/ Digital Synthesis of Complex Spectra by means of Multiplication of Non-linear Distorted Sine Waves
ï~~ -2 - - 3 - music (Risset (5) generated clarinet-like sounds by using a simulation of a saturated amplifier). A first improvement is to use a transfer function in terms of a limited degree polynomial, so that thetinput 4sband limited, eliminating foldover (1). This method uses the fact that it is possible to calculate a transfer function of this form transforming a given input value into a given output spectrum. Conversely it is possible to calculate the output spectrum from every transfer function and input value (Shaefer (6), Suen (7)). For a given transfer function, the variation of the input amplitude (which we will now call the index fonction) leads to a variation of the spectral balance of the output. The knowledge of a transfer function allows us to determine an instrument, i.e. a class of sounds. Notes of different qualities are produced by choosing different index functions. The first part of this paper deals with the standard use of non linear distortion. Although self sufficient, this method becomes more interesting when we add amplitude modulation (multiplication of the output by a sine wave) thereby shifting the spectrum and folding it through zero (as in FM). As we shall explain, this allows us to produce spectra with formant structures, missing harmonics and inharmonic sounds. By multiplying more signals (which can be distorted or not), one can obtain more complex sounds. For example, an equivalent to the double frequency modulation (8) is presented, whereby little additional computation yields complex sounds. Some simple examples are explicitly given and some indications for producing more elaborate sounds are mentioned. THE THEORY. The starting point of calculation has been clearly exposed by Shaefer (6) for a fixed output spectrum and by Suen (7) for an evolving input. Here we give a presentation of their results in matrix form, which may be easier to understand and which is certainly easier to program. Given a non linear amplifier, the transfer function of which is F(I) and an input I(T), the output is F(I(T)). If now the input is a cosine wave of amplitude X and the transfer function a nth order polynomial, then we can write I(TM) -:X coo (-W ) (1) F(.t) =, + x + d d +d3 4.+.. +4,1" Q(A[= do + d4Xc~wA... t,x'-4w Developping cos WT in terms of cos KWT yields the following relations where H. is the amplitude of the harmonic, we obtain (2) Q(A)_ -/t t,C94tM W +...' 4,#4Mw with: (3) H H' l- (A)0 (,, a A is an (N + 1, N + 1) matrix, which is abstracted from a general 2 dimensional array. Figure 1 gives the generative relations and presents the first ten lines and columns of this matrix. A practical example is also given which shows how to calculate the output spectrum partials for a given distortion and one index value. A transfer function of degree N gives a spectrum composed of N harmonics (which is very important in digital synthesis, because this can avoid foldover, provided the sampling frequency is greater than twice the nth harmonic frequency value). There is complete independence between the odd harmonics (depending upon the values of the odd order coefficients of the polynomial) and the even ones. This can be seen in the A matrix, for equals 0 if I and J does not have the same parity. This feature in particular permits an easy control of the balande between odd and even harmonics in a spectrum. Conversely, the expression of cos(Kw) in terms of cos(wr)gives the relations between an output spectrum for a given value of the index X and the resulting di coefficients.
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