Limiting spectral distribution of the sample covariance matrix of the windowed array data
 Ehsan Yazdian^{1}Email author,
 Saeed Gazor^{2} and
 Mohammad Hasan Bastani^{3}
DOI: 10.1186/16876180201342
© Yazdian et al.; licensee Springer. 2013
Received: 13 August 2012
Accepted: 5 February 2013
Published: 6 March 2013
Abstract
In this article, we investigate the limiting spectral distribution of the sample covariance matrix (SCM) of weighted/windowed complex data. We use recent advances in random matrix theory and describe the distribution of eigenvalues of the doubly correlated Wishart matrices. We obtain an approximation for the spectral distribution of the SCM obtained from windowed data. We also determine a condition on the coefficients of the window, under which the fragmentation of the support of noise eigenvalues can be avoided, in the noiseonly data case. For the commonly used exponential window, we derive an explicit expression for the l.s.d of the noiseonly data. In addition, we present a method to identify the support of eigenvalues in the general case of signalplusnoise. Simulations are performed to support our theoretical claims. The results of this article can be directly employed in many applications working with windowed array data such as source enumeration and subspace tracking algorithms.
Introduction
The distribution of the eigenvalues of the sample covariance matrix (SCM) of data has important impact on the performance of signal processing algorithms. Over the last decade, the properties of complex Wishart matrices are used in the analysis and design of many signal processing algorithms such as in array processing. Our knowledge about the distribution of eigenvalues, eigenvectors and determinants of complex Wishart matrices and their limiting behavior is emerging as a key tool in a number of applications, e.g., in data compression and analysis of wireless MIMO channels [1, 2], array processing, source enumeration and identification [3–5], adaptive algorithms [6, 7]. The densities of the singular values of random matrices and their asymptotic behavior (as the matrix size tends to infinity) has been employed in some applications [8–10]. The eigenvalues of the SCM are often used to describe many signal processing problems. For example in [8], they are used as sufficient statistics for array source enumeration.
Let X _{1}, …, X _{ N } be N independent zero mean Gaussian random vectors with covariance matrix of A, i.e., ${\mathcal{N}}_{M}(0,\mathbf{A})$, where A is a nonnegative M × M Hermitian matrix. The SCM R _{ N } is defined as ${\mathbf{R}}_{N}=\frac{1}{N}\sum _{i=1}^{N}{X}_{i}{X}_{i}^{H}=\frac{1}{N}{\mathbf{XX}}^{H}$, where X = [X _{1},…,X _{ N }] contains N snapshots of the received data. In this article, we refer to this SCM as the SCM with rectangular window (SCMR) as all data samples have equal weights, i.e., a rectangular window is used. In this case R _{ N } has a Wishart distribution [11] and for more than four decades, it has been known that the joint probability density function (PDF) of its eigenvalues, can be expressed in terms of hypergeometric functions [12]. More recently, a simpler form of this joint PDF was derived in terms of the product of two determinants [13]. However, this form is applicable if the array is small and the eigenvalues of the covariance matrix of the observed data are distinct. Several articles have investigated the behavior of the eigenvalues of R _{ N } when M, N → ∞ assuming $\frac{M}{N}\to c>0$[14, 15]. This is a more realistic assumption than assuming M is finite and N is infinite, because in most practical applications the covariance matrix A slowly varies, hence, the effective window length could not be arbitrary long. For instance, the eigenvalue estimators that are consistent in this asymptotic regime are more robust to finite sample size than other estimators which are only guaranteed to converge for fixed M and N→∞[9]. There are many works on the distribution of eigenvalues in this asymptotic regime, such as informationplusnoise [16] and spiked models where all eigenvalues are equal excluding a small number of fixed eigenvalues (spikes) [17]. Specifically, the distribution of the largest noise eigenvalue is widely studied [18, 19].
