such as these are iterative by nature and may actually be viewed as BP on approximate graphs [BC02]. This view is not explicitly considered further in this thesis, but prefiltered sphere detection as described in section 3.3 can be considered a generalization of this idea with interference being subtracted once a sufficient level of confidence in a symbol has been reached.
3.2 Detection with Whitening
The basic idea of modeling weak interferers as Gaussian noise was introduced in connection with the system model in section 2.1 and is as such a straight-forward method of interference rejection. However, more details are required to understand how practical implementations of this scheme can be constructed.
In the generic system model, the whitening filter F was found by Cholesky factorization of the inverse noise covariance Σ−1 limited to have a bandwidth of Nm symbols. However, even for known symbols x, estimating Σ from the signalǫ=y−Hxis no easy task as the sample covariance matrix ǫǫH is rank-one. This problem can be overcome by enforcing a banded Toeplitz structure to the estimate as the true covariance matrix is known to be of this form [Chr07].
However, the complexity of such inherently iterative schemes may be too great and alternative methods operating on a smaller, better conditioned covariance matrix is of interest.
A way of achieving this is to process only a sliding-window of the received signal at a time and let the Forward/Backward Algorithm (FBA) handle interactions between the overlapping sections [Chr05b]. Letǫnn−Nm be the sliding-window at timenfrom the noise signalǫ. The length of the take-out window isNm+ 1 symbols and the window moves in steps of one symbol timeT. AsNmis typically small compared to the number of observations, the covariance ofǫnn−Nm can be reliably estimated as
ΣNm,Eh
ǫnn−Nm ǫnn−NmHi
≃ 1 N
XN n=1
ǫnn−Nm ǫnn−NmH
(3.4) Furthermore, due to the properties of the sliding-window, the resulting covari-ance estimate will be well-behaved having a Toeplitz structure (see e.g. [Chr07]
and references therein). In [Chr05b], the FBA is derived for optimal detection for such a system. The main result of this is that a special whitening filter can be derived from ΣNm such that the ordinary FBA without overlap in the observations can be applied, but due to the memory of the noise the effective channel length grows from L to L+Nm. However, by employing the smaller
covariance matrix of (3.4) in the covariance estimation, the band-constraint is effectively imposed on the covariance matrix itself and not its inverse. Hence, even assuming perfect covariance estimates, this scheme is not equivalent to filtering with the true F, but as Nm becomes ”large” compared to the true bandwidth of the covariance matrix, it will approach the result obtained from filtering with F. This is a result of Σ−N1m approaching the diagonally centered sub-matrices ofΣ−1asΣNmcaptures more and more of the structure inΣ. The
Figure 3.1: Relation betweenΣandΣNm and their respective inverses.
relation between the two covariancesΣ andΣNm and their respective inverses is illustrated by figure3.1with fully drawn lines indicatingΣ and dotted lines ΣNm. As previously mentioned in the section on the generic system model, Σ−1 is only strictly Toeplitz when disregarding the boundary conditions, i.e.
for infinite systems, and the same limitation also applies to the ”sliding” of the inverse covarianceΣ−N1m in the right-hand side of figure 3.1.
To illustrate the potential of the described whitening solution in the presence of interference, simulations of a GSM link is shown in figure3.2. The Carrier-to-Interference Ratio (CIR) is defined as the ratio between the desired signal powerC and the total interference powerI =P
iIi. The left-hand side of the figure shows the BER for a single interferer (K = 2) and the right-hand side shows performance for two interferers (K= 3) having a 10dB power difference.
The used channel model is the so-called TUx multipath model [3GP] defined for GSM and resulting in an overall channel of length L= 7. The simulations are
3.2 Detection with Whitening 29
done using perfect parameter knowledge, i.e. perfect knowledge of channel and covariance. Although classical sampling theory suggests that the oversampling factorNspsdoes not have to be higher than one [Pro95], it may be advantageous to use oversampling in practice due to various challenges such as synchroniza-tion, excess bandwidth in transmit pulseshaping, adjacent channel interference, etc. The difference in performance in figure3.2going fromNsps= 1 toNsps= 2 is due to the additional channel diversity that may be exploited from the excess bandwidth of the transmitted signals. Another point that should be stressed here is the IQ-split of the covariance matrix as introduced in section 2.2. As
−25 −20 −15 −10 −5 0
10−4 10−3 10−2 10−1 100
CIR[dB]
BER
1 Interferer
Nsps=1 Nsps=2 IQ−Whitening IQ−LMMSE BCJR
−15 −10 −5 0 5 10
10−4 10−3 10−2 10−1 100
CIR[dB]
BER
2 Interferers,II12=10dB
Nsps=1 Nsps=2 IQ−Whitening IQ−LMMSE BCJR
Figure 3.2: BER of a GSM link in the presence of interference,Nm= 3,K′= 1, SNR=40dB.
mentioned earlier, this method is capable of greatly increasing the achievable diversity for real-valued modulations such as the GMSK4 modulation used in GSM.
However, the important point here is the difference in performance between dif-ferent detectors for the same setup. The FBA derived for an AWGN channel, also known simply as the BCJR algorithm due to the authors of [BCJR74], is indicated by BCJR in the figure. Due to the AWGN assumption of the de-tector, performance lacks greatly compared to the two detectors incorporating knowledge of the noise covariance, i.e. MMSE and whitening. It should also be noted how whitening consistently outperforms the MMSE detector, but the drawback of whitening is naturally its potentially much larger complexity due to using the FBA on a channel of length ˜L=L+Nm. If the increased complexity is allowed, whitening can potentially outperform any other detector having an equal or smaller set of discrete signals in its model, e.g. the MMSE, as whiten-ing performs optimal inference in the discrete set of signals. Another advantage
4The GMSK modulation used in GSM may be well-approximated by aπ2-BPSK modulation
of this is, that it does so without the need for additional parameter estimates compared with the MMSE, e.g. channel coefficients of interfering signals. This is important in a real-life implementation as reliably estimating channel coeffi-cients of all interfering users is often virtually impossible. The real strength of whitening is therefore the capability of scaling the set of discrete signals in the model all the way from the Gaussian MMSE and to full discrete joint detection.
A natural extension of this is then to further approximate the detection in the discrete set by the use of suboptimal detectors and thus provide an even more flexible detection framework. A possible solution for approximate detection in the discrete set is discussed next.