Consider the joint estimation of the transmitted symbols in time domain A and the collective channel impulse response coefficients h, making up together the parameter vectorθ =
AH hHH
. Then the Fisher Information matrix (FIM) on the basis of the data Y in (4.11) alone is
JY= 1
σv2[HA]H[HA] . (4.19) θ is unidentifiable since indeed for a q × q mixing filter ψ(z), we have H(q)a[m] = (H(q)ψ(q)) (ψ−1(q)a[m]) = H(q)e a[m]. Hence it is impos-e sible to distinguish H(q) fromH(q). If the delay spread ofe H(q) is known and/or there are border conditions on the transmitted signal, then the frequency-selective mixture ψ(z) becomes a frequency-flat ψ. In this case, JY has a null space which is the column space of
AH −hHH
. Indeed [HA]
AH −hHH
= 0. So the multiplicative ambiguity ψ translates into an (additive) singularity in the FIM.
4.9 Simulations 63 In the case of Gaussian white symbols, the prior information on A trans-lates into an additional FIM
JA= 1 σa2
I 0 0 0
(4.20) so at this point the overall FIM is Jθ = JY+JA which will have become non-singular. This would indicate identifiability. The ambiguity in this case is indeed reduced from an unconstrained ψ to a unitary ψ. However, there is still ambiguity and hence unidentifiability. Actually, the proper treatment with Gaussian symbols does not allow presenting the FIM in the compact complex form presented here. In fact, θ needs to be doubled in size by considering separately its real and imaginary components and the associated FIM needs to be considered, in order to see the FIM nullspace corresponding to a unitary ambiguity matrix.
When now furthermore (or alternatively) a Gaussian prior for the chan-nel h is considered, then the FIM forθ gets augmented with
Jh=
0 0 0 (Cho)−1
(4.21) which will again render the overall FIM Jθ = JY+Jh(+JA) nonsingular.
So it would seem that the addition of prior information with an identical non-zero Power Delay Profile (PDP) for each of the antenna pair channels (corresponding to a nonsingular diagonal Cho) renders Jθ nonsingular and hence leads to (channel) identifiability. However this is not necessarily the case. In the case of Gaussian white symbols, and a unitary ambiguity matrix ψ, if Cho is such that (ψ⊗Ip(L+1))HCho(ψ⊗Ip(L+1)) = Cho (in which case the channel prior is insensitive to a unitary mixture), then still the Bayesian blind problem remains unidentifiable. The above condition occurs if Cho is of the form Cho =Iq⊗C for any square matrixC of sizep(L+1). Hence the regularization of the blind channel estimation problem via prior information is a tricky issue due to the multiplicative nature of the ambiguity.
4.9 Simulations
We simulate in this section both Bayesian and Variational Bayesian Blind (VBB) channel estimation techniques based on (4.12), (4.13) and (4.16), (4.18) appearing above. Moreover we simulate also a version of the Vari-ational Bayesian approach where the channel parameters are deterministic unknowns, treated as random with no prior information, so Cho = ∞I in
(4.18). We shall refer to this approach as Uniformed VBB (UVBB). In each MonteCarlo simulation we generate a Rayleigh fading channel with expo-nentially decaying power delay profile (PDP) for the channel between each transmitting and receiving antenna pair as follows: e−wn where n = 0 : L andw= 2 normally. Hence,Cho is the diagonal matrixCho =Iq⊗C⊗Ipwhere C = diag{e−wn, n = 0 : L}. As for the symbols, we generate i.i.d. Gaus-sian symbols (which are hence i.i.d. GausGaus-sian both in time and frequency domain). The performance of the different Bayesian channel estimators is evaluated by means of the Normalized MSE (NMSE) vs. SNR. The per receive antenna SNR is SNR = σa2tr{Cho}
p σ2v .
The NMSE is defined asavgavg||h||−hh||bb2||2 wherehbb=bhψis the channel estimate adjusted for blind channel estimation ambiguities. As we assume the channel length known here,ψrepresents an instantaneous mixing matrix of size (q× q). The mixing matrixψcan be obtained by minimizing the Frobenius norm of the following matrix error: min
ψ ||h′−hb′||2F whereh′= (hH[0]· · ·hH[L])H and bh′ = (bhH[0]· · ·hbH[L])H. For an unconstrained mixture, we get ψ = (hb′Hhb′)−1hb′Hh′ =UΣVH where the last expression represents the SVD of the resultingψ. In the case thatψ gets constrained to be a unitary matrix, the solution isψ =U VH, see [5].
