 Review
 Open Access
Noncommutative large entries for cognitive radio applications
 Antonia Maria Masucci^{1}Email author and
 Merouane Debbah^{2}
https://doi.org/10.1186/168714992012167
© Masucci and Debbah; licensee Springer. 2012
 Received: 24 May 2011
 Accepted: 10 May 2012
 Published: 10 May 2012
Abstract
Cognitive radio has been proposed as a solution for the problem of underutilization of the radio spectrum. Indeed, measurements have shown that large portions of frequency bands are not efficiently assigned since large pieces of bandwidth are unused or only partially used. In the last decade, studies in different areas, such as signal processing, random matrix theory, information theory, game theory, etc., have brought us to the current state of cognitive radio research. These theoretical advancements represents a solid base for practical applications and even further developments. However, still open questions need to be answered. In this study, free probability theory, through the free deconvolution technique, is used to attack the huge problem of retrieving useful information from the network with a finite number of observations. Free deconvolution, based on the moments method, has shown to be a helpful approach to this problem. After giving the general idea of free deconvolution for known models, we show how the moments method works in the case where scalar random variables are considered. Since, in general, we have a situation where more complex systems are involved, the parameters of interest are no longer scalar random variables but random vectors and random matrices. Random matrices are noncommutative operators with respect to the matrix product and they can be considered elements of what is called noncommutative probability space. Therefore, we focus on the case where random matrices are considered. Concepts from combinatorics, such as crossing and noncrossing partitions are useful tools to express the moments of Gaussian and Vandermonde matrices, respectively. Our analysis and simulation results show that free deconvolution framework can be used for studying relevant information in cognitive radio such as power detection, users detection, etc.
Keywords
 Cognitive Radio
 Primary User
 Random Matrice
 Secondary User
 Cognitive Radio Network
1 Introduction
In the last decade, recent studies [1] have shown that future communication systems should be designed to be able to adapt to their environment in order to tackle the problem of the underutilization of a precious resource such as the radio spectrum. Measurements have shown that large portions of frequency bands are not efficiently used, that is, for most of the time, large pieces of bandwidth are unoccupied or partially occupied [2]. A possible solution, introduced by Mitola [3, 4], is represented by cognitive networks, that can be thought of as selflearning, adaptive and intelligent networks. In cognitive networks, unlicensed (secondary) systems improve spectral efficiency by sensing the environment and filling opportunistically the discovered holes spectrum (or white spaces) of licensed systems (primary), which have exclusive right to operate in a certain spectrum band [5]. The current development of microelectronics allows us to suppose that these wireless systems, for which the spectrum utilization will play a key role, will be realized in the near future. These systems provide an efficient utilization of the radio spectrum based on the methodology understandingbybuilding to learn from the environment and to adapt their parameters to statistical variations in the input stimuli [6].
The current development in cognitive radio research is the result of a multidisciplinary study that allows us to analyze different aspects of cognitive radio. We identify in signal processing, game theory, information theory, random matrix theory, etc., enabling areas for the development of cognitive radio.
Signal processing plays a major role in designing cognitive wireless networks, especially in spectrum sensing to identify spectrum opportunities and in the design of cognitive spectrum access to exploit the identified spectrum holes. We refer to spectrum sensing as the process where devices look for a signal in the presence of noise for a given frequency band. Several digital signal processing techniques, such as matched filtering, energy detection, and cyclostationary feature detection are analyzed [7, 8] to improve radio sensitivity and detect the presence of primary users. In [9], it is proved that the energy detector is an efficient spectrum sensing technique when the secondary user has limited information on the primary user's waveform, i.e., only the power of the local noise is known. The authors of [10] formulate the spectrum sensing problem as a nonlinear optimization problem, minimizing the interference to the primary user and meeting the requirement of opportunistic spectrum utilization. Cooperation between users follows as a consequence of the following constraints: (1) secondary users should not interfere with the primary transmissions and they should be able to detect the primary signal even if decoding the signal may be impossible [9]; (2) secondary users are in general not aware of the exact transmission scheme used by primary users. Cooperation among all cognitive users operating in the same band reduces the detection time and increases the overall agility with which cognitive users are able to shift bands [11–13]. Cooperation is designed in [14] as joint detection among all the cooperating users and in [15] as fusion center that makes the final decision about the occupancy of the band by fusing the decisions made by all cooperating users. In [16], cooperation is analyzed for the partial CSI (channel state information) scenario at the secondary users.
