## 2 System Model and Previous Works
New Differential Detector and Non-Coherent Distance**  
Consider the received signal at \(m\)th BS antenna as in Equation (1), the conditional probability of the received signal vector \(\mathbf{y}_{m}\), given transmitted signal vector \(\mathbf{x}\), is calculated as  
\[p(\mathbf{y}_{m}|\mathbf{x})=\frac{1}{2\pi\det(R_{y})}\exp\bigg{\{}-\frac{1}{2 }(\mathbf{y}_{m}-\mathbf{\bar{y}}_{m})^{H}R_{y}^{-1}(\mathbf{y}_{m}-\mathbf{ \bar{y}}_{m})\bigg{\}} \tag{6}\]  
where \(\mathbf{\bar{y}}_{m}\) is the mean of \(\mathbf{y}_{m}\) and is given as  
\[\mathbf{\bar{y}}_{m}=E\{\mathbf{y}_{m}\}=\sqrt{\rho}\mu_{h}\mathbf{x}, \tag{7}\]  
where \(\det(R_{y})\) is the determinant of \(R_{y}\), and \(R_{y}\) is the covariance matrix of \(\mathbf{y}_{m}\), which can be calculated as  
\[R_{y} =E\bigg{\{}(\mathbf{y}_{m}-\mathbf{\bar{y}}_{m})(\mathbf{y}_{m}- \mathbf{\bar{y}}_{m})^{H}\bigg{\}}\] \[=E\bigg{\{}\rho(h_{m}-\mu_{h})\mathbf{x}\mathbf{x}^{H}(h_{m}-\mu_ {h})^{H}+\sqrt{\rho}(h_{m}-\mu_{h})\mathbf{x}\mathbf{n}_{m}^{H} \tag{8}\] \[\qquad+\sqrt{\rho}(h_{m}-\mu_{h})^{H}\mathbf{n}_{m}\mathbf{x}^{H} +\mathbf{n}_{m}\mathbf{n}_{m}^{H}\bigg{\}}=(\rho\sigma_{h}^{2}\mathbf{x} \mathbf{x}^{H}+I_{N}).\]  
Since the proposed detector aims to maximize the summation of the conditional probability of received signal vector \(\mathbf{y}_{m}\) at all BS antennas \(1\leq m\leq M\), given transmitted signal vector \(\mathbf{x}\), the estimated signal vector \(\mathbf{\hat{x}}\) can be calculated as  
\[\mathbf{\hat{x}}=\arg\max_{\mathbf{x}\in X}\ \sum_{m=1}^{M}p(\mathbf{y}_{m}| \mathbf{x}), \tag{9}\]  
where \(\chi\) presents the vector space of all possible transmitted signal vectors \(\mathbf{x}\). Since the natural logarithm function is monotonically increasing, maximizing \(p(\mathbf{y}_{m}|\mathbf{x})\) is equivalent to maximize \(\ln p(\mathbf{y}_{m}|\mathbf{x})\). Finally, the proposed detector is given as  
\[\mathbf{\hat{x}}=\arg\max_{\mathbf{x}\in X}\bigg{\{}\sum_{m=1}^{M}-(\mathbf{y }_{m}-\mathbf{\bar{y}}_{m})^{H}R_{y}^{-1}(\mathbf{y}_{m}-\mathbf{\bar{y}}_{m}) -\ln(\det(R_{y}))\bigg{\}}, \tag{10}\]  
Specially, when the channel is Rayleigh fading, we have \(\mathbf{\bar{y}}_{m}=0\), and the proposed detector becomes  
\[\mathbf{\hat{x}}=\arg\max_{\mathbf{x}\in X,x_{i}\in Q_{1}}\bigg{\{}\sum_{m=1} ^{M}-\mathbf{y}_{m}^{H}R_{y}^{-1}\mathbf{y}_{m}-\ln(\det(R_{y}))\bigg{\}}, \tag{11}\]where \(Q_{1}\) is the first quadrant. This means the number of decision values that need to be calculated is reduced by a factor of four, from \(N^{L}\) to \(\frac{N^{L}}{4}\), with \(N\)-QAM. The reduction occurs since when \(\bar{\mathbf{y}}_{m}=0\), for any codeword \(\mathbf{x}=[x_{t},x_{t+1},...,x_{t+L-1}]^{T}\) with the first symbol \(x_{t}\) belongs to the first quadrant of the constellation, there are also three other codewords which have the first symbol \(x_{t}^{\prime}\) belong to three other quadrants, that have the same estimated vector \(\hat{\mathbf{x}}\) as \(\mathbf{x}\). In other words, we only need to calculate the decision values of codewords which have the first symbol \(x_{t}\) belongs to the first quadrant of the constellation in case of Rayleigh fading.