# Bidirectional MMSE Algorithms for Interference Mitigation in CDMA Systems over Fast Fading Channels

###### Abstract

This paper presents adaptive bidirectional minimum mean-square error (MMSE) parameter estimation algorithms for fast-fading channels. The time correlation between successive channel gains is exploited to improve the estimation and tracking capabilities of adaptive algorithms and provide robustness against time-varying channels. Bidirectional normalized least mean-square (NLMS) and conjugate gradient (CG) algorithms are devised along with adaptive mixing parameters that adjust to the time-varying channel correlation properties. An analysis of the proposed algorithms is provided along with a discussion of their performance advantages. Simulations for an application to interference suppression in DS-CDMA systems show the advantages of the proposed algorithms.

Bidirectional signal processing, adaptive interference suppression, fast-fading channels, adaptive receivers.

## I Introduction

Low-complexity reception and interference suppression is essential in multiuser mobile systems if battery power is to be conserved, data-rates improved, and quality of service enhanced. Conventional adaptive schemes fulfil many of these requirements and have been a significant focus of the research literature [1, 2]. However, in time-varying fading channels commonly associated with highly mobile systems, adaptive techniques encounter problems when estimating and tracking the variations of parameters such as the receive filter and channel state information (CSI). The development of cost-effective parameter estimation and tracking techniques for highly dynamic channels remains a very challenging problem.

Existing strategies to enhance the performance of estimation techniques include the use of optimized convergence parameters in conventional adaptive algorithms to extend their ability to deal with fading and improve their convergence and tracking performance [3, 4, 5, 6]. However, the stability of adaptive step-sizes and forgetting factors is a concern unless they are constrained to lie within a predefined region [4]. Furthermore, the fundamental problem of demodulating the transmitted data symbols whilst suppressing multiuser interference (MUI) remains. Approaches to avoid and/or improve the tracking and estimation of fading coefficients have been reported in [7, 8, 9, 10, 12]. Although a channel might be highly time-variant, adjacent fading coefficients can be approximately equal and have a significant level of correlation. These properties can be exploited to obtain a sequence of faded symbols where the primary purpose of the receive filter is to suppress interference and track the ratio between successive fading coefficients, dispensing with the estimation of the fading coefficients themselves. However, this scheme has a number of limitations due to the fact that only one correlation time instant is employed, which results in instability and a difficulty to track highly time-variant signals.

In this paper, a bidirectional minimum mean-square error (MMSE) based interference suppression scheme for highly-dynamic fading channels is presented. The channel correlation between adjacent time instants is exploited to improve the robustness, tracking and convergence performances of existing adaptive schemes. Bidirectional normalized least mean squares (NLMS) and conjugate gradient (CG) algorithms are devised along with a mixing strategy that adaptively weights the contribution of the considered time instants. An analysis of the proposed schemes is given and establishes the factors behind their behaviour and improved performance. The proposed schemes are applied to DS-CDMA systems and simulations show that they significantly outperform existing schemes.

The remainder of the paper is structured as follows, Section II states the problem and explains the motivation of the work. The algorithmic implementations of the proposed bidirectional methods are given in Section IV, and analysis of the proposed algorithms in Section V. Simulations and performance evaluation are given in Section VI and conclusions in Section VII.

## Ii DS-CDMA Signal Model and Problem Statement

Consider the uplink of a multiuser DS-CDMA system with users, processing gain and multipath channels with paths. The received signal after chip-pulsed matched filtering and sampling at the chip rate is given by

(1) |

where , and and are the spreading sequence and signal amplitude of the user, respectively. The matrix models the channel propagation effects for the user, corresponds to the transmitted symbol of the the th user, is the ISI vector and is the noise vector.

The design of linear receivers consists of processing the received vector with the receive filter with coefficients that provide an estimate of the desired symbol as follows

(2) |

where the detected symbol is given by , where is a function that performs the detection according to the constellation employed. It is also possible to use non-linear receiver techniques. The problem we are interested in solving in this work is how to estimate the parameter vector of the receive filter in fast time-varying channels.

## Iii Proposed Bidirectional Processing Strategy

Adaptive parameter estimation techniques have two primary objectives: estimation and tracking of the desired parameters. However, in fast fading channels the combination of these two objectives places unrealistic demands on conventional estimation schemes. Differential techniques reduce these demands by relieving the adaptive filter of the task of tracking fading coefficients. This is achieved by posing an optimisation problem where the ratio between two successive received samples is the quantity to be tracked. Such an approach is enabled by the presumption that, although the fading is fast, there is a significant level of correlation between the adjacent channel samples

(3) |

where is the channel coefficient of the desired user. The interference suppression of the resulting filter is improved in fast fading environments compared to conventional adaptive filters but only the ratio of adjacent fading samples is obtained. Consequently, differential modulation, where the ratio between adjacent symbols is the data carrying mechanism, are suited to differential MMSE schemes.

