## 3 Proposed Uplink NOMA  
### Proposed Scheme  
Unlike orthogonal multiple access systems, channel estimates in uplink NOMA systems might not be accurate even in high SNR environments due to the interference from other devices, and it is desirable to alleviate the performance degradation caused by channel estimation errors. In this paper, we propose an uplink NOMA scheme in which every other symbols after symbol repetition are complex-conjugated before multiplying the spreading pattern, as described in Figure 2.  
Let \(\mathbf{I}_{even}\) (\(\mathbf{I}_{odd}\)) be the \(L\times L\) matrix in which the even (odd) diagonal elements are 1 and the others are 0. Notice that \(\mathbf{I}_{even}+\mathbf{I}_{odd}=\mathbf{I}\), where \(\mathbf{I}\) is the \(L\times L\) identity matrix. In the proposed scheme, the transmitted signal \(\mathbf{t}_{k}\) for the \(k^{th}\) device can be written as follows:  
\[\mathbf{t}_{k}=\mathbf{I}_{odd}\mathbf{s}_{k}x_{k}+\mathbf{I}_{even}\mathbf{ s}_{k}x_{k}^{*} \tag{17}\]If an inaccurate channel estimate \(e^{i(\theta_{m}+\Delta\theta_{m})}\) with phase error \(\Delta\theta_{m}\) is used instead of \(e^{i\theta_{m}}\), the channel compensated symbol can be expressed as:  
\[ \begin{array}{l}\mathbf{I}_{odd}e^{-j(\theta_{m}+\Delta\theta_{m})}\mathbf{s }_{m}^{H}\mathbf{r}+\mathbf{I}_{even}(e^{-j(\theta_{m}+\Delta\theta_{m})} \mathbf{s}_{m}^{H}\mathbf{r})^{*}\\ =L\sqrt{P_{m}}x_{m}\cos(\Delta\theta_{m})\\ +\sum\limits_{k=1,k\
eq m}^{K}\sqrt{P_{k}}(I_{odd}s_{m}^{H}I_{odd}s_{k}e^{j( \theta_{k}-\theta_{m}-\Delta\theta_{m})}+\mathbf{I}_{even}\mathbf{s}_{m}^{T} \mathbf{I}_{even}\mathbf{s}_{k}^{*}e^{-j(\theta_{k}-\theta_{m}-\Delta\theta_{ m})})x_{k}\\ +(\mathbf{I}_{odd}\mathbf{s}_{m}^{H}e^{-j(\theta_{m}+\Delta\theta_{m})}+ \mathbf{I}_{even}\mathbf{s}_{m}^{T}e^{j(\theta_{m}+\Delta\theta_{m})}) \mathbf{n}\end{array}  \tag{18}\]  
since:  
\[\mathbf{I}_{odd}\mathbf{s}_{m}^{H}\mathbf{I}_{odd}\mathbf{s}_{m}e^{-j\Delta \theta_{m}}+\mathbf{I}_{even}\mathbf{s}_{m}^{T}\mathbf{I}_{even}\mathbf{s}_{m }^{*}e^{j\Delta\theta_{m}}=\frac{L}{2}e^{-j\Delta\theta_{m}}+\frac{L}{2}e^{j \Delta\theta_{m}}=L\cos(\Delta\theta_{m}). \tag{19}\]  
Let us define:  
\[\rho_{L/2}\equiv E\Big{\{}\Big{|}\mathbf{I}_{odd}\mathbf{s}_{m}^{H}\mathbf{I} _{odd}\mathbf{s}_{k}\Big{|}_{2}\Big{\}}=E\Big{\{}\Big{|}\mathbf{I}_{even} \mathbf{s}_{m}^{H}\mathbf{I}_{even}\mathbf{s}_{k}\Big{|}_{2}\Big{\}}(m\
eq k) \tag{20}\]  
and:  
\[\eta\equiv\frac{\rho_{L}}{2\rho_{L/2}}. \tag{21}\]  
In Equation (21), \(\eta\) may depend on the characteristics of the spreading patterns. Note that, if random sequences are used for spreading patterns, then Equation (1) holds and \(\eta=1\). From Equation (18), the SINR of each real signal in a QPSK-modulated symbol with the channel estimation phase error \(\Delta\theta_{m}\) using the proposed scheme, denoted as \(\gamma_{proposed}(\Delta\theta_{m})\), can be written as:  
\[ \begin{array}{l}\gamma_{proposed}(\Delta\theta_{m})=\frac{L^{2}P_{m}}{2\rho_{L/ 2}\sum\limits_{k=1,k\
eq m}^{K}P_{k}+L\omega^{2}}\cos^{2}(\Delta\theta_{m})\\ =\frac{L^{2}P_{m}}{\eta\rho_{L}\sum\limits_{k=1,k\
eq m}^{K}P_{k}+L\omega^{2}} \cos^{2}(\Delta\theta_{m})\end{array}  \tag{22}\]  
Figure 2: Proposed uplink NOMA. and the BER before channel decoding for the proposed scheme with \(\Delta\theta_{m}\), denoted as \(p_{proposed}(\Delta\theta_{m})\), can be written as:  
\[p_{proposed}(\Delta\theta_{m})=Q\left(\sqrt{\frac{L^{2}P_{m}}{\eta\rho_{L}\sum \limits_{k=1,k\
eq m}^{K}P_{k}+L\sigma^{2}}}\cos(\Delta\theta_{m})\right). \tag{23}\]  
If \(\eta\) is substantially smaller than \(1\), and channel estimation is perfect, i.e., \(\Delta\theta_{m}=0\), then:  
\[p_{proposed}^{(\eta<1)}(0)=Q\left(\sqrt{\frac{L^{2}P_{m}}{\eta\rho_{L}\sum\limits _{k=1,k\
eq m}^{K}P_{k}+L\sigma^{2}}}\right)<p_{perfect} \tag{24}\]  
and the proposed scheme does not need to be used. On the other hand, if \(\eta\) is \(1\) or close to \(1\), then Equation (23) can be rewritten as:  
\[