# Differential Characteristics Based Iterative Multiuser Detection for Wireless Sensor Networks
## 3 Multiuser Detection Based on Differential Characteristics
### Differential Characteristics Calculation  
On the basis of the decision function of optimal multiuser detection, let:  
\[F(\mathbf{b})=\frac{1}{2}\mathbf{b}^{\mathrm{T}}\mathbf{ARAb}-\mathbf{b}^{ \mathrm{T}}\mathbf{Ay}=\frac{1}{2}\sum_{i=1}^{K}\sum_{j=1}^{K}A_{i}A_{j}\rho_{ ij}b_{i}b_{j}-\sum_{i=1}^{K}b_{i}A_{i}y_{i} \tag{10}\]  
From (10), it is clear that function \(F(\mathbf{b})\) is a quadratic form of vector \(\mathbf{b}\), which means this is a non-linear equation with the order of two. And it is not easy to solve the minimum of \(F(\mathbf{b})\) and the corresponding value of \(\mathbf{b}\). In addition, the function \(F(\mathbf{b})\) embraces cross-component \(b_{i}b_{j}\) (\(i,j=1,\,2,\,\ldots\), \(K\), \(i\
eq j\)), and we can't judge a bit independently by \(F(\mathbf{b})\). Hence, it is not appropriate for function \(F(\mathbf{b})\) to be the characteristic function. Figure 1: Diagram of the proposed multiuser detection algorithm. In this case, it is better to calculate the derivative of function \(F(\mathbf{b})\) to decrease the order of characteristic function. Calculating the partial derivation of (10), we get:  
\[\frac{\partial F}{\partial\mathbf{b}}=\mathbf{ARAb-Ay} \tag{11}\]  
By expanding (11), we get the \(K\)th-order linear equations given as follows:  
\[\left\{ \begin{array}{c}\frac{\partial F}{\partial b_{1}}=\sum\limits_{j=1}^{ K}A_{1}A_{j}\rho_{1j}b_{j}-A_{1}y_{1}\\ \frac{\partial F}{\partial b_{2}}=\sum\limits_{j=1}^{K}A_{2}A_{j}\rho_{2j}b_{ j}-A_{2}y_{2}\\ \hskip 28.452756 pt\cdots\\ \frac{\partial F}{\partial b_{K}}=\sum\limits_{j=1}^{K}A_{K}A_{j}\rho_{kj}b_{ j}-A_{K}y_{K}\end{array} \right. \tag{12}\]  
Because \(F(\mathbf{b})\) is a discrete equation, we cannot just setting the partial derivative described above equal to 0 and get the stationary point of \(F(\mathbf{b})\). Let \(L(b_{i})=\sum\limits_{j=1}^{K}A_{i}A_{j}\rho_{ij}b_{j}-A_{i}y_{i}\), \(i=1\), \(2\),..., \(K\). Substituting the candidate sequence \(\mathbf{b}\) into (12) (Results of conventional suboptimal multiuser detectors can be employed as the candidate sequence \(\mathbf{b}\). In this paper, the output of matched filter as shown in (4) is employed. ), there will be two situations as discussed below:  
1. No wrong bit in \(\mathbf{b}\) Based on the theory of extreme value, if multiple access interference is the only interference resource without AWGN and the elements in \(\mathbf{b}\) are all correct, the result of (12) strictly equals 0. Substituting (3) into \(L(b_{k})\), we can see that: \[L(b_{k})=-A_{k}n_{k}\sim N\Big{(}0,\,A_{i}^{2}N_{0}^{2}/2\Big{)}\quad k=1,2, \cdots,K\] (13) Apparently, \(L(b_{k})\) is a zero-mean Gaussian random variable. 2. Wrong bit exists in \(\mathbf{b}\) Suppose \(b_{i}\) (\(i=\)1, 2,..., \(K\)) is the wrongly received bit (in other words, \(-b_{i}\) is the correct bit), the other bits are correct. Substituting it to the \(i\)th equation of (12), we get: \[ \begin{array}{ll}L(b_{i})&=\sum\limits_{j=1}^{K}\quad A_{i}A_{j}\rho_{ij}b_{ j}-A_{i}y_{i}+A_{i}^{2}\rho_{ii}b_{i}=\sum\limits_{j=1}^{K}\quad A_{i}A_{j}\rho_{ij}b_{ j}-A_{i}y_{i}+A_{i}^{2}\rho_{ii}(-b_{i})+2A_{i}^{2}\rho_{ii}b_{i}\\ &j\
eq i\\ &=2A_{i}^{2}N_{0}b_{i}-A_{i}n_{i}\end{array} \] (14) As for \(k\) which is not equal to \(i\), we get: \[L(b_{k})=2A_{i}A_{k}\rho_{ik}b_{i}-A_{k}n_{k}\quad k=1,2,\cdots,K\quad k\
eq i.\] (15) From the above, we can get: \[L(b_{i})\,\sim N(2A_{i}^{2}b_{i},\frac{A_{i}^{2}N_{0}}{2})\] (16) \[L(b_{k})\,\sim N(2A_{i}A_{k}\rho_{ik}b_{k},\frac{A_{k}^{2}N_{0}}{2}) \quad k=1,2,\cdots,K\quad k\
eq i\] (17) According to (16) and (17), we can see when the \(i\)th bit of \(\mathbf{b}\) is wrong, then we get \(|L(b_{i})|>>|L(b_{k})|\), \(k=1\), \(2\),..., \(K\), \(k\
eq i\) with a large probability. Therefore, the function \(L(\mathbf{b})\)can obviously differentiate the wrong bits and the right bits through the absolute value of it. In addition, \(L(\mathbf{b})\) is a \(K\)th-order linear equations which can get the result of \(L(\mathbf{b})\) without complex computations.