Performance and Complexity Analysis of Blind FIR Channel Identification Algorithms Based on Deterministic Maximum Likelihood in SIMO Systems

Abstract

We analyze two algorithms that have been introduced previously for Deterministic Maximum Likelihood (DML) blind estimation of multiple FIR channels. The first one is a modification of the Iterative Quadratic ML (IQML) algorithm. IQML gives biased estimates of the channel and performs poorly at low SNR due to noise induced bias. The IQML cost function can be “denoised” by eliminating the noise contribution: the resulting algorithm, Denoised IQML (DIQML), gives consistent estimates and outperforms IQML. Furthermore, DIQML is asymptotically globally convergent and hence insensitive to the initialization. Its asymptotic performance does not reach the DML performance though. The second strategy, called Pseudo-Quadratic ML (PQML), is naturally denoised. The denoising in PQML is furthermore more efficient than in DIQML: PQML yields the same asymptotic performance as DML, as opposed to DIQML, but requires a consistent initialization. We furthermore compare DIQML and PQML to the strategy of alternating minimization w.r.t. symbols and channel for solving DML (AQML). An asymptotic performance analysis, a complexity evaluation and simulation results are also presented. The proposed DIQML and PQML algorithms can immediately be applied also to other subspace problems such as frequency estimation of sinusoids in noise or direction of arrival estimation with uniform linear arrays.

Appendix A: Asymptotic Performance Study of DIQML and PQML

1.1 A.1 Asymptotic behavior of PQML (M→∞)

We prove here that PQML needs a consistent initialization in order to give a consistent estimate of the channel.

1.1.1 A.1.1 Inconsistent Initialization

The element (i,j) of the “Hessian” ${\mathcal{P}}(h)$ of the PQML cost function with λ=1 (introducing the generalized eigenvalue does not change the following arguments much, see the next subsection also) can be written as

(40)

Recall that $\mathrm{E} \boldsymbol{Y}\boldsymbol{Y}^{H} = \boldsymbol {X}\boldsymbol{X}^{H} + {\sigma_{v}^{2}}I = {\mathcal{T}}(h^{o})AA^{H}{\mathcal{T}}^{H}(h^{o}) +{\sigma_{v}^{2}}I$. Asymptotically both terms ${\mathcal{P}}_{1}(h)$ and ${\mathcal{P}}_{2}(h)$ differ from their expected value by ${\mathcal{O}}_{p}(\frac{1}{\sqrt{M}})$, and

(41)

(42)

Let $\mathrm{E} {\mathcal{P}}_{i}(h) = \mathrm{E} {\mathcal{P}}_{i1}(h) + \mathrm{E} {\mathcal{P}}_{i2}(h) , i=1,2$, be a decomposition in signal and noise terms. Note that $\mathrm{E} {\mathcal{P}}_{12}(h) = \mathrm{E} {\mathcal{P}}_{22}(h)$ so that we have cancellation of the noise terms in $\mathrm{E} {\mathcal{P}}(h)$. For h≠αh ^o, for any α∈ℂ, $\mathrm{E}{\mathcal{P}}(h) \neq \mathrm{E}{\mathcal{P}}_{11}(h)$ (i.e. the noise-free IQML Hessian) because of $\mathrm{E}{\mathcal{P}}_{21}(h)$, the signal contribution in $\mathrm{E}{\mathcal{P}}_{2}(h)$. So, since $\mathrm{E}{\mathcal{P}}_{21}(h) h^{o} = {\mathcal{O}}(1)$ if $h-\alpha h^{o}={\mathcal{O}}(1)$ for any α∈ℂ, an iteration of PQML yields asymptotically an inconsistent estimate for an inconsistent initialization.

