The multi-dimensional matched lter,w(q;q;1), and the metric polynomial, (q;q;1), can be tuned in a few dierent ways.
4.4.1 Direct MMSE Tuning
We assume here that a short training sequence of the transmitted signal d(t) is known and we want to tune the multi-dimensional matched lter,
w(q;q;1), and the metric polynomial(q;q;1) using this training sequence and the corresponding received signal samplesy(t).
In [71] and [121], generalizations of the direct tuning approach developed in [116] are presented. The coecients of a feedforward lter, w(q;q;1),
4.4. Tuning the Multi-Dimensional Matched Filter 171
d(t)
y(t) ..........................-
-
- -
6 w(q;q;1)
(q;q;1)
+ P
;
e(t)
Figure 4.2: MMF lter tuning.
and the coecients of a non-causal feedback lter, (q;q;1), are tuned to minimize the mean square error (MSE) of the error signal,
e(t) =w(q;q;1)y(t);(q;q;1)d(t): (4.24) See Figure 4.2. The polynomial row vector
w(q;q;1) = [w1(q;q;1) ::: wM(q;q;1)] (4.25) is a MISO FIR lter and(q;q;1) is a double sided, complex conjugate sym- metric3, non-causal FIR-lter with the middle coecient,0, constrained to be equal to one. That is,
(q;q;1) =;nq+n + ::: + 1 + ::: +nq;n (4.26) with ;k =Hk. By this minimization, noise whitening and matched lter- ing will be performed by w(q;q;1), while(q;q;1) will contain the overall impulse response [121].
We can from this see that the multidimensionalmatched lter, as all matched lters, maximizes the peak-to-noise ration (PNR), i.e. it maximizes the peak in the impulse response of channel for the desired signal over the noise vari- ance. In other words, it maximizes
0
E[e(t)eH(t)] : (4.27)
3If one so desires one can relax the requirement that the metric polynomial should be complex conjugate symmetric.
This is equivalent to minimizing the variance in the error signale(t) E[e(t)eH(t)] (4.28) while constraining0 to one (or any other value >0).
It is natural to choose the structure of the feedforward lter, w(q;q;1), consistent with an ideal MMF with a truncated noise plus interference spec- trum operator, bH(q) ~R;nn1(q;q;1). The spectrum operator, ~R;nn1(q;q;1), here represents a truncated version ofR;nn1(q;q;1). The lterw(q;q;1) will thus be non-causal or anti-causal, sincebH(q) is anti-causal and ~R;nn1(q;q;1) is either a matrix constant or a double-sided polynomial matrix. It is also natural to choose the number of coecients in(q;q;1) consistent with the structure chosen forw(q;q;1).
The estimates ^w(q;q;1) and ^(q;q;1) can be found either adaptively or by solving a system of equations formed directly from the training data.
Convergence to the ideal solution will result in an error signal which is white, with minimal variance.
When the true lter orders are used and the training sequence is long enough, the MMF will be given by the estimate ^w(q;q;1), up to a multiplicative constant, and the corresponding metric to be used in the Viterbi algorithm will be given by the ^(q;q;1), also up to a multiplicative constant [121].
A problem arises if the available training sequence is short. If for instance the number of training symbols is smaller than the number of coecients in the lters, then the coecients cannot be determined uniquely. A regularization of the equations can then be introduced. By adding articial noise into the system of equations, a solution can be computed, but it will in general be inferior to the true matched lter. However, if the number of antennas is small, say M = 2, then a short training sequence can suce to properly tune the MLSE with a direct method.
By adjusting the number of coecients inw(q;q;1) and(q;q;1), the ability to combat a temporally colored interference can be obtained. Adding more coecients increases the temporal noise whitening ability as well as the matched ltering capability of the lter at the price of more degrees of freedom and a longer memory in the metric.
