Decision feedback equaliser design using support vector machines

The paper derives a minimum bit error rate (BER) solution for the decision feedback equaliser (DFE) that employs a linear combination of the channel observations and the past decisions. We show that by using a geometric translation the DFE is reduced to a simpler linear equaliser. A BER expression for the linear equaliser is obtained under the assumption of linearly separable decision regions, and a method is proposed to optimally set the linear-combiner coefficients of the DFE. This minimum BER solution is superior to the usual minimum mean square error (MSE) solution.

In a companion paper, we have studied the zero-forcing (ZF) transceiver with decision feedback equalizer (DFE) over slowly time-varying narrowband multiinput multioutput (MIMO) channels. The space-time generalized triangular decomposition (ST-GTD) was used for the design of ZF-DFE transceivers. The space-time geometric mean decomposition (ST-GMD) ZF transceiver minimizes both the arithmetic mean square error (MSE) at the feedback detector and the average uncoded bit error rate (BER) in moderate high signal-to-noise ratio (SNR). This paper addresses the design problem of DFE transceiver without zero-forcing constraint. In the first part, a channel independent temporal precoder is superimposed on the conventional block-wise GMD-based minimum mean square error (MMSE) DFE transceiver to take advantage of the temporal diversity. In the second part, ST-GTD is applied for the design of MMSE DFE transceivers. With accurate channel prediction and space-time powerloading, the proposed ST-GMD MMSE transceiver minimizes the arithmetic MSE at the feedack detector, and maximizes Gaussian mutual information. For practical applications, the ST-GTD MMSE transceiver which does not require channel prediction but shares the same asymptotic BER performance with the ST-GMD MMSE system is also developed. In the convex region, our analysis shows that the proposed MMSE transceivers has better BER performance than the conventional GMD-based MMSE transceiver; the average BERs of the proposed systems are nonincreasing functions of the ST-block size. The superior performance of ST-GMD MMSE transceiver over the ST-GMD ZF transceiver is also verified analytically.

Block data transmission is an efficient scheme to mitigate the inter-symbol interference (ISI) caused by dispersive channels. Recent research work attempt to obtain optimal designs of precoders given various linear and non-linear receivers. In this paper we focus on the systems with optimal precoder designs based on the minimization of bit error rate (BER) which is the more meaningful criterion in data communications. The linear receivers employed in these systems include zero-forcing (ZF) and minimum mean square error (MMSE) equalizers. Recently, block decision feedback equalization (DFE) has also been suggested to be used in the design of such block data transmission systems. We also revisit the jointly optimum designs of the precoder coupled with the DFE receivers. First we derive the block-data error rate for all of these different systems with their corresponding optimal precoders. Then we rank their performance accordingly. Simulation results are also obtained to verify our analysis. Later, we evaluate the computational complexity on the various systems and draw some conclusions on their relative merits. Finally, we derive the lower and upper bounds of bit error rate (BER) considering error propagation for systems employing nonlinear equalizers.

Differential Evolution (DE) is a novel but powerful optimizer. It has been proved that the DE is superior to Genetic Algorithm (GA) in many real life applications. This paper proposed a memetic Differential Evolution assisted Minimum Bit Error Rate (MBER) linear detector for Multiple-input and multiple-output (MIMO) multi-user system. Compared with GA based MBER and other linear MIMO detectors such as minimum mean square error (MMSE) and zero forcing(ZF)detectors, the proposed scheme shows a better performance in terms of the bit error rate.

The paper considers the problem of constructing adaptive minimum bit error rate (MBER) neural network equalisers for binary signalling. Motivated from a kernel density estimation of the bit error rate (BER) as a smooth function of training data, a stochastic gradient algorithm called the least bit error rate (LBER) is developed for adaptive nonlinear equalisers. This LBER algorithm is applied to adaptive training of a radial basis function (RBF) equaliser in a channel intersymbol interference (ISI) plus co-channel interference setting. A simulation study shows that the proposed algorithm has good convergence speed, and a small-size RBF equaliser trained by the LBER can closely approximate the performance of the optimal Bayesian equaliser. The results also demonstrate that the standard adaptive algorithm, the least mean square (LMS), performs poorly for neural network equalisers because the minimum mean square error (MMSE) is clearly suboptimal in the equalisation setting.

The performance of linear filters degrade drastically when applied to mitigate intersymbol interference caused by channels with frequent nulls in their spectral characteristics like, e.g., time-dispersive radio channels. In such cases, the well-known decision feedback equalizer (DFE) is one possible nonlinear approach to improve the quality of the receiver. However, adapting the DFE filter coefficients to equalize time-varying channels is computationally intense, especially if the dimension of the observation vector is very high. In this paper, we apply the conjugate gradient (CG) algorithm to a conventional minimum mean square error (MMSE) DFE in order to reduce its computational complexity. Moreover, we compare its performance to the one of MMSE DFE versions which are based on the computationally efficient least-mean-square (LMS) or recursive least-squares (RLS) algorithm, respectively. The analysis additionally includes a detailed investigation of computational complexity with respect to the required number of floating point operations (FLOPs). Simulation results when applied to a digital communications system show the ability of the CG based MMSE DFE to outperform either the LMS and the RLS based MMSE DFE although its computational complexity is even smaller in most of the cases.

We investigate the equalization and channel identification for space-time block coded signals over a frequency-selective multiple-input multiple output (MIMO) channels. The equalization is considered by taking into account the cyclostationarity of space-time block coded signals. The minimum mean square error (MMSE) solutions are derived for linear and decision feedback (DF) equalizers. Due to the redundancy induced by space-time block coding, a blind estimation for MIMO channels is investigated by using the subspace method. Together with pilot symbols, the semi-blind channel estimation can be used and it can result in less (bit error rate) performance degradation of the MMSE linear and DF equalizers.