From a game theoretic point of view spectrum sharing may be considered as a competition. The importance of studying cognitive radio networks in a game theoretic framework is multifold. By modeling dynamic spectrum sharing between users as a game, users behaviors and actions can be analyzed in a formalized structure, where the theoretical results in game theory can be fully applied [17, 18]. The optimization of spectrum usage is generally a multiobjective optimization problem, which is very difficult to analyze and to solve. Moreover, game theory provides us game models that predict convergence and stability of networks [19]. In [20], a gametheoretic adaptive channel allocation scheme is proposed for cognitive radio networks. In particular, a game is formulated to analyze the selfish and cooperative behaviors of the players. The players of this game were the wireless nodes and their strategies were defined in terms of channel selection. In [21], the convergence dynamics of the different types of games in cognitive radio systems is studied. Then, a game theory framework is proposed for distributed power control to achieve agility in spectrum usage in a cognitive radio network.
Information theory is used to characterize the achievable rates in a cognitive radio network under different assumptions on how the secondary systems interfere with the primary ones. Fundamental understanding on the capacity of the cognitive systems are provided in [22–26]. Using recent results on random matrix theory, the authors of [27, 28] propose a new method for signal detection in cognitive radio, based on the eigenvalues of the covariance matrix of received signal at the secondary users. In [29], a spectrum sensing technique that relies on the use of multiple receivers to infer on the structure of the received signals using random matrix theory is proposed. The authors show that their technique is quite robust and does not require the knowledge of signal or noise statistics. These methods do not require any prior information on the primary signal or on the noise power. In [30, 31] two hypothesis tests allowing to detect the presence of an unknown transmitter using several sensors are proposed and random matrix theory is used to provide the error associated with both tests.
We recognize as a crucial point of cognitive radio development understanding how much it is possible to infer from the network with the knowledge of just few observations. In the current study, we use free probability theory, through the concept of free deconvolution, to handle the problem of retrieving useful information from the network with a limited number of observations. Free deconvolution, based on the moments method, has shown to be a interesting tool to attack this problem.
where S = [s_{1},..., s_{ K } ] is an n × K i.i.d zero mean Gaussian vector with variance $\frac{1}{K}$. In cognitive random networks, the number of samples K is of the same order as n, due to the high mobility of the network and to the fact that the statistics are considered to be the same within a K number of samples. Because of this, the use of classical asymptotic signal processing techniques is not more efficient since they require a number of samples K >> n. Therefore,our main problem consists in retrieving information within a window of limited samples. In this sense, free probability theory, through the concept of free deconvolution, is a very appealing framework for the study of cognitive networks. The main advantage of free deconvolution framework is that it provides us with helpful techniques to obtain useful informations from a finite number of observations. The deconvolution framework comes here from the fact that we would like to invert Equation (1) and express R with respect to $\widehat{\mathbf{R}}$, since we can only have access to the sample covariance matrix. This is not possible, however, one can compute the eigenvalues of R knowing only the eigenvalues of $\widehat{\mathbf{R}}$.
In the following, the general idea of free deconvolution is presented. We show how the moments method works in the case where scalar random variables are considered. However, since in practical situations systems are more complex, the parameters of interest are no longer scalar random variables and they need to be represented by random matrices. Therefore, we analyze the case where random matrices are considered. We analyze moments method for matrices which show the freeness property and we show that it can be used to propose algorithmic methods to compute moments of finite Gaussian random matrices. Moreover, we analyze the case of matrices for which freness does not hold: Vandermonde, Hankel, Toeplitz. In the end, we present applications showing how the moments method approach can be used for studying cognitive radio: power detection, users detection, etc. In last section, we discuss our results presenting conclusions and open problems.
2 Information plus noise model
with M and N K × n independent random matrices. We are interested in retrieving information about the transmitted signal from the received signal, more explicitly to obtain the eigenvalues of MM^{ H } from the eigenvalues of YY^{ H } and NN^{ H } . This is exactly the goal of deconvolution.
 (1)
Can one derive the eigenvalue distribution of A from those of A + B and B? If feasible in the large nlimit, this operation is named additive free deconvolution,
 (2)
Can one derive the eigenvalue distribution of A from those of AB and B? If feasible in the large nlimit, this operation is named multiplicative free deconvolution.