where {w _{ i } ≥ 0,i = 1,…,N} is a nonnegative sequence. Hereafter, we refer to R _{ N } as the SCM. The SCMR is obtained using a rectangular window, i.e., where w _{ i } is nonzero and constant for i = 1,…,N. These weights allow to flexibly emphasize or deemphasize some of the observations. For example smaller weights for old data samples allows to improve the agility of the algorithms. For instance in cognitive radio, it is important to detect the activities of users and the idle channels as fast as possible, thereby reducing the detection time and improving the agility of the system [20, 21]. Among all windows, the exponential window, w _{ i } = w _{0} p ^{ i }, is commonly used. Two reasons for this popularity are (1) this window allows to develop fast recursive algorithms which are considerably less expensive in terms of computational complexity, thereby facilitate the realtime implementation of these algorithms (e.g. see [22, 23]) and (2) allows to forget the old data, thereby improving the tracking ability in nonstationary environments. For instance exponentially windowed data is used in most of the existing subspace tracking algorithms[24, 25]. That is because only a rankone update is required for each new data vector to update the underlying SCM, which leads to simple low cost subspace tracking algorithms.
In this article, we study the effects of windowing on the distribution of the eigenvalues of the SCM. In this case, the SCM in (1) has a doubly correlated Wishart distribution [26–30]. We must note that, there are numerous research results for the case of Wishart matrices, however, the spectral properties in the doubly correlated case has not been sufficiently studied.
where λ _{1}, λ _{2}, … ,λ _{ M } are eigenvalues of A and #{.} denotes the cardinality of a set. Note that, in this definition all eigenvalues of A are assumed to be real. Although this formulation is less explicit than the joint PDF of eigenvalues, it describes the statistical behavior of the eigenvalues. In many practical cases A is a random matrix and the e.s.d. F ^{ A }(x) is a random function which converges almost surly to a deterministic cumulative distribution function as the dimension of the system grows. In such cases, lim M → ∞ F ^{ A }(x) is referred to as the limiting spectral distribution (l.s.d.) of A.
In recent years, some results have been obtained on the limiting behavior of the e.s.d. of correlated Wishart matrices. In this article, for the white noise case, we study the behavior of eigenvalues of the SCM. In particular for the exponential window, we extend the results previously demonstrated in [31] and give more details along with the proofs of the required theorems. We then consider the case of signal plus noise and present a method to determine the support of eigenvalues. The main contributions in this article are

A method is proposed to approximate the spectral distribution of the SCM using arbitrary windows with that of an equivalent Wishart Distribution. For the especial case of white noise (noise only), this approximation is the Marchenko–Pastur (M–P) distribution, which is the known distribution for the case of a rectangular window.

In Theorem 2, we derive an accurate and explicit equation for the l.s.d. of the SCM of noiseonly data for the exponential window. Many simulations are performed to show the accuracy of this l.s.d.

In Theorem 3 we present a systematic method to compute the support of eigenvalues in the signal plus noise data case using an exponentially weighted window. In addition to the results, we follow up a different and novel approach in proving this theorem compared with the existing proof for the rectangular window case where the Stieltjes transform m(z) has the explicit inverse [15]. This approach can be easily utilized for other window types where the Stieltjes transform is expressed explicitly or implicitly as a function of z.
The demonstrated results provide a key step toward characterization of the distribution of eigenvalues in the general Covariance matrix of windowed data. The results of this work are useful in the design and implementation of robust algorithms using windowed snapshots. Our derivations in Theorems 2 and 3 can be directly used to design unbiased eigenvalue and eigenvector estimators. These estimators are important especially because the exponential window is used in numerous applications. They can be used as a basis to improve the performance and accuracy of many existing algorithms which are based on exponentially windowed data, in many fields such as subspace tracking, DOA estimation and source enumeration.
The remainder of this article is organized as follows: Section 2 introduces the system model and some important mathematical tools. We derive an approximation for the Stieltjes transform of l.s.d. of eigenvalues of weighted windowed array data in Section 3. Asymptotic spectrum of the eigenvalues in noiseonly data case is analyzed in Section 4. The signal plus noise case is studied in Section 5. Section 6 provides simulation results. Finally, we conclude this work and suggest future works in Section 7.
2 System model for windowed SCM
where U = [U _{1},…,U _{ N }] is an M × N matrix contains i.i.d. zeromean unitvariance complex Gaussian entries and ${\mathbf{W}}_{N}\stackrel{\Delta}{=}\text{diag}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}({w}_{1},\dots ,{w}_{N})$. The matrix R _{ N } has a doubly correlated Wishart distribution. In practice, it is very complex to directly characterize the e.s.d. of R _{ N } thus, we use the Stieltjes transform of this distribution and indirectly characterize the behavior of the eigenvalues. Then, in the asymptotic regime as M,N→∞ given $\frac{M}{N}\to c>0$, the inverse transform of the limit gives the l.s.d of SCM.
