## 2 Background  
In our CEVTI system, we consider the case of CDMA, in which there are \(N\) antennas (which we term users in the rest of this paper) in the base station. At first, each user will send a signal to its antenna within communication range. The signal experiences relay block fading [32]. We assume that the transmitted signal of each antenna is uniformly distributed. Please note that the transmitted signal is first coded and preprocessed. The code rate of each user is \(R_{c}\). We use \(C_{i}\) to denote the codeword set of user \(i\) after interleaving. Each codeword then is multiplexed with pilot symbols, the number of which is \(L_{p}\).  
In this section, we describe the background notations of CEVTI, assuming that in the wireless sensor network, there are \(M\) mobile users and \(N\) antennas for each BS. We use \(d_{i}\) to denote the transmitted symbol of mobile user \(i\).  
Suppose that each antenna has the state value of \(\{\pm 1\}\), and that we have defined \(d_{i}\) to denote the transmitted symbol of user \(i\), i.e., \(d_{i}\in\{\pm 1\}\). Therefore, \(d_{i}[l]\) is the \(l\)th transmitted signal, and column vector \(\mathbf{d}_{i}=[\mathbf{d}_{i,p},\mathbf{d}_{i,c}]^{T}=[d_{i}[0],...,d_{i}[L-1]]\) is the symbol vector transmitted by user \(i\). We use \(\mathbf{d}_{i,p}\) to denote the pilot symbol of user \(i\), the length of which is \(L_{p}\), and \(\mathbf{d}_{i,c}\in\mathcal{C}_{k}\) to denote the codeword of user \(i\), the length of which is \(L_{c}\). Obviously \(L=L_{c}+L_{p}\). For ease of reference, we treat both \(\mathbf{d}_{i}\) and \(\mathbf{d}_{i,c}\) as the codeword vectors of user \(i\). The transmitted symbol of user \(i\) is then modulated by a normalized sequence embedding \(s_{i}[l]\) of length \(N_{c}\). Through a channel, the transmitted signal then experiences channel gain, including noise, fading, shifting, etc. However, we assume that the channel gain remains constant in one block time and differs in different blocks.  
The vector \(d_{i}\in\mathcal{C}_{k}\) denotes the combination of the pilot symbol vector and the codeword vector; the length of the latter is \(L_{c}\). Therefore, the dimension of \(d_{i}\) is \(L\) (we may also refer to \(d_{i}\) as a codeword to simplify notation). Every symbol is first processed by a signature waveform, and then is further altered by the transmitted channel, including channel gains, block fading etc. We assume the within each block, consisting of \(L\) symbols, the channel influence is constant, while between blocks, channel gains are different.  
Next, we use \(\mathbf{r}[l]=[r_{1}[l],...,r_{N_{L}}[l]]^{T}\) to indicate the receiving signal at interval \(l\). So that the received signal after experiencing coding and channel gain is:  
\[\mathbf{r}[l]=\mathbf{S}[l]\mathbf{A}\mathbf{d}[l]+\mathbf{w}[l]=\mathbf{S}[l ]\mathbf{D}[l]\mathbf{a}+\mathbf{w}[l] \tag{1}\]  
where \(\mathbf{S}[l]\) is the spreading sequences of all users at signaling interval \(l\), of which \(\mathbf{a}\triangleq[a_{1},...,a_{N}]^{T}\) is the channel coefficient, and \(\mathbf{A}=\mathrm{diag}\{a_{1},...,a_{N}\}\) is the matrix of diagonalized channel coefficient, of which \(a_{i}\) is the channel coefficient value of user \(i\). Assuming a Rayleigh-fading channel and white Gaussian noise distribution, we have \(\mathbf{a}\sim\mathcal{N}(\mathbf{0},\mathbf{\Sigma_{a}})\).