1.1.2 A.1.2 Consistent Initialization

Assume h is a consistent estimate of h ^o, i.e. h=h ^o+Δh, where typically $\Delta h = {\mathcal{O}}_{p}(\frac{1}{\sqrt{M}})$. We get

(43)

whereas the other term in $\mathrm{E}{\mathcal{P}}(h)$, $\mathrm {E}{\mathcal{P}}_{11}(h)$, can be verified to be of order 1. So ${\mathcal{P}}_{21}(h)$ is asymptotically negligible: with a consistent initialization, the role of ${\mathcal{P}}_{2}$ is to remove the noise contribution in ${\mathcal{P}}_{1}$. Apart from terms in ${\mathcal{O}}_{p}(\frac{1}{\sqrt{M}})$, ${\mathcal{P}}$ becomes asymptotically equivalent to the noise-free IQML Hessian, so the estimation of h is consistent. So an iteration of PQML yields asymptotically a consistent estimate for a consistent initialization.

1.2 A.2 Performance of DIQML and PQML

We consider the following general generalized eigenvalue problem for blind channel estimation:

$$ \min_{h,\lambda} h^H \bigl\{ \widehat {F}\bigl(\boldsymbol{Y}, h^c\bigr) - \lambda \widehat{G} \bigl(\boldsymbol{Y},h^c \bigr) \bigr\} h $$

(44)

subject to $\widehat{F}(\boldsymbol{Y}, h^{c}) - \lambda \widehat {G}(\boldsymbol{Y},h^{c})\geq0$ and constraints on h. h ^c is a consistent estimate of h. $\widehat{F}(\boldsymbol{Y},h^{c})=\frac{1}{M}{\mathcal{Y}}^{H} {\mathcal{R}}^{+}(h^{c}) {\mathcal{Y}}={\mathcal{P}}_{1}(h^{c})$ for DIQML and PQML, $\widehat{G}(\boldsymbol{Y},h^{c})=\frac{1}{M}{\mathcal{D}}(h^{c})$ for DIQML and $\widehat{G}(\boldsymbol{Y},h^{c})=\frac{1}{M}{\mathcal{B}}^{H}(h^{c}) {\mathcal{B}}(h^{c})={\mathcal{P}}_{2}(h^{c})$ for PQML. It can be shown that the channel estimation performance given by (44) is asymptotically unchanged when one replaces $\widehat{F}(\boldsymbol {Y}, h^{c})$ and $\widehat{G}(\boldsymbol{Y},h^{c})$ by $\widehat{F}(\boldsymbol{Y})=\widehat{F}(\boldsymbol{Y}, h^{o})$ and $\widehat{G}(\boldsymbol{Y})=\widehat{G}(\boldsymbol{Y},h^{o})$, respectively (since ${\mathcal{O}}(\|\Delta h^{c}\|^{2}) = {\mathcal{O}}_{p}(\frac{1}{M})$). Asymptotically, we also have

$$ \left \{ \everymath{\displaystyle} \begin{array}{l} \widehat{F}(\boldsymbol{Y}) = F^o + {\mathcal{O}}_p\biggl(\frac {1}{\sqrt{M}}\biggr), \\ [12pt] \widehat{G}(\boldsymbol{Y}) = G^o + {\mathcal{O}}_p\biggl(\frac {1}{\sqrt{M}}\biggr), \end{array} \right . $$

(45)

where $F^{o}(h^{c}) = \mathrm{E} \widehat{F}(\boldsymbol{Y},h^{c})$, $G^{o}(h^{c}) = \mathrm{E} \widehat{G}(\boldsymbol{Y},h^{c})$ and F ^o=F ^o(h ^o), G ^o=G ^o(h ^o). Although we will not need this, one may also remark that $F^{o}(h^{o}) = \lim_{M\rightarrow\infty}F^{o}(h^{o}) + {\mathcal{O}}(\frac{1}{M})$ and similarly for G ^o.

1.2.1 A.2.3 Asymptotic Expression for Δλ

The solution of (44) for λ and h is the minimal generalized eigenvalue and corresponding eigenvector of $\widehat{F}(\boldsymbol{Y})$ and $\widehat{G}(\boldsymbol{Y})$.

$$ \widehat{F}(\boldsymbol{Y}) \hat{{h}}- \hat{\lambda} \widehat {G}( \boldsymbol{Y}) \hat{{h}}= 0 \quad \Rightarrow\quad \hat{\lambda}= \frac{\hat{{h}}^H \widehat{F}(\boldsymbol{Y}) \hat{{h}}}{\hat{{h}}^H \widehat{G} (\boldsymbol{Y}) \hat{{h}}} . $$

