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This paper focuses on symbol detection for a multi‐user single‐input multiple‐output (MU‐SIMO) uplink using supervised machine learning techniques. An extreme learning machine (ELM) is chosen as the equalizer at the receiver end due to its super‐fast learning ability and minimum training error. To tune the parameters, a pilot based online training for an ELM network is proposed, and those parameters are further used to detect the unknown transmitted symbols. Conventionally, the channel matrix is estimated using known pilot symbols explicitly and then the transmitted symbol is recovered using a linear equalizer. Here, instead of directly estimating the channel coefficients, the symbols are detected using an ELM based direct equalizer followed by a hard decision rule. To reduce the pilot requirement of the proposed model, a method is discussed where the channel is estimated using a small number of pilot symbols and the ELM network is trained using a separate training dataset along with the channel estimate. Furthermore, the performance comparison between the proposed ELM based equalizers, linear equalizers, and nonlinear detection techniques is shown. [ABSTRACT FROM AUTHOR]

We consider a space-time coded system in a correlated Rayleigh flat fading environment with imperfect channel estimation. Two receiver structures are considered: a suboptimum receiver and the optimum maximum likelihood receiver. The system performance of both receivers is analyzed in terms of the pairwise error probability. Index Terms--Channel estimation, correlated Rayleigh flat fading, pairwise error probability (PEP), receive correlation, space-time block coding.

A transmit-receive diversity system in correlated Rayleigh fading in which the receiver estimates the channel through pilot symbols, and feeds this information back to the transmitter through a feedback path, is considered. The imperfect channel state information (CSI) is used by the transmitter to obtain the transmit weight vector for data transmission. The optimum receiver in the maximum-likelihood (ML) sense obtained from the conditional distribution of the received signal vector, conditioned on the imperfect CSI and the transmit weight vector, is derived for the system. For the case of M-ary phase-shift keying (MPSK), an analytical expression for the conditional symbol error probability (SEP), conditioned on the channel estimate and the transmit weight vector, is obtained, with the transmit weight vector chosen to minimize this conditional SEP. For the receive-only and transmit-only correlation scenarios with ill-conditioned eigenvalues of the receive and transmit covariance matrices (that is, some of the eigenvalues are very small), we derive expressions for the diversity gain. Numerical results are presented to compare the performance of our receiver with that of a conventional receiver in case of exponentially correlated fading. These results show that the optimum receiver typically has about a 0.5-dB gain over a conventional receiver when the correlation coefficient exceeds 0.5 and the number of receive antennas is much larger than the number of transmit antennas. Index Terms--Correlated rayleigh fading, diversity gain, imperfect channel state information (CSI), optimum receiver, transmit-receive diversity.

The average of the product of two Ganssian Q-functions having arguments as different scaled versions of the same Nakagami distributed fading gain magnitude is derived in closed-form. Both the cases when (1) the fading parameter is an integer, and (2) the fading parameter is an odd multiple of one-half, are considered. The results are applied to evaluating in closed-form (a) the symbol error probability (SEP) of rectangular quadrature amplitude modulation with maximal-ratio combining in independent Nakagami fading, and (b) the outage probability of a two-user synchronous code-division multiple-access system having signal-to-noise ratio (SNR) imbalance among users in Nakagami fading. It is found in application (a) that with increase in the degree of fading, the robustness to decision distance imbalance between in-phase and quadrature signals increases. In addition, there exists an optimum decision distance ratio at which the SEP has a minimum. In application (b), we find that the outage probability increases sharply with SNR imbalance factor at low SNR imbalance and tends to reach saturation as the imbalance increases. Index Terms--Gaussian Q-function, maximal-ratio combining (MRC), Nakagami fading, outage probability, rectangular quadrature amplitude modulation (QAM), symbol error probability (SEP), synchronous code-division multiple-access (CDMA) system.

This paper investigates detection mechanisms for time hopping pulse position modulated (TH-PPM) multiple access systems in both additive white Gaussian noise and fading environments. In such systems, the raw data is not linearly related to the received signal. One technique is to express the pulse position modulated signal as the sum of linearly modulated signals with the input being a nonlinear function of the raw data. This reduces the system to an equivalent pulse amplitude modulated system with a significant increase in complexity. We present two new lower complexity detectors. The first detector, which is a frame-level maximum likelihood (ML) detector, uses the properties of the pulse correlation matrix to develop a lower complexity detection method. The second detector, which is a chip-level ML detector, relies on the knowledge of the time hopping sequence to despread the received signal. The detection is then done on a chip-by-chip basis. The performance of these detectors is compared in terms of the bit error rate for the user with weakest power and the average number of correctly decoded users. Index Terms--Bit error rate (BER), correctly decoded users, multiuser detection, pulse position modulation (PPM), time hopping (TH).

