# An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing
## 3 Matrix Applications in Signal Estimation Theory
### Random Matrix Theory in Signal Estimation and Detection  
Suppose we are given a parameter to be estimated with a \"population size\" of \(N\) and \(n\) observations of the population. Many estimation methods proposed in literature rely on the asymptotic behavior that the number of observations is much larger than the population size, i.e., \(n/N\rightarrow\infty\). However, in modern signal processing, this assumption sometimes is not met in practice. For instance, in biological networks, the estimate generally consists of a large number of DNAs and the population size may also grow as the number of observations increases. Moreover, in radar applications, a large number of sensors are used to detect a fast moving object with observations available only during a short time period. Therefore, under such a set-up, the proposed estimators may perform poorly if the condition \(n/N\rightarrow\infty\) is not satisfied. Recently, a novel estimation method based on the large dimensional random matrix theory has been proposed to cope with this situation. The random matrix theory originated from the work of [PERSON] [29] in 1955 and was later generalized to a large extent by [PERSON] and [PERSON] [30] in 1967. Recently, a considerable amount of work based on random matrix theory has emerged in the literature of signal processing, wireless communications and information theory. Consider a simple example from [31], where \(\mathbf{x}\in\mathbb{C}^{N}\) is a complex random vector with zero mean and covariance matrix \(\mathbf{R}\). Given \(n\) observations of \(\mathbf{x}\), we will consider a classic estimator of \(\mathbf{R}\) in the form of the sample covariance matrix \(\mathbf{R}_{n}=\frac{1}{n}\sum\limits_{i=1}^{n}\mathbf{x}_{i}\mathbf{x}_{i}^{ H}\). Because of the law of large numbers, \(||\mathbf{R}_{n}-\mathbf{R}||_{2}\xrightarrow{\text{a.s.}}0\) as \(n\rightarrow\infty\), where \(||\cdot||_{2}\) denotes the spectral norm of a matrix, i.e., the absolute value of the largest eigenvalue if the matrix is Hermitian, and \(\xrightarrow{\text{a.s.}}\) stands for almost sure convergence. By taking the special case where \(\mathbf{R}=\mathbf{I}_{N}\), it was shown in [31] that if \(n\rightarrow\infty\) but \(n/N\) does not tend to zero, i.e., \(n/N=1/2\), the spectral norm of \(\mathbf{R}_{n}-\mathbf{I}_{N}\) is at least greater than 1. The problem lies in the fact that the random eigenvalue distribution \(F^{\mathbf{R}_{n}}\) of \(\mathbf{R}_{n}\) does not converge to a Dirac mass measure at 1 [31]. However, it is observed that in many scenarios, it converges to a limit distribution as \(N,n\rightarrow\infty,N/n=c\). In the special case where \(\mathbf{R}=\mathbf{I}_{N}\), the limit distribution is referred to as the Marcenko-Pastur distribution [30]. The derivation of the eigenvalue distribution for such large dimensional matrices brings the most fundamental tool of random matrix theory, namely the Stieltjes transform, and it is also a necessary prior step for parameter estimation and detection problems. Furthermore, the knowledge of the eigenvalue distribution for large dimensional matrices alone plays a paramount role in some wireless communication problems, such as computing or approximating the capacity of multi-antenna based channels. The interested reader can refer to [32; 33; 34; 35] for more details about applications of random matrix theory in wireless communications. In the following, we present a parameter estimation problem where the conventional sample covariance method fails for comparable population size and number of observations. In this case, an estimation method based on the spectral properties of large dimensional random matrices, referred to as the G-estimation method [36] can be employed to improve the performance. Rather than analyzing the complicated mathematical details of the random matrix theory and G-estimation method, our aim is instead to illustrate how the random matrix theory can be applied in a typical application.