Definition 1
Hence, in order to characterize the asymptotic distribution of the sample eigenvalues, we alternatively characterize the asymptotic behavior of the corresponding Stieltjes transform, and then use the Stieltjes inversion formula in (5) to obtain l.s.d. of SCM f ^{ R }(x). We use the following theorem which gives the Stieltjes transform of the correlated Wishart matrix [29] and is the basis for derivations in this article.
Theorem 1
Proof 1
See[29]for proof. Similar results are also demonstrated in[26], [28]with some differences in the assumptions on correlation matrices. □
We emphasize that (6) and (7) give the exact distribution in the asymptotic regime as M,N → ∞ with $\frac{M}{N}\to c>0$. Since in practice, the array dimension and/or sample size are usually finite numbers, this method gives a deterministic approximation for the actual sample eigenvalue distribution.
where ${a}_{\pm}={\sigma}^{2}{\left(1\pm \sqrt{c}\right)}^{2}$ and ${\pi}_{a,b}\left(x\right)=\left\{\begin{array}{cc}1& \phantom{\rule{1em}{0ex}}a\le x\le b\\ 0& \phantom{\rule{1em}{0ex}}\text{otherwise}\end{array}\right.$.
3 Effective length of a window
In this section, we define the effective length of a window which allows to approximate the distribution of the eigenvalues of windowed SCM with that of a rectangular window with an equivalent length, assuming that the covariance matrix of data A satisfies the assumptions of Theorem 1. In several existing articles some intuitive equivalent length are defined simply to extend the previously existing results for the rectangular case in order to analyze the behavior of the eigenvalues in the weighted window cases [22], [23].
where $E\left\{.\right\}=\int \left(.\right)d{F}^{\mathbf{W}}\left(w\right)$ and $\leftO\right\le \frac{\leftm{}^{I}\right{\sigma}^{2}{}^{I+1}}{1\beta}E\left\{{w}^{I+1}\right\}$.
where using E{w ^{3}} < sup{w ^{2}}E{w} and E{w ^{2}} < sup{w}E{w} it is easy to show that the approximation error is bounded by ${\sigma}^{2}E\left\{w\right\}\frac{2{\beta}^{2}}{1\beta}$.
Definition 2
with all coefficients equal to w _{ e }. The average weight w _{ e } is a scale parameter for the eigenvalues of covariance matrix of the received data. Although we have derived the effective length for the noise only data, our results reveal that this effective window length gives accurate results for the signal plus noise case.
For the white noise data, the l.s.d. of SCM can be approximated by the M–P distribution defined in (9) by substituting c and σ ^{2}, with c _{ e } and w _{ e } σ ^{2}, respectively. Note that the effective window length is always smaller than the number of samples N. This approximation can be intuitively interpreted as a Wishart approximation where the effect of “windowing” is approximated with a rectangular window with an effective number of samples of N _{ e } and the covariance matrix of the received data is scaled to A _{ e }=w _{ e } A”.
which is not a function of N. As expected the effective length of the window increases as the forgetting factor p approaches one.
4 Spectral analysis of noiseonly data
In this section, for the windowed data case, the l.s.d. of the SCM is characterized more accurately. In practice, the array dimension and the effective window length are both finite. However, we are interested in the impact of the weights of the window f ^{ W }(w), on the limiting distribution of the eigenvalues as M,N→∞ employing Theorem 1. We use two approaches to model f ^{ W }(w), Discrete and Continuous. The former considers f ^{ W }(w) as a finite sum of discrete masses at the coefficients of the window. The discontinuous distribution function modeling is useful to analyze the support of eigenvalues and its connectivity. The latter approach, approximates f ^{ W }(w) as a continuous function allowing to derive some explicit equations for the Stieltjes transform.
The following lemma is the key to determine these intervals on real axis [15].
