Figure 8.5:  ROC curve for single-source detection, K = 1, N = 4, M = 8, SNR = −3 dB, FAR range of practical interest.

The integral in the numerator is then extended on its first column, the remainder of the matrix having a Vandermonde determinant. What remains is then an integral form of the g₁ parameter. Hence the result.

□

The scenario where K ≥ 1 unfolds similarly. The final theorem can be found in [2]. The receiver operating characteristics (ROC) curve of the Neyman-Pearson test against that of the energy detector is provided in Figure 8.5 for N = 4, M = 8 and σ² =3 dBm. We observed a significant performance gain in terms of detection rate incurred by the Neyman-Pearson test compared to the classical energy detector.

This completes this section on hypothesis testing. In the following section, we go beyond the hypothesis test and move to the question of parameter inference in a slightly more complex data plus noise model than above.

8.5.2      Parameter Estimation

We consider a similar scenario as in the previous section, where now the K sources use different transmit powers P₁,…,P_K, which the receiver is entitled to infer from successive observations.

Consider K data sources which are transmitting data simultaneously. Transmitter k ∈ {1,…, K} has power P_k and has space dimension n_k, e.g., is composed of n_k antennas. We denote $n\underset{\_}{\underset{\_}{\Delta }}{\displaystyle {\sum }_{k=1}^K{n}_k}$ the total number of transmit dimensions. Consider also a sink or receiver with space dimension N, N > n. Denote ${\mathbf{\text{H}}}_k\in {ℂ}^{N\times n_k}$ the multidimensional filter matrix between transmitter k and the receiver. We assume that the entries of $\sqrt{N}{\mathbf{\text{H}}}_k$ are independent and identically distributed with zero mean, unit variance and finite fourth order moment. At time instant m, transmitter k emits the signal ${\text{x}}_k^{\left(m\right)}\in {ℂ}^{n_k}$, with entries assumed to be independent, independent along m, k, identically distributed along m, and have all zero mean, unit variance and finite fourth order moment (the ${\text{x}}_k^{\left(m\right)}$ need not be identically distributed along k). Assume further that at time instant m the receive signal is impaired by additive white Gaussian noise with entries of zero mean and variance σ², denoted σw^(m)∈ ℂ^N. At time m, the receiver therefore senses the signal y^(m) ∈ ℂ^N defined as

$y^{\left(m\right)}={\displaystyle \sum _{k=1}^K\sqrt{P_k}{H}_k{\text{x}}_k^{\left(m\right)}}+\sigma w^{\left(m\right)}.$

Assuming the filter coefficients are constant over at least M consecutive sampling periods, by concatenating M successive signal realizations into Y = [y⁽¹⁾,…, y^(M)] ∈ ℂ^N×M, we have

$Y={\displaystyle \sum _{k=1}^K\sqrt{P_k}{H}_k{X}_k}+\sigma W,$

where ${\mathbf{\text{X}}}_k=\left[{\text{x}}_k^{\left(1\right)},\dots {\text{x}}_k^{\left(M\right)}\right]\in {ℂ}^{n_k\times M}$, for every k, and W = [w⁽¹⁾,…, w^(M)] ∈ ℂ^N×M. This can be further rewritten as

$Y=H{P}^{\frac{1}{2}}X+\sigma W,$

(8.31)

where P ∈ ℝ^n×n is diagonal with first n₁ entries P₁, subsequent n₂ entries P₂, etc. and last n_K entries P_K, H = [H₁,…, H_K] ∈ ℂ^N×n and $\mathbf{\text{X}}={\left[{\mathbf{\text{X}}}_1^T,\dots {\mathbf{\text{X}}}_K^T\right]}^T\in {ℂ}^{n\times M}$. By convention, we assume P₁ ≤ …≤ P_K.

Our objective is to infer the values of the powers P₁,…, P_K from the realization of a single random matrix Y. This is successively performed from different approaches in the following sections. We first consider the conventional approach that assumes n small, N much larger than n, and M much larger than N. This will lead to a simple although largely biased estimation algorithm. This algorithm will be improved using Stieltjes transform approaches in the same spirit as in Section 8.4.

Conventional Approach

The first approach assumes numerous sensors in order to have much diversity in the observation vectors, as well as an even larger number of observations, in order to create an averaging effect on the incoming random data. In this situation, let us rewrite (8.31) under the form

$Y=\left(H{P}^{\frac{1}{2}}\sigma I_N\right)\left(\begin{array}{c}X\\ W\end{array}\right).$

(8.32)

We shall denote λ₁ ≤ … ≤ λ_N the ordered eigenvalues of $\frac{1}{M}{\mathbf{\text{YY}}}^{†}$ (the non-zero eigenvalues of which are almost surely different).

