Minimum Mean Square Error Decision Feedback Equalizer Research Articles

Finite-Order Filter Designs of Source and Multiple Full-Duplex Relays for Cooperative Communications in Presence of Frequency- Selective Fading and Inter-Relay Interference

This paper considers the finite-order filter design problem for the source and multiple full-duplex (FD) relays in a cooperative communication system under frequency-selective fading channels. The goal is to optimize the source and relay filters such that the end-to-end signal-to-interference-plus-noise ratio (SINR) of minimum mean-squared error decision-feedback equalizer (MMSE-DFE) can be maximized. The resultant design problem is very difficult since we need to deal with self interference (SI), inter-relay interference (IRI), and inter-symbol interference (ISI) at the same time. Novel designs are then proposed to overcome the difficulty in this work. Transforming the signals into the frequency domain and using some optimization techniques, we first theoretically derive the power spectrum of the source filter and the spectrums of the relay filters. Then, the finite-order design is developed to approach the derived spectrums. Based on the weighted least-square (WLS) criterion, the Steiglitz-McBride method is exploited to obtain the filter coefficients in finite lengths. Numerical results demonstrated that complete removal of SI may not be a good strategy; preserving a certain amount of SI at each relay instead provides better SINR performance.

Finite-Order Source and Relay Filtering Design for Wideband Full-Duplex Amplify-and-Forward Relaying Networks

Compared with half-duplex relaying, the full-duplex relay (FDR) system provides higher spectral efficiency due to the concurrent transmission and reception at the relay node. As known, the full-duplex operation will introduce the self-interference (SI) that significantly degrades the system performance. Conventionally, SI is treated as a harmful signal that needs to be removed from the system as completely as possible. The conventional design concept, however, may not be efficient since SI is in fact a delayed version of the desired signal. Specifically, it is possible to have further performance improvement if SI can be exploited appropriately. In this paper, we will investigate the source/relay filter design problem where the source and relay are considered as finite impulse response (FIR) and infinite impulse response (IIR) filters, respectively. The design goal is to optimize the end-to-end performance for the linear minimum mean-square error (MMSE) and nonlinear MMSE decision-feedback equalizers. To reduce the implementation cost, we further propose a finite-order filter design for the source and relay precoders. Simulations demonstrate that our designs outperform the conventional ones.

Iterative Decision Feedback Equalization Using Online Prediction

In this article, a new category of soft-input soft-output (SISO) minimum-mean square error (MMSE) finite-impulse response (FIR) decision feedback equalizers (DFEs) with iteration-wise static filters (i.e. iteration variant) is investigated. It has been recently shown that SISO MMSE DFE with dynamic filters (i.e. time-varying) reaches very attractive operating points for high-data rate applications, when compared to alternative turbo-equalizers of the same category, thanks to sequential estimation of data symbols [1]. However the dependence of filters on the feedback incurs high amount of latency and computational costs, hence SISO MMSE DFEs with static filters provide an attractive alternative for computational complexity-performance trade-off. However, the latter category of receivers faces a fundamental design issue on the estimation of the decision feedback reliability for filter computation. To address this issue, a novel approach to decision feedback reliability estimation through online prediction is proposed and applied for SISO FIR DFE with either a posteriori probability (APP) or expectation propagation (EP) based soft feedback. This novel method for filter computation is shown to improve detection performance compared to previously known alternative methods, and finite-length and asymptotic analysis show that DFE with static filters still remains well-suited for high-spectral efficiency applications.

Bridging the Gap Between MMSE-DFE and Optimal Detection of MIMO Systems

In this paper, we propose a novel low-complexity, near-optimal soft-input soft-output detector for $\text {N}\times \text {M}$ multiple-input multiple-output (MIMO) systems. Our algorithm is based on the combination of minimum mean square error decision feedback equalization (MMSE-DFE) and conditional optimization. In a round-robin fashion, one symbol is detected using exhaustive search in such a way that all $\text {N}\times (\text {M}-1)$ submatrices of the baseband channel matrix are considered and the one with the best metric is chosen. This search over all columns of the channel matrix, which can be performed in parallel, has the advantages of improving the performance of the hard-output version of the detector, and refining the list of candidates for efficient implementation of the soft-output detector for MIMO systems with error correcting codes. In particular, it is shown that the error performance of the soft-output system is comparable to that of the list sphere decoder (LSD) but with much smaller list size, and hence smaller complexity than the latter. We also analyze the performance and complexity of the proposed algorithm and discuss different techniques to further reduce its complexity without affecting the performance. Finally, the obtained theoretical results are validated via simulations.