Lemma 1 ([32], Lemma 6.1)
where z(m) is the inverse function of m(z). Also conversely, for any real m in the domain of z(m) if $\frac{\mathit{\text{dz}}\left(m\right)}{\mathit{\text{dm}}}>0$ then x = z (m) is outside the support of F.
4.1 Discrete distribution function approach
In many signal processing applications the white noise subspace is separated from the signal subspace based on the eigenvalues of the SCM. Such a fragmentation of the support of noise eigenvalues misleads the subspace based algorithms and leads to noise eigenvalues to be mistaken as signal ones.
Under this connectivity condition, the support of eigenvalues is the interval [x _{ l } = z(m _{ l }),x _{ u }=z(m _{ u })], which can be calculated, numerically. Our simulations show that this condition is satisfied for popular window types especially for N _{ d } ≫ 1 used in practice. Figure 2 shows a typical case for c > 1 where $\frac{\mathit{\text{dz}}\left(m\right)}{\mathit{\text{dm}}}=0$ has an even number of realvalued solutions (counting multiplicities) which we denote them by ${m}_{1}^{}\le {m}_{1}^{+}<\cdots <{m}_{q}^{}\le {m}_{q}^{+}$ (in addition to m _{ l },m _{ u }). Each pair of these solutions determines a subinterval for the support of eigenvalues, i.e., we have ${S}_{F}=[{x}_{l},{x}_{u}]\left\{\right[{x}_{1}^{},{x}_{1}^{+}]\cup \cdots \cup [{x}_{q}^{},{x}_{q}^{+}\left]\right\}$, where ${x}_{i}^{}=z\left({m}_{i}^{}\right),{x}_{i}^{+}=z\left({m}_{i}^{+}\right)$. Reducing c or reducing the gap between weight values {w _{ i }} makes the support more compact at the expense of using more temporal samples.
4.2 Continuous function approach
The goal of this approach is to find closed form expressions of Stieltjes integrals of the l.s.d. This approach could be used for any window shapes. However, we start with the triangular window and then consider the exponential window which are more popular. Here, we model the function f ^{ W }(w) with a continuous distribution and evaluate (18) to found the Stieltjes transform.
for $m\in (\frac{1}{2c{\sigma}^{2}},\infty )$ and m≠0.
is a continuous function, independent of window size N and satisfies the assumptions of Theorem 1. Thus, this theorem is applicable to the exponential window truncated at some large integer N.
for all $m\in \left(\frac{1}{{c}_{0}{\sigma}^{2}},\infty \right)\setminus \left\{0\right\}$ where ${n}_{0}=\frac{1}{ln\left(p\right)}$.
One can use the same method as in the discrete distribution function approach and identify the support of the distribution S _{ F }. However, the function z(m) in (26) is simple and the following theorem gives the explicit distribution.
Theorem 2
and upper and lower boundaries of the support are
respectively, where ω _{ k }(x) is the branch of Lambert W function ^{b}[33]with k=−1 and k=0.
Proof 2
According to the Lemma 1, boundaries of the support of eigenvalues are the real solutions of z ^{′}(m) = 0, i.e., with some simple calculations, are the solutions of
Using (26), the boundaries z(m _{−}) and z(m _{+}) are obtained as in (28a) and (28b) which determine the support of eigenvalues as the interval $\left[z\right({m}_{}),z({m}_{+}\left)\right]\subset \mathbb{R}$.
Dropping the real terms inside the brackets and applying some simplifications, we obtain (27). □
Remark 1
In contrast to these methods for the exponential window, we derive an accurate explicit closed form expression which can be easily employed in many applications such as in signal processing and economy.
5 Spectral analysis of signal plus noise data
In this section, we consider the case of white noise plus some signal sources, i.e., where the eigenvalues of A are not equal. In the general case, let λ _{ q } > ⋯ > λ _{1} >0 denote the set of q distinct eigenvalues of the covariance matrix and the multiplicity of λ _{ ℓ } is denoted by k _{ ℓ } (we must have $M=\sum _{\ell =1}^{q}{k}_{\ell}$). For example suppose a real phased array communication system with q − 1 independent signals impinging on it simultaneously on the same frequency band from different directions where q < M. The smallest eigenvalue λ _{1} can be interpreted as the noise eigenvalue and other q − 1 larger eigenvalues are referred to as signal eigenvalues. In the asymptotic regime, when N,M are growing large, we assume that $\frac{{k}_{\ell}}{M}\to {\alpha}_{\ell}>0$, where α _{ ℓ },ℓ= 1, …,q are multiplicity ratios of eigenvalues. In this case the spectral distribution of the matrix A in Theorem 1 can be expressed as sum of Dirac delta functions, i.e. $d{F}^{\mathbf{A}}\left(a\right)=\sum _{i=1}^{q}{\alpha}_{i}\delta (a{\lambda}_{i})\mathit{\text{da}}$.