In recent times, there has been growing interests in integration of voice, data and video traffic in wireless communication networks. With these growing interests, WCDMA has immerged as an attractive access technique. The performance of WCDMA system is deteriorated in presence of multipath fading environment. The paper presents space-time coded minimum mean square error (MMSE) Decision Feedback Equalizer (DFE) for wideband code division multiple access (WCDMA) in a frequency selective channel. The filter coefficients in MMSE DFE are optimized to suppress noise, intersymbol interference (ISI), and multiple access interference (MAI) with reasonable system complexity. For the above structure, we have presented the estimation of BER for a MMSE DFE using computer simulation experiments. The simulation includes the effects of additive white Gaussian noise, multipath fading and multiple access interference (MAI). Furthermore, the performance is compared with standard linear equalizer (LE) and RAKE receiver. Numerical and simulation results show that the MMSE DFE exhibits significant performance improvement over the standard linear equalizer (LE) and RAKE receiver.

Adaptive beamforming for binary phase shift keying communication systems

The paper investigates the application of the multilayer perceptron structure to the packet wise adaptive decision feedback equalization of a M-ary QAM signal through a TDMA indoor radio channel in the presence of intersymbol interference (ISI) and additive Gaussian noise. Since the multilayer neural networks are capable of producing complex decision regions with arbitrarily nonlinear boundaries, this would greatly improve the performance of conventional decision feedback equalizer (DFE), where the decision boundaries of conventional DFE's are linear. Nevertheless, the applications of the traditional multilayer neural networks have been limited to real valued signals. To tackle this difficulty, a neural based DFE is proposed to deal with the complex QAM signal over the complex valued fading multipath radio channel without performing time consuming complex valued backpropagation training algorithms, while maintaining almost the same computational complexity as the original real valued training algorithm. Moreover, this neural based DFE trained by packet wise backpropagation algorithm would approach an ideal equalizer after receiving a sufficient number of packets. Results show that the neural based DFE provides a superior bit error rate performance relative to the conventional mean square DFE, especially in poor signal to noise ratio conditions.

Compared to wired channels, time dispersive radio channels possess more frequent nulls in their spectral characteristics. Thus, the performance of linear filters to compensate intersymbol interference degrades dramatically. The well-known nonlinear decision feedback equalizer (DFE) is one approach to improve this behavior. However, in systems with observations of high dimensionality, the optimum DFE structure is computational intensive. In this paper, we apply the method of the multi-stage Wiener filter (MSWF) to a conventional minimum mean square error (MMSE) DFE in order to reduce computational complexity. The application of the new algorithm to an Enhanced Data rates for GSM Evolution (EDGE) system demonstrates the ability to outperform the even more computational intensive linear Wiener filter (WF).

We considers the beamforming assisted multiple-antenna receiver for space-division multiple-access based multiuser systems that employ high-throughput quadrature amplitude modulation (QAM) signalling. The bit error ratio (BER) expression as the function of the beamformer's weight vector is derived, and the minimum BER (MBER) beamforming receiver is then obtained as the solution of the resulting optimisation problem that minimises the MBER criterion. A simplified conjugate gradient algorithm, which has previously demonstrated its effectiveness in solving the minimum symbol error ratio (MSER) optimisation problem, is employed to solve this MBER optimisation. For high-order QAM, although the bit decision is an inherently more complicated procedure than making a symbol decision, it turns out that the computational complexity of computing the MBER solution is similar to that of computing the MSER solution. As expected, our simulation results show that both the MBER and MSER systems achieve the same BER performance, and they significantly outperform the standard minimum mean squares error based solution.

A minimum bit error rate (MBER) multiuser detector designed for space division multiple access (SDMA) aided orthogonal frequency division multiplexing (OFDM) systems is proposed. It is shown that the MBER detector outperforms the minimum mean squared error (MMSE) detector, since the MBER detector directly minimises the BER, while the MMSE detector minimises the mean-squared error (MSE), which does not guarantee achieving the minimum BER. When supporting two users, the proposed MBER scheme substantially outperforms the classic MMSE arrangement in the investigated propagation scenario.

The paper proposes a receiver scheme for multicarrier-CDMA (MC-CDMA) that while not making use of the well known technique of guard interval is able to combat the effects of intersymbol as well as intercarrier interference (ISI, ICI). This is achieved by means of an adaptive multiple input multiple output (MIMO) fractionally spaced (FS) decision feedback equaliser (DFE) receiver scheme performing joint detection (JD). The forward section of the equaliser is designed to operate at a fractional rate, while the feedback section is symbol spaced. The analysis and design of the MIMO FS/DFE scheme is made for the minimum mean square error (MMSE) criterion. In this context, the steady state behaviour is analysed and the receivers performance are computed by means of the residual mean square error (MSE). Using Gaussian approximation an expression of the bit error rate (BER) for binary phase shift keying (BPSK) modulation is derived for both the down link (DL) and up link (UL) communication scenarios.

The combination of multitone modulation with direct sequence spectrum spreading (DS/SS) has been introduced in the past. The performance of a correlation receiver has been evaluated for a multipath channel and in the presence of an additional multiple access interference. We analyze the problem of decision feedback equalization (DFE) for such a system. In order to understand the potential of the system with equalization, we first study the steady-state behavior of the equalizer for a minimum mean square error (MMSE) criterion. The investigation is carried out for a receiver made of a bank of filters matched to both the symbol shape and the channel, and for a two path channel. Assuming transmission of binary phase shift keying (BPSK) symbols, an exact expression of the bit error probability is obtained in the form of an integral. Then adaptive least mean square (LMS) and recursive least square (RLS) structures are derived. The performance of the adaptive RLS algorithm is demonstrated by means of computer simulations.