Lower Bounds and Approximations for the Information Rate of the ISI Channel

We consider the discrete-time intersymbol interference (ISI) channel model, with additive Gaussian noise and fixed independent identically distributed inputs. In this setting, we investigate the expression put forth by Shamai and Laroia as a conjectured lower bound for the input–output mutual information after application of a minimum mean-square error decision-feedback equalizer receiver. A low-signal to noise ratio (SNR) expansion is used to prove that the conjectured bound does not hold under general conditions, and to characterize inputs for which it is particularly ill-suited. One such input is used to construct a counterexample, indicating that the Shamai–Laroia expression does not always bound even the achievable rate of the channel, thus excluding a natural relaxation of the original conjectured bound. However, this relaxed bound is then shown to hold for any finite entropy input and ISI channel, when the SNR is sufficiently high. We derive two conditions under which the relaxed bound holds, involving compound channel capacity and quasiconvexity arguments. Finally, new simple bounds for the achievable rate are proven, and compared with other known bounds. Information-estimation relations and estimation–theoretic bounds play a key role in establishing our results.

Iterative MMSE-DFE and Error Transfer for OFDM in Doubly Selective Channels

An iterative design of minimum mean square error decision feedback equalizer (MMSE-DFE) to suppress the intercarrier interference (ICI) of orthogonal frequency division multiplexing systems in the time-variant channels is analyzed in this paper. The channel is assumed to vary in a linear fashion within a frame. The error power transfer is investigated in parallel with the equalizing process, showing the gain in iterations and aiding the MMSE decision. Iteratively, the ICI components are estimated and canceled from the receive symbol, providing an MMSE estimate of the source symbol with an increasing reliability. It is shown that based on the error power transfer analyzed in this paper, ICI components can be significantly removed via the iterative MMSE-DFE, leaving the error composed of noise and negligible residue of ICI. Simulations have verified the accuracy of the error power estimation.

The Decision Delay in Finite-Length MMSE–DFE Systems

This paper investigates the performance of the minimum mean square error (MMSE) decision feedback equalization system and optimizes its key parameter, decision delay, to minimize the system MSE. For the first typical scenario where the feedback filter length $$N_b$$Nb is larger than or equal to the channel order $$v$$v, we present an analytical expression of the system MSE in terms of decision delay. A few results in previous literature can be regarded as special cases of this expression. For the second typical scenario where feedback filter length $$N_b$$Nb is less than the channel order $$v$$v, it is difficult to obtain the explicit solutions. We prove that at a given decision delay, the MSE for $$N_b< v$$Nb

Channel Equalization of WCDMA Downlink System Using Finite Length MMSE-DFE

The performance of WCDMA system deteriorates in the presence of multipath fading environment. Fading destroys the orthogonality and is responsible for multiple access interference (MAI). Though conventional rake receiver provides reasonable performance in the WCDMA downlink system due to path diversity, but it does not restores the orthogonality. Linear equalizer restores orthogonality and suppresses MAI, but it is not efficient, since its performance depends on the spectral characteristics of the channel. To overcome this, Minimum Mean Square Error- Decision Feedback Equalizer (MMSE-DFE) with a linear, anticausal feedforward filter, causal feedback filter and simple detector is proposed in this paper. The filter taps of finite length DFE is derived using the cholesky factorization theory, capable of suppressing noise, Intersymbol Interference (ISI) and MAI. This paper describes the WCDMA downlink system using finite length MMSE-DFE and takes into consideration the effects of interference which includes additive white gaussian noise, multipath fading, ISI and MAI. Furthermore, the performance is compared with conventional rake receiver and MMSE and the simulation results are shown. Keywords - MMSE, MMSE-DFE, rake receiver, WCDMA

