## 2 System Model and Previous Works
To calculate the proposed non-coherent distance, we propose the distance from \(\mathbf{x}_{1}\) to \(\mathbf{x}_{2}\) and the distance from \(\mathbf{x}_{2}\) to \(\mathbf{x}_{1}\) as in Equations (12) and (13) based on the proposed detector, Equation (10), as below:  
\[d(\mathbf{x}_{1}\rightarrow\mathbf{x}_{2})=||[-\rho(c_{h}\mathbf{x}_{1})^{H}R _{x_{1}}^{-1}(c_{h}\mathbf{x}_{1})-\ln(\det(R_{x_{1}}))]-[-\rho(c_{h}\mathbf{ x}_{1})^{H}R_{x_{2}}^{-1}(c_{h}\mathbf{x}_{1})-\ln(\det(R_{x_{2}}))]||; \tag{12}\]  
\[p(\mathbf{y}_{m}|\mathbf{x})=\frac{1}{2\pi\det(R_{y})}\exp\bigg{\{}-\frac{1}{2 }(\mathbf{y}_{m}-\mathbf{\bar{y}}_{m})^{H}R_{y}^{-1}(\mathbf{y}_{m}-\mathbf{ \bar{y}}_{m})\bigg{\}} \tag{6}\]0  
\(R_{x_{1}},R_{x_{2}}\)and\(c_{h}\) in Equations (12) and (13) are calculated as  
\[p(\mathbf{y}_{m}|\mathbf{x})=\frac{1}{2\pi\det(R_{y})}\exp\bigg{\{}-\frac{1}{2 }(\mathbf{y}_{m}-\mathbf{\bar{y}}_{m})^{H}R_{y}^{-1}(\mathbf{y}_{m}-\mathbf{ \bar{y}}_{m})\bigg{\}} \tag{6}\]1  
The Equation (12) is the non-coherent distance from \(\mathbf{x}_{1}\) to \(\mathbf{x}_{2}\), which is based on the assumption that we did send the codeword \(\mathbf{x}_{1}\) but the detector wrongly estimated that \(\mathbf{x}_{2}\) was sent. Inversely, the Equation (13) is the non-coherent distance from \(\mathbf{x}_{2}\) to \(\mathbf{x}_{1}\) in which, the \(\mathbf{x}_{2}\) was actually sent but the detector wrongly estimated that \(\mathbf{x}_{1}\) was sent. In other words, Equations (12) and (13) can be used by likelihood estimator as a distance between two codewords \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\). The larger the values of Equations (12) and (13) are, the less chance the detector wrongly estimates between \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\). Eventually, the proposed non-coherent distance is calculated as  
\[p(\mathbf{y}_{m}|\mathbf{x})=\frac{1}{2\pi\det(R_{y})}\exp\bigg{\{}-\frac{1}{2 }(\mathbf{y}_{m}-\mathbf{\bar{y}}_{m})^{H}R_{y}^{-1}(\mathbf{y}_{m}-\mathbf{ \bar{y}}_{m})\bigg{\}} \tag{6}\]2  
In the differential encoding part, we apply the look-up table algorithm for DQAM as in [14; 15] by using the proposed non-coherent distance in Equation (15). The main contribution of these above steps and equations are summed up as follow. * Equation (10) is our proposed detector which is derived based on the conditional probability of received signal and contains channel statistical information, so that it adapts better with the change in environment. * Equation (11) is a simplified detector when the channel is Rayleigh fading. * Equations (12) and (13) are derived based on Equation (10), and they are used to evaluate the distance between two signal vectors (a small distance means that it is easy to wrongly decode between two signal vectors). * \(d(\mathbf{x}_{1}\rightarrow\mathbf{x}_{2})\) is distance from \(\mathbf{x}_{1}\) to \(\mathbf{x}_{2}\) which is used to evaluate how likely it is to wrongly estimate \(\mathbf{x}_{1}\) as \(\mathbf{x}_{2}\). Similar explanation is applied to \(d(\mathbf{x}_{2}\rightarrow\mathbf{x}_{1})\). * Eventually, we use Equation (15) to take the minimum value between \(d(\mathbf{x}_{1}\rightarrow\mathbf{x}_{2})\) and \(d(\mathbf{x}_{2}\rightarrow\mathbf{x}_{1})\) as the distance between \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\).