# Spectral Envelope Transformation in Singing Voice for Advanced Pitch Shifting
## 2 Parametric Model of the Spectral Envelope
### Description of the Spectral Envelope Model  
The term spectral envelope refers to a real function that passes through the prominent spectral peaks of the spectrum. For harmonic signals, these prominent peaks are normally the harmonics, but the main problem estimating the spectral envelope is revealed when some of the harmonics are missing or weak, suggesting that the spectral envelope should not go through these. Many existing approaches to parameterize the spectral envelope of speech or singing voice are based on linear prediction coding (LPC) [15], whereby the vocal track acoustic response is modeled by an N-order all-pole filter [16]. However, the LPC model has systematic errors in high-pitched  
Figure 1: Diagram of the analysis of the envelope behavior and our pitch shifting scheme with spectral envelope transformation. signals due to the strong dependency between the optimal order of the filter and the fundamental frequency [4]. Furthermore, manipulating the coefficients to achieve specific timbre transformations is not simple, so this technique was discarded to achieve our goal, as well as other further approaches, like line spectral frequencies (LSF) [17] or cepstrum-based methods [4]. An indispensable requirement for using a reliable model is to ease the control of parameters; therefore, we opted for a formant-based model developed in a previous work [14]. The spectral envelope of this model is inspired by some speech/singing synthesis systems, like [18; 19], and it is based on several resonator filters in parallel determining the acoustic formants, plus a certain slope serving as the glottal source. With a source filter and a set of five parallel resonators, we modeled the spectrum of a sung vowel in the frequency band of [0,5000] Hz. The resonators correspond to the glottal resonator \(R_{GP}\) and the first four formants of the vocal tract (\(R_{1}\) to \(R_{4}\)). Thus, the envelope is defined as:  
\[E\left(f\right)_{dB}=Source\left(f\right)_{dB}+Resonances\left(f\right)_{dB} \tag{1}\]  
where the source and the resonances are expressed as:  
\[Source\left(f\right)_{dB}=Gain_{dB}+SlopeDepth_{dB}\left(e^{\alpha\cdot f}-1\right) \tag{2}\]  
\[Resonances\left(f\right)_{dB}=20 log_{10}\left(\left|R_{GP\left(f\right)}+\sum _{i=1}^{4}R_{i}\left(f\right)\right|\right) \tag{3}\]  
where \(R_{i}(f)\) is the frequency response of resonator \(i\). Our approach is modeled with four-pole resonators each consisting of two identical second-order filters in cascade with two complex pole pairs \(p_{1}=p_{2}=\rho_{i}\cdot e^{\theta_{i}}\) and \(p_{3}=p_{4}=\rho_{i}\cdot e^{-\theta_{i}}\), defined by:  
\[R_{x}(z)=\frac{K\cdot\left(1-2\cdot\rho_{i}\cdot cos(\theta_{i})+\rho_{i}^{2} \right)}{\left(1-2\cdot\rho_{i}\cdot cos(\theta_{i})z^{-1}+\rho_{i}^{2}\cdot z ^{-2}\right)^{2}\cdot\left(1-\rho_{i}\right)^{2}} \tag{4}\]  
with:  
\[K=10^{-4}\qquad\rho_{i}=e^{-\pi B_{i}/f_{s}}\qquad\theta_{i}=\frac{2\cdot\pi \cdot f_{i}}{F_{s}} \tag{5}\]  
In these expressions, \(f_{i}\) is the central frequency of resonator \(i\), \(B_{i}\) is the 6-dB bandwidth and \(F_{s}\) is the sampling rate, all of them expressed in Hz. Table 1 shows the ranges of values for the different parameters in our model. \(f_{1}\), \(f_{2}\) and \(f_{3}\) will be considered to define the formants \(F1\), \(F2\) and \(F3\), respectively. Automatically estimating the formant frequencies and bandwidths is an important task for our scheme. The existing algorithms observed [20; 21] suffer from substantial errors or mis-adjustments, requiring human intervention to achieve a reliable annotation of sounds [22]. However, we decided to develop a completely automatic system, which allows a precise estimation of formants by a