# Exploring the Effectiveness of the Phase Features on Double Compressed AMR Speech Detection
## 4 Magnitude and Phase Spectra of AMR-Encoded Speech Signals
Figure 2 displays the average difference phase spectra \(\Delta\theta_{\text{O,SC}}(k)\) (left panel) and \(\Delta\theta_{\text{SC,DC}}(k)\) (right panel) calculated using the AMR-NB codec across various bit rates. When computing \(\Delta\theta_{\text{SC,DC}}(k)\), which represents the average difference phase spectra between the SC and DC signals, the first compression bit rate (BR1) of the DC signals is set to match the value used for obtaining the SC signal (BR = BR1 = 5.9 kbps). This adjustment allows us to assess the effect of the second compression bit rate (BR2) on the phase spectrum. From the figure, it is observed that compressing the narrowband speech signal considerably affects the phase spectrum above 500 Hz, and the difference between the phase spectrum of the original and SC signals becomes larger as the compression bit rate increases. For the DC signals in turn, compressing the signal for the second time notably affects the phase spectrum at higher frequencies. Figure 3 shows the average phase spectra \(\Delta\theta_{\text{O,SC}}(k)\) and \(\Delta\theta_{\text{SC,DC}}(k)\) computed using the AMR-WB codec. Similar to the observations with AMR-NB-coded signals (as shown  
Figure 2: Average difference phase spectra between the original (uncompressed) and SC speech signals (**left**) and between the SC and DC speech signals (**right**). The AMR-NB codec is used to generate the SC and DC signals. For the DC signals, the first compression bit rate (BR1) matches with the compression bit rate of the SC speech signal (BR = 5.9 kbps) to asses the effect of second compression bit rate (BR2) on DC signals. in Figure 2), compressing the speech signal for the first time using the AMR-WB codec has a considerable impact on the phase spectrum. As the compression bit rate increases, the phase spectrum exhibits notable variations compared to the uncompressed signal spectrum. These variations become more pronounced when the signal is compressed for the second time, particularly at higher frequencies. Based on the average difference phase spectra visualized in Figures 2 and 3, it is evident that the phase spectra of the SC and DC signals exhibit different behaviors. Therefore, the phase spectrum can serve as a discriminative feature between SC and DC signals. In this study, we explore the effect of the STFT phase spectrum on DC AMR speech detection by proposing the use of three different phase spectrum-based feature representations:  
* STFT Phase Spectrum: In this approach, the STFT phase spectrum \(\theta(k,t)\) is unwrapped to obtain a continuous function of \(k\), in order to remove the discontinuities by adding an integer multiple of \(2\pi\) to the phase spectrum at each DFT bin \(k\). The unwrapped phase spectrum is used as the feature for DC AMR speech detection. * Cosine Normalized Phase Spectrum (CosPhase): After unwrapping the phase spectrum, the dynamic range of the phase spectrum may exhibit large variations, making it challenging to model the phase information accurately. To address this, we apply the cosine function to the unwrapped phase spectrum, resulting in features within the range \([-1,\,1]\). * CosPhase with discrete cosine transform (DCT + CosPhase): To obtain a more compact representation of the phase information, DCT is applied to the CosPhase features to obtain uncorrelated features. In addition to the three phase spectrum-based features described above, we also utilize the magnitude spectrum (spectrogram) representation of the speech signal as a baseline feature in our experiments, enabling us to compare the performance of the phase spectrum features.