The glottal source

An extensive study into glottal modelling is given in Flanagan [164], which describes various mass/spring/damper systems. These models can be somewhat difficult to model in discrete-time systems, so instead we adopt models that simply generate a time-domain function that has the properties described above. One such model [368], [376] 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 of N and Ni, in Equation (11.32) by To, and T and in Equation (11.31) by the impulse function M [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—12 dB per octave. It is this, combined with the radiation effect, that gives all speech spectra their characteristic spectral slope. 

While Equation (11.32) gives a reasonable approximation of the glottal volume-velocity signal, it does not model the secondary effects of jitter, shimmer and ripple. More realism can be added to the glottal signal by the addition of zeros into the transfer function, giving the expression 

We have defined (7(z) as a voliune-velocity signal, mainly for purposes of developing the vocal-tract transfer function. While, in reahty, the radiation R z) occurs after the operation of V (z), we aren t restricted to this interpretation mathematically. As we shall see, it is often usefiil to combine U(z) and R(z) into a single expression. The effect of this is to have a system wherein the radiation characteristic is apphed to the glottal-flow waveform before it enters the vocal tract. This is equivalent to measuring the pressure waveform at the glottis, and, if we adopt the radiation characteristic of Equation (11.29), 

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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... 

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... 

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