Appending Y ∈ ℂ^N×M into the larger matrix Y̲ ∈ ℂ^(N+n)×M

$\underset{\_}{Y}=\left(\begin{array}{cc}H{P}^{\frac{1}{2}}& \sigma I_N\\ 0& 0\end{array}\right)\left(\begin{array}{c}X\\ W\end{array}\right),$

we recognize that, conditioned on $\text{H,}\frac{1}{M}{\underset{\_}{\mathbf{\text{YY}}}}^{†}$ is a sample covariance matrix, for which the population covariance matrix is

$T\triangleq \left(\begin{array}{cc}HP{H}^{†}+{\sigma }^2{I}_N& 0\\ 0& 0\end{array}\right)$

and the random matrix

$\left(\begin{array}{c}X\\ W\end{array}\right)$

has independent (non-necessarily identically distributed) entries with zero mean and unit variance. The population covariance matrix T, whose upper left entries also form a matrix unitarily equivalent to a sample covariance matrix, clearly has an almost sure limit spectral distribution as N grows large for fixed or slowly growing n. Extending Theorem 8.2.9 and Theorem 8.3.3 to c = 0 and applying them twice (once for the population covariance matrix T and once for $\frac{1}{M}{\underset{\_}{\mathbf{\text{YY}}}}^{†}$, we finally have that, as M, N, n → ∞ with M/N → ∞ and N/n → ∞, the distribution of the largest n eigenvalues of $\frac{1}{M}{\underset{\_}{\mathbf{\text{YY}}}}^{†}$ is asymptotically almost surely composed of a mass σ² + P₁ of weight lim n₁/n, a mass σ² + P₂ of weight lim n₂/n, etc. and a mass σ² + P_K of weight lim n_K/n. As for the distribution of the smallest N − n eigenvalues of $\frac{1}{M}{\mathbf{\text{YY}}}^{†}$, it converges to a single mass in σ².

If σ² is a priori known, a rather trivial estimator of P_k is then given by

$\frac{1}{n_k}{\displaystyle \sum _{i\in {\mathcal{N}}_k}\left({\text{λ}}_i-{\sigma }^2\right),}$

where ${\mathcal{N}}_k\left\{{\displaystyle {\sum }_{j=1}^{k-1}{n}_j+1,\dots ,}{\displaystyle {\sum }_{j=1}^k{n}_j}\right\}$ and we recall that λ₁ ≤ … ≤ λ_N are the ordered eigenvalues of $\frac{1}{M}{\mathbf{\text{YY}}}^{†}$.

This means in practice that P_K is asymptotically well approximated by the averaged value of the n_K largest eigenvalues of $\frac{1}{M}{\mathbf{\text{YY}}}^{†}$, P_{K − 1} is well approximated by the averaged value of the n_K−1 eigenvalues before that, etc. This also assumes that σ² is perfectly known at the receiver. If it were not, observe that the averaged value of the N − n smallest eigenvalues of $\frac{1}{M}{\mathbf{\text{YY}}}^{†}$ is a consistent estimate for σ². This therefore leads to the second estimator ${\widehat{P}}_k^{\infty }$ for P_k, that will constitute our reference estimator,

${\widehat{P}}_k^{\infty }=\frac{1}{n_k}{\displaystyle \sum _{i\in {\mathcal{N}}_k}\left({\text{λ}}_i-{\widehat{\sigma }}^2\right)},$

where

${\widehat{\sigma }}^2=\frac{1}{N-n}{\displaystyle \sum _{i=1}^{N-n}{\text{λ}}_i.}$

Incidentally, although not derived on purpose, the refined (n, N, M)-consistent estimator of Section 8.5.2 will appear not to depend on a prior knowledge of σ². Note that the estimation of P_k only relies on n_k contiguous eigenvalues of $\frac{1}{M}{\mathbf{\text{YY}}}^{†}$, which suggests that the other eigenvalues are asymptotically uncorrelated from these. It will turn out that the improved (n, N, M)-consistent estimator does take into account all eigenvalues for each k, in a certain manner.

The Stieltjes Transform Method

The Stieltjes transform approach relies heavily on the techniques from Mestre, established in [24] and introduced in Section 8.4. We provide hereafter only the main steps of the method. The details can be found in [23].

Limiting spectrum of B_N. In this section, we prove the following result.

Theorem 8.5.3. Let ${\mathbf{\text{B}}}_N=\frac{1}{M}{\mathbf{\text{YY}}}^{†}$, with Y defined as in (8.31). Then, for M, N, n growing large with limit ratios M/N → c, N/n_k → c_k , 0 < c, c₁, …, c_K < ∞, the empirical spectral distribution $F^{{\mathbf{\text{B}}}_N}$ of B_N converges almost surely to the distribution function F, whose Stieltjes transform m_F(z) satisfies, for z ∈ ℂ⁺,

$m_F\left(z\right)=c{m}_{\underset{\_}{F}}\left(z\right)+\left(c-1\right)\frac{1}{z},$

(8.33)

where m_F̲(z) is the unique solution with positive imaginary part of the implicit equation in m_F̲ ,

$\frac{1}{m_{\underset{\_}{F}}}=-{\sigma }^2+\frac{1}{f}-{\displaystyle \sum _{k=1}^K\frac{1}{c_k}\frac{P_k}{P_{k}f}}$

(8.34)

in which we denoted f the value

$f\left(1-c\right)m_{\underset{\_}{F}}-cz{m}_{\underset{\_}{F}}^2.$

Proof. First remember that the matrix Y in (8.31) can be extended into the larger sample covariance matrix Y̲ ∈ ℂ^(N+n)×M

$\underset{\_}{Y}=\left(\begin{array}{cc}H{P}^{\frac{1}{2}}& \sigma I_N\\ 0& 0\end{array}\right)\left(\begin{array}{c}X\\ W\end{array}\right).$

From Theorem 8.2.9, since H has independent entries with finite fourth order moments, we have that the e.s.d. of HPH† converges weakly and almost surely to a limit distribution G as N, n₁,…, n_K → ∞ with N/n_k → c_k > 0. For z ∈ ℂ⁺, the Stieltjes transform m_G(z) of G is the unique solution with positive imaginary part of the equation in m_G,

$z=-\frac{1}{m_G}+{\displaystyle \sum _{k=1}^k\frac{1}{c_k}\frac{P_k}{1+P_k{m}_G}.}$