(46)

We denote $\hat{{h}}=h^{o}+\Delta h$, and $\hat{\lambda}=\lambda^{o} +\Delta\lambda$, where $\Delta h \stackrel{M \rightarrow\infty}{\longrightarrow} 0$, $\Delta\lambda\stackrel{M \rightarrow\infty}{\longrightarrow} 0$. We have $\lambda^{o}= \frac{h^{o H} F^{o} h^{o}}{h^{o H} G^{o} h^{o}}$ and, performing a series expansion, we get

$$ \Delta\lambda= \frac{h^{o H} [\widehat{F}(\boldsymbol{Y})-\lambda^o\widehat {G}(\boldsymbol{Y})] h^{o}}{h^{o H} G^o h^{o}} + {\mathcal{O}}_p\biggl( \frac{1}{M}\biggr) . $$

(47)

1.2.2 A.2.4 Asymptotic Expressions for Δh and $C_{\Delta h\Delta h}=\mathrm{E}(\hat{h}-h^{o}) (\hat{h}-h^{o})^{H}$

After substitution of the solution for λ, the estimation problem for h becomes

$$ \min_{h} \bigl\{ h^H \bigl\{ \widehat{F}(\boldsymbol{Y}) - \hat{\lambda }(\boldsymbol{Y})\widehat{G}(\boldsymbol{Y}) \bigr\} h = {\mathcal{F}} (h) \bigr\} . $$

(48)

The estimation of h is performed under constraints ${\mathcal{K}}(h_{R})=0$ with tangent subspace ${\mathcal{M}}_{h_{R}^{o}}$ at $h_{R}=h_{R}^{o}$. Let ${\mathcal{V}}_{R}^{o}$ be a matrix whose columns form an orthonormal basis of ${\mathcal{M}}_{h_{R}^{o}}$. Then locally we can write $\Delta h_{R} = {\mathcal{V}}_{R}^{o} \theta$ where θ are the unconstrained parameter variations. A Taylor series expansion of ${\mathcal{F}}(h)$ at h ^o in terms of θ gives

(49)

Optimization of (49) up to second order w.r.t. θ gives for $\Delta h_{R} = {\mathcal{V}}_{R}^{o} \theta$

$$ \Delta h_R = {\mathcal{V}}_R^o \biggl({ \mathcal{V}}_R^{o T} \frac{\partial^2 {\mathcal{F}}(h^o)}{\partial h_R\partial h_R^T} {\mathcal{V}}_R^o \biggr)^{-1}{\mathcal{V}}_R^{o T} \frac{\partial {\mathcal{F}}(h^o)}{\partial h_R} $$

(50)

assuming that the matrix inverse exists (which will be the case here). The expression becomes easier to work with when expressed in terms of complex quantities (see [12]):

$$ \Delta h = {\mathcal{V}}^o \biggl({\mathcal{V}}^{o H} \frac{\partial}{\partial h^*} \biggl(\frac{\partial{\mathcal{F}}(h^o)}{\partial h^*} \biggr)^H {\mathcal{V}}^o \biggr)^{-1}{\mathcal{V}}^{o H} \frac{\partial {\mathcal{F}}(h^o)}{\partial h^*} . $$

(51)

For the constraints (5), (6) or equivalent, the columns of ${\mathcal{V}}^{o}$ form a basis for the orthogonal complement of h ^o. We shall also require

$$ \left \{ \everymath{\displaystyle} \begin{array}{l} J^{(1)}_{hh} = \mathrm{E} \biggl(\frac{\partial{\mathcal{F}}(h^o)}{\partial h^* } \biggr) \biggl(\frac{\partial{\mathcal{F}}(h^o)}{\partial h^* } \biggr)^H ,\\ [12pt] J^{(2)}_{hh} = \mathrm{E} \frac{\partial}{\partial h^*} \biggl(\frac{\partial{\mathcal{F}}(h^o)}{\partial h^*} \biggr)^H . \end{array} \right . $$

(52)