Comparison of Orthogonal Frequency-Division Multiplexing and Pulse-Amplitude Modulation in Indoor Optical Wireless Links

We evaluate the performance of three direct-detection orthogonal frequency-division multiplexing (OFDM) schemes in combating multipath distortion in indoor optical wireless links, comparing them to unipolar M-ary pulse-amplitude modulation (M-PAM) with minimum mean-square error decision-feedback equalization (MMSE-DFE). The three OFDM techniques are DC-clipped OFDM and asymmetrically clipped optical OFDM (ACO-OFDM) and PAM-modulated discrete multitone (PAM-DMT). We describe an iterative procedure to achieve optimal power allocation for DC-OFDM. For each modulation method, we quantify the received electrical SNR required at a given bit rate on a given channel, considering an ensemble of 170 indoor wireless channels. When using the same symbol rate for all modulation methods, M-PAM with MMSE-DFE has better performance than any OFDM format over a range of spectral efficiencies, with the advantage of (M-PAM) increasing at high spectral efficiency. ACO-OFDM and PAM-DMT have practically identical performance at any spectral efficiency. They are the best OFDM formats at low spectral efficiency, whereas DC-OFDM is best at high spectral efficiency. When ACO-OFDM or PAM-DMT are allowed to use twice the symbol rate of M-PAM, these OFDM formats have better performance than M-PAM. When channel state information is unavailable at the transmitter, however, M-PAM significantly outperforms all OFDM formats. When using the same symbol rate for all modulation methods, M-PAM requires approximately three times more computational complexity per processor than all OFDM formats and 63% faster analog-to-digital converters, assuming oversampling ratios of 1.23 and 2 for ACO-OFDM and M-PAM, respectively. When OFDM uses twice the symbol rate of M-PAM, OFDM requires 23% faster analog-to-digital converters than M-PAM but OFDM requires approximately 40% less computational complexity than M-PAM per processor.

Comparison of Orthogonal Frequency-Division Multiplexing and ON–OFF Keying in Direct-Detection Multimode Fiber Links

We compare the performance of several direct-detection orthogonal frequency-division multiplexing (OFDM) schemes to that of ON-OFF keying (OOK) in combating modal dispersion in multimode fiber links. We review known OFDM techniques, including dc-clipped OFDM (DC-OFDM), asymmetrically clipped optical OFDM (ACO-OFDM) and pulse-amplitude modulated discrete multitone (PAM-DMT). We describe an iterative procedure to achieve optimal power allocation for DC-OFDM and compare analytically the performance of ACO-OFDM and PAM-DMT. We also consider unipolar <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary pulse-amplitude modulation ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -PAM) with minimum mean-square error decision-feedback equalization (MMSE-DFE). For each technique, we quantify the optical power required to transmit at a given bit rate in a variety of multimode fibers. For a given symbol rate, we find that unipolar <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -PAM with MMSE-DFE has a better power performance than all OFDM formats. Furthermore, we observe that the difference in performance between <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -PAM and OFDM increases as the spectral efficiency increases. We also find that at a spectral efficiency of 1 bit/s/Hz, OOK performs better than ACO-OFDM using a symbol rate twice that of OOK. At higher spectral efficiencies, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -PAM performs only slightly better than ACO-OFDM using twice the symbol rate, but requires less electrical bandwidth and can employ analog-to-digital converters at a speed only 81% of that required for ACO-OFDM.

