# Engineering Acoustics/source-filter theory

The source-filter theory (Fant 1960) hypothesizes that an acoustic
speech signal can be seen as a source signal, filtered with the
resonances in the cavities of the vocal tract downstream from the
glottis or the constriction. This simple model for speech synthesis is
based on an assumption that the dynamics of the system is linear and
separable into three main blocks: a glottal energy (soure), the vocal
tract (filter) and the effect of modeling radiation sound that are
independent(As shown in the figure on the right).

The glottal source roughly matches the subglottal systems, while vocal
tract (VT) corresponds to the supra-glottal system. The radiation block
can be considered as a converter, which converts volume velocity in to
acoustic pressure.In general, the radiation characteristic R(f) and the
spectrum envelop of the source function S(f) for the glottal source are
smooth and monotonic functions of frequency. The transfer function T(f),
however, is usually characterized by several peaks corresponding to
resonances of the acoustic cavities that form the vocal tract.
Manipulating the shape of these cavities results in the positions and
amplitudes of the peaks. Figure on the left qualitatively shows the
configuration of vocal tract corresponding to a vowel. The forms of the
source spectrum S(f), the transfer function T(f), the radiation
characteristic R(f), and the sound pressure pr(f) is shown in each case.

The transfer function T(f) is determined by applying the theory of sound
propagation in tubes of arbitrary shape. For frequencies up to 5000 Hz,
the cross dimensions of vocal tract are less than a wavelength of the
sound. Therefore, the sound propagation can be considered as plane waves
parallel to the axis of the tube and the vocal tract can be viewed as an
acoustic tube of varying diameter.

## vocal tract transfer function

The vocal tract is approximated as an acoustic tube of a given length
composed of a number of sections with different section areas. This is
equivalent to the modelling of the sampled vocal tract transfer function
(H(s)) as a superposition of a given number of spectral poles and zeros,
which in spectral domain can be represented by

$H(s)= k\begin{matrix} \prod_{i=1}^N \frac{s-s_{ai}}{s-s_i}\end{matrix}$

Where K is constant, s~a1~,s~a2~,..are the zeros of H(s), s~1~,s~2~,\...
are the poles. for this equation, the poles and zeros mostly occur in
complex conjugate pairs, and the real parts of these complex frequencies
are much less than the imaginary parts which means that the peak energy
lost in one cycle is much less than the energy stored in one cycle of
the oscillation. Therfeore, the poles of H(s) can be expressed as below:

$\frac{1}{k_n}\begin{matrix} \prod_{i=1}^N \frac{s_i{s_i}^*}{(s-s_i)(s-{s_i}^*)}\end{matrix}$

Where Kp is a constant and the stars indicate complex conjugates.
Natural frequencies of vocal tract are represented by poles and the
imaginary parts indicate formant frequencies which refers to frequencies
at which oscillations happen in the absence of excitation and the real
parts give the rates of decay of theses oscillations. In the other
words, depending on the shape of the acoustic tube (mainly influenced by
tongue position), a sound wave travelling through it will be reflected
in a certain way so that interferences will generate resonances at
certain frequencies. These resonances are called
formants . Their location
largely determines the speech sound that is heard.

## Acoustic interpretation of transfer function

!Vocal tract as tubes with varying cross
section\|300px

According to acoustics of tubes, Pressure and volume velocity of a tube
at the end of tube (x=L) can be related to the variables at beginning of
the tube (x=0). The following transfer matrix expresses the acoustical
relationship between two sides of a tube in frequency domain:

$$\begin{bmatrix}
P_0 \\
U_0\\
\end{bmatrix}
=T(\omega)* \begin{bmatrix}
P_L \\
U_L \\
\end{bmatrix},   T(\omega)=
\begin{bmatrix}
 a_{11} & a_{22} \\
 a_{21} & a_{22}\\
 \end{bmatrix}$$

$$a_{11}=cos(KL), a_{21}=jsin(KL)/(\rho*c), a_{21}=\rho*c*jsin(KL), a_{22}=cos(KL)$$

where K is the wave number, and L is the length of the tube.The aboved
mentioned relation can be used to calculate the state of the wave field
at one location, having on hand te state of the field at another
location.

Since the vocal tract can be considered as n tubes with different cross
section (see figure on the right), the transer function can be used to
relate the states between glottis and the radiate sound:

$$T(\omega)=T_1(\omega)*T_2(\omega)*T_3(\omega)*....*T_n(\omega)$$

and the overall equation for vocal tract becomes:

$$\begin{bmatrix}
P_g \\
U_g\\
\end{bmatrix}
=T(\omega)* \begin{bmatrix}
P_r \\
U_r \\
\end{bmatrix}, where P_r=U_r*Z_{rad}$$

In this equation, Z~rad~ is the radiation impedance. Up to a frequency
of about 6000 Hz, the acoustic radiation impedance can be written
approximately as:

$$Z_{rad}=\rho*c/A_m*(\pi*f^2/c^2*A_m)*K_s(f)+j*2*\pi*f*\rho*(0.8a)/A_m$$

where A is the area of the mouth opening, a is the effective radius, and
K~s~(f)is adimensionless frequency dependant factor that accounts for
baffling effect of the head.

