# Performance of Differential-Phase OFDM Systems over Selective Multipath Fading Channels
## 5 BER of the OFDM-DPSK System over Selective Fading Channels  
In the differential OFDM system, the average probability of error is evaluated by taking the SNR and SIR into consideration, where the total average signal to noise/interference ratio in the presence of the selective channel and AWGN is given by  
\[\overline{\gamma}_{s_{k}} = \frac{\mathcal{S}}{\mathcal{I}+\sigma_{k}^{2}}=\left[\left(\frac{ \mathcal{S}}{\mathcal{I}}\right)^{-1}+\left(\frac{\mathcal{S}}{\sigma_{k}^{2} }\right)^{-1}\right]^{-1} \tag{42}\] \[= \left[(SIR)_{k}^{-1}+(SNR)_{k}^{-1}\right]^{-1}\]  
where \((SIR)_{k}\) is evaluated as in (37). The symbol error rate of the \(k\)-th subchannel; \(P_{s_{k}}(\overline{\gamma}_{s_{k}})\), in the OFDM-DPSK frame, is evaluated by substituting \(\overline{\gamma}_{s_{k}}\) in the formula of the probability of error. If Gray bit mapping is applied to the DPSK scheme, then the bit error rate in the \(k\)-th subchannel is given by \(P_{s_{k}}(\overline{\gamma}_{s_{k}})/\log_{2}M\). The probability of an error bit appearing in the whole frame can be obtained by averaging \(P_{s_{k}}(\overline{\gamma}_{s_{k}})\) over the whole frame [16]  
\[\overline{P}_{b}=\frac{1}{N\log_{2}M}\sum_{k=0}^{N-1}P_{s_{k}}(\overline{ \gamma}_{s_{k}}) \tag{43}\]  
The conditional probability of a symbol error \(M\)-ary DPSK in the additive white Gaussian noise channel is obtained as [14]  
\[P_{s_{k}}(\gamma_{s_{k}})=\frac{\sin\left(\frac{\pi}{M}\right)}{2\pi}\int_{- \pi/2}^{\pi/2}\frac{e^{-\gamma_{s_{k}}(1-\cos(\pi/M)\cos(\varphi))}}{1-\cos( \pi/M)\cos\varphi}d\varphi \tag{44}\]  
For BDPSK (\(M=2\)), (44) reduces to the following form  
\[P_{s_{k}}(\gamma_{s_{k}})=\frac{1}{2}e^{-\gamma_{s_{k}}} \tag{45}\]  
when Gary code mapping is used, the bit error probability performance of \(M\)-ary DPSK is related to the symbol error probability by  
\[P_{b}(\gamma_{t_{k}})\simeq\frac{P_{s_{k}}(\gamma_{s_{k}})}{\log_{2}M} \tag{46}\]  
where \(\gamma_{s_{k}}=\gamma_{t_{k}}\log_{2}M\). The average probability of error can be evaluated as  
\[P_{s}(\overline{\gamma}_{s_{k}})=\int_{0}^{\infty}P_{s}(\gamma_{s_{k}})f_{ \gamma_{t_{k}}}(\gamma_{s_{k}})d\gamma_{s_{k}} \tag{47}\]  
The average symbol error probability of \(M\)-ary DPSK over the Nakagami-\(m\) fading channel is given by substituting (11) and (44) into the integral form in (47) and making some substitutions. The symbol error rate is given by  
\[P_{s}(\overline{\gamma}_{s_{k}})=\frac{\sin(\frac{\pi}{M})}{2\pi}\int_{-\pi/2 }^{\pi/2}\frac{d\varphi}{(1-\cos(\pi/M)\cos\varphi)(1+\frac{\overline{\gamma} _{s_{k}}}{m}(1-\cos(\pi/M)\cos\varphi))^{m}} \tag{48}\]  
The average bit error probability performance of \(M\)-ary DPSK is obtained by substituting (48) into (46), that is  
\[P_{b}(\overline{\gamma}_{t_{k}})=\frac{\sin(\frac{\pi}{M})}{2\pi\log_{2}M} \int_{-\pi/2}^{\pi/2}\frac{d\varphi}{(1-\cos(\pi/M)\cos\varphi)(1+\frac{ \overline{\gamma}_{t_{k}}\log_{2}M}{m}(1-\cos(\pi/M)\cos\varphi))^{m}} \tag{49}\]The bit error probability performance of binary DPSK over Nakagami-_m_ fading can be obtained by substituting \(M=2\) in (49), and the result is  
\[P_{b}(\overline{\gamma}_{b_{k}})=\frac{1}{2}\left[\frac{m}{m+\overline{\gamma}_{ b_{k}}}\right]^{m} \tag{50}\]  
Another well known BER formula is for QDPSK which is obtained by substituting \(M=4\) in (49)  
\[P_{s}(\overline{\gamma}_{s_{k}})=\frac{1}{2\sqrt{2}\pi}\int_{-\pi/2}^{\pi/2} \frac{d\varphi}{(1-\frac{1}{\sqrt{2}}\cos\varphi)(1+\frac{\overline{\gamma}_{ b_{k}}}{m}((1-\frac{1}{\sqrt{2}}\cos\varphi))^{m}} \tag{51}\]  
and the average probability of an error bit in the OFDM-DPSK system can be evaluated as in (43).