Additive Synthesis, Aplitude Modulation and Frequency Modulation Pro Eduardo R Miranda Varèse-Gastproessor eduardo.iranda@btinternet.co Electronic Music Studio TU Berlin Institute o Counications Research http://www.kgw.tu-berlin.de/
Topics: Additive Synthesis Aplitude Modulation (and Ring Modulation) Frequency Modulation
Additive Synthesis The technique assues that any periodic waveor can be odelled as a su sinusoids at various aplitude envelopes and tie-varying requencies. Works by suing up individually generated sinusoids in order to or a speciic sound.
Additive Synthesis eg21
Additive Synthesis eg24
A very powerul and lexible technique. But it is diicult to control anually and is coputationally expensive. Musical tibres: coposed o dozens o tie-varying partials. It requires dozens o oscillators, noise generators and envelopes to obtain convincing siulations o acoustic sounds. The speciication and control o the paraeter values or these coponents are diicult and tie consuing. Alternative approach: tools to obtain the synthesis paraeters autoatically ro the analysis o the spectru o sapled sounds.
Aplitude Modulation Modulation occurs when soe aspect o an audio signal (carrier) varies according to the behaviour o another signal (odulator). AM = when a odulator drives the aplitude o a carrier. Siple AM: uses only 2 sinewave oscillators. eg23
Coplex AM: ay involve ore than 2 signals; or signals other than sinewaves ay be eployed as carriers and/or odulators. Two types o AM: a) Classic AM b) Ring Modulation
Classic AM The output ro the odulator is added to an oset aplitude value. I there is no odulation, then the aplitude o the carrier will be equal to the oset. a i = = a c a a c i eg22
I the odulation index is equal to zero, then there is no odulation. I it is higher than zero then the carrier will take an envelope with a sinusoidal variation.
In classic siple AM, the spectru o the output contains 3 partials: at the requency o the carrier + two sidebands, one below and one above the carrier s requency value. Sidebands = subtract the requency o the odulator ro the carrier and add the requency o the odulator to the carrier.
Aplitudes - The carrier requency reains unchanged - The sidebands are calculated by ultiplying the aplitude o the carrier by hal o the value o the odulation index, E.g. is i = 1, the sidebands will have 50% o the aplitude o the carrier. a = a c i ap _ sidebands = a c (0.5 i)
Ring Modulation The aplitude o the carrier is deterined entirely by the odulator signal. I there is no odulation, then there is no sound eg23
When both signals are sinewaves, the resulting spectru contains energy only at the sidebands. The energy o the odulator is split between the 2 sidebands. The requency o the carrier is not present. RM distorts the pitch o the signal; original pitch is lost.
The ultiplication o 2 signals is also a or o RM.
Both classic AM and RM can use signals other than sinusoids, applying the sae principles. Great care ust be taken in order to avoid aliasing distortion (above 50% o the sapling rate).
Frequency Modulation Modulation occurs when soe aspect o an audio signal (carrier) varies according to the behaviour o another signal (odulator). FM = when a odulator drives the requency o a carrier. Vibrato eect, good exaple to illustrate the principle o FM, with the dierence that vibrato uses sub-audio as the odulator (below 20 Hz). Siple FM: uses only 2 sinewave oscillators. eg25
Siple FM The output o the odulator is oset by a constant, represented as c. I the aplitude o the odulator is equal to zero, then there is no odulation. In this case the output o the carrier will be a siple sinewave at requency c. In the aplitude o the odulator is greater than zero, then odulation occurs. The output ro the carrier will be a signal whose requency deviates proportionally to the aplitude o the odulator. FM1
The aplitude o the odulator is called requency deviation, and is represented as d. The paraeters o the siple FM algorith are: Frequency deviation = d Modulator requency = Carrier aplitude = a c Oset carrier requency = c
I is kept constant whilst increasing d, then the period o the carrier s output will increasingly expand and contract proportionally to d. I d is kept constant whilst increasing, then the rate o the deviation will becoe aster. FM2
The spectru o siple FM sounds The spectru is coposed o the carrier requency ( c ) and a nuber o partials (called sidebands) on either side o it, spaced at a distance equal to the odulator requency ( ). The sideband pairs are calculated as ollows, where k is an integer, greater than zero, which corresponds to the order to the partial counting ro c : c c + k k
The aplitude o the partials are deterined ostly by the requency deviation (d). I d = 0 then the power o the signal resides entirely in the oset carrier requency ( c ). Increasing the value o d produces sidebands at the expense o the power in c. The greater the value o d, the greater the nuber o generated partials and the wider the distribution o power between the sidebands
Modulation index helps to control the nuber o audible sidebands and their respective aplitudes: i = d d = i As i increases ro zero, the nuber o audible partials also increases and the energy o c is distributed aong the. The nuber o sideband pairs with signiicant aplitude can generally be predicted as i = 1. Exaple i i = 3 then there will be 4 pairs o sidebands surrounding c. FM3
Estiating the aplitude o the partials c ay oten be the ost proinent partial in an FM sound; in this case it deines the pitch. The aplitudes o the partials are deined by a set o unctions: Bessel unctions. They deterine scaling actors or pairs o sidebands, according to their position relative to c.
