Realizable MIMO Decision Feedback Equalizers:
Structure and Design
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
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
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.
PhD Thesis by Claes Tidestav.
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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- Report version (March 1998):
- Postscript (uncompressed)
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