We consider a space-time block coded system having L time slots and using decision feedback channel estimation with a finite window size. The channel matrix estimate is updated every L time slots by: 1) a least-squares (LS) estimate and 2) a minimum mean-square error (MMSE) estimate. Using a Markov model of the decision process, we analyze the decoding error probability (DEP) of this scheme in Rayleigh fading. The transition probability matrix is obtained by simulation of the decision process for a system using the Alamouti code. From this matrix, the DEP can be computed for any time shift. We find from computations that the DEP increases with increase in the time shift but tends to saturate with time to a steady-state value. It is also observed that both the state and the transition probabilities tend to saturate with the increase in average signal-to-noise ratio per diversity branch. Index Terms--Decision feedback channel estimation (DFCE), decoding error probability (DEP), Markov model, Rayleigh fading, space-time block coding.

In this letter, we analyze the performance of the minimum mean-square error (MMSE) detector for a code-division multiple-access system employing binary phase-shift keying. The performance is measured in terms of the probability distribution and the average number of correctly decoded users. We consider the case of equipower users having equicorrelated signature waveforms. The distribution of the number of correctly decoded users is expressed as an integral over sum of products of Gaussian Q-functions. From it, a closed-form expression for the average number of correctly decoded users is derived. Index Terms--Correctly decoded users, equicorrelated signature waveforms, equipower users, minimum mean-square error (MMSE) detector, multiuser detection.

In a communication system using receive diversity and linear combining in the presence of cochannel interference (CCI), optimum combining (OC) is known to give the best error performance since it maximizes the instantaneous signal-to-interference-plus-noise ratio of the combiner output, and consequently, in the presence of Gaussian interference plus noise, it minimizes the error rate. However, this is based on the assumption that a perfect estimate of the channel is available. Channel estimation methods in reality use some overhead. When the channel is time-invariant, the estimation error decreases with increase in the amount of overhead, like the number of pilot symbols. With the growing need for high data rate applications, the amount of overhead that can be allocated for the estimation of the channel needs to be reduced, and the channel estimation error cannot be ignored, in this situation, replacing the channel by its imperfect estimate in the OC weight vector no longer results in an optimum scheme. We have to find an optimum scheme based on the channel estimation method and the detection criterion, which results in what we call optimized diversity combining (ODC). Here we focus on ODC resulting from a pilot symbol based maximum likelihood (ML) channel estimation method applied to a correlated flat Rayleigh fading channel in the presence of CCI and additive noise. The channel is randomly time-invariant during the reception of pilot and data symbols. The decision rule, which is optimum in the ML sense, is derived using concepts of Gaussian and Wishart statistics. Numerical results show that ODC can perform significantly better than OC with imperfect channnel estimates by appropriate choice of system parameters. Index Terms--Characteristic function, imperfect channel estimation, maximum likelihood estimate, optimized diversity combining (ODC), probability density function (pdf), pseudo-Wishart distribution, symbol error probability, Wishart distribution.

Consider a wireless communication system in flat fading with N transmit and M receive antennas using space-time block coding, where N x 1 code vectors are transmitted over L symbol intervals, resulting in an N x L code matrix. A least-squares estimate (LSE) as well as a minimum mean-square estimate (MMSE) of the M x N channel matrix is obtained from a sequence of pilot code vectors. For the case of linear square (i.e., with N = L) orthogonal codes over constant envelope constellations, we obtain an expression for the exact decoding error probability (DEP) for coherent maximum-likelihood decoding. We also find the coding gain for high average signal-to-noise ratio (SNR) per diversity branch in the case of Rayleigh fading. A comparison between both channel-estimation techniques is done in terms of the average pilot-power-to-signal-power ratio (APPSPR). It is found that MMSE requires lower pilot power than LSE for the same DEP and the same average SNR per diversity branch. In addition, the error performance with LSE approaches that with MMSE, with an increase of average SNR per branch or an increase of APPSPR. Index Terms--Decoding error probability (DEP), flat fading, least-squares channel estimation, minimum mean-square channel estimation, space-time block coding.

