# Differential Characteristics Based Iterative Multiuser Detection for Wireless Sensor Networks
## 2 Several Multiuser Detection Methods for DS-UWB Systems  
Suppose \(K\) users transmit signals simultaneously in a DS-UWB system. In this binary phase shift key (BPSK) modulated DS-UWB system, the received signal can be expressed as:  
\[r(t)=\sum_{k=1}^{K}\sqrt{E_{k}}S_{k}(t)d_{k}+n(t)\quad 0\leq n\leq T \tag{1}\]where \(E_{k}\) denotes the bit energy of the \(k\)th user; \(S_{k}(t)\) denotes the unit-energy signature waveform which is constructed by the pseudo random sequences of the \(k\)th user; \(d_{k}\in\{+1,\;-1\}\) denotes the bit value of \(k\)th user; \(n(t)\) denotes the noise with power spectral density \(N_{0}\), and \(T\) denotes the bit interval. After a bank of matched filters, the output of \(k\)th user sampled at \(T\) is achieved by the following expression:  
\[y_{k}=\int\limits_{0}^{T}r(t)S_{k}(t)\mathrm{d}t=\sqrt{E_{k}}d_{k}+\sum \limits_{ \begin{array}{c}j=1\\ j\
eq k\end{array} }^{K}\sqrt{E_{j}}\rho_{jk}d_{j}+n_{k} \tag{2}\]  
where the self-correlation of user \(k\) is normalized to 1, i.e., \(\rho_{kk}=\int\limits_{0}^{T}S_{k}(t)S_{k}(t)\mathrm{d}t=1\), the cross correlation of user \(j\) and \(k\) is \(\rho_{jk}=\int\limits_{0}^{T}S_{j}(t)S_{k}(t)\mathrm{d}t\), and the noise at the output of \(k\)th user is \(n_{k}=\int\limits_{0}^{T}S_{k}(t)n(t)\mathrm{d}t\). In principle, it is assumed that \(\rho_{ii}\gg\rho_{ij},\;i\
eq j\). The matched filter outputs of the \(K\) users can be written in vector form as:  
\[\mathbf{y}=\left[y_{1}\,y_{2}\cdots y_{K}\right]^{\mathrm{T}}=\mathbf{RAd}+ \mathbf{n} \tag{3}\]  
where \(\mathbf{R}\) is the correlation matrix of the signature waveforms, i.e., \(\mathbf{R}_{ij}=\rho_{ij}\), \(\mathbf{A}=\mathrm{diag}\big{[}\sqrt{E_{1}}\,\sqrt{E_{2}}\cdots\sqrt{E_{K}} \big{]}\), \(\mathbf{d}=\left[d_{1}\,d_{2}\cdots d_{K}\right]\), and \(\mathbf{n}\) is the \(K\times 1\) zero-mean additive white Gaussian noise vector. The symbol decisions after matched filters are given by:  
\[\mathbf{b}=\mathrm{sgn}(\mathbf{y}) \tag{4}\]  
As is shown in (2), each element of \(\mathbf{b}\) is interfered by other \(K\)-1 elements of \(\mathbf{d}\) due to \(\mathbf{R}_{ij}\
eq 0\) (when \(i\
eq j\)). Thus various multiuser detection algorithms were proposed to eliminate the bad effect of multiple access interference. On the basis of matched filter, optimal multiuser detection takes advantages of maximum likelihood sequence estimation algorithm. The likelihood function can be expressed as:  
\[p(\mathbf{y}|\mathbf{b})=\exp(\frac{-\frac{1}{2}(\mathbf{y}-\mathbf{RAb})^{ \mathrm{T}}(\sigma^{2}\mathbf{R})^{-1}(\mathbf{y}-\mathbf{RAb})}{(2\pi)^{K/2} \sigma\Big{|}\mathbf{R}\Big{|}^{1/2}}) \tag{5}\]  
where \(\mathbf{I}\,\mathbf{R}\,\mathbf{I}\) denotes the determinant of \(\mathbf{R}\). the maximum likelihood symbol decisions can be written as:  
\[\mathbf{b}_{\mathrm{OMD}}=\arg\max_{\mathbf{b}}\Big{\{}2\mathbf{b}^{\mathrm{ T}}\mathbf{Ay}-\mathbf{b}^{\mathrm{T}}\mathbf{ARAb}\Big{\}} \tag{6}\]  
The above maximization problem is a combinatorial optimization problem which is known to be NP-hard: its computational complexity increases exponentially with the number of user in a WSN. This \(O(2^{K})\) implementation complexity required by optimal multiuser detection makes it impractical for real system. Consequently, various low complexity multiuser detection algorithms are proposed to achieve the tradeoff problem between computational complexity and BER performance. Optimal multiuser detection represents, however, a basis for comparison for other suboptimal detectors. Linear multiuser detection is an important class of low complexity multiuser detection algorithms. Linear multiuser detectors form the output \(\mathbf{b}_{linear}\) based on various linear filters.