/ Digital Synthesis of Complex Spectra by Means of Non-linear Distortion of Sine Waves and Amplitude Modulation

Page  1 ï~~ Digital Synthesis of Complex Spectra by Means of Non-linear Distortinn of Sine Waves and Amplitude Modulation - Daniel Arfib IRCAM Paris, France NON LINEAR DISTOk1ION OF SINE WAVES; -............................................................ THE NON LINEAR O1STOkIION OF SINE WAVES CAN BE INTRODUCED BY ANALOGY TO THE AMPLIFICATION OF A SINE WAVE BY A BAD.IHICH PRODUCES HAR:4ONIC DISTORION. THE QUALITY OF WHICH 15 DEPENDENT UPON THE INPUT AMPLITUDE AND THE TRANSFER FUNCTION OF THE AMPLIFIER. 10EXAMPLES WILL POW BE GIVEN. FOR EACH OF THEM THE TRANSFER FUNCTION AID THE DISTORTED SINE WAVE IS REPRESENTED. THE FIRST AMPLIFIER IS SATURATING. SO ITS TRANSFER FUNCTION HAS A BREAK POINT AT ONE VALUE; AS IT IS SYMMETRIC AROUND ZERO. THE OUTPUT SPECTRUM IS COMPOSED OF ONLY ODD HARMONICS; - M1 1 7 MY+ ilw rif KrMM MYr w IrrWFi +1rM+W1.:W1 11YaL1ki+4tMMrMi iMY1wY/111i + ilYl Aa i ii - - 'i iitiliiliMglMliMi " a w. irw S -. " iewl IN ADUlTION TO ADDITIVE AND SU[ISTRACTIVE SYNTHESIS. NEW RESFARCH HAS FOCUSED ON MOPE GoLHAL METHOUS WHICH DETERMINE LINE SPECTRA,WHr;E TOE LINE aly FREQUENCIES AND AMPLITUDES ARE GOVERNED HY A FEW PARAMETERS. THIS CAN BE OBTAINED IN FM (CHOWNING) OR OTHER TRIGONOMETRIC RELATIONS SUCH AS THE DISCRETE SUMMATION FORMULAE METHOD (MOOHER). AN OTHER WAY,HERE PRESENTED, IS TO COMBINE NON LINEAR DISTORTION OF SINE WAVES (ALSO CALLED WAVESHAPING) si t..fet AND AMPLITUDE MODULATION BY A S1NEVWAVF DESIGNATED THE CARRIER BY ANALOGY WITH FM. THIE FIRST PART OF THIS PAPER DEALS WITH THE STANDARD USE OF NON LINEAR DISTORTION AS CAN BE USED IN DIGITAL (HARDWARE OR SOFTWARE) PROCESSING WITH APPLICATION TO MUSICAL SOUNDS. THE SECOND PART INTRODUCES AMPLITUDE MODULATION AND DEMONSTRATES WITH EXAMPLES ITS MUSICAL EFFECT. THE THIRD PART IS ABOUT POSSIBLE DEVELOPMENTS OF THIS METHOD,SUCH AS THE DOUBLE MODULATION, WHICH PERMITS THE PERFORMANCE OF MORE ELABORATE SOUNDS WITH A SLIGHT INCRESE OF PROCESSING. ".r ""i.""wf"""""""#""s"""ws""."w""gr""w"""w""""""w"""""min= an""" THE SECOND ONE AS WELL,; PRODUCES CROSSOVER DISTORSION, AND WILL PRODUCE ODD HARMONICS * * wwwwwawwwwwwwww * * * * -* wswwwwwwwwwwwwwwww ww H