The Transfer function of the system can be calculated as fallows:

$$H(\omega)=U_r/U_g$$

Therefore the equations result in:

$$P_r(\omega)=U_g(\omega)*H(\omega)*Z_{rad}(\omega)$$

As can be seen, the equation above expresses the pressure in front of
the mouse with a source, filter and radiation characteristics of the
mouse. This equation describes the source filter theory mentioned in the
first section.

## Effects of vocal tract wall and other losses

In previous section vocal tract is modeled as a system without losses,
except the termination impedance term. However, There are some other
second order effects that are necessary for precise modelling, such as
wall effects, heat conduction and viscosity, glottal opening. These
loses can change band-widths of the resonances frequencies. Also, they
can change or shift resonance frequences.

## Resonant frequencies of air in a tube

The relationship between vocal tract shape and transfer function is
complex -we will consider the simple case of a uniform tube. The vocal
tract in a vowel can be approximated by a tube which is closed at one
end (the glottis) and open at the other (the lips). For a relatively
unconstricted vocal tract, the resonances of a 17 cm vocal tract occur
at the following frequencies:

`f= n * c / 4 * L for n = 1, 3, 5, ...`

f = formant frequency in Hz c = speed of sound 34,000 cm/s L= length of
vocal tract in cm

So the lowest formant frequency in a 17 cm. vocal tract is:

f = c / 4 \* L

# 34,000 / 4 \* 17

500 Hz

And the spacing between formants is: f = 2 \*c / 4 \* L

# c / 2 \* L (always twice the lowest f)

1000 Hz

Therefore, formant frequencies are: F~1~=500, F~2~=1500, F~3~=2500,
F~4~=3500.

## Two Tube Vocal Tract Models of Vowels

!Two tube model for vowel
/a/\|left\|300px
!Two tube model for vowel
/i/\|right\|300px

Two resonators or uniform tubes of different cross sectional areas can
be connected to approximate some vowels or consonants. In this case,
natural frequencies of the whole system are not simply the frequencies
for each tube because of acoustic coupling. figures() shows different
configuration of tubes to simulate vowels /a/, /i.

Typical values (for an adult male vocal tract for vowel /a/) are l1 =
8 cm, l2 = 9 cm with A1 = 5 cm2, A2 = 0.5 cm2. Acoustic theory predicts
that there will be resonances at 944 Hz, 1063 Hz, 2833 Hz. The narrow
and wide tubes can be considered as separate tubes with resonance
frequencies obeying the one stated in previous section for a tube.
However, the acoustic impedance at the boundary between two tubes is not
zero, thus effects the natural frequencies of the tubes. The natural
frequencies of the combined system are the frequencies for which the sum
of the reactances at the junction is zero that is:

$$-\rho*c/A_1 cot(KL_1)+\rho*c/A_2tan(KL_2)=0$$

It should be noted that when the natural frequencies of the tubes are
remote from one another, the influence of coupling is small.

Typical values for vowel /i/ in the human vocal tract are l1 = 9 cm, l2
= 8 cm, A1 = 5 cm2, A2 = 0.5 cm2. Thus, in theory, F1 = 202 Hz, F2 =
1890 Hz, F3 = 2125 Hz.

## Four Tube Vocal Tract Models of Vowels

!four tube
model\|400px

Four tube models of vowels provide a much better estimate of formant
frequencies for a wider range of vowels than do two tube models and so
are more a more popular method of modeling vowels. Such models consist
of a lip tube (tube 1) a tongue constriction tube (tube 3) and
unconstricted tubes either side of the constriction tube. This model is
controlled by three parameters. They are i) the position of the centre
of tube 3, ii) the cross-sectional area of tube 3, and iii) the ratio of
the length to the cross-sectional area at the lip section. For extreme
back constrictions tube 4 disappears whilst for extreme front
constrictions tube 2 disappears.

Calculations of resonance frequencies using the 4 tube model are quite
complex and so Fant (1960) supplied a (fairly complex) graphical
representation of the relationship between the three parameters and the
resultant formant frequencies. These graphical representations are
called nomograms. The original
versions of these nomograms supply, for a continuous range of x
constriction positions (i.e. distance from the centre of the tongue
constriction to the glottis) a continuous range of resultant F1 to F5
values. The original nomograms do this for 5 values of lip area (A1) and
for two values of tongue constriction cross-sectional area (A3). For
different vocal tract lengths, different nomograms need to be computed.

The four tube, three parameter, model provides a sufficiently accurate
prediction of most vowel sounds, but cannot model nasalisation of
vowels.

## References

1- Kenneth N. Stevens,2000, Acoustic phonetics, The MIT Press.

2- Kinsler *et al.* 2000,Fundamentals of Acoustics, John Wiley & Sons.

3- Titze, I.R. (1994). Principles of Voice Production, Prentice Hall
(currently published by NCVS.org),
.

4- James L. Flangam and Lawrence R. Rabiner,1973, Speech synthesis.