Block Diagonal GMD for Zero-Padded MIMO Frequency Selective Channels

In the class of systems with linear precoder and decision feedback equalizers (DFE) for zero-padded (ZP) multiple-input multiple-output (MIMO) frequency selective channels, existing optimal transceiver designs present two drawbacks. First, the optimal systems require a large number of feedback bits from the receiver to encode the full precoding matrix. Second, the full precoding matrix leads to complex computations. These disadvantages become more severe as the bandwidth (BW) efficiency increases. In this paper, we propose using block diagonal geometric mean decomposition (BD-GMD) to design the transceiver. Two new BD-GMD transceivers are proposed: the ZF-BD-GMD system, where the receiver is a zero-forcing DFE (ZF-DFE), and the MMSE-BD-GMD system, where the receiver is a minimum- mean-square-error DFE (MMSE-DFE). The BD-GMD systems introduced here have the following four properties: a) They use the block diagonal unitary precoding technique to reduce the required number of encoding bits and simplify the computation. b) For any block size, the BD-GMD systems are optimal within the family of systems using block diagonal unitary precoders and DFEs. As block size gets larger, the BD-GMD systems produce uncoded bit error rate (BER) performance similar to the optimal systems using unitary precoders and DFEs. c) For the two ZF transceivers (ZF-Optimal and ZF-BD-GMD) and the two MMSE transceivers (MMSE-Optimal and MMSE-BD-GMD), the average BER degrades as the BW efficiency increases. d) In the case of single-input single-output (SISO) channels, the BD-GMD systems have the same performance as those of the lazy precoder transceivers. These properties make the proposed BD-GMD systems more favorable designs in practical implementation than the optimal systems.

MMSE DFE Transceiver Design Over Slowly Time-Varying MIMO Channels Using ST-GTD

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.

Fast Kalman Equalization for Time-Frequency Asynchronous Cooperative Relay Networks With Distributed Space-Time Codes

Cooperative relay networks are inherently time and frequency asynchronous due to their distributed nature. In this correspondence, we propose a transceiver scheme to combat both time and frequency offsets for cooperative relay networks with multiple relay nodes. At the relay nodes, a distributed linear convolutive space-time coding is adopted, which has the following advantages: 1) Full cooperative diversity can be achieved using a minimum mean square error (MMSE) or MMSE decision feedback equalizer (MMSE-DFE) detector, instead of a maximum-likelihood receiver when only time asynchronism exists. 2) The resultant equivalent channel possesses some special structure, which can be exploited to reduce the equalization complexity at the destination node. By taking full advantage of such a special structure, fast Kalman equalizations based on linear MMSE and MMSE-DFE are proposed for the receiver, where the estimation of the state vector (information symbols) can be recursively taken and become very computationally efficient, compared with direct equalization. The proposed scheme can achieve considerable diversity gain with both time and frequency offsets and applies to frequency-selective fading channels.

Minimum SER Block Precoding and Equalization for Frequency-Selective Fading Channels

In this paper, we study the joint transmitter and receiver design problem over frequency-selective fading channels. Motivated by a conventional minimum mean square error decision feedback equalizer (MMSE-DFE), we present a simple approximate maximum-likelihood (ML) decision feedback equalizer at the receiver. We then perform precoding in a downlink scenario at the transmitter side to minimize the analytical symbol error rate (SER) of the proposed equalization scheme. Simulation results indicate that the proposed precoder achieves substantial performance gains over previously reported techniques.

Computationally efficient equalization for asynchronous cooperative communications with multiple frequency offsets

In cooperative communications, time and frequency synchronization is an important issue needed to be addressed in practice. Due to the nature of cooperative communications, multiple frequency offsets may occur and the traditional frequency offset compensations may not apply. For this problem, equalization for the time-varying channel has been used in the literature, where the equalization matrix inverse needs to be retaken every symbol. In this paper, we propose computationally efficient minimum mean square error (MMSE) and MMSE decision feedback equalizers (MMSE-DFE) when multiple frequency offsets are present, where the equalization matrix inverses do not need to be retaken every symbol. Our proposed equalization methods apply to linear convolutively coded cooperative systems, where linear convolutive space-time coding is used to achieve the full cooperative diversity when there are timing errors from the cooperative users or relay nodes, i.e., asynchronous cooperative communication systems.

BER Performance of Space-Time Coded MMSE DFE for Wideband Code Division Multiple Access (WCDMA)

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.