Page  2 ï~~ s 046joll, - low, * DEFINITION OF A TRANSFEN FUNCTION IN TERMS OF A POLYNOMIAL; USING A TRANSFER FUNCtION DEFINED IN TERMS OF A POLYNOMIAL OF DEGREE N INSTEAD OF LINE SEGMENTS GIVES US AN AMPLIFIER WHICH PRODUCES NO MORE THAN N HARMONICS. THEREFORE THE OUTPUT SPECTRUM IS HAND LIMITED AND THIS ELIMINATES ANY FOLDOVERyIF THE DEGREE OF THE POLYNOMIAL IS WELL CHOSEN. AN IMPORTANT FEATURE IS THE ABILITY TO COMPUTE THIS TRANFER FUNCTION WHEN THE AMPLITUDE INPUT AND OUTPUT SPECTRUM DEFINED. CONVERSELY IT IS POSSIBLE TO COMPUTE THE EVOLUTION OF THE OUTPUT SPECTRUM AS A FUNCTION OF THE EVOLUTION OF THE INPUT AMPLITUDE THE THIRD ONE IS A RECTIFIER=ITS TRANSFER FUNCTION IS SYMMETRIC ALONG THE Y AXIS, IT PRODUCES ONLY EVEN HARMONICS, THAT IS TO SAY IT SOUNDS ONE OCTAVE HIGHER. *# * * * * * * * " - --swwwwww*-------------------------------------------- FIGURE INDICATING WHAT HAPPENSI A SPECIAL CLASS OF THESE POLYNOMIALS IS THE SO-CALLED TCHEBYCHEV POLYNOMIALS.. THEY PRODUCE AN OUTPUT SPECTRUM COMPOSED OF ONLY ONE HARMONIC FOR AN UNITY VALUE OF THE AMPLITUDE INPUT, THIS PRODUCES A FREQUENCY MULTIPLICATION, BUT FOR OTHER VALUES, THE SPECTRUM IS MORE COMPLEX. FIGURE INDICATING HOW THIS MULTIPLICATION APPEARS SOUNDS THATOWe WILL NOW HEAR ARE PRODUCED BY THE DISTORTION OF A SINE WAVE OF GRADIUALLY INCREASING AMPLITUDE BY ODD ORDER TCHEBYCHEV r

Page  3 ï~~ IN A MORE MUSICAL ENVIRONMENT, A VERY EFFICIENT SCORE USING MUSIC V PROGRAM CAN GIVE RISE TO SUCH EXAMPLES - POLYNOMIALS. ONLY ODD HARMONICS ARE PRODUCED. SOUND 6 r SOUND 2.,.: NOW EVEN ORDER TCHEBYCHEV POLYNOMIALS WILL PRODUCE EVEN HARMONICS.AND SO THE SOUND APPEARS ONE OCTAVE HIGHER. SOUND 3 WE CAN NOW TAKE A LINEAR COMBINATION OF THESE FUNCTIONS TO PERFORM A STEADY STATE DEFINED IN TERMS OF HARMONIC DISTRIBUTION. THE TWO NEXT EXAMPLES END WITH SUCH A DISTRIBUTION, THE FIRST RICHER THAN THE SECOND! SOUND 4 MORE INTERSTING SOUNDS CAN HE CREATED BY CHOOSING A TRANSFER FUNCTION. AND ALLOWING THE INPUT TO -*V4W WITH TIME. THE TRANSFER FUNCTION CAN ALSO RE EVALUATED AS A POLYNOMIAL APPROXIMATION OF A KNOWM DISTORTION OR BY EXPERIMENTAL AND SOMEWHAT ARBITRARY DETERMINATION.THIS CAN LEAD TO SUCH SOUNDS$ AMPLITUDE MODULATION OF DISTORTED WAVES IF WE MULTIPLY SIGNALS PROCESSED AS BEFORE BY A PURE SINE WAVE, THE INITIAL SPECTRUM IS SHIFTED BY THE AMPLITUDE MODULATION FREQUENCY. THEREFORE COMPONENTS APPEAR WHICH ARE FOLDED THROUGH ZERO, IN THE SAME SENSE AS IN FM. IF THEY COINCIDE WITH SHIFTED