coef-ficients. It is worthy to note that for SNR less than 20 dB the algorithm always converges typically after three or four iterations while at higher SNR the convergence is guaranteed at no more than ten iterations. However, we consider that convergence is achieved when the following condition is ful-filled: σb2w,i−bσw,i2 −1
σb2w,i ≤0.1 where i denotes the number of the current iteration at which the convergence is checked. Figure 3.1 shows the performance of both SRM and NSF with and without structuring where three antennas have been utilized at the receiver and the sampled spectrum has been com-puted from just one OFDM symbol. We remark that SRM yields better performance than NSF. This is due to the fact that when we work with one OFDM symbol then SRM is a weighted version of NSF with the weight be-ing the largest eigenvalue of the sampled spectrum at each tone. However, when structuring is used, both estimators show at least 3 dB gain even at very high SNR. It is also obvious that after structuring the performance of both estimators is congruent whatever the SNR is. To elaborate more the advantage of our structuring algorithm we plot in Figure 3.2 the BER ver-sus SNR where we have used the estimated channels by various algorithms to equalize the received signal using MMSE equalizer and a hard decision decoding to extract the received bits. This result shows that our algorithm outperforms the non-structured ones by more than 2 dB at BER = 10−2.
3.7 Conclusions
To sum up, we have shown in this chapter the capability to exploit the classi-cal blind deterministic channel estimators with a great computational power saving within the cyclic prefix systems. This is accomplished by minimizing the sum of the cost functions at different tones instead of minimizing the ordinary cost function in the time domain. Moreover, we propose a spatio-temporal based algorithm to enhance the sample covariance matrix upon which a class of well-known estimators rely. The enhancement is achieved by enforcing both the rank and the FIR structure properties. The numerical simulations show that the proposed algorithm has the potential to provide a 5 dB gain (in terms of NMSE) at low to moderate SNR while it still has the capability to provide a noticeable gain at high SNR.
0 5 10 15 20 25 30
−45
−40
−35
−30
−25
−20
−15
−10
−5 0
SNR (dB)
NMSE (dB)
M = 1, L = 5, N = 128, MonteCarlo = 10000, m = 3
SRM SRM−Struct NSF NSF−Struct
Figure 3.1: The NMSE versus SNR for structured and non-structured esti-mators.
0 5 10 15 20
10−6 10−5 10−4 10−3 10−2 10−1 100
SNR (dB)
SER
M = 1, L = 5, N = 128, MonteCarlo = 10000, m = 3
SRM SRM−struct NSF NSF−struct
Figure 3.2: The BER versus SNR for structured and non-structured estima-tors.
Chapter 4
Variational Bayesian Blind and Semi-blind Channel Estimation
Blind and semi-blind channel estimation is a topic that enjoyed explosive de-velopments throughout the nineties, and then came to a standstill, probably because of perceived unsatisfactory performance. Blind channel estimation techniques were developed and usually evaluated for a given channel realiza-tion, i.e. with a deterministic channel model. Such blind channel estimates, especially those based on subspaces in the data, are often only partial and ill-conditioned. On the other hand, in wireless communications the channel is typically modeled as Rayleigh fading, i.e. with a Gaussian (prior) distri-bution expressing variances of and correlations between channel coefficients.
In recent years, such prior information on the channel has started to get exploited in pilot-based channel estimation, since often the pure pilot-based (deterministic) channel estimate is of limited quality due to limited pilots.
In this chapter we explore a Bayesian approach to (semi-)blind channel es-timation, exploiting a priori information on fading channels. In the case of deterministic unknown input symbols, it suffices to augment the classi-cal blind (quadratic) channel criterion with a quadratic criterion reflecting the Rayleigh fading prior. In the case of a Gaussian symbol model the blind criterion is more involved. The joint ML/MAP estimation of
chan-55
nels, deterministic unknown symbols, and channel profile parameters can be conveniently carried out using Variational Bayesian techniques. Variational Bayesian techniques correspond to alternating maximization of a likelihood w.r.t. subsets of parameters, but taking into account the estimation errors on the other parameters. To simplify exposition, we elaborate the details for the case of MIMO OFDM systems.
4.1 Introduction
Blind and semi-blind channel estimation techniques have been developed and are usually evaluated for a given channel realization, i.e. with a deter-ministic channel model, see [18] for an overview of such techniques. Such blind channel estimates, especially those based on subspaces in the data, are often only partial and ill-conditioned. Indeed, only part of the channel is blindly identifiable, especially in the case of MIMO channels. The type of blind channel estimation techniques we are mostly referring to here involve an FIR multichannel and are typically based on the second-order statistics of the received signal. Two types of techniques can be considered, treating the unknown input symbols as either deterministic unknowns or Gaussian white noise. In the first case, the techniques are often based on the subspace structure induced in the data by the multichannel aspect. The part of the channel that can be identified blindly is larger in the Gaussian input model case than in the deterministic input model case, but is in any case incom-plete. Many of the deterministic input approaches are also quite sensitive to a number of hypotheses such as correct channel length (filter order) and no channel zeros. In general this means that these blind channel estimates can often become ill-conditioned, when the channel impulse response is tapered (e.g. due to a pulse shape filter) or when the channel is close to having zeros. In fact this means that the blind information on the channel can be substantial, but is limited to only part of the channel.
An overview of blind channel estimation techniques can be found in [18]
for SIMO systems and in [5] for MIMO systems. Specific blind channel estimation techniques for Cyclic Prefix systems, as will be considered here, were introduced in [39], see also [29]. The concept of Bayesian blind channel estimation was introduced in [56], with in particular some considerations on identifiability issues where-as in this chapter we focus on algorithms.