/ Digital Synthesis of Complex Spectra by means of Multiplication of Non-linear Distorted Sine Waves
ï~~ - 5 4 -Li d.0 f(2A)Â~lo\ (4) C (/ B is an (N+1, N+1) matix, the generative procedure and a(10,10) subset of which is given in figure 2. A practical example indicates how to calculate the polynomial coefficients to obtain a given output spectrum for one value of the index. THE STANDARD USE OF NON LINEAR DISTORTION. The preceeding section has shown how to calculate a transfer function F(I). This allows us to produce for a given input of amplitude X an output spectrum with given values of the first N harmonics, using a polynomial of nth order. Our concern is now the evolution of the out put spectrum in terms of the evolution of the X input. With the same transfer function we can calculate every spectrum corresponding to other values of the input amplitude X (which is in fact a timbre index). Each sound produced this way has no more than N harmonics, and the evolution of the spectrum with X presents no discontinuity, though it can present individual large variations of the harmonics, including zeroing and changing of sign. As indicated earlier, the evolution of odd and even harmonics are absolutely unrelated (as a limit example, setting to zero the even order coefficients terms of the polynomial insures every spectrum to comprise only odd harmonics). For any transfer function, we can draw the evolution of a spectrum as a function of the input amplitude. A possible representation is presented in figure 3 were the evolution of the spectrum is computed for different values of the index. THE STANDARD USE OF NON LINEAR DISTORTION IN DIGITAL SYNTHESIS. To perform non linear distorsion, a sound synthesis program or a digital device requires a polynomial function generator in order to get a table corresponding to the transfer function, and the simulation of a distorting amplifier through a table look-up device. THE POLYNOMIAL FUNCTION GENERATOR. A function generator has to compute a table of values corresponding to the transfer function, the input sequence being the coefficients DO, D1, 02,...,ON of the polynomial. Due to the fact that the function is digitally stored in a limited area and its absolute value must be bound, the gene rated function must be centered and normalized; hence we have choosen to store the function in a (2M + 1) area and to compute T is a normalisation value such that the maximum value of this function is 1. It must be clear that with this definition we can relate X and J by n" X and X= 1 and so the excursion of the index of timbre X is limited to (-1,1) as stated in figure 4. This has proved not to be a drastic limitation, because one can transform the original mathematical polynomial and input by the change of variables X' = X/XMAX, 4,.)MA THE TABLE LOOK-UP UNIT GENERATOR. We now need a table look-up device (or program). It must calculate an output from two inputs corresponding to: OUT = I1-F(12). If it is cal culated without interpolation (by truncation) the table must be choosen large enough to avoid a kind of quantification noise. An other feature is that the origin of the scanning is not the first location but the M + 1 one, so the input can be negative. AN INTRUMENT FOR STANDARD USE OF NON LINEAR DISTORTION. We give here more specific details in the framework of a Music V implementation. A function generator has been written, which calculates a polynomial function the values of which are stored in a 512 locations area. Since it is not an odd value, they are effectively stored in 511 locations (figure 4), so the preceeding value of M is 255. A new unit generator has been written, which is a table look-up with interpolation. The standard Music V oscillator could have been used in a degenerated way, with a null increment, however this is cumbersome. An instrument following Mathews definitions can now be defined in figure 5.
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