In what follows, we present an approach to determine the support of eigenvalues and also the l.s.d. of exponentially weighted SCM of signal plus noise data in the asymptotic regime. The first in determining the distribution of the eigenvalues is to determine its support on the real positive axis.
The definition of the Stieltjes transform in (4) implies that for any distribution F and real x outside the support of F, m(x) is well defined and its derivative, ${m}^{\prime}\left(x\right)=\int \frac{\mathit{\text{dF}}\left(y\right)}{{(yx)}^{2}}$, is obviously real and positive. Thus, m(z) is increasing on intervals on real line outside the support of its distribution function F[15]. Therefore, the inverse function theorem proves that its inverse exists on these intervals and shall also be increasing. For the one sided correlated Wishart matrices, where the inverse of m(z) has an explicit expression, Lemma 1 shows that the converse of the above statements are also true [15], i.e. for any real m in the domain of z(m), if $\frac{\mathit{\text{dz}}\left(m\right)}{\mathit{\text{dm}}}>0$ then x = z (m) is outside the support of the distribution. Therefore, the support of eigenvalues is a Borel subset of ${\mathbb{R}}^{+}$ for which z(m) is increasing which can be determined by simply plotting the inverse function z(m) for real m. Paul and Silverstein ([29], page 2) suggested the same method for doubly correlated Wishart matrices if there exists an explicit inverse z = z(m) for the limiting Stieltjes transform m(z). Unfortunately, for nonrectangular windows, the inverse of m(z) in general has no explicit expression [29]. Fortunately, by introducing two auxiliary variables u and h in what follows for the exponential window, we found z(h) which implicitly expresses z as a function of m. Then, we prove that the same method can be extended for the exponential window case, while the main difference here is that we are able to use the implicit expressions to determine this Borel set. Although the exponential window case is studied in this article, the same approach may be used for some other window types, to determine the support of eigenvalues.
Proposition 1
The auxiliary variable h, as a function of u and z, has some interesting properties as:
(1) h always lies in the subset ${D}_{h}\subset {\mathbb{C}}^{+}$ for all $z\in {\mathbb{C}}^{+}$.
Proof 3
The first property can is simply implied from (46) as the imaginary part of h and u have the same sign. Using (45) and (46), we can easily find (47). The third property is proved as follows. The constraint in (48) is obtained from Im{u} ∈ (0,Π) and (46). According to Theorem 1, for any $z\in {\mathbb{C}}^{+}$, there is a unique u∈D _{ u }, satisfying (44). The unique pair (z,u) gives an h in ${\mathbb{C}}^{+}$ according to (45). In order to prove the uniqueness of h, suppose that h _{1} and h _{2} in ${\mathbb{C}}^{+}$ satisfy (47) and (48). Thus, (46) yields u _{1},u _{2}∈D _{ u } satisfying (44). In addition, we must have u _{1} = u _{2} since for any $z\in {\mathbb{C}}^{+}$, there exists a unique u _{1}∈D _{ u }. Thus for z and u _{1} = u _{2}, (45) yields that h _{1} = h _{2}. □
Although z(h) in (47) is defined only for h∈D _{ h }, it is an analytic function for all $h\in \mathbb{C}\setminus \left\{0,\frac{1}{{\lambda}_{1}},\dots ,\frac{1}{{\lambda}_{q}}\right\}$. In addition note that $z\left(h\right)=\frac{1}{h}$ at the roots of $\sum _{i=1}^{q}\frac{{\alpha}_{i}}{{\lambda}_{i}h+1}1=0$.
for h∈D _{ h }. Similar to z(h), the complex function m _{ h }(h) is an analytic function for all $h\in \mathbb{C}$ except at the set of real values $\left\{0,\frac{1}{{\lambda}_{1}},\dots ,\frac{1}{{\lambda}_{q}}\right\}$ and the points where z(h) = 0.