LINES, THE AMPLITUDE MODULATED SIGNAL'S FUNDAMENTAL FREQUENCY IS EQUAL TO THE ONE OF THE INITIAL SIGNAL. THIS MEANS THAT THE SPECTRUM IS FORMANT LIKE; IN THIS CASE THE PHASE OF THE AMPblTUDE MODULATION WAVE INFLUENCES THE SPECTRUM, THE NEXT SOUND SHOWS SUCCESSIVE SHIFTS OF AN ORIGINAL DISTORTED WAVE: SOUND IF THE RATIO OF THE FUNDAMENTAL FREQENCY OF THE DISTORTED WAVE BY THE AMPL1TUDE MODULATION FREQUENCY IS A UOTIENT OF LOW ORDER PRIME NUMBERS, WE CREATE SOUNDS WITH MISSING HARMONICS: i... SOUND SOUND 5 THERE IS AN ABSOLUTE INDEPENDENCE BETWEEN THE EVOLUTION OF ODD AND EVEN HARMONICS:ODD ONES ARE DEPENDING UPON 00 ORDER COEFFICIENTS OF THE POLYNOMIAL THE SAME IS TRUE FOR EVEN ONES. THAT MEANS FOR EXAMPLE THAT IF WE SET TO ZERO EVEN ORDER COEFFICIENTS ONLY ODD HARMONICS WILL BE PRODUCED. IF THIS RATIO IS NOT CLOSE TO A RATIONAL 'SIMPLE FRACTION, INHARMONICITY IS PRODUCED;...:. w SOUND WE SEE THAT WE HAVE THE AVANTAGES OF THE CLASSICAL FM METHOD, WITH A NEW IMPORTANT POSSIBILITY! THE ABILITY TO HAVE DIFFERENT DISTORTION. AND SO DIFFERENT TIMBRES, GIVES US A NEW DEGREE OF FREEDOM. W

Page  4 ï~~ DEVELOPMENTS OF NON LINEAR DISTORTION; IF THE WAVEFORM OF THE AMPLITUDE IS NOT A SINE WAVE BUT A COMPLEX WAVE, THE OUTPUT SPECTRUM IS THE CONVOLUTION OF THE TwO INPUT SPECTRA$ IF THIS COMPLEX WAVE 15 IN FACT AN OTHER DISTOHED SINE WAVE, WE HAVE THEN A DOUBLE MODULATION WHICH CAN YIELD MORE COMPLEX SOUNDS. IT CAN BE UNDERSTOOD - EF'r AAS A SYSTEM OF MULTIPLE CARRIERS, EACH OF THEM HAVING ITS OWN S'PWEWM oFwMIuh - DEVELOPMENT. MANY SPECIAL ARRANGEMENTS CAN BE FOUND, t4 ONE EXAMPLE IS NOW PRESENTED; SOUND CONCLUSION; FO CONCLUDE, WE CAN EXAMINE SOME IMPLEMENTATIONS OF THIS METHOD. AS IT ONLY REDUIRES A SINE GENERATOR, A POLYNOMIAL TABLE CONSTRUCTION AND A TABLE LOOK-UP, IT IS VERY EFFICIENT TO USE THIS METHOD IN A COMPUTER SYNTHESIS PROGRAM; THE HARDWARE IMPLEMENTATION MAY NOT BE COMPLICATED AND A RELATIVELY LOW-POWER MINICOMPUTER HAS BEEN USED TO SIMULATE A MONOPHONIC VOICE IN REAL TIME, THE SCANNING OF THE OUTPUT WAVEFORM BEING PERFORMED BY HARDWARE. A CURRENTLY RELEVANT PROBLEM IS HOW TO RELATE MUSICAL OR PSYCHO ACOUSTICAL DATA TO PARAMETERS SUCH AS TUE POLYNOMIAL TRANSFER FUNCTION AND FREOUENCY SPECTRUM. SOME MATHEMATICAL PROGRAMS CAN HELP. BUT AN INTUITIVE APPRECIATION IS OFTEN REQUIRED. MORE DETAILS ABOUT IT WILL SOON BE PUBLISHED.