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 [35], 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 [810]. 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.,
Some signal processing algorithms process a batch of data together and deal with the SCMR. In addition, the existing results in literature about the behavior of the eigenvalues mainly consider the rectangular window. However in a number of practical signal processing algorithms, the SCM is estimated by applying a window as follows
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 [2630]. 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.
Manipulating the joint PDF of the eigenvalues which is a very complex function is
not practical, particularly for large matrices. An alternative approach used in the
literature, is to employ the following empirical spectral distribution (e.s.d.) of
a square matrix
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
We assume that
where U = [U_{1},…,U_{N}] is an M × N matrix contains i.i.d. zeromean unitvariance complex Gaussian entries and
Definition 1
[15]Stieltjes transformm(z),
The inverse Stieltjes transform formula is as follows:
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
For a finite length window with length ofN, consider the matrix defined by
where e(z) is the unique solution of the following equation in
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
To show how this method works, we now consider the simplest case (where the distribution is well known) using a rectangular window and white Gaussian noise, i.e., W = I_{N × N} and A = σ^{2}I_{M × M}. In this case, we have dF^{W}(w) = δ(w − 1)dw and dF^{A}(x) = δ(x − σ^{2})dx, where δ(x) is the Dirac delta function. Thus with straightforward manipulations of (6) and (7), the Stieltjes transform is found to be the solution of
In this case, as expected the e.s.d. of the SCMR,
where
Now, let us consider an arbitrary window and white noise A = σ^{2}I_{M×M}. In this case from (6), (7) and dF^{A}(x) = δ(x − σ^{2})dx, we obtain
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].
Consider a window w_{i}>0 of length N and denote
where
Since 0 < c ≪ 1, for I = 2 and defining
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
Definition 2
The expression (12) represents the M–P distribution as in (8) for a rectangular window of length
with all coefficients equal tow_{e}. The average weightw_{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”.
Now, we compute the effective length of the triangular and the exponential windows.
A triangular window is defined by
The exponential window is very popular in signal processing applications due to its
simple implementation and is defined by w_{i}=w_{0}p^{i} for i=1,2,…, where p∈(0,1) and w_{0} is a normalization constant. We note that the exponential window is inherently an
infinite length window. Interestingly, in Theorem 1 the window length and the array
dimension jointly tend to infinity where
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.
Let S_{F} denote the support of the function F^{R}(x) and
The following lemma is the key to determine these intervals on real axis [15].
Lemma 1 ([32], Lemma 6.1)
For any c.d.f.F, letS_{F}denote its support and
wherez(m) is the inverse function ofm(z). Also conversely, for any realmin the domain ofz(m) if
This simply means that the support S_{F}, is the union of intervals on the vertical axis where z(m) is increasing for real values of m. According to (10), for noise only data z(m) can be written as follows
4.1 Discrete distribution function approach
Suppose the window consists of N_{d} distinct weights w_{i},i=1,…,N_{d}, each with multiplicity
Figure 1, represent a typical case of the function on the righthand side of (19). Lemma 1
states that the support of the distribution of eigenvalues is the complement of the
set of all values x ∈ R^{+} for which x = z(m) is increasing for real values of m, i.e.,
Figure 1. A typical representation of the function z(m) in (19) versus
For c < 1, i.e., where the length of the window is more than the array dimension,
Figure 2. A typical representation of the function z(m) in (19) versus
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.
In fact, it is desirable that the support of eigenvalues be as compact as possible.
To avoid such an undesirable fragmentation, the equation
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
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 a triangular window
Substituting f^{W}(w) in (18), we get
for
Again, we first use Lemma 1 and determine the support of eigenvalues (by plotting
z(m) for real m and finding the intervals on the vertical axis where z(m) is not increasing). Figure 3 plots the lower and upper boundaries of support of eigenvalues for a triangular window
for different values of c. It can be seen that the discrete distribution
Figure 3. Upper (values on the right) and Lower (values on the right) boundaries of the support of eigenvalues using the triangular window versus c .
For the exponential window, first we introduce the new parameter γ as the ratio of smallest to largest weights of the truncated exponential window.
The coefficients of the window can be redefined as as a function of γ as
where ⌊.⌋ is the floor function. This increasing staircase function takes values on
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.
Substituting f^{W}(w) in (18), in the asymptotic regime of Theorem 1 as γ→0, such that
for all
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
For the exponentially weighted window, the l.s.d. of SCM,f^{R}(x), is given by
and upper and lower boundaries of the support are
respectively, whereω_{k}(x) is the branch of Lambert W function^{b}[33]withk=−1 andk=0.