5.1 Support of eigenvalues
Theorem 3
For the exponentially weighted window defined in Theorem 2, under the assumptions of Theorem 1, the complement of support of eigenvalues, is the set of values of x = z(h) on the vertical axis where $\frac{\mathit{\text{dz}}\left(h\right)}{\mathit{\text{dh}}}>0$ for some $h\in \mathbb{R}$, where z(h) is defined in (47).
Proof 4
Let S _{ F } denotes the support of the function F ^{ R }(x) and ${S}_{F}^{c}$ shows its complement. To prove Theorem 3, first we show that for any $x\in {S}_{F}^{c}$, there exist a $h\in \mathbb{R}$ where $\frac{\mathit{\text{dz}}\left(h\right)}{\mathit{\text{dh}}}>0$. Then, we prove the converse, i.e. if $\frac{\mathit{\text{dz}}\left(h\right)}{\mathit{\text{dh}}}>0$ for some $h\in \mathbb{R}$, then x=z(h) is a real number outside the support of eigenvalues.
From (5), we see that S _{ F } consists of points on the real axis where Im {m(x+i y)} tends to a positive number when y → 0^{+}. Thus to find S _{ F }, we must determine such subintervals on the real axis, or equivalently we can determine ${S}_{F}^{c}$ by finding the intervals on the real axis where lim y → 0^{+}{m(x+i y)} is real. Consider any x _{1} and x _{2} such that $({x}_{1},{x}_{2})\subset {S}_{F}^{c}\subset {\mathbb{R}}^{+}$. According to the definition of Stieltjes transform in (4), m(z) and $u\left(z\right)={c}_{0}(1+\mathit{\text{zm}}(z\left)\right)={c}_{0}\int \frac{\mathrm{\lambda dF}\left(\lambda \right)}{\lambda z}$ are both real and well defined for any z ∈ (x _{1}, x _{2}). In addition $\frac{\mathit{\text{du}}}{\mathit{\text{dz}}}={c}_{0}\int \frac{\mathrm{\lambda dF}\left(\lambda \right)}{{(\lambda z)}^{2}}$ is nonnegative on this interval. Thus u(z) is a real invertible function on (x _{1},x _{2}), and its inverse z(u) is also real and increasing on the interval $\left(u\right({x}_{1}),u({x}_{2}\left)\right)\in \mathbb{R}$, i.e. $\frac{\mathit{\text{dz}}}{\mathit{\text{du}}}>0$.
Lemma 2
For any given $z\in {\mathbb{R}}^{+}$, the function h(u,z) in (45) is monotonically increasing versus $u\in \mathbb{R}$.
Proof
Defining $h(0,z)=\frac{\gamma 1}{z}=\underset{u\to 0}{\mathrm{lim}}h(u,z)$, the function h(u,z) is continuous for all $u\in \mathbb{R}$ and all $z\in {\mathbb{R}}^{+}$, and for all $z\in {\mathbb{R}}^{+}$ we have □
Since $\frac{\mathit{\text{dz}}}{\mathit{\text{dh}}}=\frac{\mathit{\text{dz}}}{\mathit{\text{du}}}\frac{\mathit{\text{du}}}{\mathit{\text{dh}}}$ and $\frac{\mathit{\text{du}}}{\mathit{\text{dh}}}=\frac{{c}_{0}}{M}\sum _{i=1}^{q}\frac{{k}_{i}{\lambda}_{i}}{{({\lambda}_{i}h+1)}^{2}}$ are positive for all $h\in \mathbb{R}$, Lemma 2 implies that the signs of $\frac{\mathit{\text{dz}}}{\mathit{\text{du}}}$ and $\frac{\mathit{\text{dz}}}{\mathit{\text{dh}}}$ are identical. Thus if z is an increasing function of $u\in \mathbb{R}$, it is also an increasing function of $h\in \mathbb{R}$ as well, and vice versa, i.e., the intervals for which z is increasing versus u is equal to the intervals for which z is increasing versus h. This proves the direct part of the theorem.
