## 2 Conventional Uplink NOMA
### Channel Estimation Errors  
If pilot signals are also superimposed on the shared resource, the channel estimation performance can be degraded unless large amount of resources are allocated to pilots or the number of concurrently random-accessing devices is small. Assuming that same spreading patterns are used for pilot symbols, the transmitted signal \(\mathbf{t}_{k}^{(p)}\) for the pilots of the \(k^{th}\) device can be written as:  
\[\mathbf{t}_{k}^{(p)}=\mathbf{s}_{k}x_{k}^{(p)} \tag{9}\]  
where \(x_{k}^{(p)}\) is the pilot signal with unit magnitude. The BS receives signals from the \(K\) devices, expressed as:  
\[\mathbf{r}^{(p)}=\sum\limits_{k=1}^{K}\sqrt{P_{k}}e^{\mathrm{i}\theta_{k}}t_{ k}^{(p)}+\mathbf{n}^{(p)} \tag{10}\]  
where \(\mathbf{r}^{(p)}\) is the received signal vector and \(\mathbf{n}^{(p)}\) is the corresponding noise with variance \(\sigma^{2}\). The BS performs de-spreading for the \(m^{th}\) device, written as follows:  
\[ \begin{array}{l}\left(x_{m}^{(p)}\right)^{*}\mathbf{s}_{m}^{H}\mathbf{r}^{ (p)}=\left(x_{m}^{(p)}\right)^{*}\mathbf{s}_{m}^{H}\bigg{(}\sum\limits_{k=1}^ {K}\sqrt{P_{k}}e^{\mathrm{i}\theta_{k}}t_{k}^{(p)}+\mathbf{n}^{(p)}\bigg{)}\\ =\sum\limits_{k=1}^{K}\sqrt{P_{k}}e^{\mathrm{i}\theta_{k}}s_{m}^{H}s_{k}(x_{m }^{(p)})^{*}x_{k}^{(p)}+\left(x_{m}^{(p)}\right)^{*}\mathbf{s}_{m}^{H} \mathbf{n}^{(p)}\\ =L\sqrt{P_{m}}e^{\mathrm{i}\theta_{m}}+\sum\limits_{k=1,k\
eq m}^{K}\sqrt{P_{ k}}s_{k}^{H}s_{k}e^{\mathrm{i}\theta_{k}}\big{(}x_{m}^{(p)}\big{)}^{*}x_{k}^{(p)}+ \left(x_{m}^{(p)}\right)^{*}\mathbf{s}_{m}^{H}\mathbf{n}^{(p)}\end{array}  \tag{11}\]  
It may not be easy to achieve satisfactory results with a single pilot symbol, and channel estimate values obtained from multiple repeated pilots can be combined with filtering. The SINR of the channel estimate using \(N\) pilot symbols with equal weights of filtering can be expressed as follows:  
\[\gamma^{(p)}=\frac{2 NL^{2}P_{m}}{\rho_{L}\sum\limits_{k=1,k\
eq m}^{K}P_{k}+L \sigma^{2}} \tag{12}\]  
If \(N\) is large, then the SINR of the channel estimate can be improved and accurate channel estimates can be obtained at the expense of the waste of resources for pilots. Note that, if \(N\) is small and \(K\) is large, then the channel estimate might not be accurate even when \(P_{m}/\sigma^{2}\) is large, i.e., signal-to-noise ratio (SNR) is large. Unlike orthogonal multiple access transmissions, satisfactory channel estimates may not be obtained with increasing SNR, and it is necessary to consider the effect of channel estimation errors in uplink NOMA systems. Suppose that an inaccurate channel estimate \(e^{j(\theta_{m}+\Delta\theta_{m})}\) with phase error \(\Delta\theta_{m}\) is used instead of \(e^{j\theta_{m}}\) for the \(m^{th}\) device. In this case, the channel compensated signal in Equation (6) needs to be rewritten as follows:  
\[e^{-j(\theta_{m}+\Delta\theta_{m})}\mathbf{s}_{m}^{H}\mathbf{r}=L\sqrt{P_{m}}e ^{-j\Delta\theta_{m}}x_{m}\ +\sum\limits_{k=1,k\
eq m}^{K}\sqrt{P_{k}}s_{m}^{H}s_{k}e^{j( \theta_{k}-\theta_{m}-\Delta\theta_{m})}x_{k}+e^{-j(\theta_{m}+\Delta\theta_{m })}\mathbf{s}_{m}^{H}\mathbf{n} \tag{13}\]  
Due to the channel estimation phase error \(\Delta\theta_{m}\) in the range of \(|\Delta\theta_{m}|<\pi/4\), one of the two real parts in a QPSK symbol is decreased while the other is increased.