The techniques generally used to compute the operation of deconvolution in the large nlimit are the moments method [35] and the Stieltjes transform method [36]. Each of these methods has its advantages and its drawbacks. The moments method only works for measures with moments and characterizes the convolution only by giving its moments but it is easily implementable and, in many applications, one needs only a subset of the moments depending on the number of parameters to be estimated. Instead, the Stieltjes transform method works for any measure and it allows, when computations are possible, to recover the densities. Unfortunately, this method works only in very few cases, since the operations which are necessary are almost always impossible to implement in practice and combining patterns of matrices naturally leads to more complex equations for the Stieltjes transform and can only be performed in the large nlimit.
We analyze the concept of free deconvolution based on the moments method which uses the empirical moments of the eigenvalue distribution of random matrices to obtain information about the eigenvalues. The moments method has shown to be a fruitful technique in both the asymptotic and the finite setting to compute deconvolution, as well as the simplest patterns, sums and products, and products of many independent matrices.
3 Moments method
3.1 Scalar case
Therefore to obtain the distribution of X from the ones of X +Y and Y one can compute the cumulants of X by the formula c_{ n } (X) = c_{ n } (X + Y) − c_{ n } (Y) and then deduce the moments of X from its cumulants.
then, we obtain $\mathbb{E}\left[{X}^{n}\right]=\frac{\mathbb{E}\left[{\left(XY\right)}^{n}\right]}{\mathbb{E}\left[{Y}^{n}\right]}$. Therefore, using the moments approach, we can compute the moments of X.
The moments method for scalar random variables seems to be very straightforward, however in general we have more complex situations. The generalization to multiuser multiantenna communication systems has dramatically changed the nature of wireless communication problems. Furthermore, multidimensional stochastic problems need to be solved since cognitive devices are required to be simultaneously smarter and able to collaborate with one another. The random parameters in these problems are no longer scalar random variables but potentially vectors and matrices. The computation of deconvolution for random matrices is more complex than the scalar case and it is explained in the following.
3.2 Historical perspective
with x ≤ 2. In this way, the moments approach is shown to be a useful method for computing the eigenvalues distribution of classical known matrices.
When more than one matrix is considered, the concept of asymptotic freeness [38] leaves us to compute the eigenvalue distribution of sums and products of random matrices.
3.3 Free probability framework
Free probability theory [38] was introduced by Voiculescu in the 1980s in order to attack some problems related to operator algebras and it can be considered as a generalization of classical probability theory to noncommutative algebras. The analogy between the concept of freeness and the independence in classical probability leaves us to work with noncommutative operators like matrices that can be considered elements in what is called a noncommutative probability space. The algebra of Hermitian random matrices is a particular case of such a probability space, for which the random variables, i.e., the random matrices, do not commute with respect to the matrix product.
Given A, B n × n hermitian and asymptotically free random matrices such that their eigenvalues distributions converge to some probability measure µ_{ A } and µ_{ B }, respectively, then the eigenvalue distributions of A + B and AB converge to a probability measure which depends on µ_{ A } and µ_{ B }, called additive and multiplicative free convolution, and denoted by µ_{ A } ⊞ µ_{ B } and µ_{ A } ⊠ µ_{ B }, respectively.
Additive free deconvolution: The additive free deconvolution of a measure ρ by a measure ν is (when it exists) the only measure µ such that ρ = µ ⊞ ν. In this case, µ is denoted by µ = ρ ⊟ ν.
Multiplicative free deconvolution The multiplicative free deconvolution of a measure ρ by a measure ν is (when it exists) the only measure µ such that ρ = µ ⊠ ν. In this case, µ is denoted by µ = ρ ⊠ ν.
which means that we can express the moments of A + B and the moments of AB as a function of the moments of A and the moments of B. In other words, the joint distribution of A + B and the joint distribution of AB depend only on the marginal distributions of A and B.
Hence, when, for n → ∞, the moment ${m}_{\mathbf{A}}^{n,p}$ converges almost surely to an analytical expression ${m}_{\mathbf{A}}^{p}$ that depends only on some specific parameters of A (such as the distribution of its entries).^{b} Therefore, in the finite setting one is still able by recursion to express all the moments of A with respect only to the moments of A + B and B, or AB and B.
