## 2 Conventional Uplink NOMA  
### System Model  
In this paper, we consider \(K\) devices concurrently transmitting packets to a single resource of a BS. The signals from the \(K\) devices can be distinguished by user-specific pseudo-noise (PN) spreading patterns of length \(L\), and we assume a sufficiently large number of spreading patterns so that the collision probability of two or more devices selecting the same spreading pattern is negligible. Let \(\mathbf{s}_{k}=\left[s_{k,1},\cdots,s_{k,L}\right]^{T}\) (\(\left|s_{k,l}\right|=1\)) be the spreading pattern vector for the \(k^{th}\) device \((k=1,\cdots,K)\). If a random-like phase sequence is used for the pattern, in other words, \(s_{k,l}=e^{i\phi_{k,l}}\), where \(\phi_{k,l}\) is a random variable uniformly distributed from 0 to \(2\pi\), then the average correlation value of two different spreading patterns \(\rho_{L}\equiv E\big{\{}|\mathbf{s}_{m}^{H}\mathbf{s}_{k}|_{2}\big{\}}\)\((m\
eq k)\) can be expressed as follows:  
\[\rho_{L}^{(random)}\equiv E\big{\{}|\mathbf{s}_{m}^{H}\mathbf{s}_{k}|^{2} \big{\}}=E\Bigg{\{}\left|\sum_{l=1}^{L}e^{i(\phi_{k,l}-\phi_{m,l})}\right|^{2} \Bigg{\}}=L(m\
eq k) \tag{1}\]  
A spreading pattern can be designed so that the average correlation value can satisfy Equation (1) or the average correlation value can be further reduced by smartly designing the spreading patterns, in other words, \(\rho_{L}\leq L\). Hence, we can say that \(E\big{\{}\big{|}\mathbf{s}_{m}^{H}\mathbf{s}_{k}\big{|}_{2}\big{\}}=L^{2}\) for \(m=k\) and \(E\big{\{}\big{|}\mathbf{s}_{m}^{H}\mathbf{s}_{k}\big{|}_{2}\big{\}}=\rho_{L} \leq L\) for \(m\
eq k\). Figure 1: Conventional uplink NOMA. An uplink NOMA system considered in this paper is assumed to perform the procedure shown in Figure 1. After channel coding and QPSK modulation, the modulated symbols are repeated by \(L\) times and multiplied by a user-specific PN sequence of length \(L\). The spread symbols can be transmitted using narrowband transmission, orthogonal frequency division multiplexing (OFDM), or other transmission scheme. If OFDM is used for the transmission, inverse fast Fourier transform (IFFT) is followed by cyclic prefix (CP) addition. Let \(x_{k}\) be the QPSK-modulated data signal with \(|x_{k}|=1\) for the \(k^{th}\) device \((1\leq k\leq K)\). The transmitted signal vector of the \(k^{th}\) device, denoted as \(\mathbf{t}_{k}=\left[t_{k,1},\cdots,t_{k,L}\right]^{T}\), can be written as follows:  
\[\mathbf{t}_{k}=\mathbf{s}_{k}x_{k} \tag{2}\]  
The BS receives superimposed signals from the \(K\) devices, written as  
\[\mathbf{r}=\sum\limits_{k=1}^{K}\sqrt{P_{k}}e^{i\theta_{k}}t_{k}+\mathbf{n} \tag{3}\]  
where \(\mathbf{r}\) is the received signal vector, \(P_{k}\) is the received signal power of the \(k^{th}\) device, \(\mathbf{n}\) is the complex noise vector with variance \(\sigma^{2}\), and \(\theta_{k}\) is the phase of the channel for the \(k^{th}\) device, which can be modeled as a random variable uniformly distributed from 0 to \(2\pi\). In this paper, we assume a successive interference cancellation technique starting from the device with the strongest received signal power, assuming that \(P_{k}\)'s can be accurately measured using the pilots at the BS. A successive interference cancellation technique has lower complexity than other complicated reception schemes such as maximal likelihood or parallel interference cancellation techniques. The index for the first decoding device, written as \(m\), is given by:  
\[m=\underset{k=1,\cdots,K}{\operatorname{argmax}}P_{k} \tag{4}\]  
The performance of the first decoded device is important, since it affects the performance of other devices as well. If the decoding process for the first device is failed, the interference cannot be cancelled out from the received signal, and the next device needs to be decoded with lower signal-to-interference-plus-noise ratio (SINR). In order to decode the data from the \(k^{th}\) device, the BS performs de-spreading, expressed as follows:  
\[