However, limiting the optimization process to two adjacent samples exposes the differential MMSE process to the negative effects of uncorrelated samples

(4) |

and does not exploit the correlation that may be present between two or more adjacent samples, i.e.

(5) |

To address these weaknesses, we propose a bidirectional MSE cost function based on adjacent received data vectors so that the number of channel scenarios under which an algorithm performs reliable estimation and tracking is increased and performance improved. Termed the bidirectional MMSE, due to the use of multiple and adjacent time instants, it can exploit the correlation between successive received signals and reuse data [2].

The optimization problem of the proposed scheme is given by

(6) |

where is the expected value of the filter, is a weighting factor used in the cost function to address problems with uncorrelated fading coefficients and . Note that the time instants of interest have been altered to avoid the use of future samples. In addition to (6), an output power constraint is required to avoid the trivial zero correlator solution

(7) |

In fast-fading channels, the correlation between the considered time instant is unlikely to be equal. Therefore variable weighting or mixing of the cost function will be required to obtain improved performance. However, the setting of the weights is problematic if they are to be fixed. An adaptive scheme is preferable which can take account of the time-varying channels. The errors extracted from the cost function (6) are chosen as the metric for this approach. This provides an input to the weighting factor calculation process that is directly related to the optimization in (6). The time-varying mixing factors are given by

(8) |

where and the individual errors terms are calculated for and

(9) |

The forgetting factor, , is user defined and, along with the normalization by the total error, , and , ensures and a convex combination at each time instant.

## Iv Proposed Bidirectional Algorithms

In this section, the proposed bidirectional MMSE-based algorithms based on (6) are derived. In particular, we concentrate on the case where and develop bidirectional NLMS and CG adaptive algorithms. Let us consider the following cost function

(10) |

The time-varying mixing factors are adjusted by 8 where and the individual errors terms are given by

(11) |

The forgetting factor, , is user defined and, along with the normalization by the total error, , and , ensures and a convex combination at each time instant.

### Iv-a Bidirectional NLMS Algorithm

We first devise a low-complexity bidirectional NLMS algorithm that iteratively computes the solution of (LABEL:eq:cost3). The instantaneous gradient of (LABEL:eq:cost3) is taken with respect to , and the errors terms of (11) are incorporated to yield the update equation

(12) |

where is the step-size and the adaptive mixing parameters have been included. The normalization factor, , is given by

(13) |

where is an exponential forgetting factor [9]. The enforcement of the constraint is performed by the denominator of (12) and ensures that the the filter does not tend towards a zero correlator. The complexity of this algorithm is , which corresponds to roughly times that of the NLMS.

### Iv-B Bidirectional Conjugate Gradient Algorithm

Due to the incongruous form of the bidirectional formulation and the conventional matrix inversion lemma based recursive least-squares (RLS) algorithm, an alternative bidirectional CG algorithm is now derived. We begin with the time-averaged autocorrelation and crosscorrelation structures and from (LABEL:eq:cost3)

(14) |

and

(15) |

respectively. After some algebraic manipulations with the terms, the final correlation structures are given by

(16) |

(17) |

where the adaptive mixing factors have been included. Inserting these structures into the standard CG quadratic form yields

(18) |

From [13], the unique minimizer of (18) is also the minimizer of

(19) |

At each time instant, a number of iterations of the following method are required to reach an accurate solution, where the iterations are indexed with the variable . At the time instant the gradient and direction vectors are initialized as

(20) |

and

(21) |

respectively, where the gradient expression is equivalent to those used in the derivation of the NLMS algorithm. The vectors and are orthogonal with respect to such that for . At each iteration, the filter is updated as

(22) |

where is the minimizer of such that

(23) |

The gradient vector is then updated according to

(24) |

and a new CG direction vector found

(25) |

where

(26) |

ensures the orthogonality between and where . The iterations (22) - (26) are repeated until .

## V Analysis of the Proposed Algorithms

The form of the bidirectional MSE cost function precludes the application of standard MSE analysis. Consequently, we concentrate on the signal to interference plus noise ratio (SINR) of the proposed NLMS algorithm to analyze its performance.

### V-a SINR Analysis

To begin, we convert the SINR expression given by

(27) |

where and are the signal and interference and noise correlation matrices, into a form amenable to analysis. Substituting in the filter error weight vector, , where is the instantaneous standard MMSE receiver, and taking the trace of the expectation yields

(28) |

where , , and . From (28) it is clear that we need to pursue expressions for and to reach an analytical interpretation of the bidirectional scheme.