(8.35)

The almost sure convergence of the e.s.d. of HPH† ensures the almost sure convergence of the e.s.d. of the matrix $\left({\mathbf{\text{HPH}}}_0^{†}+{\sigma }^2{\text{I}}_N{}_0^0\right)$. Since m_G(z) evaluated at z ∈ ℂ⁺ is the Stieltjes transform of the l.s.d. of HPH† + σ²I_N evaluated at z + σ², adding n zero eigenvalues, we finally have that the e.s.d. of $\left({\mathbf{\text{HPH}}}_0^{†}+{\sigma }^2{\text{I}}_N{}_0^0\right)$ tends almost surely to a distribution H whose Stieltjes transform m_H(z) satisfies

$m_H\left(z\right)=\frac{c_0}{1+c_0}{m}_G\left(z-{\sigma }^2\right)-\frac{1}{1+c_0}\frac{1}{z},$

(8.36)

for z ∈ ℂ⁺, where we denoted by c₀ the limit of the ratio N/n, i.e., $c_0={\left(c_1^{-1}+\dots +c_K^{-1}\right)}^{-1}$.

As a consequence, the sample covariance matrix $\frac{1}{M}{\underset{\_}{\mathbf{\text{YY}}}}^{†}$ has a population covariance matrix which is not deterministic but whose e.s.d. has an almost sure limit H for increasing dimensions. Since X and W have entries with finite fourth order moment, we can again apply Theorem 8.2.9 and we have that the e.s.d. of ${\underset{\_}{\mathbf{\text{B}}}}_N\underset{\_}{\underset{\_}{\Delta }}\frac{1}{M}{\underset{\_}{\mathbf{\text{Y}}}}^{†}\underset{\_}{\mathbf{\text{Y}}}$ converges almost surely to the limit F̲ whose Stieltjes transform m_F̲(z) is the unique solution in ℂ⁺ of the equation in mF̲

$\begin{array}{ccc}z& =& -\frac{1}{m\underset{\_}{F}}+\frac{1}{c}\left(1+\frac{1}{c_0}\right){\displaystyle \int \frac{t}{1+t{m}_{\underset{\_}{F}}}dH\left(t\right)}\\ & =& -\frac{1}{m_{\underset{\_}{F}}}+\frac{1+\frac{1}{c_0}}{c{m}_{\underset{\_}{F}}}\left(1-\frac{1}{m_{\underset{\_}{F}}}{m}_H\left(-\frac{1}{m_{\underset{\_}{F}}}\right)\right)\end{array}$

(8.37)

for all z ∈ ℂ⁺.

For z ∈ ℂ⁺, m_F̲(z) ∈ ℂ⁺. Therefore −1/m_F̲(z) ∈ ℂ⁺ and one can evaluate (8.36) at −1/m_F̲(z). Combining (8.36) and (8.37), we then have

$z=-\frac{1}{c}\frac{1}{c{m}_{\underset{\_}{F}}{\left(z\right)}^2}{m}_G\left(-\frac{1}{m_{\underset{\_}{F}}\left(z\right)}-{\sigma }^2\right)+\left(\frac{1}{c}-1\right)\frac{1}{m_{\underset{\_}{F}}\left(z\right)},$

(8.38)

where, according to (8.35), m_G(−1/m_F(z) − σ²) satisfies

$\frac{1}{m_{\underset{\_}{F}}\left(z\right)}=-{\sigma }^2+\frac{1}{m_G\left(-\frac{1}{m_{\underset{\_}{F}}\left(z\right)}-{\sigma }^2\right)}-{\displaystyle \sum _{k=1}^K\frac{1}{c_k}\frac{P_k}{1+P_k{m}_G\left(-\frac{1}{m_{\underset{\_}{F}}\left(z\right)}-{\sigma }^2\right)}.}$

(8.39)

Together with (8.38), this is exactly (8.34), with $f\left(z\right)=m_G\left(-\frac{1}{m_{\underset{\_}{F}}\left(z\right)}-{\sigma }^2\right)=\left(1-c\right)m_{\underset{\_}{F}}\left(z\right)-cz{m}_{\underset{\_}{F}}{\left(z\right)}^2$.

Since the eigenvalues of the matrices B_N and B̲_N only differ by M − N zeros, we also have that the Stieltjes transform m_F(z) of the l.s.d. of B_N satisfies

$m_F\left(z\right)=c{m}_{\underset{\_}{F}}\left(z\right)+\left(c-1\right)\frac{1}{z}.$

(8.40)

This completes the proof of Theorem 8.5.3.

□

For further usage, notice here that (8.40) provides a simplified expression for m_G (−1/m_F̲(z) − σ²). Indeed we have,

$m_G\left(-1/m_{\underset{\_}{F}}\left(z\right)-{\sigma }^2\right)=-z{m}_F\left(z\right)m_{\underset{\_}{F}}\left(z\right).$

(8.41)

Therefore, the support of the (almost sure) l.s.d. F of B_N can be evaluated as follows: for any z ∈ ℂ⁺, m_F(z) is given by (8.33), in which m_F̲(z) is solution of (8.34); the inverse Stieltjes transform formula (8.4) allows then to evaluate F from m_F(z), for values of z spanning over the set {z = x + iy, x > 0} and y small.

Multisource power inference. In the following, we finally prove the main result of this section, which provides the G-estimator P̂₁,…, P̂_K of the transmit powers P₁,…, P_K.