To prove the converse part, consider that Theorem 3 implies that $\frac{\mathit{\text{dz}}\left({h}_{0}\right)}{d{h}_{0}}$ is real and nonnegative for some ${h}_{0}\in \mathbb{R}$. Since z(h) and m _{ h }(h) are both real at point h = h _{0}, it is sufficient to show that the point h _{0} belongs to the boundary of D _{ h }. In this case, as the function m(h) is continuous in the complex plane (excluding few points as stated after (49)), we conclude that lim_{ y → 0} ^{+} Im {m(h _{0}+i y)} = Im {m(h _{0})} = 0. To show that h _{0} is on the boundary of D _{ h }, we prove that the points in the vicinity of h _{0} in the positive complex plane, belong to D _{ h }. Let {h _{ n }} be any complex sequence with positive imaginary part converging to h _{0} as N → ∞. Since z(h) is continuous, the sequence {z _{ n }} = {z(h _{ n })} exists and converges to z(h _{0}).
Lemma 3
Let z(h) be an analytic function of h over an open set G, and h(t) ∈ G be a differentiable curve at t. Then if $\frac{\mathit{\text{dz}}\left(h\right)}{\mathit{\text{dh}}}$ is a positive real number, we have $arg\left\{\frac{d}{\mathit{\text{dt}}}z\left(h\right)\right\}=arg\left\{{h}^{\prime}\right(t\left)\right\}$.
Proof 5
This lemma is obtained from the Chain rule; since z(h(t)) is differentiable at t and $\frac{d}{\mathit{\text{dt}}}z\left(h\right(t\left)\right)={h}^{\prime}\left(t\right){z}^{\prime}\left(h\right)$. Thus for positive real $\frac{\mathit{\text{dz}}\left(h\right)}{\mathit{\text{dh}}}$, the argument of $\frac{d}{\mathit{\text{dt}}}z\left(h\right)$ and h ^{ ′ }(t) are the same. □
We use Lemma 3 which implies that if $\frac{\mathit{\text{dz}}\left(h\right)}{\mathit{\text{dh}}}$ is positive and real at the point h = h _{0} then $arg\left\{\frac{d}{\mathit{\text{dt}}}z\left(h\right)\right\}=arg\left\{{h}^{\prime}\right(t\left)\right\}$ for any differentiable curve h(t) at t = t _{0} where h(t _{0}) = h _{0}, i.e. the slope of the curve h(t) in the complex plane is the same as the slope of z(h(t)) at h(t) = h _{0}. Now for sufficiently large n, consider the line L _{ n } = h(t) = (1 − t)h _{0} + t h _{ n }, 0 ≤ t ≤ 1 in ${\mathbb{C}}^{+}$ which originates from h _{0} and ends at h _{ n }. The transformation of L _{ n }, z(L _{ n }), is also a line in the positive imaginary part of complex plane with the same slope as L _{ n }, as we have supposed that $\frac{\mathit{\text{dz}}\left(h\right)}{\mathit{\text{dh}}}\approx {z}^{\prime}\left({h}_{0}\right)$ for the points on L _{ n }. Thus the point z _{ n } also lies in the positive complex plane. In the other words, for sufficiently large n, the sequence {z _{ n }} lies in ${\mathbb{C}}^{+}$ ; hence the sequence {h _{ n }} is in D _{ h }. Finally, we conclude that the Stieltjes transform is defined on any such sequences and the sequence of Stieltjes transform {m(z _{ n })} = {m _{ h }(h _{ n })} is also in ${\mathbb{C}}^{+}$ for those values of n and converges to m _{ h }(h _{0}) which is a real number. Thus z(h _{0}) is outside the support of eigenvalues. □
Remark 2
We must note that we use a different approach in proving Theorem 3 comparing with proof exists for the rectangular window case where the Stieltjes transform m(z) has the explicit inverse[15]. This approach is very simple and can be used in other cases where the Stieltjes transform is expressed explicitly or implicitly as a function of z.