Substituting the filter error weight vector into the filter update expression of (12) yields a recursive expression for

(29) |

where the terms are the error terms of (11) when the optimum filter is used. Using the direct averaging approach of Kushner [14], the solution to the stochastic difference equation of (29) can be approximated by the solution to a second equation [2], such that

(30) |

where and are correlations matrices. Specifically, are autocorrelation matrices given by

(31) |

and cross-time-instant correlation matrices, given by

(32) |

Using (30) and the independence assumptions of , and , we arrive at an expression for

(33) |

where . Following a similar method, an expression for can also be reached

(34) |

At this point we study the derived expression to gain an insight into the operation of the bidirectional algorithm and the origins of its advantages over the conventional differential scheme. Equivalent expressions for the existing differential NLMS scheme are given by

(35) |

The bidirectional scheme has a number of additional correlation terms compared to the existing scheme. Evaluating the cross-time-instant matrices with regards to the independence assumptions yields

(36) |

From the expression above it is clear that underlying factor that governs the SINR performance of the algorithms is the correlation between the considered time instants, and similarity between data-ruse and the use of and . Accordingly, it is the additional correlation factors of the bidirectional algorithm that enhance its performance compared to the conventional techniques, confirming the initial motivation behind the proposition of the bidirectional approach. Lastly, the expressions of (36) can be seen as the factors that influence the number of considered time instants.

Central to the performance of the bidirectional schemes are the correlation factors and the related assumption of . Examining the effect of the fading rate on the value of shows that at fading rates of up to , where is the normalized fading parameter. Consequently, after a large number of received symbols with high total receive power

(37) |

due to the decreasing significance of the identity matrix. This indicates that the expected value of the SINR of the bidirectional scheme, once have stabilized, should be similar to the differential scheme. A second implication is that the bidirectional scheme should converge towards the MMSE level due to the equivalence between the bidirectional scheme and the MMSE solution. Fig. 1 illustrates the analytical performance using the above expressions.

The correlation matrices are calculated via ensemble averages prior to commencement of the algorithm and . In Fig. 1 one can see the convergence of the simulated schemes to the analytical and MMSE plots, validating the analysis. Due to the highly dynamic nature of the channel, using the expected values of the correlation matrix alone cannot capture the true transient performance of the algorithms. However, the convergence period of the analytical plots within the first 200 iterations can be considered to be within the coherence time and therefore give an indication of the transient performance relative to other analytical plots. Using this justification and the aforementioned analysis, it is clear that the advantages brought by the bidirectional scheme are predominantly in the transient phase due to the additional correlation information supplied by and and their analogy with data reuse algorithms. This observation is supported by the similar forms of the analytical and simulated results and their subsequent convergence.

## Vi Simulations

We apply the proposed bidirectional adaptive algorithms to interference suppression in the uplink of the DS-CDMA system described in Section II. Simulations are averaged over packets and the parameters are specified in each plot. Conventional schemes use BPSK modulation and the differential schemes employ differential phase shift keying where the sequence of data symbols to be transmitted by user are given by where is the unmodulated data.

The BER performance of existing and bidirectional schemes is illustrated in Fig. 2. The existing RLS and proposed CG algorithms converge to near the MMSE level with the bidirectional scheme providing a clear performance advantage. However, the NLMS schemes have a slower convergence performance due to their reduced adaptation rate compared to the CG algorithms.

Fig. 3 illustrates the performance of the proposed CG and existing RLS algorithms as the fading rate is increased, where the SINR is normalised by the instantaneous SNR. The conventional schemes are unable to cope with fading rates in excess of and begin to diverge at the completion of the training sequence. The bidirectional scheme outperforms the differential schemes but the performance begins to decline once fading rates above are reached. The increase in performance of the bidirectional scheme can be accounted for by the increased correlation information supplied by the matrices and and effective data reuse. A second benefit of the bidirectional scheme is the improved performance at low fading rate. The introduction of the mixing factors into the bidirectional algorithm improves performance further, especially at higher fading rates. An explanation for this can be established by referring back to the observations on the correlation factors . Although fading rates of may be fast, the assumption is still valid. Consequently, and equal weighting is optimum. However, as the fading rate increases beyond this assumption breaks down and the correlation information requires unequal weighting for optimum performance, a task fulfilled by the adaptive mixing factors.

## Vii Conclusions

We have presented bidirectional MMSE-based parameter estimation algorithms that exploit the time correlation of rapidly varying fading channels. The ratio between successive received vectors is tracked using correlation information gathered at adjacent time instants to avoid tracking of the faded or unfaded symbols. The results show that the proposed algorithms applied to interference suppression in DS-CDMA systems significantly outperform existing algorithms.

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