Theorem 8.5.4. Let B_N ∈ ℂ^N×N be defined as ${\mathbf{\text{B}}}_N=\frac{1}{M}{\mathbf{\text{YY}}}^{†}$ with Y defined as in (8.31), and λ = (λ₁,…, λ_N), λ₁ ≤ … ≤ λ_N, be the vector of the ordered eigenvalues of B_N. Further assume that the limiting ratios c₀, C₁,…,C_K, c and P are such that the cluster mapped to P_k in B_N does not map another P_i, i ≠ k. Then, as N, n, M grow large, we have

${\widehat{P}}_k-P_k\stackrel{\text{a}.\text{s}.}{\to }0,$

where the estimate P̂_k is given by

•  if M ≠ N,

${\widehat{P}}_k=\frac{NM}{n_k\left(M-N\right)}{\displaystyle \sum _{i\in {\mathcal{N}}_k}\left({\eta }_i-{\mu }_i\right),}$

•  if M = N,

${\widehat{P}}_k=\frac{N}{n_k\left(N-n\right)}{\displaystyle \sum _{i\in {\mathcal{N}}_k}{\left({\displaystyle \sum _{j=1}^N\frac{{\eta }_i}{{\left({\text{λ}}_j-{\eta }_i\right)}^2}}\right)}^{-1},}$

in which ${\mathcal{N}}_k\left\{{\displaystyle {\sum }_{j=1}^{k-1}{n}_j+1,\dots ,}{\displaystyle {\sum }_{j=1}^k{n}_j}\right\},{\eta }_1\le \dots \le {\eta }_N$ are the ordered eigenvalues of the matrix diag $\left(\text{λ}\right)-\frac{1}{N}\sqrt{\text{λ}}{\sqrt{\text{λ}}}^{†}$ and μ₁ ≤ … ≤ are the ordered eigenvalues of the matrix diag $\left(\text{λ}\right)-\frac{1}{M}\sqrt{\text{λ}}{\sqrt{\text{λ}}}^{†}$.

Remark 8.5.1. We immediately notice that, if N < n, the powers P₁,…,P_l , with l the largest integer such that $N-{\displaystyle {\sum }_{i=l}^K{n}_i}<0$, cannot be estimated since clusters may be empty. The case N ≤ n turns out to be of no practical interest as clusters always merge and no consistent estimate of either P_i can be described.

Proof. The approach pursued to prove Theorem 8.5.4 relies strongly on the original idea of [22], which was detailed for the case of sample covariance matrices in Section 8.4. From Cauchy’s integration formula,

$P_k=c_k\frac{1}{2\pi i}{\displaystyle \underset{c_k}{\oint }\frac{1}{c_k}\frac{\omega }{P_k-\omega }}d\omega =c_k\frac{1}{2\pi i}{\displaystyle \underset{c_k}{\oint }\frac{1}{c_r}\frac{\omega }{P_r-\omega }}d\omega$

(8.42)

for any negatively oriented contour C_k ⊂ ℂ, such that P_k is contained in the surface described by the contour, while for every i ≠ k, P_i is outside this surface. The strategy is very similar to that used for the sample covariance matrix case in Section 8.4. It comes as follows: we first propose a convenient integration contour C_k which is parametrized by a functional of the Stieltjes transform m_F (z) of the l.s.d. of B_N. We proceed to a variable change in (8.42) to express P_k as a function of m_F(z). We then evaluate the complex integral resulting from replacing the limiting m_F(z) in (8.42) by its empirical counterpart ${\widehat{m}}_F\left(z\right)=\frac{1}{N}\text{tr}{\left({\mathbf{\text{B}}}_N-z{\text{I}}_N\right)}^{-1}$. This new integral, whose value we name P̂_k, is shown to be almost surely equal to P_k in the large N limit. It then suffices to evaluate P̂_k, which is just a matter of residue calculus.

Similar to Section 8.4, it turns out that the clusters generated in the spectrum of B_N can be mapped to one or many power values P_k. In what follows, we assume that the clusters are disjoint so that no holomorphicity problem arises. We can prove the following (given in detail in [23] and [28]). There exist $x_{k_F}^{\left(l\right)}$ and $x_{k_F}^{\left(r\right)}$ outside the support of F, on either side of cluster k_F (i.e., the cluster in F that is uniquely mapped to P_k) such that m_F̲(z) has limits $m_{\underset{\_}{F},k_G}^{\left(l\right)}\underset{\_}{\underset{\_}{\Delta }}{m}_F^{\circ }\left(x_{k_F}^{\left(l\right)}\right)$ and $m_{\underset{\_}{F},k_G}^{\left(r\right)}\underset{\_}{\underset{\_}{\Delta }}{m}_F^{\circ }\left(x_{k_F}^{\left(r\right)}\right)$, as $z\to x_{k_F}^{\left(l\right)}$ and $z\to x_{k_F}^{\left(r\right)}$, respectively, with $m_F^{\circ }$ the analytic extension of m_F̲ in the points $x_{k_F}^{\left(l\right)}\in ℝ$ and $x_{k_F}^{\left(r\right)}\in ℝ$. These limits $m_{\underset{\_}{F},k_G}^{\left(l\right)}$ and $m_{\underset{\_}{F},k_G}^{\left(r\right)}$ are on either side of cluster k_G (i.e., the cluster in G mapped uniquely to P_k) in the support of −1/H, and therefore $-1/m_{\underset{\_}{F},k_G}^{\left(l\right)}-{\sigma }^2$ and $-1/m_{\underset{\_}{F},k_G}^{\left(l\right)}-{\sigma }^2$ are on either side of cluster k_G in the support of G.

