## 2 Theoretical Background
### Cepstral Analysis  
A somehow basic but effective description of the cepstrum (that can be applied to both its power and complex definitions) is the \"spectrum of the spectrum on logarithmic amplitude axis\" [39], as it is computed as the inverse transform of the logarithm of its estimated spectrum. First described in [37], its original and main use is for pitch detection and determination [40], but in the years it has found plenty of applications in speech processing, such as for the estimation of harmonics-to-noise ratios [41]. Resorting to the cepstrum rather than the standard frequency spectrum gives some interesting advantages, mainly the possibility to easily discern and eventually edit off uninteresting harmonics. Some slightly different definitions exist, such as e.g., the discrete complex cepstrum with unwrapped phase, which are useful for time-frequency analysis purposes [42]. Preserving the phase information, the complex extension of the cepstrum allows for reconstructing the original time-dependent signal; however, for the purpose of this research, the definition of the power cepstrum is more convenient. Analytically, this can be defined by resorting to the direct and inverse z-transform definitions, as ([43], Chapter 24)  
\[f\triangleq\left[\mathcal{Z}^{-1}\left\{\log\left(\left|H(z)\right|^{2} \right)\right\}\right]^{2}=\left[\frac{1}{2\pi j}\oint_{\Gamma_{c}}\log\left( \left|H(z)\right|^{2}\right)z^{n-1}dz\right]^{2} \tag{1}\]  
where \"\(\sim\)\" denotes the power cepstrum of function \(f\), \(\mathcal{Z}^{-1}(\cdot)\) is the inverse z-transform operator, \(H(z)=\mathcal{Z}(f)\) represents the z-transform of an ordered, discretised time sequence \(f=x[n]\), and \(n\) indicates the specific time instant; the contour of integration, \(\Gamma_{c}\), lies within an annular region where \(H(z)\) is defined as analytic and single-valued. Evaluating Equation (1) on the unit circle, the problem is reconducted to the Fourier domain, and it can be restated as  
\[\hat{f}=\left[\frac{1}{2\pi}\int_{-\pi}^{\pi}\log\left(\left|H(\omega)\right|^ {2}\right)e^{j\omega t}dt\right]^{2} \tag{2}\]  
For an angular frequency \(\omega=2\pi f\) and the Discrete Fourier Transform \(H(\omega)\). The basic idea exploited is the concept of homomorphic deconvolution, i.e., separating different components of one given signal, if \(f\) itself is made up by a convolution of some distinct underlying signal components. With respect to the simple, more common Fourier Transform in the frequency domain, this process allows for better discerning between the content of interest and unnecessary/unrelated influences, disturbances or echoes, as convolution in time domain becomes summation in cepstral domain. Thus, thanks to the frequency-warping, the cepstral response simultaneously incorporates properties from both the time and the frequency domains. Please note that, due to direct and inverse transform, the measurement unit is the same as the departing one i.e., time expressed in seconds; nevertheless, \(f\) cannot be considered to be an ordered time series, because of the warping. Further details can be found in [15]. However, the cepstrum of a signal maintains its same dimensionality, without any gain in conciseness with respect to raw THs. Thus, specific features, i.e., the Cepstral Coefficients (CCs), have to be extracted from it.