# Co-Efficient Vector Based Differential Distributed Quasi-Orthogonal Space Time Frequency Coding
## 4 Pairwise Error Probability Analysis
The non-coherent ML decoder could thus be simplified as:  
\[\mathbf{\hat{V}}^{k+1}=\arg\max_{\mathbf{V}^{k+1}\in\mathbf{V}}\middle|\mathbf{\chi}^{k}+{V^{k+ 1}}^{H}\mathbf{\chi}^{k+1}\right\| \tag{28}\]  
where \(\left\|\cdot\right\|\) was the Frobenius norm. The Chernoff bound on the PEP of mistaking \(\mathbf{V}^{k+1}\) by \(\hat{V}^{k+1}\) could be given as [33]. \[PEP\left(\mathbf{V}^{k+1}-\mathbf{\hat{V}}^{k+1}\right)=\frac{1}{2}\left\{\frac{\text{ det}\Big{[}\lambda\big{(}{I_{\text{PT}}}+\text{Y}\mathbf{X}\mathbf{\Lambda}\mathbf{R}\mathbf{ \Lambda}^{H}\mathbf{X}^{H}\big{)}+(1-\lambda)\Big{(}{I_{\text{PT}}}+\text{Y}\hat{ }\mathbf{X}\mathbf{\Lambda}\mathbf{R}\mathbf{\Lambda}^{H}\hat{\mathbf{X}}^{H}\Big{)}\Big{]}}{\frac{ \text{det}\Big{(}{I_{\text{PT}}}+\text{Y}\mathbf{X}\mathbf{\Lambda}\mathbf{R}\mathbf{\Lambda}^{H} \mathbf{X}^{H}\Big{)}\cdot\text{det}^{(1-\lambda)}\Big{(}{I_{\text{PT}}}+\text{Y} \hat{}\mathbf{X}\mathbf{\Lambda}\mathbf{R}\mathbf{\Lambda}^{H}\hat{\mathbf{X}}^{H}\Big{)}}\right\} \tag{29}\]where \(\mathbf{X}\) and \(\hat{\mathbf{X}}\) were two distinct codewords, \(\hat{\mathbf{X}}=\left[I_{\text{PT}}{}^{T},\hat{\mathbf{V}}^{k+1}{}^{T}\right]\) and \(\lambda=E\Big{\{}\text{exp}\Big{(}\lambda\Big{[}\text{ln}p\Big{(}\mathbf{\gamma} \Big{|}\mathbf{\psi}^{k+1}\Big{)}-p\Big{(}\mathbf{\gamma}\Big{|}\hat{\mathbf{V}}^{k+1}\Big{)} \Big{]}\Big{)}\Big{\}}\) was used to get the tightest bound. By simple algebraic manipulation (29) could be simplified as  
\[\begin{array}{ll}PEP(\mathbf{V}^{*}(k+1)&-\hat{\mathbf{V}}^{-}(k+1))\\ &=1/2\{\text{det}[I_{\text{\text{\text{\text{\text{\text{\text{\text{\text{ \text{\text{\text{\text{\text{\text{\text{\texttexttext  
also independently simulated for \(P=4\) cooperating nodes in coherent networks that can acquire perfect CSI. The results for \(BPSK\) are presented in Figure 3.  
From the results of the simulated schemes, it is evident that, compared to the unitary matrices AF protocol, the performance of the differential DF scheme using co-efficient vectors is slightly better than that obtained using unitary matrices in the low SNR region while the performance improves significantly in the high SNR region. For example, at \(10^{-4}\) BER, the performance gain is about 1.2 dB. A notable difference between the designs is observed in terms of the computation complexity of the decoder at the relay nodes and the destination. For designs utilizing unitary matrices, a simple pairwise decoder recovers the information bits at the relays and the destination by conducting \(c=2^{m}\) comparisons, \(m\) is the number of bits. In contrast, designs utilizing co-efficient vectors rely on an exhaustive search over all possible elements of the co-efficient vector. The most complex method is to perform a full search over all possible combinations in the co-efficient vector set. This equates to \(c=2^{2m}\) comparisons to recover the elements of the co-efficient vector set, with a further \(1\leq c\leq 2^{2m}\) comparisons required for mapping and inverse mapping at the encoder and decoder, respectively. It is thus clear that the decoding complexity limitation of the co-efficient vector based design is due to the constellation size. To counter this, suitable optimum detection schemes, such as those proposed in [37; 38], which are independent of constellation size, can be utilized. Next, we analyse the performance of our proposed scheme in frequency-selective and time-selective channels. Specifically, we set our simulation parameters for different Doppler frequencies \(f_{D}\) ranging from 40 Hz to 200 Hz and different root mean square delay spreads \(\tau_{\text{rms}}\) between 0.1 \(\upmu\)s and 4 \(\upmu\)s, the Doppler frequencies correspond to mobile speeds between 22 km/h and 108 km/h. For the quasi-orthogonal codes, the rotation angles are set to \(\theta/M\), where \(M\) is the constellation size.