Consider any continuously differentiable complex path Γ_F ,_k with endpoints $x_{k_F}^{\left(l\right)}$ and $x_{k_F}^{\left(r\right)}$, and interior points of positive imaginary part. We define the contour C_F,k as the union of Γ_F ,_k oriented from $x_{k_F}^{\left(l\right)}$ to $x_{k_F}^{\left(r\right)}$ and its complex conjugate${\Gamma }_{F,k}^{*}$ oriented backwards from $x_{k_F}^{\left(r\right)}$ to $x_{k_F}^{\left(l\right)}$. The contour C_F ,_k is clearly continuous and piecewise continuously differentiable. Also, the support of cluster k_F in F̲ is completely inside C_F,k, while the support of the neighboring clusters is away from C_F,k. The support of cluster k_G in H is then inside −1/m_F̲(C_F,k), and therefore the support of cluster k_G in G is inside C_Gk ≜ −1/m_F(C_Gk) − σ². Since m_F̲ is continuously differentiable on $\mathbb{C}$ ℝ (it is in fact holomorphic [21]) and has limits in $x_{k_F}^{\left(l\right)}$ and $x_{k_F}^{\left(r\right)}$ is also continuous and piecewise continuously differentiable. Going one last step in this process, we finally have that P_k is inside the contour C_k ≜ −1/m_G(C_G,k), while P_i, for all i ≠ k, is outside C_k. Since m_G is also holomorphic on $\mathbb{C}$ ℝ and has limits in $-1/m_F^{\circ }\left(x_{k_F}^{\left(l\right)}\right)-{\sigma }^2$ and $-1/m_F^{\circ }\left(x_{k_F}^{\left(r\right)}\right)-{\sigma }^2$, C_k is a continuous and piecewise continuously differentiable complex path, which is sufficient to perform complex integration [30].

Recall now that P_k was defined as

$P_k=c_k\frac{1}{2\pi i}{\displaystyle \underset{c_k}{\oint }{\displaystyle \sum _{r=1}^K\frac{1}{c_r}}\frac{\omega }{P_r-\omega }}d\omega .$

With the variable change ω = − 1/m_G (t), this becomes

$\begin{array}{lll}{P}_k\hfill & =\hfill & c_k\frac{1}{2\pi i}{\displaystyle {\oint }_{{\mathcal{C}}_{G,k}}{\displaystyle \sum _{r=1}^K\frac{1}{c_r}}\frac{-1}{1+P_r{m}_G\left(t\right)}}\frac{{m^{\prime }}_G\left(t\right)}{m_G{\left(t\right)}^2}dt\hfill \\ \hfill & =\hfill & c_k\frac{1}{2\pi i}{\displaystyle {\oint }_{{\mathcal{C}}_{G,k}}\left(m_G\left(t\right)\left[-\frac{1}{m_G\left(t\right)}{\displaystyle \sum _{r=1}^K\frac{1}{c_r}}\frac{-1}{1+P_r{m}_G\left(t\right)}\right]+\frac{c_0-1}{c_0}\right)}\frac{{m^{\prime }}_G\left(t\right)}{m_G{\left(t\right)}^2}dt.\hfill \end{array}$

From Equation (8.35), this simplifies into

$P_k=\frac{c_k}{c_0}\frac{1}{2\pi i}{\displaystyle \underset{{\mathcal{C}}_{G,k}}{\oint }\left(c_{0}t{m}_G\left(t\right)+c_0-1\right)}\frac{{m^{\prime }}_G\left(t\right)}{m_G{\left(t\right)}^2}dt.$

(8.43)

Using (8.38) and proceeding with the further change of variable t = −1/m_F̲(z) − σ², (8.43) becomes

$P_k=\frac{c_k}{c_0}\frac{1}{2\pi i}{\displaystyle \underset{{\mathcal{C}}_{G,k}}{\oint }\left(1+{\sigma }^2{m}_{\underset{\_}{F}}\left(z\right)\right)}\left[-\frac{1}{z{m}_{\underset{\_}{F}}\left(z\right)}-\frac{{m^{\prime }}_{\underset{\_}{F}}\left(z\right)}{m_{\underset{\_}{F}}{\left(z\right)}^2}-\frac{{m^{\prime }}_F\left(z\right)}{m_F\left(z\right)m_{\underset{\_}{F}}\left(z\right)}\right]dz.$

(8.44)

This whole process of variable changes allows us to describe P_k as a function of m_F (z), the Stieltjes transform of the almost sure limiting spectral distribution of B_N, as N → ∞. It then remains to exhibit a relation between P_k and the empirical spectral distribution of B_N for finite N. This is what the subsequent section is dedicated to.

