# An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing

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## Abstract

**:**

## 1. Introduction

## 2. Matrix Theory in Modern Wireless Communication Systems

#### 2.1. SVD for Modeling MIMO Channels

#### 2.2. Matrix Representation of OFDM

#### Matrix Analysis of OFDM

**Λ**is a diagonal matrix whose diagonal elements stand for the eigenvalues of $\mathbf{H}$ and $\mathbf{U}$ is a unitary matrix whose rows correspond to the columns of DFT matrix $\mathbf{W}$ in (5). In other words, $\mathbf{U}={\mathbf{W}}^{H}$. Thus, one can infer the following relations:

## 3. Matrix Applications in Signal Estimation Theory

#### 3.1. Cholesky Decomposition for Whitening the Noise

**Σ**is not diagonal) and $\mathbf{\theta}$ represents the unknown vector to be estimated. Knowledge of the probability density function (PDF) $p(\mathbf{x};\mathbf{\theta})$ lies at the basis for several very well-known and used estimators such as the minimum variance unbiased estimator (MVUE) and the maximum likelihood estimator (MLE) [12]. In the present application, $p(\mathbf{x};\mathbf{\theta})$ stands for a multivariate Gaussian PDF of the form:

**Σ**to be full rank and consider its Cholesky decomposition, i.e., $\mathsf{\Sigma}=\mathbf{D}{\mathbf{D}}^{T}$, and multiply with the Cholesky factor ${\mathbf{D}}^{-1}$ both sides of (10), we obtain

#### 3.2. SVD for Least-Squares Based Estimation

**Σ**can be represented as

#### 3.3. Matrix Inverse Lemma for Derivation of the Recursive Least-Squares Filter

- (1)
- Choose a large positive number α, and initialize ${\mathsf{\Sigma}}_{0}=\alpha \mathbf{I}$ and ${\widehat{\mathbf{h}}}_{0}=\mathbf{0}$.
- (2)
- For $n=1,2,\cdots ,N$, sequentially update$$\begin{array}{cc}& {\mathbf{k}}_{n}=\frac{{\mathsf{\Sigma}}_{n-1}{\mathbf{a}}_{n}}{1+{\mathbf{a}}_{n}^{H}{\mathsf{\Sigma}}_{n-1}{\mathbf{a}}_{n}},\hfill \\ & \u03f5\left(n\right)=x\left(n\right)-{\mathbf{a}}_{n}^{H}{\widehat{\mathbf{h}}}_{n-1},\hfill \\ & {\widehat{\mathbf{h}}}_{n}={\widehat{\mathbf{h}}}_{n-1}+\u03f5\left(n\right){\mathbf{k}}_{n},\hfill \\ & {\mathsf{\Sigma}}_{n}={\mathsf{\Sigma}}_{n-1}-{\mathbf{k}}_{n}{\mathbf{a}}_{n}^{H}{\mathsf{\Sigma}}_{n-1}.\hfill \end{array}$$

#### 3.4. Matrix Theory in Sensor Array Signal Processing

**Φ**, which can be addressed by exploring the eigenvalue decomposition of the sensor array covariance matrix. The interested reader is referred to [14,26,28] for more details about ESPRIT and other matrix factorization based algorithms for sensor array estimation.

#### 3.5. Random Matrix Theory in Signal Estimation and Detection

## 4. Matrices in Image Processing

#### 4.1. Block Transform Coding

#### 4.2. Wavelet Representation

## 5. Compressive Sensing

**Ψ**is the basis matrix with columns $[{\mathbf{\psi}}_{0},\cdots ,{\mathbf{\psi}}_{N-1}]$. The signal $\mathbf{x}$ is called K-sparse if only K coefficients of $\mathbf{s}$ are non-zero and the remaining $N-K$ coefficients are roughly zero (or can be truncated to zero). Thus, in order to process the transform coding, the locations of the K non-zero coefficients must be coded, which results in an overhead. Moreover, the $N-K$ zero coefficients still need to be computed even though they will be discarded.

**Φ**such that $\mathbf{y}=\mathsf{\Phi}\mathbf{x}$ preserves most of the information embedded in $\mathbf{x}$. Second, find an algorithm to recover $\mathbf{x}$ from only M samples of $\mathbf{y}$.

#### 5.1. Good Sensing Matrices

**Φ**is good for compressive sensing and reconstruction. The reconstruction seems to be impossible at first glance since the problem is ill-conditioned ($M<N$). However, it is now well-known that one can reconstruct the sparse signal from a limited number of measurements if the sensing matrix satisfies the restricted isometry property (RIP) [39,40,41]. A matrix $\mathsf{\Phi}\in {\mathbb{R}}^{M\times N}$ is said to satisfy the RIP of order $K<M$ with isometry constant ${\delta}_{K}\in (0,1)$ if

**Φ**are i.i.d. Gaussian random variables with zero means and variance $1/N$, and $M\ge cKlog(N/K)$ where c is a small constant, then

**Φ**satisfies RIP with a very high probability. Therefore, the signal can be recovered from only $M\ge cKlog(N/K)\ll N$ variables. Other available selections for robust sensing matrices are discussed in [42,44].

#### 5.2. Compressive Signal Reconstruction

**Φ**, basis matrix

**Ψ**and compressed signal $\mathbf{y}$. The ultimate goal of compressive sensing is to find the possible sparsest representation of the signal. Therefore, the ideal reconstruction problem can be expressed via the ${l}_{0}$ norm as follows:

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Wang, X.; Serpedin, E.
An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing. *Algorithms* **2016**, *9*, 68.
https://doi.org/10.3390/a9040068

**AMA Style**

Wang X, Serpedin E.
An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing. *Algorithms*. 2016; 9(4):68.
https://doi.org/10.3390/a9040068

**Chicago/Turabian Style**

Wang, Xu, and Erchin Serpedin.
2016. "An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing" *Algorithms* 9, no. 4: 68.
https://doi.org/10.3390/a9040068