## 5 Performance Analysis and Results
### Visual Quality Metrics
The \(BER\) and \(PSNR\) performances of the proposed \(FFT\), \(FrFT\) and \(MPFrFT\) coded OFDM schemes are tabulated at different values of \(E_{b}N_{0}(0\)to10 dB) in Tables 5 and 6. The results given in Table 5 are plotted in Figures 6 and 7, respectively. In Figure 6, at \(E_{b}N_{0}=8\)dB, the FFT-OFDM BER performance is \(8.60\times 10^{-4}\), the FrFT-OFDM BER performance is \(8.21\times 10^{-4}\) and the MPFrFT-OFDM BER performance is \(8.59\times 10^{-4}\). In Figure 6, at \(E_{b}N_{0}=8\)dB, the FFT-OFDM PSNR performance is \(30.65\)dB, the FrFT-OFDM PSNR performance is \(30.8548\)dB and the MP-FrFT-OFDM PSNR performance is \(30.6579\)dB. The proposed FrFT-OFDM system gains PSNR performance improvement by about \(0.1969\)dB compared with the proposed MPFrFT-OFDM system, but a large key space is achieved by the MPFrFT OFDM system than the FrFT-OFDM system. \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{2}{c}{**Noise Var. (\(\sigma^{2}\))**} & \multicolumn{1}{c}{\(0.01\)} & \(0.05\) & \(0.10\) & \(0.15\) & \(0.20\) \\ \hline \multirow{4}{*}{\(\begin{bmatrix}\text{Detected}\\ \text{Image}\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}\text{Decrypted}\\ \text{Image}\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}\text{Decrypted}\\ \text{Image}\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}\text{Decrypted}\\ \text{Image}\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}\text{Decrypted}\\ \text{Image}\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}\text{Decrypted}\\ \text{Image}\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}\text{Decrypted}\\ \text{Image}\end{bmatrix}\)} \\ \cline{2-3}  & \(PSNR(\text{dB})\) & \(Inf\) & \(28.6397\) & \(14.515\) & \(8.8606\) & \(6.56\times 10^{-2}\) \\ \hline \multirow{4}{*}{\(\begin{bmatrix}\text{Detected}\\ \text{Image}\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}\text{Decrypted}\\ \text{Image}\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}\text{Decrypted}\\ \text{Image}\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}\text{Decrypted}\\ \text{Image}\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}\text{Decrypted}\\ \text{Image}\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}\text{Decrypted}\\ \text{Image}\end{bmatrix}\)} \\ \cline{2-3}  & \(PSNR(\text{dB})\) & \(Inf\) & \(28.6397\) & \(14.515\) & \(8.8606\) & \(6.56\times 10^{-2}\) \\ \hline \end{tabular}
\end{table} 
Table 4: AWGN channel effect at zero mean \(\mu=0\) and over certain ranges of noise variances (\(\sigma^{2}=0.01,0.05,0.10,0.15,0.20\)). \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{2}{c}{\( \begin{array}{c}\text{\bf{E}}_{b}N_{0}\\ \text{\bf{(dB)}}\end{array} \)} & \multicolumn{2}{c}{\(\begin{array}{c}\text{\bf{(}\ref{BER}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{ }\text{}\text{}\text{}\text{}\text{}\text{}\text  
Different \(E_{b}N_{0}\) values \(2,8\) and \(16\)dB are chosen in Table 7 in order to highlight the visual quality metric performance of the proposed systems under the Salt and Pepper noise effect. In medical ultrasound imaging, Speckle is a granular interference that inherently exists in and degrades the quality of the medical images. It results from the coherence of backscattered signals from various distributed targets [4]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(E_{b}N_{0}\) (dB)} & \multicolumn{3}{c}{PSNR (dB)} \\ \cline{2-4}  & **FFT** & **FrFT** & **MPFrFT** \\ \hline
0 & 9.0709 & 9.5156 & 7.7798 \\ \hline
2 & 15.0297 & 15.6164 & 13.1508 \\ \hline
4 & 21.7511 & 22.6765 & 19.9282 \\ \hline
6 & 27.5973 & 28.3270 & 26.6475 \\ \hline
8 & 30.6554 & 30.8548 & 30.6579 \\ \hline
10 & 31.1966 & 31.1123 & 31.2009 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: \(PSNR\) performances of the proposed \(FFT\), \(FrFT\) and \(MPFrFT\) coded OFDM under Salt & Pepper noise, noise density \(d=0.02\).