# Performance of Differential-Phase OFDM Systems over Selective Multipath Fading Channels
## 6 Numerical Results and Conclusion  
In this section, numerical examples are discussed to simulate the performance of the OFDM-DPSK system over the multipath fading Nakagami-_m_ channel in the presence of AWGN. For this purpose, we will use the physical parameters of the IEEE 802.11a standard system that employs the 64 subcarriers OFDM scheme, as shown in Table 1. This systems has a bandwidth of 20 MHz and, consequently, the subchannel bandwidth is \(\Delta f=312.5\) kHz. The useful data duration in the OFDM frame is \(T_{s}=3.2\) us and the FFT is performed every \(T_{s}=50\) ns. Different values for the guard period will be used to evaluate the signal-to-interference ratio. The power delay profile showing the delay spread versus the path gain, as shown in Table 2, is used to simulate the system. The delay spread in the table has an RMS value of \(\tau_{rms}=129\) ns. Equation (37) can be used to calculate \((SIR)_{k}\) for the system in Tables 1 and 2 for the \(k\)-th branch of the FFT processor. The average SIR for the OFDM signalling interval can be calculated as  
\[SIR=\frac{1}{N}\sum_{k=0}^{N-1}(SIR)_{k} \tag{52}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Setting** \\ \hline Bandwidth (\(BW\)) & 20 MHz \\ \# of subchannels & \(N=64\) \\ Subcarrier separation (\(\Delta f\)) & \(312.5\) kHz \\ Symbol duration (\(T_{s}\)) & 50 ns \\ Data duration (\(T_{s}\)) & \(3.2\) μs \\ Guard period (\(T_{g}\)) & \(200,200,600,\) and \(800\) ns \\ FFT size & Same as the number of subchannels \\ Fading & Multipath selective \\ Severity parameter \(m\) & different integer _m_\(\geq 1\) \\ Power Delay Profile & exponential \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters for IEEE 802.11a standard. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Delay Spread (ns)** & **Normalized Path Gain** & **Path Gain (dB)** \\ \hline
0 & 1.00 & 0 \\
100 & 0.4893 & \(-3.10\) \\
200 & 0.2394 & \(-6.21\) \\
300 & 0.1171 & \(-9.31\) \\
400 & 0.0573 & \(-12.42\) \\
500 & 0.0280 & \(-15.53\) \\
600 & 0.0137 & \(-18.63\) \\
700 & 0.0067 & \(-21.74\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of Power delay profile: delay spread in (ns) and path loss (dB). The average SIR when the guard period \(T_{g}=200,400,600\) and \(800\) ns is shown in Table 3.  
The previous formulas are tested and the results are reported in this section. The channel parameters in Tables 1-3 are used in the simulation. Different numbers of carriers \((N)\), severity parameters \((m)\), guard period intervals \((T_{g})\), and differential modulation volume \((M)\) are used to study the system performance. Figure 4a shows the average probability of error of the binary differential OFDM with \(64\) subcarriers and different guard period durations and Nakagami severity parameter. The results clearly show that the system performance improves significantly as the guard duration increases. However, significant increase to the guard duration will negatively impact the system data rate. Practical considerations and extreme caution in selecting the proper guard duration should be applied. Figure 4b illustrates the probability of error at \(T_{g}=400\) ns and a bandwidth of \(20\) MHz, however the number of subcarriers is varied from \(N=8\) to \(1024\). The results indicate that the system performance is in favor of a higher number of subcarriers. We notice that the difference in the probability of error is slightly smaller when the number of subcarriers increases from \(256\) to \(1024\). However, utilizing more channels makes the frequency-selective fading almost negligible. The same parameters are repeated for Quadrature differential OFDM (\(M=4\)) and the results are reported  
Figure 4: OFDM–BDPSK system with **(a)**\(N=64\) and different guard period durations and severity parameter \((m)\) values **(b)**\(T_{g}=400\) ns and \(BW=20\) MHz with different numbers of subcarriers and severity parameter \((m)\) values.