## 3 Materials and Methods
### Method for Generating the Sonic Watermark
#### 3.2.2 The Audio Signal Generator Blocks
This is likely to happen in the case of the proposed sonic watermark because the ticking sound (i.e., the masker) has a wide bandwidth, as shown in Section 3.1.2. This is an argument for having narrow bandwidth chirp signals, placed at high frequencies where the critical bands are larger. To improve the auditory masking performance even more, only one chirp signal should be placed in each critical band. Auditory filters can be characterized using the equivalent rectangular bandwidth (ERB). In this convenient approach, the filters are treated as rectangular band-pass filters [28; 29; 30]. The ERB can be computed, for young listeners and moderate sound levels, according to [31], with the following formula: \[\text{ERB}(f)=6.23\cdot f^{2}+93.39\cdot f+28.52,\] (3) where \(f\) is the central frequency of the filter in kHz. The results in [31] are given for frequencies between 124 Hz and 6.5 kHz and show an increase of the ERB with the frequency. Later results shown in [32] propose another mathematical relation for estimating the ERB: \[\text{ERB}(f)=24.7\cdot(4.37\cdot f+1).\] (4)The values for ERB obtained using (4) are smaller than the ones estimated by (3). Therefore, for the proposed sonic watermark, the bandwidth of the chirp signals is chosen to be 240 Hz and they are distributed one in each critical band using (3), minimizing the chances that two or more chirp signals will be placed in the same critical band. Based on (3), if the first chirp signal occupies the frequency interval from 8 kHz to 8.24 kHz, the 9 th chirp will still be in the audio bandwidth, up to 17.2 kHz. Neither (3) or (4) give accurate results for frequencies over 10 kHz, so the worst-case estimation was considered, using (3) (i.e., the chirp signals may be spread more than the minimum necessary, thus improving the masking performance). The frequency ranges of the nine chirp signals that are used in the proposed system, designed according to the above-mentioned five principles, are summarized in Table 1. All nine signals have the same bandwidth, equal to 240 Hz, and the same duration, equal to 45 ms. The ENF measurement system considered in this design, presented in Section 3.2.1, delivers the ENF value with a resolution equal to \(10^{-3}\) Hz. The received value is processed according to the flow diagram illustrated in Figure 7 to obtain the nine bits (from \(b_{1}\) to \(b_{8}\)) that are used further to control the presence or absence of the nine chirp signals that are mixed with the ticking sound. If **B** is a row vector containing the values of the nine bits, and **C** is a matrix containing on each row the samples of each chirp signal \(\mathbf{c}_{i}\) with \(i=\overline{0,8}\), the output signal of this block, denoted with \(\mathbf{C}_{\text{out}}\), is a mixture of chirp signals that can be obtained using the following relations:  
\[ \begin{gathered}\mathbf{B}=[b_{0},b_{1},b_{2},b_{3},b_{4},b_{5},b_ {6},b_{7},b_{8}],\\ \mathbf{c}_{i}=[c_{0},c_{1}\cdots c_{iL-1}]^{\mathsf{T}},\ i= \overline{0,8},\ L=45\cdot 10^{-3}\cdot Fs,\\ \mathbf{C}=[\mathbf{c}_{0},\mathbf{c}_{1},\mathbf{c}_{2},\mathbf{ c}_{3},\mathbf{c}_{4},\mathbf{c}_{5},\mathbf{c}_{6},\mathbf{c}_{7},\mathbf{c}_{8}]^{ \mathsf{T}},\\ \mathbf{C}_{\text{out}}=\mathbf{BC}\cdot\mathbf{w}(L),\end{gathered}  \tag{5}\]  
where \(Fs\) is the sampling frequency, \([\cdot]^{\mathsf{T}}\) is the transposition operator, and \(\mathbf{w}(L)\) is a Hamming window with the length equal to \(L\). This method of encoding the ENF is based on the information illustrated \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Chirp Signal Identifier** & **Frequency Range (kHz)** & **Chirp Signal Identifier** & **Frequency Range (kHz)** \\ \hline \(\mathbf{c}_{0}\) & 8–8.24 & \(\mathbf{c}_{5}\) & 12.6–12.84 \\ \(\mathbf{c}_{1}\) & 8.72–8.96 & \(\mathbf{c}_{6}\) & 13.92–14.16 \\ \(\mathbf{c}_{2}\) & 9.54–9.78 & \(\mathbf{c}_{7}\) & 15.34–15.58 \\ \(\mathbf{c}_{3}\) & 10.46–10.7 & \(\mathbf{c}_{8}\) & 16.96–17.2 \\ \(\mathbf{c}_{4}\) & 11.48–11.72 & & \\ \hline \hline \end{tabular}
\end{table}