Post on 20-Nov-2015
description
Synthesis Techniques
Juan P Bello
Synthesis It implies the artificial construction of a complex body by
combining its elements. