# Audio Steganalysis Estimation with the Goertzel Algorithm
## 2 Materials and Methods
### Steganalyzer Process
In this step, the entire audio vector is scanned to detect the variation in pitch, which can represent the data inserted in the original audio vector. From the set of data entered, given \(x^{i}\), we initially develop the first sequence as shown below:  
\[X[k]=\sum_{d=0}^{N-1}x^{i}[d]e^{-\frac{2\pi kd}{N}},k=0,1,\cdots,N-1 \tag{1}\]  
Multiplying the right side of the equation by \(e^{-\frac{2\pi kd}{N}}=1\), and rearranging the terms yields the following equation [17]:  
\[X[k]=\sum_{d=0}^{N-1}x^{i}[d]e^{-\frac{2\pi kd(d-N)}{N}},k=0,1,\cdots,N-1 \tag{2}\]  
where \(x^{i}[d]\) represents audio vector analysts and \(N\) is the number of sampling intervals of the signal, \(d\) is the position of the current element of the audio vector to be analyzed and \(k\) represents the \(k\)th element of the FFT. The size of the block \(N\) is equivalent to the number of points in the FFT. This value controls the frequency resolution for which the Goertzel algorithm is used, which is also known as the sample interval width [19; 20]. Step 1. Frequency scanning: In this step, the audio vector is divided to obtain the sample interval from Equations (5) and (6), where \(N\) can take values of 10, 100, 1000, or 44,100. In this paper, we choose \(N=100\), so the interval width would be 441 samples per second. To detect the frequency interval, these must be multiples of the sample rate/\(N\). The value of \(N\) does not necessarily have to be a multiple of two, enabling Goertzel algorithm flexibility with respect to the FFT [19; 20]. See Figure 2.  
\[X_{m}[k]=\sum_{d=0}^{N-1}X[d+k]e^{-\frac{2\pi kd}{N}} \tag{3}\]  
Figure 2: Block diagram of the proposed stegoanalyzer. \[X_{m}[k+1]=\sum_{d=0}^{N-1}X[d+k+1]e^{-\frac{2\pi md}{N}} \tag{4}\]  
Substituting \(p=d+1\) into \(X_{m}^{k+1}\) yields:  
\[X_{m}[k+1]=\sum_{d=0}^{N-1}X[p+k]e^{-\frac{(2\pi m(p+1)}{N}} \tag{5}\]  
By factorizing the domains of the exponential factor, we obtain:  
\[X_{m}[k+1]=e^{i2\pi m/N}\sum_{p=0}^{N-1}X[p+k]e^{-2\pi mnp/N}+X[k+N]e^{-\frac{2 \pi mN}{N}}\cdot e^{\frac{2\pi m}{N}}-X[k]e^{j2\pi m/N} \tag{6}\]  
Finally, due to the periodicity of the exponential term \(\left(e^{j2\pi m}=1\right)\), by substituting in \(X_{m}^{k}\), the previous expression is simplified to apply the sliding window:  
\[X_{m}[k+1]=e^{j2\pi m/N}[X_{m}[k]+X[k+N]-X[k]] \tag{7}\]  
where \(X_{m}[k+1]\) represents the component in the frequency of the possible stegoaudio analyzed, \(m\) represents the vector to be analyzed, \(k\) represents the set of samples to be analyzed, \(d\) represents the current analyzed sample, and \(N\) represents the total length of the vector. Step 2. In this step, the statistical analysis is performed. The coefficients of the high-order statistical models, such as asymmetry, kurtosis, and variance, of \(\overrightarrow{X}\) are obtained from the time/frequency decompositions extracted from the previous step, with \(n=1,2,\ldots,N\). These coefficients provide the characteristics of the statistical distribution of the possible stegoaudio and thus obtain a double validation of the tone frequency that has been identified in the previous step. * Linear Prediction  
Linear prediction analysis is an advanced technology and has been used in many fields. As an important tool utilized in the audio signal process, it has been one of the most popular and effective methods, especially for evaluating the basic speech features such as fundamental tone, formant, spectrum, lower-speed transmissions, voice storage, etc. [5; 21]. We propose using linear prediction to verify the error between the components of the possible stegoaudio from the following expression [21; 22; 23]:  
\[X[k]=\sum_{d=0}^{N-1}x^{i}[d]e^{-\frac{2\pi kd}{N}},k=0,1,\cdots,N-1 \tag{1}\]0  
where \(\overrightarrow{LP}\) is the linear condition, \(Q\) is the stegoaudio, and \(\overrightarrow{w}\) is given by the transposed vector \(Q\). The weights \(w_{k}\) of the coefficients are selected to minimize the estimation of the squared error. The coefficients are selected to minimize the function of the squared error.