Both (B and VBB) algorithms are initialized by using (for mh
n) noisy perturbations of the true channels. In the first iteration of (4.16) we use Rh
n =mh
nmH
hn, henceCh
n = 0. In Figure 4.1 we can notice how close the performance of both the Variational Bayesian and the Bayesian algorithms is since both fully exploit the prior information that exists about the channel and the symbols. However, we can notice that the UVBB method (with + marker, also called ”Deterministic” in the legend) lags behind the normal Variational Bayesian (with * marker) where the prior information is taken into consideration. This is an expected result since the more information we exploit the better performance we get. However, at higher SNR the performance of the deterministic blind algorithm converges to that of the Bayesian blind algorithms. Also this result is expected since at very high SNR the contribution of prior information becomes negligible.
Whereas Fig. 4.1 uses a unitary ψ, Fig. 4.2 uses an unconstrained ψ, which leads to reduced NMSE since more prior information is exploited.
At least, Fig. 4.1 shows that the exploitation of the white character of the symbols as we do here leads to a reduced unidentifiability of ψ to just a unitary mixing matrix.
In Fig. 4.3 the OFDM block lengthN gets increased from 20 (as in the
4.9 Simulations 65
0 5 10 15 20 25 30 35 40 45 50
−40
−30
−20
−10 0 10 20
SNR
NMSE
L = 5, q = 2, N = 20, MonteCarlo = 586, p = 3
Bayesian Variational Bayesian Deterministic
Figure 4.1: NMSE vs. SNR for B, VBB, UVBB algorithms, for N = 20, unitary ψ.
0 5 10 15 20 25 30 35 40 45 50
−35
−30
−25
−20
−15
−10
−5 0
SNR (dB)
NMSE (dB)
L = 5, q = 2, N = 20, MonteCarlo = 100, p = 3
Bayesian Variational Bayesian Deterministic
Figure 4.2: NMSE vs. SNR for B, VBB, UVBB algorithms, for N = 20, unconstrained ψ.
0 5 10 15 20 25 30 35 40 45 50
−45
−40
−35
−30
−25
−20
−15
−10
−5 0
SNR
NMSE
L = 5, q = 2, N = 100, MonteCarlo = 100, p = 3
Bayesian Variational Bayesian Deterministic
Figure 4.3: NMSE vs. SNR for B, VBB, UVBB algorithms, forN = 100, unconstrainedψ.
0 5 10 15 20 25 30 35 40 45 50
−40
−35
−30
−25
−20
−15
−10
−5 0
SNR
NMSE
L = 5, q = 2, N = 20, MonteCarlo = 100, p = 3
Bayesian Variational Bayesian Deterministic
Figure 4.4: NMSE vs. SNR for B, VBB, UVBB algorithms, for N = 20, unconstrainedψ,w= 0.5 in PDP.
4.9 Simulations 67 previous two figures) to 100. The result is that the prior information intro-duced by a Bayesian approach only helps at low SNR, as could be expected.
The other noticeable effect is that the Variational approach outperforms the non-Variational version over a wide SNR range.
In Fig. 4.4 finally, an exponential PDP with much shorter time constant (w= 0.5) is used, as compared to w= 2 in the previous three figures. The result is that the prior information only helps at very high SNR.
Part II
SIMO FIR Systems
69
Chapter 5
Bayesian Blind FIR Channel Estimation Algorithms
Blind channel estimation techniques were developed and usually evaluated for a given channel realization, i.e. with a deterministic channel model. On the other hand, in wireless communications the channel is typically mod-eled as Rayleigh fading, i.e. with a Gaussian (prior) distribution expressing variances of and correlations between channel coefficients. In this chapter, we explore a Bayesian approach to blind channel estimation, exploiting a priori information on fading channels. We mainly focus on joint ML/MAP estimation of channels and symbols on one hand, and on ML/MAP esti-mation of channels with elimination of symbols on the other hand. As a consequence, a unified framework in addition to three new Bayesian estima-tors are introduced where their performance is compared by simulations to three existing non-Bayesian estimators. In the same context, we provide an insightful discussion of the accurate way of deriving the Bayesian Cramer Rao bound (BCRB) with an emphasis on its singularity.