## 5 Experiments
### Spectrum-Based Attack  
Since the watermarking method embeds the message by doubling the intDCT coefficients of the high-frequency bands, the spectral values of the high-frequency parts in the stego data have changed. Specifically, as shown in Figure 1, when we embed the data with the conventional method, we can observe a clear border of the data-embedded spectral band in the spectrogram. Therefore, we tried to detect the border of the spectral band by spectrum-based attack. If the proposed methods can deteriorate the performance of spectrum-based attack, it means that the proposed methods have better robustness traits than the conventional method. We implemented the spectrum-based attack method as follows. Let \(P_{n}(t)\) be the power spectrogram of a speech signal obtained by the short-time Fourier transform, where \(n\) is the number of the frequency bin and \(t\) is the frame number (\(1\leq t\leq T\) and \(1\leq n\leq N\)). Then, we calculate the spectral discontinuity \(D_{n}\) as follows. \[D_{n}=\frac{1}{T}\sum_{t}\lvert P_{n}(t)-P_{n+1}(t)\rvert \tag{7}\]  
When a spectrogram has a clear change at the same frequency for all frames, the absolute discontinuity value becomes large. Figure 5 shows examples of the discontinuity values for the cover data and stego data in which the data are embedded by the conventional method. We can confirm that the stego data have a discontinuity at the 96 th frequency bin, which is not observed in the cover data. Figure 5: Cont. Next, we analyze the data based on the distribution of the discontinuity values. We do not use the values of the lower frequency bins because the values of that frequency band fluctuate more than that of the higher frequency. Let the cut-off frequency bin number be \(n_{c}\). Then, we calculate the mean and variance of the discontinuity values as follows. \[\mu=\frac{1}{N-n_{c}+1}\sum_{n=n_{c}}^{N}D_{n} \tag{8}\]  
\[\sigma^{2}=\frac{1}{N-n_{c}}\sum_{n=n_{c}}^{N}(D_{n}-\mu)^{2} \tag{9}\]  
We checked the log-likelihood of the peak value of \(D_{n}\) according to the mean and variance. If the peak value is far from the mean value, the log-likelihood becomes small, showing that a suspicious border pattern is found. \[n_{p}=\arg\max_{n_{c}\leq n\leq N}|D_{n}| \tag{10}\]  
\[L=\log N\Big{(}D_{n_{p}};\mu,\sigma^{2}\Big{)} \tag{11}\]  
If the value of \(L\) is lower than a threshold, the data are determined to be a data-embedded signal. According to the above-mentioned method, we tried to detect the presence of a watermark. We used the 12 speech signals, which are the same as the previous experiment. For each watermarking method, we prepared 12 watermarked signals and 12 without watermarks. Then, we calculated the log-likelihood \(L\) according to Equation (11). Using the log-likelihood values, we calculated the Equal Error Rate (EER) by controlling the detection threshold so that the False Acceptance Rate (FAR) and False Rejection Rate (FRR) became equal. EER is widely used to evaluate the performance of attack methods [48]. If the EER is low, it means that the spectrum-based attack method can detect the watermark precisely. Since our goal is to conceal the watermark, the EER for the proposed method should be high. Details of the FAR and FRR of the results of the steganalysis results of conventional work with fixed expansion, the proposed solution 1, and the proposed solution 2 are listed in Figure 6. The FAR and FRR of the original 12 tracks and the stego tracks are mixed for computation to estimate whether there are data embedded by the proposed steganalysis  
Figure 5: Examples of the discontinuity values for the cover and stego data. methodology. The threshold for EER is determined when FAR and FRR are equal. The larger the EER threshold is, the more effective the proposed method is for anti-steganalysis.