Uppsala universitet

Realizable MIMO Decision Feedback Equalizers: Structure and Design

Claes Tidestav , Anders Ahlén and Mikael Sternad

IEEE Transactions on Signal Processing, January 2001, pp. 121-133.
© 2001 IEEE

During the last three decades, decision feedback equalizers (DFE:s) have been used in digital communication to suppress intersymbol interference (ISI), i.e. to remove the effects of a frequency selective communication channel. The DFE constitutes a good compromise between performance and complexity: It provides much better performance than a linear equalizer, and it has a much lower complexity than the optimum detector, the maximum likelihood sequence estimator (MLSE).

A DFE consists of two filters and a decisions non-linearity. The ISI corrupted measurements are input to the feedforward filter. From the output of the feedforward filter, the output of the feedback filter is subtracted to remove the effect of residual ISI caused by the already detected symbols. A hard decision is then made to decide what symbol was transmitted. This decision is fed into the feedback filter to remove its effect on future symbol estimates. The coefficients of the feedforward and feedback filters are adjusted according to a criterion, the two most common being the zero-forcing (ZF) criterion and the minimum mean square error (MMSE) criterion. With a zero-forcing equalizer, all intersymbol interference is removed, whereas with an MMSE equalizer, the mean square difference between the transmitted signal and a soft signal estimate is minimized.

During the last few years, channels with several inputs and/or outputs have gained increased interest. Such channels occur in many areas, e.g. in cellular communication systems where antenna arrays are used to improve the detection. Oversampled channel models can also be formulated as a channel with several outputs. With a detector based on a model with multiple inputs and/or outputs, it is possible to suppress not only intersymbol interference, but also co-channel interference, i.e. interference from other signals.

A multiple input-multiple output (MIMO) DFE is a DFE where both the feedforward and the feedback filter have multiple inputs and multiple outputs. The DFE is an attractive compromise between complexity and performance also in the MIMO case. Studies of MIMO DFE:s are based on one of two principles: either a DFE with a non-causal feedforward filter or a DFE whose structure is fixed prior to the design.

In this paper, we present a generalized DFE with several inputs and outputs, which minimizes the mean square error under the constraint of realizability. The resulting DFE utilizes multivariable IIR filters with optimal filter degrees, and its parameters can be obtained from closed form design equations. In the limit, when the smoothing lag tends to infinity, we also obtain the non-realizable MMSE DFE. Furthermore, we introduce the existence of a zero forcing MIMO DFE as a criterion for near-far resistance of the corresponding MMSE DFE. Our derivations are based on a discrete time system model, where the multivariable channel may have an infinite impulse response, and where the noise is described by a multivariate ARMA model.

We present and discuss optimum multivariable decision feedback equalizers (DFE:s). The equalizers are derived under the constraint of realizability, requiring causal and stable filters and finite smoothing lag. The design is based on a discrete-time channel model, where a digital signal passes through a dispersive multivariable channel with infinite impulse response. The additive noise is described by a multivariate ARMA model. Both minimum mean square error (MMSE) and zero-forcing (ZF) DFE:s are derived, under the assumption of correct past decisions.

For the MMSE DFE, the optimal structure is obtained, and it is noted that the conventional structure, with FIR filters in both the feedforward and the feedback links is optimal only under rather restrictive conditions. Simple design equations on closed form are also presented.

Conditions for the existence of a ZF DFE are presented, and we suggest that the existence of a ZF DFE guarantees near-far resistance of the corresponding MMSE DFE.

Simulations indicate that it may be advantageous to use a DFE with optimal structure as opposed to the conventional structure. However, in some cases, the conventional structure is close to optimal, and in these cases, the performance degradation is small for the conventional DFE. Also, the performance improvement of the optimum DFE is reduced when error propagation is taken into account.

Related publications:
PhD Thesis by Claes Tidestav.
ICASSP'99 paper, summarizing the results.
Paper in IEEE-IT on scalar optimum realizable DFE:s which use IIR filters.
Paper in IEEE-COM on MIMO FIR DFE:s and their application for reuse within cell .

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