We will give a characterization of free deconvolution in terms of free cumulants, which are polynomials in the moments with a nice behaviour with respect to the freeness. The nomenclature comes from classical probability theory where corresponding objects are well known. There exists a combinatorial description of these classical cumulants, which depends on partitions of sets. In the same way, free cumulants can also be described combinatorially, the only difference to the classical case is the replacement of partitions by the so called noncrossing partitions [39].
Definition 3.3 A partition π of a set {1, 2,..., n} is a decomposition in subsets V_{ i }: π = {V_{1},..., V_{ r } } such that${\bigcup}_{i=1}^{r}{V}_{i}=\left\{1,\dots ,n\right\}$, with V_{ i } ≠ ∅ and V_{ i } ∩ V_{ j } = ∅ for all i ≠ j.
The set of all partitions of {1, 2,..., n} is denoted by $P\left(n\right)$, and V_{ i } are called blocks of π.
where ${c}_{\pi}={\prod}_{i=1}^{\left\pi \right}{c}_{\left{\pi}_{i}\right}$ when π = {π_{1},..., π_{ π } }.
Definition 3.4 A partition π of {1,..., n} is noncrossing if whenever we have four numbers 1 ≤ i < k < j < l ≤ n such that i and j are in the same block, k and l are in the same block, we also have that i, j, k, l belong to the same block.
where V_{ i } is the cardinality of the block V_{ i } . From (6) it follows that the first p cumulants can be computed from the first p moments, and viceversa.
The following characterization allows us to compute easily the additive free convolution using free cumulants.
The implementation of additive free deconvolution is based on the following steps: for the two matrices (A + B) and B, we first compute the free cumulants, then, considering the relation between the cumulants and the moments, we can obtain information about the distribution of the eigenvalues of A.
Hence, we can compute multiplicative free convolution by the following characterization.
In this case also, by a simple recursion, one can express D_{ p } from M_{ p } . It is clear how the operation of deconvolution can be viewed as operating on the moments: explicit expression for the moments of the Gram matrices associated to our models (sum or product of a deterministic matrix and a complex standard Gaussian matrix) are found, and are expressed in terms of the moments of the matrices involved. Hence, deconvolution means to express the moments, in this case of the deterministic matrices, in function of the moments of the Gram matrices.
Similar results are found when the Gaussian matrices are assumed to be square and selfadjoint. The implementation of the results is also able to generate the moments of many types of combinations of independent Gaussian and Wishart random matrices.
The algorithms are based on iterations through partitions and permutations and they give us rather complex expression. However, the author of [41] have generated Matlab codes based on concepts as partitions and permutations in order to implement the above results.
(the roots of which provide the eigenvalues of the associated matrix) can be fully characterized since its n − k coefficient is given by (− 1) ^{ k } ∏_{ k }(λ_{1},..., λ_{ n } ). In this way the entire characteristic polynomial can be computed, and the eigenvalues can also be found.
4 Non free case
In recent works, deconvolution, based on the moments method, has been analyzed when n → ∞ for some particular matrices A and B. For instance, when A is a random Vandermonde matrix and B is a deterministic diagonal matrix [43], or when A and B are two independent random Vandermonde matrices [44]. The authors in [43] developed analytical methods for finding moments of random Vandermonde matrices with entries on the unit circle and provide explicit expressions for the moments of the Gram matrix associated to the models considered. The explicit expressions found for the moments are useful for performing deconvolution. In these cases the moments technique has been shown to be very appealing and powerful in order to derive the exact asymptotic moments of "non free matrices". This type of matrices occurs in cognitive radio [45].
where the phases ω_{1},..., ω_{ M } are i.i.d. random variables in [0, 2π].
The asymptotic behaviour of random Vandermonde matrices is analyzed when N and M are large, both go to infinity at a given ratio $\frac{M}{N}\to c$, with c constant. The scaling factor $\frac{1}{\sqrt{N}}$ and the assumption that the entries lies on the unit circle guarantee that the analysis will give limiting asymptotic behaviour.
exists, we call it a Vandermonde mixed moment expansion coefficient.
