## 2 System and Channel Models
### Channel Model for Mobile Wireless Sensor Networks  
For purposes of analysis, in this paper, we consider a flat Rayleigh fading channel model with path loss, transmit correlation and receive correlation. The results can be extended to frequency selective fading channels with the use of orthogonal frequency division multiplexing (OFDM). Under these assumptions, the received signal at a target MS can be represented in discrete-time as:  
\[\mathbf{y}\ =\ \mathbf{H}\mathbf{s}+\mathbf{n} \tag{1}\]  
where \(\mathbf{y}\) is the \((N\times 1)\) received signal vector, \(\mathbf{s}\) is the \((M\times 1)\) transmitted signal vector, \(\mathbf{n}\) is the \((N\times 1)\) independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian noise vector with variance \(N_{0}\) and \(\mathbf{H}=[\mathbf{h}_{1}\mathbf{h}_{2}\cdots\mathbf{h}_{M}]\) is a \((N\times M)\) channel matrix, where \(\mathbf{h}_{m}=[h_{m,1}h_{m,2}\cdots h_{m,N}]^{T}\) is the channel impulse response (CIR) vector from the \(m\)-th transmit antenna of the assigned SEN to the MS. Here, we omit MS and time indices without loss of generality. Note that the columns of \(\mathbf{H}\) are a function of the transmit antennas and sensor nodes chosen for transmission (this is not explicitly indicated in the notation, but should be apparent in the manuscript). The channel correlation is assumed to have a Kronecker structure [36]. Further, the channels between different sensor nodes are assumed to be uncorrelated. With these assumptions, the correlation between  
Figure 1: Mobile wireless multimedia sensor networks (WMSNs) with multi-node (MN) multiple input and multiple output (MIMO). the CIR's of the \(m_{1}\)-th transmit/\(n_{1}\)-th receive and \(m_{2}\)-th transmit/\(n_{2}\)-th receive antenna pairs can be represented [36] as:  
\[E\{h_{m_{1},n_{1}}h_{m_{2},n_{2}}^{*}\}= \begin{cases}0,&\text{different SEN}\\ \sigma_{m}^{2}\eta_{\text{SEN}m,m_{1},m_{2}}\rho_{\text{SEN}m,n_{1},n_{2}},& \text{same SEN}m\text{ ($m$=1,2)}\end{cases}  \tag{2}\]  
where \(E\{\}\) is the expectation operator and \(\eta_{\text{SEN}m,m_{1},m_{2}}\)\((\rho_{\text{SEN}m,n_{1},n_{2}})\) is the transmit (receive) antenna correlation between antenna \(m_{1}\) and \(m_{2}\) of SEN\(m\) (\(n_{1}\) and \(n_{2}\) of a target MS), where \(\eta_{\text{SEN}m,m_{1},m_{2}}\)=1 (\(\rho_{\text{SEN}m,n_{1},n_{2}}\)=1) when \(m_{1}=m_{2}\) (\(n_{1}=n_{2}\)). Then, the correlation matrix, \(\mathbf{Q}\), of the CIR vector can be decomposed into transmit and received correlation matrices, \(\mathbf{R}_{t}\) and \(\mathbf{R}_{r}\)[36], as:  
\[\mathbf{Q} =\{vec(\mathbf{H})vec(\mathbf{H})^{H}\} \tag{3}\] \[=E\begin{cases} \begin{vmatrix}\sigma_{1}^{2}&\sigma_{1,2}&\cdots& \sigma_{1,M}\\ {(\sigma_{1,2})}^{*}&\sigma_{2}^{2}&\cdots&\sigma_{2,M}\\ \vdots&\vdots&\ddots&\vdots\\ {(\sigma_{1,M})}^{*}&{(\sigma_{2,M})}^{*}&\cdots&\sigma_{M}^{2}\\ \end{vmatrix} \end{vmatrix}\otimes\mathbf{R}_{r}\] \[=\mathbf{R}_{t}\otimes\mathbf{R}_{r}\]  
where \(vec(\mathbf{X})\) is a vector made by stacking columns of matrix \(\mathbf{X}\), \(\otimes\) is the Kronecker product and:  
\[E\{\mathbf{h}_{m}\mathbf{h}_{m}^{H}\}=\sigma_{m}^{2}\mathbf{R}_{r} \tag{4}\] \[E\{\mathbf{h}_{m_{i}}\mathbf{h}_{m_{j}}^{H}\}=\sigma_{m_{i},m_{j }}\mathbf{R}_{r}= \begin{cases}0_{N,N},&\text{from different SEN}\\ p_{1}^{2}\eta_{\text{SEN}1,m_{i},m_{j}}\mathbf{R}_{r},&\text{from SEN1}\\ p_{2}^{2}\eta_{\text{SEN}2,m_{i},m_{j}}\mathbf{R}_{r},&\text{from SEN2}\end{cases} \]  
where \(0_{A,B}\) is the \((A\times B)\) zero matrix and \(p_{1}^{2}\) and \(p_{2}^{2}\) represent the average power from each sensor node. Note that \(p_{1}\) and \(p_{2}\) account for the path gain difference between the two sensor nodes. In most SN MIMO channel models, \(p_{1}=1\). In the MN case, however, the path gain must be retained to account for differences in loss due to different distances between sensor nodes.