Proof 2
According to the Lemma 1, boundaries of the support of eigenvalues are the real solutions ofz^{′}(m) = 0, i.e., with some simple calculations, are the solutions of
Denotingy= ln(1+c_{0}σ^{2}m)−c_{0}−1, we obtain
This equation has two real solutionsm_{− }and m_{+ }expressed using Lambert W function as
Using (26), the boundariesz(m_{−}) andz(m_{+}) are obtained as in (28a) and (28b) which determine the support of eigenvalues as the interval
To obtain the l.s.d. of SCM, we should findm(z) with positive imaginary part for allz∈[z(m_{−}),z(m_{+})]. In (26), we denote
and obtain
Therefore, the solutions are
According to (16), (33), for the values ofzin the interval of the obtained support on the real axis, due to the properties of
the complex logarithm function, the imaginary part ofvis in [−Π,Π], thus only the branches withk = 0 andk = − 1 are acceptable solutions. It is easy to see that forz ∈ [z(m_{−}),z(m_{+})], the expression on the righthand side of (34) belongs to
Using the inverse formula in (18), the l.s.d of SCM is
Dropping the real terms inside the brackets and applying some simplifications, we obtain (27). □
We can define a second effective window length by employing and comparing the boundaries
of the support of eigenvalues in (28a) and in (9) for a rectangular window which is
only in terms of σ^{2} and c. Equating the length of the supports in (9) (28a), i.e., a_{+}−a_{−} = x_{+}−x_{−}, we can find a rectangular window to match the support as same as that of the exponential
window and define the length of this rectangular window as another effective length
for the exponential window. In some array signal processing applications, the effective
length of the exponential window has been considered to be
Figure 4. Effective length of the exponential window as a function of forgetting factor p.
Remark 1
In the economic literature, other methods have been proposed to approximate the spectral
density function of exponentially weighted financial covariance matrices for Portfolio
Optimization[34], [35]. These methods that are used in other articles (e.g., in[36], [37]) are based on numerical calculations rather than developing some closed form expressions.
Pafka et al.[34]supposed that the density of the eigenvalues is aproximated by
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
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,
From (6) and (7) we obtain
Substituting (25) in (39), we get
Substituting dF^{A}(a) in (7) changes the integral to a summation and we obtain e(z) as
According to Theorem 1, for any
This expression gives the implicit relation between m and z, which cannot be sorted to express m as an explicit function of z or conversely, z as a function of m. Defining the auxiliary variable/function u = c_{0} (1+zm(z)) which provides a bijective relation between e and m for all z≠0, we have
This equation reveals that the imaginary parts of u and e have the same signs. In addition since γ <1, c_{0} is real and using the properties of the complex logarithm function in (43), we deduce
that u always lies in a strip of the positive complex plane where its imaginary part is
less than Π, i.e., the domain of u is defined as D_{u} = {u0 < Im{u}<Π}. Equation (43) also provides a bijective relation between e and u, therefore according to Theorem 1 for any
Defining the second auxiliary variable/function as
and define D_{h} as its range for all
Proposition 1
The auxiliary variableh, as a function ofuandz, has some interesting properties as:
(1) halways lies in the subset
(2) forh∈D_{h}, zcan be explicitly expressed as a function ofh
(1) For any
Proof 3
The first property can is simply implied from (46) as the imaginary part ofh and uhave 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
Although z(h) in (47) is defined only for h∈D_{h}, it is an analytic function for all
Also using (46) and (47) we express m as a function of h as follows
for h∈D_{h}. Similar to z(h), the complex function m_{h}(h) is an analytic function for all
The inverse Stieltjes transform in (5) reveals that the l.s.d. depends on the behavior
of m(z) in the vicinity of the real axis, i.e. for
Figure 5. The range of the function h(z) for
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 ofx = z(h) on the vertical axis where
Proof 4
LetS_{F}denotes the support of the functionF^{R}(x) and
From (5), we see thatS_{F}consists of points on the real axis where Im {m(x+iy)} tends to a positive number wheny → 0^{+}. Thus to findS_{F}, we must determine such subintervals on the real axis, or equivalently we can determine
Lemma 2
For any given
Proof
Defining
Since
To prove the converse part, consider that Theorem 3 implies that
Lemma 3
Let z(h) be an analytic function ofh over an open set G, and h(t) ∈ Gbe a differentiable curve at t. Then if
Proof 5
This lemma is obtained from the Chain rule; since z(h(t)) is differentiable at t and
We use Lemma 3 which implies that if
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 transformm(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
Employing Theorem 3 and plotting z(h) for h < 0 one can determine the support of eigenvalues of the SCM in the asymptotic regime.
