## 2 Problem Statement  
### Some Typical MUD Algorithms for DS-UWB Systems  
Here, we let a \(K\)-user synchronous DS-UWB system under the additive white Gaussian noise (AWGN) channel, which could also be regarded as the results of the processing of rake receivers in the multipath channel. For this reason, we only give the multiuser DS-UWB model in the AWGN case, while it can be generalized to the multipath case easily. In this multiuser DS-UWB system, each information symbol is spread over multiple pulses using pseudo random sequences (PRS). The BPSK modulation is employed by each user and the transmitted signal of the \(k\)th user can be expressed as:  
\[S_{\
u}^{(k)}(t)=\sum\limits_{j=\infty}^{\infty}\sum\limits_{n=0}^{N-1}b_{j}^{ (k)}p_{n}^{(k)}w_{\
u}(t-jT_{f}-nT_{c}) \tag{1}\]  
where \(\{b/^{(k)}\}\) is the information symbols of the \(k\)th user, \(\{p_{n}^{(k)}\}\) denotes the PRS assigned to the \(k\)th user, \(T_{c}\) is the pulse repetition period (namely the chip period), \(T_{f}\)is the time duration of information symbol that satisfies \(T_{f}=N_{c}T_{c}\), and \(N_{c}\) is the length of PRS. Besides, w\({}_{\
u}(t)\) represents the transmitted pulse waveform that can be characterized as the second derivative of Gaussian pulse [20,21]:  
\[w_{\
u}(t)=[1-4\pi(\frac{t}{\tau_{m}})^{2}]\cdot\exp[-2\pi(\frac{t}{\tau_{m}}) ^{2}] \tag{2}\]  
where \(\tau_{m}\) is the parameter that controls the width of this pulse. If these \(K\)- users are all active, the whole received signal of this system is  
\[r(t)=\sum\limits_{k=1}^{K}A_{k}S_{\
u}^{(k)}(t)+n(t) \tag{3}\]  
where \(A_{k}\) is the amplitude of the \(k\)th received signal and \(n(t)\) represents the received noise modeled as the AWGN, with a normal distribution \(N(0,\sigma_{n}^{2})\). According to [46], the output of a bank of single-user matched filters (MFs), called a conventional detector (\(CD\)), can be expressed in the following matrix and vector forms:  
\[y=RAb+ny \tag{4}\]  
where \(R\) denotes the \(K\times K\) normalized cross-correlation matrix with the \(i\)-\(k\)th entry \(\rho_{\alpha}=\int\limits_{0}^{T_{c}}S_{\
u}^{(i)}(t)S_{\
u}^{(k)}(t)dt\), \(A\) is a \(K\times K\) diagonal matrix with the amplitude of the \(k\)th user's received signal \(\{A_{k}\}_{k=1,2,\ldots,K\}\) along the diagonal, b = (\(b_{1}\), \(b_{2}\), \(\ldots\), \(b\kappa\))\({}^{T}\)is the \(K\times 1\) vector containing the transmitted information bits from \(K\) users and the superscript \"T\" denotes a vector transpose operation. In addition, \(n\) = (\(n_{1}\), \(n_{2}\), \(\ldots\), \(n_{K}\))\({}^{T}\) is the \(K\times 1\) zero-mean Gaussian random noise vector with \(n_{i}=\int\limits_{0}^{T_{c}}n(t)S_{\
u}^{(k)}(t)dt\) and its covariance matrix equal to  
\[E[\texttt{nb}^{T}]=\sigma_{*}^{2}R \tag{5}\]  
Due to the crude assumption that the MAI can be modeled as a zero-mean Gaussian random variable (called \"Gaussian approximation\") for CD [47], its performance and multiuser capacity are limited. To improve this, [PERSON] proposed the optimum multiuser detection (OMD), which is based on the maximum likelihood (ML) criterion [35]. The objective function of OMD is given as:\[\hat{\mathbf{b}}=\arg\quad\max_{\mathbf{k}\in[-1,+1]^{k}}(2b^{T}\,dy-b^{T}\,ARAb) \tag{6}\]  
From Equation (6), it is obvious that the selection of this optimal solution \(\hat{\mathbf{b}}\) in the \(K\)-dimensional Euclidean solution space is generally a Non-deterministic Polynomial (NP) hard problem [37]. That is, the computational complexity of OMD is \(O\) (\(2^{K}\)), which grows exponentially with the number of active users. For this reason, despite its significant performance and capacity gains over CD, it is useless in practice. As a result, several sub-optimal MUD algorithms have been proposed to make the tradeoff between performance and complexity. Linear MUD algorithm is one kind of sub-optimal approach, which processes the output of CD through multiplying by a matrix \(M\).