# Co-Efficient Vector Based Differential Distributed Quasi-Orthogonal Space Time Frequency Coding
## 4 Pairwise Error Probability Analysis
For different channel conditions, the performance of our proposed differential DQSTFC scheme is compared with the differential quasi-orthogonal DSTC-OFDM and DSFC schemes (whose parameters are simulated in our environment). We study the effects of Doppler spread and delay spread on the aforementioned coding schemes using different simulation parameters. We first set \(\tau_{rms}\) to a low value of 0.1 \(\upmu\)s such that the effect of delay spread is negligible. To study the influence of Doppler spread, we investigate BER performance at different Doppler frequencies. The SNR is fixed at 12 dB, the symbols are chosen from a QPSK constellation, and all the coding schemes have the same transmission rate of 2 bits/s/Hz. Figure 3: Comparison of Unitary Matrices and Co-efficient Vector Designs. From Figure 4, it can be seen that for values of \(f_{D}\) between 78 Hz and 135 Hz, the BER performance of our proposed differential DQSTFC scheme is better than that of the differential quasi-orthogonal DSTC-OFDM and DSFC schemes. This is because at such values of \(f_{D}\) the coherence time \(t_{c}\) and coherence bandwidth \(b_{c}\) of the channel are large enough to satisfy the requirements of constant channel gain across adjacent time slots and adjacent subcarriers. The BER performance of DSTC-OFDM is better than that of our proposed scheme only at values of \(f_{D}\) below 78 Hz. At such values of \(f_{D}\), the coherence time is large, and \(t_{c}\geq\Gamma\) can be satisfied. Thus, DSTC-OFDM schemes, which require constant channel gain in the temporal dimension for the entire duration of \(\Gamma=4\) symbols, have the best BER performance. When the Doppler spread increases and \(f_{D}\) is between 78 Hz and 135 Hz, the coherence time reduces and \(t_{c}\geq\Gamma\) can no longer be satisfied; however, the coherence time is still large enough to satisfy \(t_{c}\geq\Gamma/2\). Thus, the BER performance of DSTC-OFDM degrades beyond that of our proposed scheme. When the Doppler spread is severe and \(f_{D}\) is above 135 Hz, coding in the temporal dimension introduces a significant amount of inter-symbol interference; thus, our proposed scheme and the DSTC-OFDM scheme experience performance degradation. However, the BER performance of the DSFC scheme is better in this situation because the delay spread is low and the coherence bandwidth is large enough to satisfy \(b_{c}\geq\Gamma\). Thus, DSFC schemes that require constant channel gain in the frequency dimension for the entire duration of \(\Gamma=4\) symbols has the best BER performance. In contrast with DSTC-OFDM and DSFC schemes, where the requirements for constant channel gain must be satisfied for the entire duration of \(\Gamma=4\) symbols, our scheme only requires constant channel gain during the transmission of \(\Gamma=2\) symbols. Therefore, we can conclude that for cooperative networks operating in environments where CSI cannot be acquired, our proposed scheme is robust against a practical range of Doppler spread. We then set \(f_{D}\) to a fixed value of 50 Hz such that the influence of Doppler spread is low and compare the performance of all the aforementioned coding schemes at different levels of delay spread. The SNR is fixed at 12 dB, and all the coding schemes have identical transmission rate of 2 bits/s/Hz. From Figure 5, we observe that our proposed scheme outperforms the other coding schemes when the value of \(\tau_{rms}\) is between 0.8 \(\upmu\)s and 3.2 \(\upmu\)s. This is because at such values, the coherence time and coherence bandwidth of the channel is large enough to satisfy the requirements of constant channel gain across adjacent time slots and adjacent sub-carriers. When the delay spread is low and \(\tau_{rms}\) is below 0.8 \(\upmu\)s, the coherence bandwidth is large and \(b_{c}\geq\Gamma\) can be satisfied. Thus, DSFC schemes that require  
Figure 4: Co-efficient Vector DQSTFC in Frequency Selective Fading Channels. constant channel gain across \(\Gamma\) sub-carriers outperform the DSTC-OFDM scheme and our proposed DQSTFC scheme.