## 2 System Model and Previous Works  
Similar to [12; 13], we also consider an uplink massive MIMO system consisting of a single-antenna user and a base station equipped with a large number of antennas \(M(M\gg 1)\). We consider the TDD mode and the block fading model which are popular used in research on massive MIMO system [1; 3; 5; 7; 8], in which the channel is unchanged in one coherent interval \(T\). The received signal vector at the \(m\)th BS antenna is modeled as [1]  
\[\mathbf{y}_{m}=\sqrt{\rho}h_{m}\mathbf{x}+\mathbf{n}_{m}, \tag{1}\]  
where \(\rho\) presents the average signal-to-noise ratio (SNR); \(\mathbf{x}=[x_{t},x_{t+1},...,x_{t+L-1}]^{T}\) is the transmit signal vector with length \(L(L\leq T)\) and \(E[||\mathbf{x}||^{2}]=L\), where elements of \(\mathbf{x}\) are taken from the conventional QAM constellation as shown in Figure 1a, \(\mathbf{n}_{m}\) is the additive white Gaussian noise vector at the \(m\)th BS antenna whose entries follow \(CN(0,1)\); and \(h_{m}\) is the channel coefficient with \(CN(\mu_{h},\sigma_{h}^{2})\). An example of the block fading model and signal vector is illustrated as in Figure 1b. Figure 1: Conventional 16-QAM constellation and illustration of block fading. (**a**) 16-QAM constellation; (**b**) an illustration of block fading length \(T=7\) and signal vector length \(L=3\). For simplicity, we normalize the channel so that \(\mu_{h}^{2}+\sigma_{h}^{2}=1\). Since Rayleigh and Rician fading models are very popular in evaluating system performance on both massive and regular MIMO system [17; 18], we focus on these two models with \(\mu_{h}=\sqrt{\frac{K_{r}}{1+K_{r}}}\) and \(\sigma_{h}=\sqrt{\frac{1}{1+K_{r}}}\)[12]. Thus, the channel vector from a user to a BS now can be modeled as  
\[\mathbf{h}=\mu_{h}\mathbf{h_{LOS}}+\sigma_{h}\mathbf{h_{NLOS}}, \tag{2}\]  
where \(K_{r}\) represents the Rician factor; In a special case, when \(K_{r}=0\), the channel becomes a Rayleigh fading channel. Additionally, \(\mathbf{h_{LOS}}=[1,\exp^{-j\pi\sin(\theta)},...,\exp^{-j\pi(M-1)\sin(\theta)}] ^{T}\in C^{M\times 1}\) with the arrival angle \(\theta\); is the light-of-sight (LOS) component when the antenna spacing is a half of wavelength. \(\mathbf{h_{NLOS}}\in C^{M\times 1}\) denotes the non-light-of-sight (NLOS) component whose elements follow i.i.d Gaussian variables with zero mean and unit variance. In [12; 13], the authors considered two consecutive \(t\)th and \((t-1)\)th instants, with channel vectors \(\mathbf{h}_{t}\); and \(\mathbf{h}_{t-1}\), and assumed \(\mathbf{h}_{t}\approx\mathbf{h}_{t-1}\). The received signal vector at the \(t\)th instant is given as  
\[\mathbf{y}_{t}=\sqrt{\rho}\mathbf{h}_{t}x_{t}+\mathbf{n}_{t}. \tag{3}\]  
where \(x_{t}\) is taken from a 16-DQAM constellation based on [19]. With a very large number of BS antennas \(M\), they have  
\[ \begin{split}\lim_{M\rightarrow\infty}\frac{1}{M}\mathbf{h}_{t}^ {H}\mathbf{h}_{t-1}&=1,\lim_{M\rightarrow\infty}\frac{1}{M} \mathbf{n}_{t}^{H}\mathbf{h}_{t-1}&=0,\\ \lim_{M\rightarrow\infty}\frac{1}{M}\mathbf{h}_{t}^{H}\mathbf{n} _{t-1}&=0,\lim_{M\rightarrow\infty}\frac{1}{M}\mathbf{n}_{t}^{H} \mathbf{n}_{t-1}&=0,\end{split}  \tag{4}\]  
Eventually, the signal symbol at the \(t\)th instant can be detected as  
\[r_{t}=\frac{1}{M}\mathbf{y}_{t}^{T}\mathbf{y}_{t-1}^{*}=\rho x_{t}x_{t-1}^{*} \text{ for very large }M, \tag{5}\]  
in which \(x_{t}x_{t-1}^{*}\) can be mapped back to the information symbol by the encoding rule of [19]. However, the authors of [12] did not propose a detector for when the number of BS antennas \(M\) is finite. The authors of [13] generalized the detector in [12] as ([13], Equation (6)), which can be applied for any value of \(M\). After that, they proposed a new two consecutive-symbol detector based on the conditional probability and a new non-coherent distance as ([13], Equations (9) and (10)). The new distance in [13] is used for look-up table algorithm in [14; 15] for differential encoding.