Note that if ${\mathcal{F}}$ would have been the log likelihood function, then $J^{(1)}_{hh}=-J^{(2)}_{hh}$, but this equality does not hold here. We now obtain

(53)

For the quadratic problem in (48), we have (using (47) and the fact that ΔF and ΔG have zero mean):

$$ J^{(2)}_{hh} = \mathrm{E} \bigl(\widehat{F}(\boldsymbol{Y}) - \hat{\lambda}(\boldsymbol{Y}) \widehat{G}(\boldsymbol{Y} ) \bigr)= F^o - \lambda^o G^o +{\mathcal{O}}\biggl( \frac{1}{M}\biggr), $$

(54)

where we shall neglect the last term.

1.2.3 A.2.5 Application to DIQML and PQML

Specializing to DIQML and PQML, we get first of all $F^{o} - \lambda^{o} G^{o} = \frac{1}{M} {\mathcal{X}}^{H} {\mathcal{R}}^{+} {\mathcal{X}}$. To show the relation of this expression to the CRB, consider for any h, h′:

(55)

where ${\mathcal{T}}(h)A = {\mathcal{A}}h$. Hence $\sigma_{v}^{-2} {\mathcal{X}}^{H} {\mathcal{R}}^{+} {\mathcal{X}}= \sigma_{v}^{-2} {\mathcal{A}}^{H} P^{\perp}_{{\mathcal{T}}(h^{o})}{\mathcal{A}}$, which is the Fisher information matrix for deterministic ML. As F ^o−λ ^o G ^o admits h ^o as unique eigenvector corresponding to the eigenvalue zero, and ${\mathcal{V}}^{o}$ spans the orthogonal complement of h ^o,

$$ {\mathcal{V}}^o \bigl({{\mathcal{V}}^o}^H J^{(2)}_{hh} {\mathcal{V}}^o \bigr)^{-1}{ \mathcal{V}}^{o H} = \bigl(F^o - \lambda^o G^o \bigr)^{+} , $$

(56)

the Moore–Penrose pseudo-inverse of F ^o−λ ^o G ^o. Hence

$$ \Delta h = \bigl(F^o - \lambda^o G^o \bigr)^{+} \frac{\partial{\mathcal{F}}(h^o)}{\partial h^*} $$

(57)

neglecting ${\mathcal{O}}_{p}(\frac{1}{M})$ terms. Now, using (47), we also get

(58)

which leads to

$$ \Delta h = M \bigl({\mathcal{X}}^H {\mathcal{R}}^+ {\mathcal{X}} \bigr)^+ \biggl[I - \frac{G^o h^o h^{o H}}{h^{o H} G^o h^o} \biggr] \bigl(\widehat{F}- \lambda^o \widehat{G} \bigr)h^o $$

(59)

neglecting ${\mathcal{O}}_{p}(\frac{1}{M})$ terms. For DML, the same kind of analysis gives [12]:

$$ \Delta h_{\mathrm{DML}} = M \bigl({\mathcal{X}}^H {\mathcal{R}}^+ { \mathcal{X}} \bigr)^+ \bigl(\widehat{F}- \lambda^o \widehat{G} \bigr)h^o, $$

(60)

where $\widehat{F}(\boldsymbol{Y})$ and $\widehat{G}(\boldsymbol {Y})$ are the same as in the PQML case. So the estimate $\hat{{h}}$ given by DIQML and PQML is different from the DML estimate (though the difference with PQML is only ${\mathcal{O}}_{p}(\frac{1}{M})$). From (59), we see that the channel estimation performance depends on the matrix

$$ {\mathcal{W}}= \mathrm{E} \bigl\{ \bigl(\widehat{F}-\lambda^o \widehat{G} \bigr)h^o h^{o^H} \bigl(\widehat{F}- \lambda^o \widehat{G} \bigr)^H \bigr\} . $$

(61)

Recall that for both DIQML and PQML, $\widehat{F}(\boldsymbol{Y}) = \frac{1}{M}{\mathcal{Y}}^{H} {\mathcal{R}}^{+}(h^{o}) {\mathcal{Y}}$, $F^{o} = \frac{1}{M}{\mathcal{X}}^{H} {\mathcal{R}}^{+}(h^{o}) {\mathcal{X}}+\frac{\sigma_{v}^{2}}{M}{\mathcal{D}}(h^{o})$.