These quantities do not behave exactly as cumulants, but rather as weights which tell us how a partition in the moment formula we present should be weighted. In this respect, the formulas presented for the moments are different from classical or free momentcumulant formulas, since these do not perform this weighting. The limits K_{ρ,ω}may not always exist, and necessary and sufficient conditions for their existence seem to be hard to find. In [43], it has been proved that the limit in (12) exists if the density of ω is continuous. The calculation is based on combinatorial computation using crossing partitions since the matrices are not free.
exist when$\frac{M}{N}\to c>0$, P_{ n }= lim_{N→∞}tr_{ M }(P^{ n }), ${P}_{\rho}={\prod}_{i=1}^{k}{P}_{{W}_{i}}$.
Remark 1. The fact that all moments exist is not enough to guarantee that there exists a limit probability measure having these moments. However, it is proved in [46] that the Carleman's condition is satisfied.
Uniform phase distribution plays an important role or Vandermonde matrices.
exists ∀ρ. Moreover, K_{ρ,u}satisfies the following properties

0 ≤ K_{ρ,u}≤ 1;

K_{ρ,u}are rational numbers ∀ ρ;

K_{ρ,u}= 1 ⇔ ρ is noncrossing partition;

Let${V}_{\omega ,n}=\underset{N\to \infty}{\text{lim}}\mathbb{E}\left[\mathsf{\text{t}}{\mathsf{\text{r}}}_{M}{\left({\mathbf{V}}_{\omega}^{H}{\mathbf{V}}_{\omega}\right)}^{n}\right]$(with V_{ ω }a Vandermonde matrix with phase distribution ω), then V_{u,n}≤ V_{ω,n}.
The importance of uniform phase distribution is also expressed by the following theorem.
The behaviour of Vandermonde matrices is different when the density of ω has singularities and depends on the density growth rates near the singularities points. Indeed, for the case of generalized Vandermonde matrices, whose columns do not consist of uniformly distributed power, it is possible to define mixed moment expansion coefficients but the formulas are more complex.
When many independent Vandermonde matrices are considered, the following relations hold.
exist when $\frac{M}{N}$, with σ= {σ_{1}, σ_{2}} = {{1, 3,... }, {2, 4,... }}, and "≤"denotes the refinement order, i.e., any block of ρ is contained within a block of σ.
In [44], the authors generalize the above results replacing convergence in distribution with almost sure convergence in distribution.
Such matrices are applied to cognitive radio in [45], where authors consider a scenario with a primary and a secondary user wish to communicate with their corresponding receivers simultaneously over frequency selective channels is considered. Under realistic assumptions that the primary user is ignorant of the secondary user's presence and that the secondary transmitter has no side information about the primary's message, the authors propose a Vandermonde precoder that cancels the interference from the secondary user by exploiting the redundancy of a cyclic prefix.
4.1 Toeplitz and Hankel matrices
The same strategy used to compute the moments of Vandermonde matrices can be used for Hankel and Toeplitz matrices.
where X_{ i } are i.i.d., realvalued random variables with unit variance.
Similar results hold for Hankel matrices.
where X_{ i } are i.i.d., realvalued random variables with unit variance.
Toeplitz and Hankel matrices are structured matrices used for compressive wideband spectrum sensing schemes [47, 48] and for direction of arrival estimation [49].
5 Application
5.1 Power estimation
5.2 Understanding the network in a finite time
In cognitive MIMO networks, one must learn and control the "black box" (for instance the wireless channel) with multiple inputs and multiple outputs within a fraction of time and with finite energy. The fraction of time constraint is due to the fact that the channel (black box) changes over time. Of particular interest is the estimation of the capacity within the window of observation.
where R_{ Y } is the covariance of the output signal and R_{ N } is the covariance of the noise. Therefore, one can fully describe the information transfer in the system by knowing only the eigenvalues of R_{ Y } and R _{ N }. Unfortunately, the receiver has only access to a limited number L of observations of y and not to the covariance of R _{ Y }. However, in the case where x and n are Gaussian vectors, y can be written as $\mathbf{y}={\mathbf{R}}_{Y}^{\frac{1}{2}}\mathbf{u}$ where u is an i.i.d standard Gaussian vector. The problems falls therefore in the realm of inference with a correlated Wishart model $\left(\frac{1}{L}{\sum}_{i=1}^{L}{\mathbf{y}}_{i}{\mathbf{y}}_{i}^{H}={\mathbf{R}}_{i}^{\frac{1}{2}}\frac{1}{L}{\sum}_{i=1}^{L}{\mathbf{u}}_{i}{\mathbf{u}}_{i}^{H}{\mathbf{R}}_{Y}^{\frac{1}{2}}\right)$.