4.4. Tuning the Multi-Dimensional Matched Filter 173
4.4.2 Indirect MMF Tuning
In the indirect approach the channel,b(q;1), is rst estimated. This can be done with one of the methods described in Chapter 2. An advantage with this method, compared to the direct method, is that we can take advantage of a prioriknowledge when estimating the channel.
The noise plus interference spectrum operator, Rnn(q;q;1), can, for exam- ple, be estimated using the residuals from the identication procedure. If we have a relatively large number of antennas and the amount of training data is small, we may choose to only estimate the spatial spectrum of the noise plus interference, i.e. the coecient matrix for lag zero in Rnn(q;q;1). A good option is to model the noise plus interference as an AR process and use this in the spectrum estimation [7]. Modeling the noise plus interference spectrum as an AR process helps to catch some useful temporal aspects of the noise plus spectrum and it ts well into the MLSE algorithm. The re- sulting MMF lter and the metric will then be FIR lters with nite lengths as in (4.20) and in (4.21). As the number of antennas or the order of the AR noise model is increased, the number of parameters in the model can however become large compared to the number of available equations. This can make them potentially dicult to estimate accurately, especially if the SNR is not high enough.
If the spectrum operator is invertible, estimates of the MMF,w(q;q;1), and of the metric polynomial,(q;q;1), can then be formed as
^
w(q;q;1) = ^bH(q) ^R;nn1(q;q;1) (4.29)
^(q;q;1) = ^bH(q) ^R;nn1(q;q;1)^b(q;1) (4.30) where the \hat" marks quantities derived from the estimated channel, ^b(q;1), or the estimated noise spectrum operator, ^Rnn(q;q;1). When using an AR model for the noise the MMF-lter and the metric polynomial take on the simple forms in (4.20) and in (4.21). If the lter ^R;nn1(q;q;1) is a double sided IIR lter, it will have to be truncated. Otherwise the metric in the Viterbi would have innite memory.
If we use joint FIR channel and AR noise model estimation as described in Section 2.9, we can from the least squares estimates, ^N(q;1) and ^bN(q;1), of N(q;1) andbN(q;1) in (2.121) and the estimate ^Rr^r^ from (2.123), form the multidimensional matched lter ^w(q;q;1) and the metric polynomial
^(q;q;1) as
^
w(q;q;1) = ^bHN(q) ^R;^r^r1N^(q;1) (4.31) and the metric polynomial
^(q;q;1) = ^bHN(q) ^R;^r^r1^bN(q;1) (4.32) An interesting question to study, is if the indirect methods can handle a case with very low signal-to-interference ratio (SIR). It could be suggested, that very poor SIR would make estimation of the channels to the individual antenna elements non-feasible. Although the quality of the estimated chan- nels may be compromised, the simulations for the scenario presented here do not show that the indirect methods suer much from this. This issue will be discussed further in Chapter 7.
4.4.3 Indirect MMSE Tuning
An alternative indirect way of tuning the multi-dimensional matched lter is to perform the minimization of the MSE of the error signale(t) in (4.24), but instead of forming the systems of equations directly from data, we form them from the estimated channel, ^b(q;1), and the estimated noise spectrum operator, ^Rnn(q;q;1). The number of matrix coecients of ^Rnn(q;q;1) used, and the structure and length of the lters w(q;q;1) and (q;q;1), aect the temporal noise whitening and matched ltering capabilities. By constraining the lter structures, the memory length in the Viterbi algorithm can be controlled. This will aect the complexity of the Viterbi algorithm.
This indirect version of the MMSE tuning will have an advantage over the direct version. If we cannot obtain a good estimate of the space-time covari- ance matrix of the noise plus interference, we may here restrict the algorithm to use only the estimate of the spatial covariance of the noise plus interfer- ence or for example an estimated AR spectrum. This can result in better performance.
If the same structure is used and the same estimate of the channels and the noise plus interference spectrum is used, then this indirect method and the ordinary indirect MMF tuning discussed in Section 4.4.2 are equivalent.