The function z(h) has asymptotes at
Figure 6 shows a typical representation of the support of eigenvalues in the signal plus noise
case when c_{0} = 0.1 and the covariance matrix has four distinct eigenvalues 5, 3, 2, 1 with multiplicities
α_{1} = α_{2} = α_{3} = 0.1 and α_{4} = 0.7. It can be studied that in general, z(h)→ + ∞ as h→0^{−} and z(h)→0^{ + } as h→−∞ and also analogous with the rectangular window case [38] the number of extrema of z(h) (counting the multiplicities) is even and are the solutions of
Figure 6. Support of eigenvalues in the signal plus noise case using exponential window with c_{0} = 0.1 for four distinct eigenvalues λ_{4} = 5,λ_{3} = 3,λ_{2} = 2,λ_{1} = 1 with multiplicitiesα_{1} = α_{2} = α_{3} = 0.1 andα_{4} = 0.7.
Figure 7 illustrates the same curves for c_{0} = 0.4, i.e., the forgetting factor p is reduced compared with Figure 6. We observe that the smaller the forgetting factor of the exponential window the
larger the width of the subintervals associated to distinct eigenvalues. In some cases,
some of adjacent subintervals may overlap, e.g. in Figure 7, the support associated to λ_{4} = 5 and λ_{3} = 3 have overlap whereas the two smaller ones are separable. Figure 8 shows
Figure 7. Support of eigenvalues in the signal plus noise case using exponential window with c_{0} = 0.4 for four distinct eigenvaluesλ_{4} = 5, λ_{3} = 3,λ_{2} = 2,λ_{1} = 1 with multiplicitiesα_{1} = α_{2} = α_{3} = 0.1 andα_{4} = 0.7.
Figure 8. The range of the function h(z) for
Figure 9, demonstrates the support of l.s.d. of SCM identified using Theorem 3 for c_{0} ∈ {0.1,0.3} and λ_{2} ∈ [1,4] with multiplicity of α_{2} = 0.1 and λ_{1} = 1 with multiplicity of α_{1} = 0.9. We observe that for large values of λ_{2}, the support associated with two eigenvalues are disjoint intervals. However, these two disjoint intervals become connected as the distance between λ_{2} and λ_{1} reduces. In practice, the value of c_{0} determines the window shape and has an important impact on the width of these intervals and on the location of the breakpoint. The location of breakpoint determines the capability of the window to identify two distinct eigenvalues. Figure 9 illustrates that the larger the value of c_{0}, the smaller the breakpoint of the support, i.e., by increasing p, we may be able to separate closer eigenvalues.
Figure 9. Support of eigenvalues of the exponentially weighted windowed data forc_{0} = 0.25, λ_{1} = 1 and λ_{2}∈ [1,4].
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
6 Simulation results
In Figure 10, we plot the density functions and a histogram to show the accuracy of the derived
l.s.d.’s in this article for an array with a finite dimension M = 20 and an exponential window with p = 0.975. In this case we have c_{0} = −M ln(p) = 0.5. In addition, in all our simulations, we used γ = 10^{−8}; thus according to the definition of γ in the truncated exponential window, we have
Figure 10. Distribution of eigenvalues using the exponential window for M = 20 and p = 0.975 .
In Figure 11, the l.s.d of exponentially windowed data is plotted for different values of p ∈ {0.95,0.97,0.98,0.99,0.995} and M = 20. We observe that as p tends to one, the eigenvalues become more concentrated around their true values. This is because the effective length of the window increases as p approaches 1.
Figure 11. Distribution of eigenvalues using the exponential window for M = 20 and p ∈ {0.95,0.97,0.98,0.99,0.995} .
Figure 12 shows the spectral distribution for an exponentially windowed SCM in a case where
the eigenvalues are 12, 7, 3,1 with the same multiplicity ratios
Figure 12. Distribution of eigenvalues of exponentially windowed data forc_{0} = 0.1 andc_{0} = 0.4 where the covariance matrix has 4 distinct eigenvalues 12 , 7 , 3 , 1 with the same multiplicity ratios
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
^{b}The Lambert W function [33], ω (x) is also called the Omega function and is the solution of ωe^{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
Competing interests
The authors declare that they have no competing interests.
Acknowledgments
This work is supported in part by Iran Telecommunication Research Center (ITRC).
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