4.2 Acoustics of Speech Production

Transcription

1 4.2 Acoustics of Speech Production Acoustic phonetics is a field that studies the acoustic properties of speech and how these are related to the human speech production system. The topic is vast, exceeding the scope of this handout, but here we try to cover the essentials. The most important acoustic properties of the vocal tract are the resonancies that originate in a manner similar to that of, e.g., wind instruments, i.e. as resonating standing waves in an air tube. If the tube is uniform having one closed and the other open, the standings waves will occur so that the change of pressure is smallest in the closed and highest in the open as illustrated in figure 1. If the length of the tube is l, then the wavelengths (λ) are 4l, 4 3 l, 4 5 l, 4 7 l,... The length of the vocal tract of a typical adult is about 17 cm, and the speed of sound in the air (denoted by c) is assumed to be 340 m/s. The resonance frequencies can be calculated using the basic equation of wave motion, c = fλ, so in this case the frequencies, f, are m/s, m/s in other words, the odd harmonics of 500 Hz., 5 340m/s, Hz, 1500Hz, 2500Hz,... Figure 1: Standing waves in a half-closed tube. The figure shows the change of pressure which is zero at the closed and highest at the open. The human vocal tract is not an uniform tube but still, in vowels, there occurs roughly one formant per 1 kilohertz, as in the case of an uniform tube seen above. Only the formant frequencies are not harmonically related anymore but have been displaced according to the shape of the vocal tract. Solving formant frequencies based on the form of a vocal tract is not in general analytically possible (numerical solutions can be estimated, though). The relation between the vocal tract model and 24

2 reality can also be questioned. There are several facts that should be taken into account when aiming at an accurate model of speech production, such as various sound sources, time and space variations of the vocal tract, nasal coupling, radiation at the lips (i.e., the spread of the sound wave to the ambient air), different energy losses, rotational air flows, and so on. In fact, a formal theory of sound propagation for the complete configuration with all such considerations does not exist. However, much insight into speech production can be gained through studying simplified analog models. A particularly interesting and useful approach is to restrict the consideration to a model consisting of several concatenated uniform tubes. This allows an analytical solution with rather high accuracy and moderate effort. Let us first study a lossless uniform tube and assume that the travelling waves in the tube are planar as in figure 2. In tubes like this, there occur only pressure waves moving at the speed of sound in either direction, and their superpositions, i.e. sums of multitude of waves. Figure 2: In a uniform lossless tube, there occur only sums of pressure waves moving at the speed of sound in either direction. The wave is presumed to be planar. When concatenating two uniform tubes, the pressure waves still move at the speed of sound inside both tubes, but at the juction point also reflection occurs. Let us denote the cross-sectional area of the 25

3 left tube as S n and of the right tube as S n+1. The reflection coefficient k n is defined as follows: k = S n S n+1 S n + S n+1. Notice that 1 < k n < 1. The reflection coefficient diplays how large part of the pressure wave moving in the tube is reflected back at the junction. For example, of the wave travelling right in tube 1, a portion expressed by k 1 is reflected back to the tube 1, and the rest (1 k 1 ) passes on to tube 2. Correspondingly, of the wave travelling left in tube 2, a portion expressed by k 1 is reflected back. Let us now move on to the discrete-time model. Assume that the tubes are of equal length and the sampling period is equal to the time interval required by the sound wave to travel the length of one tube. The reflection of the pressure waves may be expressed as [ fn+1 (z) b n (z) ] [ (1 kn )z = 1 k n k n z 2 (1 + k n )z 1 ] [ fn (z) b n+1 (z) where b n (z) is the z-transform of a pressure wave moving backwards and f n (z) is the same for forward moving wave. The reflection coefficient k n represents the junction between tubes n and n + 1. The indices of the tubes start from the left. These are called Kelly-Lochbaum equations and they can be expressed as a flowchart as illustrated in Figure 3. We can now catenate several tubes and solve the transfer function of the system by using the equations above. In speech processing it is customary to think that glottis is on the left and mouth on the right. We still have to model the boundaries of the system somehow, for instance f 0 (z) = x(z) + gb 0 (z), where x(z) is the z-transform of a glottis signal and g tells how much of the pressure wave is reflected back at the glottis. Correspondingly, the of the vocal tract is often modeled as b N (z) = 0, where N is the index of the last tube, meaning that there is no reflection coming into the last tube (lips). Take now the forward travelling wave of the last tube as a an output of the system, y(z) = f N (z). Notice that the system is a linear filter since the system consists purely of delays, sums and multiplications by constants. The realization of a filter like this is called lattice structured and it has been proved useful in adaptive filtering for instance. Below we give a Matlab script of a realization of the filter described above (any bugs can be reported to the lecturer). The code can also be found on the web page of the course. % Demo on Kelly-Lochbaum equations clear close all %S = [ ]; % surface areas of the tubes. The last is out there S = [ ]; len = 0.03; % the length of a tube, m v = 340; % the speed of sound, m/s k = (S(1:-1)-S(2:))./(S(1:-1)+S(2:)); % reflection coefficients ], 26

4 Figure 3: Kelly-Lochbaum equations expressed as flowchart. fs = round( v/len); % sampling frequency x = zeros(1,1000); x(1) = 1; % let s calculate impulse response of length 1000 g = 0.5; % reflection at the glottis d = 0.95; % loss coefficient (quite arbitrary) F0 = zeros(1,length(s)); % forward travelling waves before delay, % first is glottis F1 = zeros(1,length(s)-1); % forward travelling waves after delay B0 = zeros(1,length(s)); % backward travelling waves after delay B1 = zeros(1,length(s)); % backward before delay for n = 1:length(x), F0old = F0; F1old = F1; 27

5 B0old = B0; B1old = B1; B0(1) = B1old(1); F0(1) = B0(1)*g+x(n); % glottis-excitation + reflection F1(1) = F0old(1); if ( length(b0) < 3), B1(1) = 0; else B1(1) = B1(2)*d*(1+k(1)) + F1(1)*d*k(1); for i = 2:length(F1), B0(i) = B1old(i); F0(i) = F1(i-1)*d*(1-k(i-1)) + B0(i)*d*(-k(i-1)); F1(i) = F0old(i-1); B1(i) = B1(i+1)*d*(1+k(i)) + F1(i)*d*k(i); b02(n) = B0(2); B0() = 0; % no reflection back F0() = F1()*d*(1-k()); y(n) = F0(); % save the output figure freqz(y,1,1024,fs); title( Frequency response ) 28