We present closed-form lower bounds for the performance of multihop transmissions with nonregenerative relays over not necessarily identically distributed Nakagami-m fading channels. The end-to-end signal-to-noise ratio is formulated and upper bounded by using an inequality between harmonic and geometric means of positive random variables (RVs). Novel closed-form expressions are derived for the moment generating function, the probability density function, and the cumulative distribution function of the product of rational powers of statistically independent Gamma RVs. These statistical results are then applied to studying the outage probability and the average bit-error probability for phase- and frequency-modulated signaling. Numerical examples compare analytical and simulation results, verifying the tightness of the proposed bounds. Index Terms--Average bit-error probability (ABEP), Gamma random variables (RVs), multihop relayed communications, Nakagami-m fading, outage probability.

We analyze the error performance of minimum selection generalized selection combining (MS-GSC), in which the minimum number of diversity branches are selected such that their combined signal-to-noise ratio (SNR) is above a given threshold. A flat Rayleigh fading channel with independent and distinctly distributed branch SNRs is considered. By transforming the ordered instantaneous branch SNRs to their differences, we derive the distribution of the number of selected branches in closed form. We then modify the derivation of this distribution to get the characteristic function (cf.) of the combiner output SNR. This cf. is used to obtain the symbol error probability for different coherent digital modulation schemes. Index Terms--Independent and distinctly distributed Rayleigh fading, minimum selection generalized selection combining (MS-GSC), number of selected branches, symbol error probability.

We analyze the error performance of an improved hybrid selection/maximal-ratio combining (H-S/MRC) technique called minimum selection H-S/MRC in flat Rayleigh fading for coherent digital modulation schemes. Here the minimum number of diversity branches are selected such that their combined signal-to-noise ratio is above a given threshold. We derive a closed-form expression for the distribution of the number of selected branches. This distribution is used to obtain the symbol error probability. Index Terms--Flat Rayleigh fading, minimum selection H-S/MRC (MS-H-S/MRC), symbol error probability (SEP).

Consider a Rake receiver for coherent binary antipodal signaling with: 1) a delayed received signal configuration; 2) weight estimation by matched filtering using the reference signal along with the decisions of the previous M symbol intervals; and 3) predetection maximal-ratio combining (MRC). The weight estimation errors here are not independent of the additive noise, and do not fit into the Gaussian weighting error model for MRC. Here we analyze the error performance of the receiver by obtaining the conditional symbol error probability, conditioned on past decisions, from the characteristic function of the decision variable, and getting the unconditional error probability (UEP) for a block of M consecutive symbols using a Markov model of the decision process. The channel is Rayleigh fading with independent and identically distributed branch gains. Results show that the error performance of the Gaussian distributed weighting error model is a bound for that of multiple symbol weight estimation by matched filtering, and the steady state UEP decreases with increase of M, but the amount of decrease reduces as M increases. Index Terms--Binary antipodal signaling, independent and identically distributed (i.i.d.) Rayleigh fading, Markov model, multiple symbol weight estimation, predetection maximal-ratio combining (MRC), Rake reception, symbol error probability (SEP), unconditional error probability (UEP).

The pseudo-Wishart distribution arises when a Hermitian matrix generated from a complex Gaussian ensemble is not full-rank. It plays an important role in the analysis of communication systems using diversity in Rayleigh fading. However, it has not been extensively studied like the Wishart distribution. Here, we derive some key aspects of the complex pseudo-Wishart distribution. Pseudo-Wishart and Wishart distributions are treated as special forms of a Wishart-type distribution of a random Hermitian matrix generated from independent zero-mean complex Gaussian vectors with arbitrary covariance matrices. Using a linear algebraic technique, we derive an expression for the probability density function (pdf) of a complex pseudo-Wishart distributed matrix, both for the independent and identically distributed (i.i.d,) and non-i.i.d. Gaussian ensembles. We then analyze the pseudo-Wishart distribution of a rank-one Hermitian matrix, The distribution of eigenvalues of Wishart and pseudo-Wishart matrices is next considered. For a matrix generated from an i.i.d, Gaussian ensemble, we obtain an expression for the characteristic function (cf) of eigenvalues in terms of a sum of determinants. We present applications of these results to the analysis of multiple-input multiple-output (MIMO) systems, The purpose of this work is to make researchers aware of the pseudo-Wishart distribution and its implication in the case of MIMO systems in Rayleigh fading, where the transmitted signals are independent but the received signals are correlated. The results obtained provide a powerful analytical tool for the study of MIMO systems with correlated received signals, like systems using diversity and optimum combining, space-time systems, and multiple-antenna systems. Index Terms--Characteristic function, diversity and optimum combining, multiple-input multiple-output (MIMO) systems, probability density function (pdf), pseudo-Wishart distribution, Wishart distribution.

