Assignment 8: Tube Resonances

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1 Linguistics 582 Basics of Digital Signal Processing Assignment 8: Tube Resonances Reading: Stevens, K. (1989). On the quantal nature of speech. Journal of Phonetics, 17, Read pp ONLY. Johnson, K. Chapter 6 Exercise: The vocal tract slide trombone: nomograms Stevens (1989) analyzes the resonances of idealized tube configurations that correspond in essential ways to the shapes of the vocal tract in the production of vowels. The tube configurations involve the concatenation of up to four cavities or sub-tubes, each characterized by a length and an area. back tube (Lback, Aback) constriction tube (Lcons, Acons) front tube (Lfront, Afront) lip tube (Llips, Alips) The upper figure on the following page shows one tube configuration (without a lip tube), and the lower one, traditionally called a nomogram, shows how resonances depend on the location the constriction cavity (represented by the length of the back cavity), as it slides from the back of the tube to the front, for 3 different values of Acons. The goal of this exercise to write functions that can be used to reproduce the nomograms by sliding a constriction in steps, calculating formant frequencies of each step, and also playing the sound that would be produced for each step by using the synthesize function. The basic script for producing the nomogram is called playtubes, and is shown on the next page. The script first sets some constants, then changes the value of Lback in a loop and calls a function called tuberes to calculate values of the formant frequencies (which are accumulated in a matrix and plotted as a nomogram at the end of the loop), a function called vt_transfer, to calculate an approximation to the vocal tract transfer function from the formant frequencies, and synthesize_only, that you used for this week s homework, that plays the synthesized sound, but suppresses any plottting. At each step through the loop, the vocal tract transfer function returned by vt_transfer is plotted, along with the area function, calculated by makeareas function (see below). The area function is a description of the tube as composed of a number of equal sections from the larynx to the lips, each of which has a particular area. At each step, the sound that would be produced by filtering a buzz through the formants is also played out.

2 Aback Acons Afront Lback Lcons Lfront

3 % playtubes.m % Louis Goldstein % 12 October 2009 % Slide constriction along VT, calculate resonances, F,BW,transfer function, and sound % plot nomogram % set constants L = 16; Lcons = 2; Afront = 3; Acons =.2; Aback = 3; srate = 10000; sectionsize =.5; f0 = 100; dur =.5; % initialize nomogram Fall = []; % begin loop with Lback=1cm; % end when the Lfront=1cm; for Lback = 1:sectionsize:(L-(Lcons+1)) Lfront = L - (Lback + Lcons); [A dist] = makeareas (sectionsize, Lback, Lcons, Lfront, Aback, Acons, Afront) figure (1) subplot (2,1,1), plot (dist, A, 'o-'); ylabel ('Area in cm') title ('Distance from Glottis') F = tuberes (srate, Lback, Lcons, Lfront, Aback, Acons, Afront); if length(f)<4 F(4) = 5000; end BW = getbandwidths(f); % uses empirical approximation of Fant (1984) [h,w] = vt_transfer(srate,f,bw); subplot (2,1,2), plot (w*srate/(2.*pi), 10*log(abs(h))) ylabel ('DB') xlabel ('Frequency in Hz') title ('Vocal Tract Transfer Function') out = syn_noplot(srate,f,bw,f0,dur); Fall = [Fall F(1:4)']; pause (1) end % plot nomogram figure (4) Lback = 1:sectionsize:(L-(Lcons+1)) plot (Lback, Fall) ylim = [0 4000]; xlabel ('Length of Back Cavity in cm'); ylabel ('Formant Frequency in Hz');

4 (1) Complete playtubes.m by writing the function tuberes. Test it to make sure it works. (a) tuberes.m To keep things relatively simple for this version, restrict the function to the case in which there is no separate lip tubelet, thus the tuberes function should look like this: function F = tuberes(srate, Lback, Lcons, Lfront, Aback, Acons, Afront) input arguments: srate sampling rate in Hz Lback length of back cavity in cm Lcons length of constriction cavity in cm Lfront length of front cavity in cm Aback area of back cavity in cm^2 Acons area of constriction cavity in cm^2 Afront area of front cavity in cm^2 output argument: F vector of formant frequencies How are the formants calculated? 1. For each of the 3 tubelets, calculate the lowest 4 uncoupled natural resonances. The back tube is effectively closed at both ends. The constriction tube is effectively open at both ends The front tube is effectively closed at one end, open at the other 2. Calculate the Helmholtz resonance of the back cavity and constriction cavity system, as below: 3. Concatenate all resonances into a single vector and sort from lowest to highest frequency. To sort, you can use the Matlab sort function. 4. Now approximate the effects of coupling. Check each successive pair of formants to see if they are separated by at least ΔF Hz (see below). If not, set the higher of the pair to the lower value plus ΔF. ΔF = f = c 2π Acons Aback Lback Lcons c 2 2π 2 Lcons L Fn Acons A where

c = 35000 cm/sec (speed of sound) L = max(lback, Lfront) A = min(aback, Afront) Fn = the frequency of the lower member of the pair 5.

