Glottal source

So far, we have developed a transfer function V(z) which is defined as the ratio of the volume velocity of the lips over the voliune velocity at the glottis. In practice however, when we measure sound, we normally in fact measure pressure signals, as this is what most microphones are designed to respond to. Most microphones exist in the far-field, that is at some distance fi om the lips, and hence the signal is influenced by a radiation impedance from the hps. This can be modelled by another transfer function, R(z), which, from Equation 11.4 is defined as 

Modelling radiation accurately soon becomes very complicated, so for our purposes, we use a very simple model that has been shown to do a reasonable job at approximating the radiation. This takes the form of a function which differentiates the signal, and this can be modelled by a FIR filter with a single zero  

We qualitatively described in Section 7.1.2 the process of how the glottis produces sound. In voiced speech, the vocal folds undergo a cycle of movement which gives rise to a periodic soince 

An extensive study into glottal modelling is given in Flanagan [164], which describes various mass/spring/damped systems. These models can be somewhat difficult to model in discrete time systems, so instead we adopt models which simply generate a time domain function which has the properties described above. One such model [376] [368] is given by  

The most important control element in the glottal source is of course the rate at which the cycles occur. In Equation 11.30, this is determined by the positions oiN and N2, in Equation 11.32 by Tq, Te and 7b and in Equation 11.31 this is determined by the impulse function u n. In the last case, the glottal volume velocity function can be thought of as a low pass filtering of an impulse stream. From these expressions and empirical measurements, it is known that this low pass filter creates a roll off of about -12dB per octave. It is this, combined with the radiation effect, that gives all speech spectra their characteristic spectral slope. 

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By definition, H gives the transfer function and frequency response for a unit impulse. In reality of course, the vocal tract input for vowels is the quasi-periodic glottal waveform. For demonstration piuposes, we will examine the effect of the /ih/ filter on a square wave, which we will use as a (very) approximate glottal source. We can generate the output waveforms y[n] by using the difference equation, and find the fi equency response of this vowel from //(e/ ). The input and output in the time domain and frequency domain are shown in figure 10.26. If the transfer function does indeed accurately describe ihe frequency behaviour of the filter, we should expect the spectra oiy[n, calculated by DFT to match H eJ )X(eJ ). We can see fiom figure 10.26 that indeed it does. 

For example, in the case of vowels, speech is produced by the glottal source waveform travelling through the pharynx, and as the nasal cavity is shut off, the waveform progresses through the oral cavity and is radiated into the open air via the lips. Hence as filters connected in series are simply multiplied in the z-domain, we can write the system equation for vowels as ... 

Where U(z) is the glottal source, with P(z), 0(z) and R(z) representing the transfer functions of the pharynx, the oral cavity and the lips respectively. As P z), 0 z) linearly combine, it is normal to define a single vocal tract transfer function V(z) = P z)0 z), such that Equation 11.1 is written... 

Figure 11.15 Plot of idealised glottal source, as volume velocity over time. The graph shows the open phase, where air is flowing through the glottis, the short return phase, where the vocal folds are snapping shut, and the closed phase, where the glottis is shut, and the volume velocity is zero. The start of the closed phase is the instant of glottal closure. ...

How do we now find the glottal source signal from the residual x[n l Recall that in the z-domain, the main filter expression is... 

The second problem concerns just how accurate our model of articulation should be. As we saw in our discussion on tube models, there is always a balance between the desire to mimic the phenomenon accurately and with being able to do so with a simple and tractable model. The earliest models, were more or less those described in Chapter 11, but since then a wide number of improvements have been made many along the lines described in Section 11.5. These have included modelling vocal tract losses, source-filter interaction, radiation from the lips, and of course improved glottal source characteristics REFS. In addition many of these have attempted to be models of both the vocal tract itself and the controls within it, such that many have models for muscle movement and motor control. 

Lu, H.-L. and Smith, J. O. (2000). Glottal source modelling for singing voice synthesis. Proceedings of the 2000 International Computer Music Conference, pp. 90-97. 

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