication theory, especially to the construction of signal sets for optical data links     

Optimal and suboptimal broad-band source location estimation

Two maximum-likelihood (ML) estimators are considered for direction-of-arrival (DOA) estimation of broadband sources with unknown spectral parameters. One is based on the assumption that the sources radiate stochastic-Gaussian signals and therefore is called the stochastic-Gaussian ML (SGML) estimator; the other, using estimates of the actual signals (not their assumed distribution), is called the conditional ML (CML) estimator. Neither is efficient if the source spectral parameters are completely arbitrary and unknown, but the problem can be avoided for a version of the SGML estimation if the signal and noise spectra are known to satisfy certain smoothness conditions. While this version of the SGML is formally superior to the CML, it is demonstrated that the performance difference is small with underconditions not infrequently encountered in practice. When these are satisfied, the computationally simpler CML can be used without significant loss. The required conditions become more stringent as the source separation decreases or correlation between sources increases. A closed-form analytic expression is obtained for the small-error variance of the CML estimator of the DOA of the nth source in the presence of N-1 other sources     

The axisymmetric modes in a longitudinal inhomogeneous SELFOC fiber

The authors consider the propagation characteristics and mode conversion of axisymmetric modes in an imperfect SELFOC fiber with longitudinal gradually varying dielectric constant K=ϵ/ϵ₀=K₀-K ₂(z)r². An analytic solution which is expressed in terms of generalized Laguerre polynomials is found