5 c = cm/sec (speed of sound) L = max(lback, Lfront) A = min(aback, Afront) Fn = the frequency of the lower member of the pair 5. Truncate the formant vector so the highest formant frequency is the (srate/2). (2) Create a new version of playtubes.m called playtubes_long.m, that will generate more realistic tube shapes for front non-low vowels (see Stevens, Figures 7 and 8). To do so, Set: Lcons=5; Acons=.3 Aback=3; Afront=1; (3) An alternative method of calculating tube resonances is shown in playtubes_direct.m, which calls the function tuberes_direct. These are shown on the last two pages. tuberes_direct works by first computing the reflection coefficients between adjacent area sections, and then using MATLAB s rc2poly function to derive filter coefficients from reflection coefficients via a lattice implementation. This calculation is more exact than the one in tuberes.m. Create a new version of playtubes_direct.m called playtubes_lips_direct.m,that will generate a nomogram for a tube configuration with a lip tubelet of 1 cm length (Llips), and an area of.3 cm 2 (Alips). (See figure below). The overall length of the vocal tract should remain 16 cm, including the lip section. Other Length and Area parameters can remain the same as in playtubes. To do this, you will need to create a new version of makeareas, called makeareas_lips, which will add two additional arguments, Alips and Llips. The main loop in playtubes will also have to be modified to accommodate the lip tubelet. Aback Acons Afront Alips Lback Lcons Lfront Llips

6 % makeareas.m % Louis Goldstein % 13 October 2009 % % make area function (Area of equal sections as a function of distance from glottis) % from the lengths and areas of tubelets and the length of sections % into which to divide the tube. % % input arguments: % sectionsize length of each section in cm % Lback length of back cavity in cm % Lcons length of constriction cavity in cm % Lfront length of front cavity in cm % Aback area of back cavity in cm^2 % Acons area of constriction cavity in cm^2 % Afront area of front cavity in cm^2 % % output arguments: % A vector containing areas of each section from larynx to lips % dist vector containing distance of each section from larynx function [A dist] = makeareas (sectionsize, Lback, Lcons, Lfront, Aback, Acons, Afront) Back_end = 1+floor(Lback/sectionsize); Cons_end = Back_end + floor(lcons/sectionsize); Front_end = Cons_end + floor(lfront/sectionsize); A(1:Back_end) = Aback; A(Back_end+1:Cons_end) = Acons; A(Cons_end+1:Front_end) = Afront; dist = [0:sectionsize:Lback+Lcons+Lfront];

7 % playtubes_direct.m same as playtubes.m, except for 2 bolded lines. % Louis Goldstein % 12 October 2009 % Slide constriction along VT, calculate resonances, F,BW,transfer function, and sound % plot nomogram % set constants L = 16; Lcons = 2; Afront = 3; Acons =.2; Aback = 3; srate = 10000; sectionsize =.5; f0 = 100; dur =.5; % initialize nomogram Fall = []; % begin loop with Lback=1cm; % end when the Lfront=1cm; for Lback = 1:sectionsize:(L-(Lcons+1)) Lfront = L - (Lback + Lcons); [A dist] = makeareas (sectionsize, Lback, Lcons, Lfront, Aback, Acons, Afront); figure (1) subplot (2,1,1), plot (dist, A, 'o-'); ylabel ('Area in cm') title ('Distance from Glottis') F= TubeRes_direct(A, dist); F = F(F<srate/2); BW = getbandwidths(f); [h,w] = vt_transfer(srate,f,bw); subplot (2,1,2), plot (w*srate/(2.*pi), 10*log(abs(h))) ylabel ('DB') xlabel ('Frequency in Hz') title ('Vocal Tract Transfer Function') out = syn_noplot(srate,f,bw,f0,dur); Fall = [Fall F(1:4)]; pause (1) end % plot nomogram figure (5) Lback = 1:sectionsize:(L-(Lcons+1)) length(lback) size(fall) plot (Lback, Fall); ylim = [0 5000]; xlabel ('Length of Back Cavity in cm'); ylabel ('Formant Frequency in Hz');

8 function [F, a, srate] = tuberes_direct(area, dist) % Louis Goldstein % Oct 13, 2009 % Calculate the resonances of an area function with % an arbitary number of sections, and an arbitary length % function [F, a, srate] = tuberesonances(area, tube_length) % Input arguments: % area vector containing areas of each section from larynx to lips % dist vector containing distance of each section from larynx % Output arguments: % F vector of formant frequencies % a vector of filter coefficients % srate sampling rate in Hz m = length(area); tube_length = dist(m) % m is the number of area sections % sampling period has to be equal to the time it would take % for a wave to travel from one section to the next % and back c = 35000; % speed of sound in cm/sec T = (tube_length/m)*2/c; % time = distance / velocity srate = 1/T; % calculate the reflection coefficient between pairs of sections for i = 1:m-1 k(i) = (area(i+1) - area(i))/(area(i+1) + area(i)); end % the last reflection coefficient is 1, since it the tube area % is assumed to be small compared to the open air k(m) =.999; % calculate a filter coefficients from the reflection coefficients a = rc2poly(k); % find formant frequencies by taking the roots of a and converting to Hz. F = formants (a, srate);

Assignment 7: Tube Resonances

Linguistics 582 Basics of Digital Signal Processing Reading: Assignment 7: Tube Resonances Stevens, K. (1989). On the quantal nature of speech. Journal of Phonetics, 17, 3-45. Read pp. 3-20. ONLY. Johnson,