Let us now define m̂_F (z) and m̂_F̲ (z) as the Stieltjes transforms of the empirical eigenvalue distributions of B_N and B̲_N, respectively, i.e.,

${\widehat{m}}_F\left(z\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^N\frac{1}{{\text{λ}}_i-z}}$

(8.45)

and

${\widehat{m}}_{\underset{\_}{F}}\left(z\right)=\frac{N}{M}{\widehat{m}}_F\left(z\right)-\frac{M-N}{M}\frac{1}{z}.$

Instead of going further with (8.44), define P_k, the “empirical counterpart” of P_k , as

${\widehat{P}}_k=\frac{n}{n_k}\frac{1}{2\pi i}{\displaystyle \underset{{\mathcal{C}}_{F,k}}{\oint }\frac{N}{n}\left(1+{\sigma }^2{\widehat{m}}_{\underset{\_}{F}}\left(z\right)\right)}\left[-\frac{1}{z{\widehat{m}}_{\underset{\_}{F}}\left(z\right)}-\frac{{{\widehat{m}}^{\prime }}_{\underset{\_}{F}}\left(z\right)}{{\widehat{m}}_{\underset{\_}{F}}{\left(z\right)}^2}-\frac{{{\widehat{m}}^{\prime }}_{\underset{\_}{F}}\left(z\right)}{{\widehat{m}}_F\left(z\right){\widehat{m}}_{\underset{\_}{F}}\left(z\right)}\right]dz.$

(8.46)

The integrand can then be expanded into nine terms, for which residue calculus can easily be performed. Denote first η₁,…, η_N the N real roots of m̂_F(z) = 0 and (μ₁, …, μ_N the N real roots of m̂_F̲(z) = 0. We identify three sets of possible poles for the nine aforementioned terms: (i) the set $\left\{{\text{λ}}_1,\dots {\text{λ}}_N\right\}\cap \left[x_{k_F}^{\left(l\right)},x_{k_F}^{\left(r\right)}\right]$, (ii) the set $\left\{{\eta }_1,\dots {\eta }_N\right\}\cap \left[x_{k_F}^{\left(l\right)},x_{k_F}^{\left(r\right)}\right]$, and (iii) the set $\left\{{\mu }_1,\dots {\mu }_N\right\}\cap \left[x_{k_F}^{\left(l\right)},x_{k_F}^{\left(r\right)}\right]$. For M ≠ N, the full calculus leads to

$\begin{array}{l}{\widehat{P}}_k=\frac{NM}{n_k\left(M-N\right)}\left[\begin{array}{llll}{\displaystyle \sum _{\begin{array}{l}1\le i\le N\\ x_{k_F}^{\left(l\right)}\le {\eta }_i\le x_{k_F}^{\left(r\right)}\end{array}}}\hfill & \eta i^{-}\hfill & {\displaystyle \sum _{\begin{array}{l}1\le i\le N\\ x_{k_F}^{\left(l\right)}\le {\eta }_i\le x_{k_F}^{\left(r\right)}\end{array}}}\hfill & \mu i^{-}\hfill \end{array}\right]\\ +\frac{N}{n_k}\left[\begin{array}{llll}{\displaystyle \sum _{\begin{array}{l}1\le i\le N\\ x_{k_F}^{\left(l\right)}\le {\eta }_i\le x_{k_F}^{\left(r\right)}\end{array}}}\hfill & {\sigma }^2{}^{-}\hfill & {\displaystyle \sum _{\begin{array}{l}1\le i\le N\\ x_{k_F}^{\left(l\right)}\le {\text{λ}}_i\le x_{k_F}^{\left(r\right)}\end{array}}}\hfill & {\sigma }^2\hfill \end{array}\right]+\frac{N}{n_k}\left[\begin{array}{llll}{\displaystyle \sum _{\begin{array}{l}1\le i\le N\\ x_{k_F}^{\left(l\right)}\le {\mu }_i\le x_{k_F}^{\left(r\right)}\end{array}}}\hfill & {\sigma }^2{}^{-}\hfill & {\displaystyle \sum _{\begin{array}{l}1\le i\le N\\ x_{k_F}^{\left(l\right)}\le {\text{λ}}_i\le x_{k_F}^{\left(r\right)}\end{array}}}\hfill & {\sigma }^2\hfill \end{array}\right].\end{array}$

(8.47)

Now, we know from Theorem 8.5.3 that ${\widehat{m}}_F\left(z\right)=\stackrel{\text{a}.\text{s}.}{\to }{m}_F\left(z\right)$ and ${\widehat{m}}_{\underset{\_}{F}}\left(z\right)=\stackrel{\text{a}.\text{s}.}{\to }{m}_{\underset{\_}{F}}\left(z\right)$ as N → ∞. Observing that the integrand in (8.46) is uniformly bounded on the compact C_F,k, the dominated convergence theorem, Theorem 16.4 of [16], ensures $\begin{array}{l}{\widehat{m}}_F\left(z\right)=\stackrel{\text{a}.\text{s}.}{\to }{m}_F\left(z\right)\frac{1}{N}\text{tr}{\left({\mathbf{\text{B}}}_N-z{\text{I}}_N\right)}^{-1}\\ {\widehat{P}}_k\stackrel{\text{a}.\text{s}.}{\to }{P}_k\end{array}$.

To go further, we now need to determine which of λ₁, …, λ_N, η₁, …, η_N and (η₁, …, η_N lie inside C_F ,_k. It can be proved, by extending Theorem 8.3.1 and Theorem 8.3.4 to the current model, that there will be no eigenvalue of B_N (or B̲_N) outside the support of F, and the number of eigenvalues inside cluster k_F is exactly n_k. Since C_F,k encloses cluster k_F and is away from the other clusters, $\left\{{\text{λ}}_1,\dots {\text{λ}}_N\right\}\cap \left[x_{k_F}^{\left(l\right)},x_{k_F}^{\left(r\right)}\right]=\left\{{\text{λ}}_{i}i\in {\mathcal{N}}_k\right\}$ almost surely, for all N large. Also, for any i ∈ {1, …, N}, it is easy to see from (8.45) that m̂_F(z) → ∞ when z ↑ λ_i and m̂_F(z) → −∞ when z ↓ λ_i. Therefore m̂_F(z) → ∞ has at least one solution in each interval (λ_i−1, λ_i), with λ₀ = 0, hence μ₁ < λ₁ < μ_{2 …} < μ_N < λ_N. This implies that, if k₀ is the index such that C_F,k contains exactly ${\text{λ}}_{k_0},\dots ,{\text{λ}}_{k_0+\left(n_k-1\right)}$, then C_F,k also contains $\left\{{\mu }_{k_0+1},\dots ,{\mu }_{k_0+\left(n_k-1\right)}\right\}$. The same result holds for ${\eta }_{k_0+1},\dots ,{\eta }_{k_0+\left(n_k-1\right)}$. When the indexes exist, due to cluster separability, ${\eta }_{k_0-1}$ and ${\mu }_{k_0-1}$ belong, for N large, to cluster k_F − 1. We are then left with determining whether ${\mu }_{k_0}$ and ${\eta }_{k_0}$ are asymptotically found inside C_F,_k.