Theorem 3 states that in order to find the support of eigenvalues, we could first find the intervals on the real line where z(h) is increasing. In a sufficiently small vicinity of these intervals on the positive imaginary part of the complex plane, it is discussed in the proof that the imaginary part of z(h) is also positive for all h in this vicinity, therefore this vicinity lies in D _{ h }. Having a closer look at Figure 5, we find that ${D}_{h}\subset {\mathbb{C}}^{+}$ approaches real axis only for some values of h which can be easily studied that these are the intervals for which z ^{′}(h)>0. Thus according to this theorem the support of eigenvalues consists of three disjoint intervals for the setting of Figure 5.
5.2 Limiting spectral distribution
In the noise only case, we find an explicit equation for the l.s.d. of the exponentially weighted SCM employing Lambert W function. However in the signal plus noise case, the l.s.d. can not be obtained explicitly and should be calculated numerically using (5) and (47). It is the same as the rectangular window case where the l.s.d of noise only data has M–P distribution, however there is no explicit equation for the signal plus noise case.
To find the imaginary part of the Stieltjes transform, one could alternatively find the complex roots with positive imaginary part of the inverse function z(m) for all z in the support of the eigenvalues, i.e., z ∈ S _{ F }. Since the imaginary parts of m(z) and h(z) have the same sign and there is no explicit expression for z(m), we find the complex roots of z(h) using (47) and (48) for any real x _{ h } = z(h)∈S _{ F }, where Re {h}∈(h _{ b−},h _{ b+}), b∈{1,…,s}. This can be done by finding ν = Im{h} for which Im {z(h)} = 0. By inserting the calculated h in (49), we obtain the Stieltjes transform for x _{ h }∈S _{ F }. Finally F ^{ R }(x) is obtained using (5). According to Proposition 1, for any $z\in {\mathbb{C}}^{+}$ there exists a unique h satisfying (47) and (48), thus the above procedure results in the desired value of h and m.
6 Simulation results
7 Conclusion
In this article the l.s.d. of SCM in the case of weighted windowed data has been studied. Defining the effective length of a window, we have approximated the distribution of the eigenvalues in the weighted window case with that of a Wishart matrix, when the number of samples are much more than array dimension. Also the connectivity condition for coefficients of the window has been developed to avoid fragmentation of the support of eigenvalues in the noise only data. For the exponential window, we have derived an exact expression for the l.s.d. of SCM which has excellent agreement with the simulation results. We have also introduced a way to analyze the support and distribution of eigenvalues in the signal plus noise data cases. The results of this work could be used in design and improvement of detectors and estimators based on weighted windowed data especially when an exponential window is employed.
Endnotes
^{a}From $\frac{w{\sigma}^{2}}{1+\mathit{\text{cwm}}{\sigma}^{2}}\frac{1}{\mathit{\text{cm}}}{\sum}_{i=1}^{I}{(\mathit{\text{cwm}}{\sigma}^{2})}^{i}=\frac{1}{\mathit{\text{cm}}}\frac{{(\mathit{\text{cwm}}{\sigma}^{2})}^{I+1}}{1+\mathit{\text{cwm}}{\sigma}^{2}}$, we get $\frac{w{\sigma}^{2}}{1+\mathit{\text{cwm}}{\sigma}^{2}}\frac{1}{\mathit{\text{cm}}}{\sum}_{i=1}^{I}{(\mathit{\text{cwm}}{\sigma}^{2})}^{i}=\frac{1}{\left\mathit{\text{cm}}\right}\frac{\mathit{\text{cwm}}{\sigma}^{2}{}^{I+1}}{1+\mathit{\text{cwm}}{\sigma}^{2}}\le \frac{\left\mathit{\text{cm}}{}^{I}\rightw{\sigma}^{2}{}^{I+1}}{1\beta}$.
^{b}The Lambert W function [33], ω (x) is also called the Omega function and is the solution of ω e ^{ ω } = z for any complex number z. This equation is not injective, thus the function ω(z) is multivalued and has a set of different branches named ω _{ k }(z) for any integer k. For real values of z, there exist two real valued branches of Lambert W function ω _{0} (z) and ω _{−1} (z) which take on real values for $z\in [\frac{1}{e},\infty )\cup [\frac{1}{e},0)$ and complex values, otherwise. The function ω _{0}(z) is referred to as the principal branch of the Lambert W function and shown by ω(z) for simplicity.
Declarations
Acknowledgments
This work is supported in part by Iran Telecommunication Research Center (ITRC).
Authors’ Affiliations
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