5.3 Users detection
and θ_{1},..., θ_{ M } are the angles of the users and are supposed to be i.i.d. and uniform on [−α, α], d is the interspacing distance between the antennas of the ULA, and λ is the wavelength of the signal.
For estimation of the number of users M, we assume that the power distribution of P is known. Based on the knowledge of the power distribution, we are able to estimate the number of users in the system. Thanks to the moments method it is possible to estimate the moments of the sample covariance matrix in (25) from the moments of the power matrix P.
where${c}_{1}=\underset{N\to \infty}{\text{lim}}\frac{N}{K}$, ${c}_{2}=\underset{N\to \infty}{\text{lim}}\frac{M}{N}$and${c}_{3}=\underset{N\to \infty}{\text{lim}}\frac{M}{K}$.
Knowing the matrix P, we can compute the moments P_{ i } . From the moments P_{ i } , using the above expressions is possible to get the moments W_{ i } of the sample covariance matrix. We consider some candidate values of the number M of users. The estimate of M is chosen as the one which minimizes the sum of the square errors between the moments W_{ i } and the moments of the observed sample covariance matrix.
5.4 Wavelength detection
6 Conclusions and open problems
In the last decade, researchers and practitioners have devised cognitive radio as a possible solution for the problem of underutilization of the radio spectrum. These theoretical advancements in cognitive radio research have set up a solid base for practical applications and even further developments. However, still open questions need to be answered. In particular, in the current work, we use free probability theory, through the concept of free deconvolution, to attack the problem of retrieving useful information from the network with a limited number of observations. Free deconvolution, based on the moments method, has shown to be a interesting tool to tackle this problem. First, we show how the moments method works in the case where scalar random variables are considered. Then, since in practical situations systems are so complex that the parameters of interest need to be represented by vectors and matrices and can not be modeled by scalar random variables, we analyze the case where random matrices are considered. We propose algorithm method to compute the moments of various models such as Gaussian and Vandermonde matrices. Matlab codes for cognitive radio is developed to implement this algorithm method. In the applications free deconvolution framework can be used for retrieving relevant information such as power with which users send information, number of users, etc.
We have analyzed how free deconvolution framework works for random matrices and how random matrices behave differently depending on their structure. Different directions of research can be followed in this framework. In Vandermonde matrix model, the deconvolution techniques have been performed taking into account only diagonal matrices. It could be interesting to address the case of general deterministic matrices. In this way, correlation between users can be considered. The knowledge of the correlation could be a relevant element to improve the cooperation among the users in a cognitive system.
The extension of free deconvolution techniques to more general functions of matrices is a hard task. The difficulty is related to the fact that up to now there is not a general hypothesis that guarantees the application of free deconvolution to any random matrix. This extension can take into account more general models that represent more realistic situations.
For future perspective we would like also to take into account a second order analysis. The study of the covariance matrices can improve the accuracy of the estimations related to the free deconvolution framework.
Declarations
Acknowledgements
This study was supported by the AlcatelLucent within the AlcatelLucent Chair on Flexible Radio at Supélec. The work of the first author has been done during her PhD at AlcatelLucent Chair on Flexible Radio at Supélec supported by Microsoft Research through its PhD Scholarship Program.
Endnotes
^{a}An algebra is unital if it contains a multiplicative identity element, i.e., an element ${1}_{A}$ with the property ${1}_{A}\cdot x=x\cdot {1}_{A}=x$ for all elements x of the algebra. ^{b}Note that in the following, when speaking of moments of matrices, we refer to the moments of the associated measure. ^{c}Let X be a random variable and ${M}_{X}\left(z\right):={\sum}_{m=0}^{\infty}\phi \left({X}^{n}\right){z}^{n}$, we define the STransform of X as${S}_{X}\left(z\right)=\frac{1+z}{z}{M}_{X}^{<1>}\left(z\right)$
where (·) < − 1 > denotes the inverse (under composition) (i.e., ${M}_{X}^{<1>}\left({M}_{X}\left(z\right)\right)={M}_{X}\left({M}_{X}^{<1>}\left(z\right)\right)=z$). ^{d}A standard complex Gaussian matrix X has i.i.d. complex Gaussian entries with zero mean and unit variance (in particular, the real and imaginary parts of the entries are independent, each with zero mean and variance 1/ 2).
Authors’ Affiliations
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