3.2 Optimal Space-Time Decision Feedback Equalizers
3.2.2 Optimal Space-Time DFE for FIR Channels with AR
The space-time DFE in Section 3.2.1 is often unnecessarily complex. A typical wireless communication channel can be modeled by an FIR channel.
This is therefore an important special case to consider. It is also practical to have decision feedback equalizers with FIR lters in the feedforward and the feedback lters, rather than IIR-lters as is the case for the space-time for DFE for ARMA channels with ARMA noise2. We will here show that if we have a space-time FIR-model for the channel of the desired signal and a space-time AR-model for the noise and interference, then the MMSE-optimal DFE will have only FIR-lters in the feedforward and feedback lters. The lters can also be computed in a simple manner.
If the noise consists mainly of co-channel interferers, then an AR model will typically not be physically motivated. A sum of interferers each propagat- ing through an FIR channel would be better described by a moving average (MA) model. From the discussion in Section 2.9.2, we can however under- stand why an AR model, even a low order AR model, for the noise plus interference can be useful when used in a decision feedback equalizer. The important observation to make is that the the AR noise model denominator does not have to model the noise particularly well, it only has to be able to suppress the noise as a part of a noise whitening lter. The use of an AR model for the noise was proposed in [7] in conjunction with a space-time MLSE. When we use an AR model for the noise in an MLSE, the memory length of the Viterbi algorithm used will be increased. This increases the complexity of Viterbi algorithm by a factor Knn, where K is the number of symbols in the alphabet and nn is the order of the AR model for the noise plus interference. Thus the complexity for the MLSE using an AR noise model increases exponentially with the order of the AR noise model.
When an AR model for the noise plus interference is used together with a space-time decision feedback equalizer, on the other hand, the increase in complexity is only linear in the order of the AR noise model. As a result we can allow AR noise models of a higher order for the space-time DFE.
2A reason for this is that if we have an IIR lter as feedback lter, then an erroneous symbol decision can cause a long error burst if the lter has poles close to the unit circle.
If we use FIR lters only, then the number of symbols an erroneous decision directly can aect will be limited by the length of the FIR feedback lter.
TheM1 received signal vector y(t) is here modeled as
y(t) =b(q;1)d(t) +N;1(q;1)M0v(t)
=b0d(t) ++bnbd(t;nb) +N;1(q;1)M0v(t) (3.26) where the M 1 polynomial column vector b(q;1) is the channel for the desired signal
b(q;1) =
2
6
4
b1(q;1) bM(...q;1)
3
7
5 (3.27)
with
bi(q;1) =bi;0+bi;1q;1+:::+bi;nbq;nb: (3.28) The noise model numerator,M0, is anMM constant matrix
M0 =
2
6
4
m11 ::: m1M
... ... ...
mM1 ::: mMM
3
7
5 (3.29)
which is assumed to be nonsingular. For the noise model denominator,
N(q;1), the transmitted symbols, d(t), and the noise sequence, v(t), we adopt the same assumptions as in Section 3.2.1.
Given the channel model (3.26), withM0 invertible, and assuming correct past decisions, ~d(t;`;1), to be fed into the feedback lter, then as shown in detail in Appendix 3.A.2, the MMSE optimal DFE with linear lters is given by
d^(t;`) =s(q;1)y(t);Q(q;1)~d(t;`;1) (3.30) where the feedforward FIR lter,s(q;1), is given by the row vector
s(q;1) =s0(q;1)M;01N(q;1): (3.31) See Figure 3.4.
The coecients of the polynomial vector s0(q;1) of degree ` can be com- puted by solving the system of equations
(B0B0H+I)sH0 =
2
6
4 b
0`
...
b 00
3
7
5 (3.32)
3.2. Optimal Space-Time Decision Feedback Equalizers 117
- -
- -
6
-
y1(t)
yM(t) s(q;1)
q;1 Q(q;1)
d^(t;`) d~(t;`) + P
;
Figure 3.4: The structure of the space-time DFE for an FIR channel with AR noise. The feedforward lter,s(q;1), is a MISO FIR lter of order`+nn and the feedback lter, Q(q;1), is a scalar FIR lter of order nb+nn;1, where `is the smoothing lag of the equalizer, nn is the order of the space- time AR noise model andnb is the order of the space-time FIR channel for the desired signal.
where
B 0 =
2
6
4 b
00 b0`
... ...
0 b00
3
7
5 (3.33)
s0 =s0;0 ::: s0;` (3.34) where b0i are the vector taps of the noise whitened channel
b
0(q;1) =b00+b01q;1+:::+b0nb0q;nb0 =M;01N(q;1)b(q;1) (3.35) with b0i = 0 if i > nb0 = nb+nn, and s0;k are the vector taps in the polynomial
s0(q;1) =s0;0+s0;1q;1+:::+s0;`q;`: (3.36) The matrix (B0B0H+I) will be nonsingular since I obviously is a full rank matrix andB0B0H is a positive semi-denite matrix, making (B0B0H +I) a full rank (and thus invertible) matrix.
The coecients of the feedback polynomial,Q(q;1) of ordernq=nb+nn;1, can be computed as
2
6
4
QH0 Q...Hnq
3
7
5=
2
6
4 b
0H`+1 b0H1
... ...
b 0H
`+nb+nn b0H nb+nn
3
7
5 2
6
4 sH0;0
...
sH0;`
3
7
5 (3.37)
whereb0i = 0 ifi > nb0 =nb+nn.
We thus see that with the channel model of (3.26) with an FIR model for the channel of the desired user and an AR model for the noise plus interference, the optimal space-time DFE takes on a particular simple form with FIR lters in the feedforward and feedback lters.
We have here assumed that the denominator in the AR model for the noise plus interference is a full matrix, with the zeros of its determinant inside the unit circle. We can however, of course, restrict the denominator to be a diagonal matrix. This can be advantageous since, as discussed in Section 2.9, a diagonal AR noise model can be easier to estimate. The space-time DFE using a diagonal denominator in the AR-model will be able to perform some coupled space-time suppression of interference, but it will not be able to suppress interferers in the space-time domain as general as the space-time DFE using a full denominator matrix.
If we further restrict all elements of the diagonal denominator matrix to be equal, then the spatio-temporal model of the noise spectrum decouples into a separate spatial model,M0, and a common temporal model, n(q;1), being the common denominator polynomial in the AR model. The resulting DFE can then only dodecoupled space-time suppression of interferers.