In this paper, expressions for multivariate Rayleigh and exponential probability density functions (pdfs) generated from correlated Gaussian random variables are presented. We first obtain a general integral form of the pdfs, and then study the case when the complex Gaussian generating vector is circular. We consider two specific circular cases: the exchangeable case when the variates are evenly correlated, and the exponentially correlated case. Expressions for the multivariate pdf in these cases are obtained in integral form as well as in the form of a series of products of univariate pdfs. We also derive a general expression for the multivariate exponential characteristic function (cf) in terms of determinants. In the exchangeable and exponentially correlated cases, cf expressions are obtained in the form of a series of products of univariate gamma cfs. The cf of the sum of exponential variates in these cases is obtained in closed form. Finally, the bivariate case is presented mentioning its main features. While the integral forms of the multivariate pdfs provide a general analytical framework, the series and determinant expressions for the exponential cfs and the series expressions for the pdfs can serve as a useful tool in the performance analysis of digital modulation over correlated Rayleigh-fading channels using diversity combining. Index Terms--Characteristic function, correlated Rayleigh fading, exchangeable variates, exponential correlation, multivariate exponential distribution, multivariate Rayleigh distribution.

We analyze the error performance of a wireless communication system employing transmit-receive diversity in Rayleigh fading. By focusing on the complex Gaussian statistics of the independent and identically distributed entries of the channel matrix, we derive a formula for the characteristic function (c.f.) of the maximum output signal-to-noise ratio. We use this c.f. to obtain a closed-form expression of the symbol error probability (SEP) for coherent binary keying. The method is easily extended to obtain the SEP for the coherent reception of M-ary modulation schemes. Index Terms--Coherent binary keying, coherent M-ary modulation, Rayleigh fading, symbol error probability (SEP), transmit-receive diversity.

We derive an upper bound and investigate some approximations on the symbol error probability (SEP) for coherent detection of M-ary phase-shift keying, using an array of antennas with optimum combining in wireless systems in the presence of multiple uncorrelated equal-power cochannel interferers and thermal noise in a Rayleigh fading environment. Our results are general and valid for an arbitrary number of antenna elements as well as an arbitrary number of interferers. In particular, the exact SEP is derived for an arbitrary number of antennas and interferers; the computational complexity of the exact solution depends on the minimum number of antennas and interferers. Moreover, closed-form approximations are provided for the cases of dual optimum combining with an arbitrary number of interferers, and of two interferers with an arbitrary number of antenna elements. We show that our bounds and approximations are close to Monte Carlo simulation results for all cases considered in this paper. Index Terms--Adaptive arrays, antenna diversity, cochannel interference, eigenvalue distribution, optimum combining, Wishart matrices.

We present an analysis of a hybrid selection/maximal-ratio combining diversity system over an evenly correlated slow frequency-nonselective Nakagami fading channel, where the correlation coefficient betweeen any pair of the diversity branch gain amplitudes is the same, and all average branch signal-to-noise ratios (SNRs) are equal. In this system, the L branches with the largest instantaneous SNR out of N available branches are selected and combined using maximal-ratio combining. From the joint characteristic function (cf) of the instantaneous branch SNRs, we obtain an expression for the cf of the combiner output SNR as a series of elementary cfs. The expression can be conveniently used to obtain the symbol error probability of coherent detection of different M-ary modulation schemes. We illustrate our methodology using M-ary phase-shift keying as an example. Index Terms--Correlated Nakagami fading, hybrid selection/ maximal-ratio combining, M-ary phase-shift keying, symbol error probability.

The bit error probability (BEP) for coherent detection of binary signals with dual-diversity predetection equal gain combining is derived using the Beaulieu series. In particular, we consider a correlated Rayleigh fading channel with unequal branch signal-to-noise ratios. The BEP expression is in terms of the power correlation coefficient of the branches, is easy to compute, and depicts clearly the effect of correlated fading on the error performance. Index Terms--Bit error probability, coherent binary keying, correlated Rayleigh fading.

We analyze the performance for the noncoherent reception of M-ary orthogonal frequency shift keying with postdetection equal gain combining over a correlated fading channel. Two kinds of correlated fading statistics are considered: 1) Nakagami fading in which the diversity branches can have unequal signal-to-noise ratios (SNRs) as well as different m-parameters and 2) Rician fading in which the diversity branches can have unequal SNRs. Using the characteristic function of the combiner output SNR, closed-form expressions for the symbol error probability are obtained. Index Terms--Correlated Nakagami and Rician fading, noncoherent M-ary FSK, postdetection EGC, symbol error probability.