Performance of DIQML

For DIQML, $\widehat{G}(\boldsymbol{Y}) = \frac{1}{M}{\mathcal{D}}(h^{o}) = G^{o}$, $\lambda^{o} = \sigma_{v}^{2}$ and hence $F^{o}-\lambda^{o} G^{o} = \frac{1}{M}{\mathcal{X}}^{H} {\mathcal{R}}^{+}(h^{o}){\mathcal{X}}$. We have

$$ {\mathcal{W}}^{\mathrm{{DIQML}}} = \frac{1}{M^2} \bigl[ {\sigma_v^2} {\mathcal{X}}^H {\mathcal{R}}^+ {\mathcal{X}}+ \sigma_v^4 {\mathcal{D}} \bigr], $$

(62)

which leads to (36).

Performance of PQML

Now $\widehat{G}(\boldsymbol{Y}) = \frac{1}{M}{\mathcal{B}}^{H}(h^{o}){\mathcal{B}}(h^{o})$, $G^{o}=\frac{\sigma_{v}^{2}}{M}{\mathcal{D}}(h^{o})$, λ ^o=1 and $F^{o}-\lambda^{o} G^{o} = \frac{1}{M}{\mathcal{X}}^{H} {\mathcal{R}}^{+}(h^{o}){\mathcal{X}}$. We get

(63)

where ${\mathcal{D}}{'}$ is defined below (38). Note that ${\mathcal{D}}{'}h^{o} = {\mathcal{D}} h^{o}$ and for any h′, $h{' H} {\mathcal{W}}^{\mathrm{{PQML}}}h^{o} = \frac{{\sigma_{v}^{2}}}{M^{2}} h{' H} {\mathcal{X}}^{H} {\mathcal{R}}^{+} {\mathcal{X}} h^{o} + \frac{\sigma_{v}^{4}}{M^{2}} \operatorname{tr} \{ {\mathcal{T}}^{H}(h{' \perp}) {\mathcal{R}}^{+} {\mathcal{T}}(h^{o \perp}) P_{{\mathcal{T}}(h^{o})} \} = 0 + 0 = 0$: ${\mathcal{W}}^{\mathrm{{PQML}}}$ has a null space spanned by h ^o. Now, for any h, h′, we have

(64)

or hence ${\mathcal{D}}{''}={\mathcal{D}}-{\mathcal{D}}{'}$, where ${\mathcal{D}}{''}$ is defined below (38) and we used the commutativity of convolution, leading to ${\mathcal{T}}(h^{\perp}){\mathcal{T}}(h^{o})={\mathcal{T}}(h^{o \perp }){\mathcal{T}}(h)$. Equations (37), (38) now follow. Note that the factor $[I - \frac{G^{o} h^{o} h^{o H}}{h^{o H} G^{o} h^{o}} ]$ in (59), which is due to Δλ, has asymptotically no effect on $C^{\mathrm{{PQML}}}_{\Delta{h}\Delta{h}}$. So asymptotically $C^{\mathrm{{PQML}}}_{\Delta{h}\Delta{h}} = C^{\mathrm{{DML}}}_{\Delta{h}\Delta{h}}$ [12]. In fact, for PQML, $\Delta\lambda= {\mathcal{O}}_{p}(\frac{1}{{M}})$, whereas $(\widehat{F}- \lambda^{o} \widehat{G} )h^{o} = {\mathcal{O}}_{p}(\frac{1}{\sqrt{M}})$. Hence also, forcing λ=1 in PQML does not influence the performance asymptotically.

Recall that the various substitutions above of $P_{{\mathcal{T}}^{H}(h^{\perp})}$ by $P_{{\mathcal{T}}(h)}^{\perp}$ are correct for all versions of h ^⊥ with p≥m that include $h^{\perp}_{\mathrm{bal},\min}$, or for ${\mathcal{T}}^{\perp}$, but for $h^{\perp}_{\min}$ (p=m−1) constitute an approximation that becomes justifiable only asymptotically (in M).