## 2 System Model  
In this section a concrete discrete-time model of a DS-CDMA system is constructed for the discussion of the problem formulation. The transmitter and receiver structures of DS-CDMA systems involve \(K\) simultaneous users. We consider the DS-CDMA model as a synchronous one, which is a base band model with fading channel [1, 6]. Assuming that the system has \(K\) users, the signal is sent by user \(k\) as follows:  
\[x_{{}_{k}}\left(t\right)=\sum_{m=0}^{M-1}b_{{}_{k}}\left(m\right)c_{{}_{k}} \left(t-m\,T\right) \tag{1}\]  
which contains the information of \(M\) symbols \(b_{{}_{lm}}\). This symbol \(b_{{}_{lm}}\) denotes the \(m\mathrm{th}\) symbol of the \(k\mathrm{th}\) user. \(c_{{}_{k}}\left(\cdot\right)\) is the \(k\mathrm{th}\) use's binary chip sequence, _i.e._, the spreading code, supported by \(\left[0,T\right)\), where \(T\) is the symbol duration. The signal passed through channel which is assumed to be fixed during one symbol duration:  
\[r\left(t\right)=\sum_{m=0}^{M-1}\sum_{k=1}^{K}A_{{}_{k}}b_{{}_{k}}\left(m \right)c_{{}_{k}}\left(t-m\,T\right)+n\left(t\right) \tag{2}\]  
where \(A_{{}_{k}}\) is the channel gain factor of the \(k\mathrm{th}\) user; \(M\) is the number of symbols per user; \(K\) is the number of users; \(n\left(t\right)\) denotes the additive white Gaussian noise with zero mean and variable variance; The chip sequence length (processing gain) is \(C=T\!\!\left/\!T_{{}_{c}}\right.\), where \(T_{{}_{c}}\) is chip duration. Since the chip sequence \(c_{{}_{k}}\left(t\right)\) is now continuous by definition, it includes not only the binary chips \(c_{{}_{k}}\left(i\right)\), but also a chip waveform \(p\left(t\right)\). More precisely:  
\[c_{{}_{k}}\left(t\right)=\sum_{i=0}^{C-1}c_{{}_{k}}\left(i\right)p\left(t-i\, T_{{}_{c}}\right) \tag{3}\]  
where \(p\left(t\right)\) is supported by \(\left[0,T_{{}_{c}}\right]\) only. This paper assumes a rectangular waveform for each user. Continuous-to-discrete time conversion of the above model can be realized by a chip-matched filter, which is a simple integrate-and-dump device. Using chip-rate sampling, which means integrating over a chip-duration \(T_{{}_{c}}\):  
\[r\left(i\right)=\int_{{}_{iT_{{}_{-}}-T_{{}_{c}}}}^{iT_{{}_{c}}}p\left(t- \left(i-1\right)T_{{}_{c}}\right)r\left(t\right)dt \tag{4}\]  
The sampled data, which have been obtained by chip-matched filtering using a processing window size of one symbol, can be written as:\[\mathbf{r}\big{(}m\big{)}=\sum_{k=1}^{K}A_{{}_{b}}{}_{k}\big{(}m\big{)}\mathbf{c}_{{} _{k}}+\mathbf{n}\big{(}m\big{)} \tag{5}\]  
The chip sequence vector \(\mathbf{c}_{{}_{k}}\) has a size of \(C\times 1\), and the noise vector \(\mathbf{n}_{{}_{m}}\) has a size of \(C\times 1\). With a simple manipulation, we can get a compact representation for the data [1,6]:  
\[\mathbf{r}\big{(}m\big{)} =A_{{}_{1}}{}^{b}_{1}\big{(}m\big{)}\mathbf{c}_{{}_{1(C\times 1)}}+...+A_{{}_{K}}{}_{b}\big{(}m\big{)}\mathbf{c}_{{}_{K(C \times 1)}}+\mathbf{n}\big{(}m\big{)}_{{}_{(C\times 1)}} \tag{6}\] \[=\big{[}\mathbf{c}_{{}_{1}}A_{1},...,\mathbf{c}_{{}_{K}}A_{{}_{K }}\big{]}_{{}_{C\times 1}}\ \mathbf{b}\big{(}m\big{)}_{{}_{(K\times 1)}}+\mathbf{n} \big{(}m\big{)}_{{}_{(C\times 1)}}\] \[=\mathbf{Gb}\big{(}m\big{)}+\mathbf{n}\big{(}m\big{)}\]  
where \(\mathbf{c}_{{}_{k}}=\big{[}c_{{}_{1k}},c_{{}_{2k}},...,c_{{}_{K}}\big{]}^{T}\), \(\mathbf{n}_{{}_{m}}=\)\(\big{[}n_{{}_{1}},n_{{}_{2}},...,n_{{}_{C}}\big{]}^{T}\), and \(\mathbf{b}\big{(}m\big{)}=\)\(\big{[}b_{{}_{1}}\big{(}m\big{)},b_{{}_{2}}\big{(}m\big{)},...,b_{{}_{K}}\big{(}m \big{)}\big{]}^{T}\). \(\big{(}\big{)}^{T}\) denotes transpose. Furthermore, we can obtain:  
\[\mathbf{X}=\mathbf{G}\mathbf{B}+\mathbf{N} \tag{7}\]  
where \(\mathbf{X}=\)\(\big{[}\mathbf{r}\big{(}1\big{)},...,\mathbf{r}\big{(}M\big{)}\big{]}\), \(\mathbf{B}=\)\(\big{[}\mathbf{b}\big{(}1\big{)},...,\mathbf{b}\big{(}M\big{)}\big{]}\), and \(\mathbf{N}=\)\(\big{[}\mathbf{n}\big{(}1\big{)},...,\mathbf{n}\big{(}M\big{)}\big{]}\).