Formants analysis

Recall that Equation 13.18 is exactly the same as the linear prediction Equation 12.16, where = fli, 02,..., Op are the predictor coefficients and x[n] is the error signal e n. This shows that the result of linear prediction gives us the same type of transfer function as the serial formant synthesiser, and hence LP can produce exactly the same range of frequency responses as the serial formant S5mthesiser. The significance is of course that we can derive the linear prediction coefficients automatically fi om speech and don t have to make manual or perform potentially errorful automatic formant analysis. This is not however a solution to the formant estimation problem itself reversing the set of Equations 13.14 to 13.18 is not trivial, meaning that while we can accurately estimate the all-pole transfer function for arbitrary speech, we can t necessarily decompose this into individual formants. 

Acero, A. Formant analysis and synthesis using hidden markov models. n Proceedings... 

Whilst the former category is aimed at the identification of the frequencies and amplitudes of the spectrum components (Figure 3.2(a)), the latter uses the estimation of the overall shape of the spectrum s amplitude envelope (Figure 3.2(b)). Short-time Fourier transform (S IFT ) and wavelet analysis are typical examples of harmonic analysis, and predictive analysis is a typical example of formant analysis. Both categories have their merits and limitations as far as sound synthesis is concerned, there is no optimum analysis technique. Some may perform better than others, according to the nature of the task at hand. 

Harmonic analysis methods, such as STFT and wavelets, are not entirely adequate for the analysis of sounds with a high proportion of non-sinusoidal components and, to a certain extent, to non-harmonic combinations of partials. The nature of these signals is not compatible with the notion that sounds are composed of harmonically related and stable sinusoids. Formant analysis proposes an alternative method of representation. The sound is represented here in terms of an overall predictive mould that shapes a signal rich in partials, such as a pulse wave or white noise. The advantage of this method is that the predictive mould does not need to specify the frequencies of the spectrum precisely any value within a certain range may qualify. 

Predictive analysis is a typical example of formant analysis. The core of this method is the interplay between two kinds of filters the all-pole filter and the all-zero filter. In electronic engineering jargon, the pole of a filter refers to a point of resonance in the spectrum and the zero of a filter to a point of attenuation. An all-pole filter is a filter that allows for several resonant peaks in the spectrum. Conversely, an all-zero filter creates various notches in the spectrum. 

The resynthesis process results from two simultaneous synthesis processes one for sinusoidal components and the other for the noisy components of the sound (Figure 3.15). The sinusoidal components are produced by generating sinewaves dictated by the amplitude and frequency trajectories of the harmonic analysis, as with additive resynthesis. Similarly, the stochastic components are produced by filtering a white noise signal, according to the envelope produced by the formant analysis, as with subtractive synthesis. Some implementations, such as the SMS system discussed below, generate artificial magnitude and phase information in order to use the Fourier analysis reversion technique to resynthesise the stochastic part. 

Bristow-Johnson, 1995] Bristow-Johnson, R. (1995). A detailed analysis of a time-domain formant-corrected pitch-shifting algorithm. J. Audio Eng. Soc., 43(5) 340-352. 

It should be clear from our exposition that each technique has inherent tradeoffs with respect to the above wish list. For example, we make many assumptions in order to use the lossless all-pole linear prediction model for all speech sounds. In doing so, we achieve a model whose parameters we can measure easily and automatically, but find that these are difficult to interpret in a useful sense. While the general nature of the model is justified, the assumptions we make to achieve automatic analysis mean that we can t modify, manipulate and control the parameters in as direct a way as we can with formant synthesis. Following on from this, it is difficult to produce a simple and elegant phonetics-to-parameter model, as it is difficult to interpret these parameters in higher level phonetic terms. 

Perform an inverse cosine transform, which gives us a mel-scaled spectrum. This differs from the analysis mel-scale cepstrum because of the liftering, but in fact the differences in the envelopes of the original and reconstructed spectra have been shown to be minor, particularly with respect to the important formant locations. 

Throughout the book, we have made statements to the effect that statistical text analysis outperforms rule methods, or that unit selection is more natural than formant synthesis. But how do we know this In one way or another, we have evaluated our systems and come to these conclusions. How we go about this is the topic of this section. 

We will now turn to the important problem of source-filter separation. In general, we wish to do this because the two components of the speech signal have quite different and independent linguistic ftmctions. The source controls the pitch, which is the acoustic correlate of intonation, while the filter controls the spectral envelope and formant positions, which determine which phones are being produced. There are three popular techniques for performing source-filter separation. First we will examine filter-bank analysis in this section, before turning to cepstral analysis and linear prediction in the next sections. 

Spectrum modelling approaches from additive to analysis-resynthesis and formant... 

Spectrum analysis is fundamentally important for spectral modelling because samples alone do not inform the spectral constituents of a sampled sound. In order to model the spectrum of sounds, musicians need adequate means to dissect, interpret and represent them. A number of methods have been created to analyse the spectrum of sounds. There are two categories of spectrum analysis harmonic and formant. 

Figure 3.2 The two categories of spectrum analysis (a) harmonic and (b) formant...

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