Design of a Fuzzy Based Outer Loop Controller for Improving the Training Performance of LMS Algorithm

Because of the fact that mobile communication channel changes by time, it is necessary to employ adaptive channel equalizers in order to combat the distorting effects of the channel. Least Mean Squares (LMS) algorithm is one of the most popular channel equalization algorithms and is preferred over other algorithms such as the Recursive Least Squares (RLS) and Maximum Likelihood Sequence Estimation (MLSE) when simplicity is the dominant decision factor. However, LMS algorithm suffers from poor performance and convergence speed within the training period specified by most of the standards. The aim of this study is to improve the convergence speed and performance of the LMS algorithm by adjusting the step size using fuzzy logic. The proposed method is compared with the Channel Matched Filter-Decision Feedback Equalizer (CMF-DFE) [1] which provides multi path propagation diversity by collecting the energy in the channel, Minimum Mean Square Error-Decision Feedback Equalizer (MMSE-DFE) [2] which is one of the most successful equalizers for the data packet transmission, normalized LMS-DFE (N-LMS-DFE) [3], variable step size (VSS) LMS-DFE [4], fuzzy LMS-DFE [5], [6] and RLS-DFE [7]. The obtained simulation results using HIPERLAN/1 standards have demonstrated that the proposed LMS-DFE algorithm based on fuzzy logic has considerably better performance than others.

Low-complexity equalization of time-varying channels with precoding

This correspondence deals with time-varying (TV) single-input multiple-output (SIMO) channels, which are both frequency selective (due to high data rate) and time selective (due to mobility). A complex exponential basis expansion model (CE-BEM) is used to model the channel. We consider a block transmission system, where on the transmit side a precoder is employed to enable the maximum available diversity for a CE-BEM channel. After direct decoding on the receive side, the resulting channel resembles a finite-impulse-response (FIR) filter on both block and symbol level. We therefore propose an equalizer that bears a structure analogous to the effective channel. In comparison with a standard block minimum mean-square error decision-feedback equalizer (BMMSE-DFE) that involves the inversion of a large-size matrix, the proposed parametric equalizer renders a similar performance but at a lower computational cost if there are multiple outputs present. Another contribution of this correspondence is a semiblind algorithm to estimate this equalizer when the channel state information is not available: the equalizer taps and the information symbol estimates are refined recursively by means of normalized least-mean-squares (NLMS) adaptation.

A unified framework for tree search decoding: rediscovering the sequential decoder

We consider receiver design for coded transmission over linear Gaussian channels. We restrict ourselves to the class of lattice codes and formulate the joint detection and decoding problem as a closest lattice point search (CLPS). Here, a tree search framework for solving the CLPS is adopted. In our framework, the CLPS algorithm is decomposed into the preprocessing and tree search stages. The role of the preprocessing stage is to expose the tree structure in a form matched to the search stage. We argue that the forward and feedback (matrix) filters of the minimum mean-square error decision feedback equalizer (MMSE-DFE) are instrumental for solving the joint detection and decoding problem in a single search stage. It is further shown that MMSE-DFE filtering allows for solving underdetermined linear systems and using lattice reduction methods to diminish complexity, at the expense of a marginal performance loss. For the search stage, we present a generic method, based on the branch and bound (BB) algorithm, and show that it encompasses all existing sphere decoders as special cases. The proposed generic algorithm further allows for an interesting classification of tree search decoders, sheds more light on the structural properties of all known sphere decoders, and inspires the design of more efficient decoders. In particular, an efficient decoding algorithm that resembles the well-known Fano sequential decoder is identified. The excellent performance-complexity tradeoff achieved by the proposed MMSE-DFE Fano decoder is established via simulation results and analytical arguments in several multiple-input multiple-output (MIMO) and intersymbol interference (ISI) scenarios.

Turbo DFE algorithm with imperfect decision feedback

In this letter, we propose a new turbo decision feedback equalizer (DFE) algorithm that takes into account decision feedback error propagation. We derive the exact mathematical expression for the probability density function (pdf) of soft decision feedback symbols, assuming that soft outputs from channel decoder are independent identically distributed Gaussian random variables with known mean and variance. We find a new set of "imperfect" turbo minimum mean-square error DFE coefficients based on the derived pdf. The proposed turbo detector outperforms other turbo equalization algorithms of similar computational complexity in terms of bit-error rate. The achieved improvement is up to 3.8 dB for severe frequency-selective channels.