Bessel unctions a c usually deines the overall loudness o the sound The aplitudes o the partials are calculated by scaling ac according to the Bessel unctions. Exaple: B 0 (i) gives the scaling or c, B 1 (i) or the irst pair o sidebands (k=1), B 2 (i) or the second pair (k=2), B 3 (i) or the third (k=3), and so on.
Bessel unctions The vertical axis is the aplitude o scaling actor according to the value o i (od. index) represented by the horizontal axis. Exaple: i i = 0 then c = ax actor and all sidebands = 0 [B 0 (0) = 1, B 1 (0) = 0, B 2 (0) = 0, B 3 (0) = 0, etc. ] B N ( i) i = d N = sideband pair
Exaple: i i = 1 then c = 0.76, 1 st pair o sidebands = 0.44, 2 nd pair = 0.11, etc. [B 0 (0) = 0.76, B 1 (0) = 0.44, B 2 (0) = 0, B 3 (0) = 0.11, B 4 (1) = 0.01, etc. ]
Negative aplitudes The Bessel unctions indicate that sidebands ay have either positive or negative aplitude, depending on i. Exaple: I i = 5, then 1 st pair o sidebands will be = -0.33 Negative aplitude does not exist: it only indicates that the sidebands are out o phase. Can be represented by plotting the downwards.
Negative aplitudes In general, the phase o the partials do not produce an audible eect Unless another partial o the sae requency happens to be present. In this case the aplitudes will either add or subtract, depending on their respective phases.
Negative requencies & Nyquist distortion I c is too low and/or the i is too high, then the odulation produce sidebands that all in the negative doain. As a rule, negative sidebands old around the 0 Hz axis and ix with the others. Relected sidebands will reverse their phase.
Negative requencies Relected sidebands will reverse their phase. Exaple: c = 440 Hz, = 440Hz, i = 3
Nyquist distortion Partials alling beyond the Nyquist liit also old over, and relect into the lower portion o the spectru.
Synthesising tie-varying spectra Modulation index i is an eective paraeter to control spectral evolution. An envelope can be eployed to tie-vary i to produce interesting spectral envelopes that are unique to FM. A partial ay increase or decrease its aplitude according to the slope the respective Bessel unction. Linearly increasing I does not necessarily increase the aplitude o the highorder sidebands linearly. FM4
Frequency ratios & sound design FM is governed by two siple ratios between FM paraeters: d : = i (od index) c : = requency ratio Freq ration is useul or achieving variations in pitch whilst aintaining the tibre virtually unchanged. I the req ratio and the od index i a siple FM instruent are aintained constant, but c is odiied then the sounds will vary in pitch, but the tibre reains unchanged. FM5
It is ore convenient to think o in ters o req ratios rather than in ters o values or c and. d : c : = i (od index) = requency ratio It is clear to see that 220 : 440 are in ratio 1:2, but not so iediate or 465.96 : 931.92. As a rule o thub, req ratios should always be reduced to their siplest or. For exaple, 4:2, 3:1.5 and 15:7.5 are all equivalent to 2:1
FM directives in ters o siple ratios FM6 FM7 FM8 FM9 FM10 FM11
Coposite FM Involves 2 or ore carrier oscillators and/or 2 or ore odulator oscillators. Produces ore sidebands, but the coplexity o the calculations or predict the spectru also increases. Basic cobinations: a) Additive carriers with independent odulators b) Additive carriers with one odulator c) Single carrier with parallel odulators d) Single carrier with serial odulators e) Sel-odulating carrier
Additive carriers with independent odulators Coposed o 2 or ore siple FM instruents in parallel. The spectru is the result o the addition o the outputs ro each instruent. FM12
Additive carriers with 1 odulator One odulator oscillator odulates 2 or ore oscillators. The spectru is the result o the addition o the outputs ro each carrier oscillator. c FM13
Single carrier with parallel odulators Modulator is the result o 2 or ore sinewaves added together. The FM orula is expanded to accoodate ultiple odulator req ( ) and od indices (i). In the case o 2 parallel odulator the sideband pairs are calculated as ollows: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 c c c c k k k k k k k k + + + + FM14
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 c c c c k k k k k k k k + + + + Each o the partials produced by one odulator oscillator (k 1 x 1 ) orges a local carrier or the other odulator oscillator (k 2 x 2 ). The aplitude scaling actor result ro the ultiplication o the respective Bessel unctions: B n (i 1 ) x B (i 2 ).
Exaple: (see Appendix I o Coputer Sound Design Book) FM15
Single carrier with serial odulators The odulating signals is a requency odulated signal. The sidebands are calculated using the sae ethod as or parallel odulators, but the aplitude scaling actors is dierent: The order o the outerost odulator is used to scale the odulations index o the next odulator: B n (i 1 ) x B (n x i 2 ). Note: no sidebands ro B (i) are generated: B 0 (i 1 ) x B 1 (0 x i 2 ) = 0. FM16
Further reading: Three Modelling Approaches to Sound Design, by E R Miranda (PDF ile tutorial3.pd) The Asterda Csound Catalogue: http://www.usic.bualo.edu/hiller/accci/