For this, we use the same approach as in [22] by noticing that, since 0 is not included in C_k , one has

$\frac{1}{2\pi i}{\displaystyle \underset{{\mathcal{C}}_k}{\oint }\frac{1}{\omega }d\omega =0.}$

Performing the same changes of variables as previously, we have

${\displaystyle \underset{{\mathcal{C}}_{F,k}}{\oint }\frac{-m_{\underset{\_}{F}}\left(z\right)m_F\left(z\right)-z{m^{\prime }}_{\underset{\_}{F}}\left(z\right)m_F\left(z\right)-z{m}_{\underset{\_}{F}}\left(z\right){m^{\prime }}_{\underset{\_}{F}}\left(z\right)}{z^2{m}_{\underset{\_}{F}}{\left(z\right)}^2{m}_F{\left(z\right)}^2}dz=0.}$

(8.48)

For N large, the dominated convergence theorem ensures again that the left-hand side of the (8.48) is close to

${\displaystyle \underset{{\mathcal{C}}_{F,k}}{\oint }\frac{-{\widehat{m}}_{\underset{\_}{F}}\left(z\right)\left(z\right)-z{\widehat{m^{\prime }}}_{\underset{\_}{F}}\left(z\right)\left(z\right)-z{\widehat{m}}_{\underset{\_}{F}}{\left(z\right)}^{\prime }\left(z\right)}{z^2{\widehat{m}}_{\underset{\_}{F}}{\left(z\right)}^2{\left(z\right)}^2}dz.}$

(8.49)

Residue calculus of (8.49) then leads to

$\begin{array}{lll}{\displaystyle \sum _{\begin{array}{l}1\le i\le N\\ {\text{λ}}_i\in \left[x_{k_F}^{\left(l\right)},x_{k_F}^{\left(r\right)}\right]\end{array}}2-}\hfill & {\displaystyle \sum _{\begin{array}{l}1\le i\le N\\ {\eta }_i\in \left[x_{k_F}^{\left(l\right)},x_{k_F}^{\left(r\right)}\right]\end{array}}1-}\hfill & {\displaystyle \sum _{\begin{array}{l}1\le i\le N\\ {\mu }_i\in \left[x_{k_F}^{\left(l\right)},x_{k_F}^{\left(r\right)}\right]\end{array}}1\stackrel{\text{a}.\text{s}.}{\to }0.}\hfill \end{array}$

(8.50)

Since the cardinalities of $\left\{i,{\eta }_i\in \left[x_{k_F}^{\left(l\right)},x_{k_F}^{\left(r\right)}\right]\right\}$ and $i,{\mu }_i\in \left[x_{k_F}^{\left(l\right)},x_{k_F}^{\left(r\right)}\right]$ are at most n_k, (8.50) is satisfied only if both cardinalities equal n_k in the limit. As a consequence, ${\mu }_{k_0}\in \left[x_{k_F}^{\left(l\right)},x_{k_F}^{\left(r\right)}\right]$ and ${\eta }_{k_0}\in \left[x_{k_F}^{\left(l\right)},x_{k_F}^{\left(r\right)}\right]$. For N large, N ≠ M, this allows us to simplify (8.47) into

${\widehat{P}}_k=\frac{NM}{n_k\left(M-N\right)}{\displaystyle \sum _{\begin{array}{l}1\le i\le N\\ {\text{λ}}_i\in \mathcal{N}k\end{array}}\left({\eta }_i-{\mu }_i\right)}$

(8.51)

with probability one. The same reasoning holds for M = N. This is our final relation. It now remains to show that the η_i and the μ_i are the eigenvalues of diag $\left(\text{λ}\right)-\frac{1}{N}\sqrt{\text{λ}}{\sqrt{\text{λ}}}^T$ and diag $\left(\text{λ}\right)-\frac{1}{M}\sqrt{\text{λ}}{\sqrt{\text{λ}}}^T$, respectively. But this is merely a consequence of [23, Lemma 1].

This concludes the proof of Theorem 8.5.4.

□

Figure 8.6:  Distribution function of the estimators ${\widehat{P}}_k^{\infty }$ and P̂_k for k ∈ {1, 2, 3}, P₁ = 1/16, P₂ = 1/4, P₃ = 1, n₁ = n₂ = n₃ = 4 antennas per user, N = 24 sensors, M = 128 samples, and SNR = 20 dB. Optimum estimator is shown in dashed lines.

