Multi User CDMA System Model

Let's have CDMA System with $K$ Users indexed by $j=1 \cdots K$. Let Each User Transmits BPSK Modulated Signal Simultaneously. Then the Transmitted Signal for one such User $j$ can be Represented as:

$$S_j(t)= \sqrt{2 P_j}\,c_j(t)\,b_j(t)\,Cos(\omega_c t+\theta_j)$$ Where,

$$c_j(t)=\sum_{n=-\infty}^{\infty}\,c_j^{(n)}\,p_{T_c}(t-n\,T_c)$$ is the $j$th User Signature Waveform, with the Signature bits

$$c_j^{(n)} \in {-1,+1}$$

$$b_j(t)=\sum_{n=-\infty}^{\infty}\,b_j^{(n)}\,p_T(t-n \, T)$$  is the BPSK Modulated Signal for User $j$


$$b_j^{(n)} \in {-1,+1}$$ and

$$T=N\,T_c$$ Where $N$ is the Spreading Factor or Processing Gain. The Received Signal at the Receiver can be Represented as

$$r(t)= \sum_{j=1}^{K} \sqrt{2 P_j}\,c_j(t-\tau_j)\,b_j(t-\tau_j)\,Cos(\omega_c t+\phi_j) + \eta(t)$$ Where $\tau_j$ is Relative Time offset and $$\tau_j \in \left [ 0 \: T \right ]$$ $\phi_j$ is Phase Offset such that $$\phi_j  \in \left [ 0 \: 2 \pi \right ]$$ and $$\phi_j=\theta_j-\omega_c \tau_j$$ $\eta(t)$ is Zero Mean AWGN Process with PSD $N_0$