# An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing
## 3 Matrix Applications in Signal Estimation Theory
### Matrix Theory in Sensor Array Signal Processing  
Sensor array signal processing focuses mainly on signal enumeration and source location applications, and presents a huge importance in many domains such as radar, sonar and underwater surveillance. The classic problem in sensor array signal processing is to detect and locate the radiating sources given the temporal and spatial information collected from the sensors. The goal of this section is to introduce the basic framework of sensor array signal processing and illustrate the fundamental role of matrix theory in representing and addressing the detection and estimation problems associated with the radiating sources. First consider the one-source-one-sensor scenario shown in Figure 5. It is proved in [26] that under the far-field and narrowband assumptions, the output of a sensor can be modeled as follows:  
\[x(t)=g(\theta)s(t)e^{-j\mathbf{r}^{T}\mathbf{k}},\]  
where \(g(\theta)\) is a function of the direction of arrival (DOA) \(\theta\), and it represents the frequency response of the sensor. Variable \(s(t)\) represents the complex envelope of the source signal, and \(\mathbf{r}\) and \(\mathbf{k}\) stand for the radius vector and the wave vector, respectively. The magnitude of \(\mathbf{k}\) is defined as \(|\mathbf{k}|=\alpha=2\pi/\lambda\), where \(\lambda\) is the wavelength. As shown in Figure 5, in a two-dimensional plane, we have \(\mathbf{k}=\alpha(\cos\theta,\sin\theta)^{T}\) and \(\mathbf{r}=(x,y)^{T}\). Thus, the output of the sensor can be simplified to  
\[x(t)=g(\theta)e^{-j\alpha(x\cos\theta+y\sin\theta)}s(t)=a(\theta)s(t), \tag{28}\]  
where \(a(\theta)=g(\theta)e^{-j\alpha(x\cos\theta+y\sin\theta)}\). For the one-source-\(L\)-sensor case, the array output vector can be expressed as  
\[\mathbf{x}(t)=\mathbf{a}(\theta)s(t), \tag{29}\]  
where \(\mathbf{x}(t)=[x_{1}(t),x_{2}(t),\cdots,x_{L}(t)]^{T}\) denotes the output of each sensor and \(\mathbf{a}(\theta)=[a_{1}(\theta),a_{2}(\theta),\cdots,a_{L}(\theta)]^{T}\) stands for the steering vector and captures the sensor array geometry information. Two common array geometries, termed as the uniform linear array (ULA) and uniform circular array (UCA), are depicted in Figure 6. For ULA, \(\mathbf{r}_{l}=[(l-1)d,0]^{T}\), and for UCA, \(\mathbf{r}_{l}=R[\cos(2\pi(l-1)/L),\sin(2\pi(l-1)/L)]^{T}\). Additionally, if we assume that all the sensors have the same frequency response, \(g_{1}(\theta)=\cdots=g_{L}(\theta)=g(\theta)\), the ULA steering vector takes the form:  
\[\mathbf{a}_{\text{ULA}}(\theta)=g(\theta)[1,e^{-j\mu d\cos\theta},\cdots,e^{- j\mu(L-1)d\cos\theta}]^{T}, \tag{30}\]  
and the UCA admits the form:  
\[ \begin{split}\mathbf{a}_{\text{UCA}}(\theta)=& g(\theta)[e^{-j\mu R\cos\theta},e^{-j\mu R[\cos(2\pi/L)\cos\theta+\sin(2 \pi/L)\sin\theta]},\cdots,\\ & e^{-j\mu R[\cos(2\pi(L-1)/L)\cos\theta+\sin(2\pi(L-1)/L)\sin \theta]}]^{T},\end{split}  \tag{31}\]  
Figure 5: One-sensor one-source array. Furthermore, if \(N\) source signals with DOA \(\theta_{n}\), \(n=1,2,\cdots,N\), are superimposed on an \(L\)-dimensional array (\(N\)-source-\(L\)-sensor), the array output vector can be expressed as  
\[\mathbf{x}(t)=\sum_{n=1}^{N}\mathbf{a}(\theta_{n})\mathbf{s}_{n}(t),\]  
or compactly as  
\[\mathbf{x}(t)=\mathbf{A}(\boldsymbol{\theta})\mathbf{s}(t), \tag{32}\]  
where \(\mathbf{x}(t)=[x_{1}(t),\cdots,x_{L}(t)]^{T}\in\mathbb{R}^{L}\), \(\mathbf{A}(\boldsymbol{\theta})=[\mathbf{a}(\theta_{1}),\cdots,\mathbf{a}( \theta_{N})]\in\mathbb{R}^{L\times N}\) and \(\mathbf{s}(t)=[s_{1}(t),\cdots,s_{N}(t)]^{T}\in\mathbb{R}^{N}\). In the presence of an additive noise \(\boldsymbol{\
u}(t)\), the sensor array equation can be expressed as:  
\[\mathbf{x}(t)=\mathbf{A}(\boldsymbol{\theta})\mathbf{s}(t)+\boldsymbol{\
u}( t). \tag{33}\]  
The central goal of array signal processing is to locate the source signals, or equivalently, to estimate the DOAs \(\boldsymbol{\theta}\) given the observed data \(\mathbf{x}(t)\) at the sensors.