We now evaluate the performance difference between the traditional and the Stieltjes transform inference methods, for K = 3 sources, P₁ = 1/16, P₂ = 1/4, N = 24 sensors, M = 128 samples, and n₁ = n₂ = n₃ = 4. This is provided in Figure 8.6 which compares the distribution function of the estimates engendered by both methods. As anticipated, we observe a significant gain in terms of bias reduction for the Stieltjes transform method.

8.6      Conclusion

Random matrix theory for signal processing is a fast growing field of research whose interest is mainly motivated by the increase of the dimensionality and complexity of today’s systems. While the first years of random matrix theory were mainly focusing on Gaussian and invariant matrix distributions, the last ten years of research were mainly targeting large dimensional matrices with independent entries. This provided interesting results in particular on the limiting spectrum of sample covariance matrices, which led to new results on inverse problems for large dimensional systems. These results are often surprisingly simple and efficient as they perform well against exact maximum likelihood solutions, even for systems of not too large dimensions. Much more is however needed from a mathematical viewpoint relative in particular to second order statistics, see e.g., [35, 36], in order to evaluate theoretically the performance of these methods as well as a generalization to more intricate random matrix structures, such as Vandermonde matrices for array processing, see e.g., [37], or unitary random matrices, see e.g., [38]. A more exhaustive account of random matrix methods as well as more details on the methods presented here can be found in [10, 17, 28].

8.7      Exercises

Exercise 8.7.1 (Sampling and Signal Energy). Based on Theorem 8.2.9, prove the Marc̆enko-Pastur law, Theorem 8.2.5.

Hint 8.7.1. Observe that the fixed-point equation in $m_{F^B}$ reduces now to a second order polynomial from which $m_{F^B}\left(z\right)$ takes an explicit form. The inverse Stieltjes transform formula 8.4 gives the expression of F^B.

Exercise 8.7.2. Let X_N ∈ ℂ^N×n be a random matrix with i.i.d. Gaussian entries of zero mean and variance 1/n. For R_N ∈ ℂ^N×N and T_N ∈ ℂ^n×n deterministic and of uniformly bounded spectral norm such that $F^{R_N}\Rightarrow F^R$ and $F^{{\mathbf{T}}_N}\Rightarrow F^T$, as N,n → ∞, determine an expression of the Stieltjes transform of the limiting eigenvalue distribution of ${\mathbf{\text{B}}}_N={\mathbf{\text{R}}}_N^{\frac{1}{2}}{\mathbf{\text{X}}}_N{\mathbf{\text{T}}}_N{\mathbf{\text{X}}}_N^{†}{\text{R}}_N^{\frac{1}{2}}$ as N/n → c.

Hint 8.7.2. Follow the proof of Theorem 8.2.9 by looking for a deterministic equivalent of $\frac{1}{N}\text{tr}\left(\mathbf{\text{A}}{\left({\mathbf{\text{B}}}_N-z{\text{I}}_N\right)}^{-1}\right)$ for some deterministic A, taken to be successively R_N and I_N. A good choice of the matrix D_N is D_N = a_NR_N.

Exercise 8.7.3. Based on the definition of the Shannon-transform and on the G-estimator for the Stieltjes transform, determine a G-estimator for

${\mathcal{V}}_{T_N}\left(x\right)=\frac{1}{N}\text{log}\text{det}\left(x{T}_N+I_N\right)$

based on the observations

$y_k=T_N^{\frac{1}{2}}{\text{x}}_k$

with x_k ∈ ℂ^N with i.i.d. entries of zero mean and variance 1/N, independent across k, for k ∈ {1,…, n}.

Hint 8.7.3. Write the expression of ${\mathcal{V}}_{{\mathbf{\text{T}}}_N}\left(x\right)$ as a function of the Stieltjes transform of T_N and operate a variable change in the resulting integral using Theorem 8.2.9.

Exercise 8.7.4. From the result of Theorem 8.3.3, propose a hypothesis test for the presence of a signal transmitted by a signal source and observed by a large array of sensors, assuming that the additive noise variance is either perfectly known or not.

Hint 8.7.4. Observe that the ratio of the extreme eigenvalues in both H₀ and H₁ hypotheses is asymptotically independent of the noise variance.

Exercise 8.7.5. For W ∈ ℂ^N×n, n < N, the n columns of a random unitarily invariant unitary matrix, w a column vector of W, prove that, if B_N is a random matrix with bounded spectral norm, function of all columns of W but w, then, as N, n → ∞ with n/N → c < 1,

$w^{†}{B}_{N}w-\frac{1}{N-n}\text{tr}\left(\left(\right)I_N-W{W}^{†}\right)B_N\stackrel{\text{a}.\text{s}.}{\to }0.$

Hint 8.7.5. write w as the normalized projection of a Gaussian vector x on the subspace orthogonal to the space spanned by the columns of W but w, i.e., w = Πx, with Π = I_N − WW^† + WW^†.

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¹We remind that a unitary matrix U ∈ ℂ^N×N is such that UU† = U^‡U = I_N.

²Throughout this work, we will respect the convention that x (be it a scalar or an Hermitian matrix) is non-negative if x ≥ 0, while x is positive if x ≥ 0.

³The Hermitian property is fundamental to ensure that all eigenvalues of X_N belong to the real line. However, the extension of the empirical spectral distribution (e.s.d.) to non-Hermitian matrices is sometimes required; for a definition, see (1.2.2) of [10].

⁴We borrow here the notation m due to a large number of contributions from Bai, Silverstein et al. [14, 15]. In other works, the notation s or S for the Stieltjes transform is used.

⁵We recall that the support Supp (F) of a real function F is the set {x ∈ℝ, ∣F(x)∣ > 0}.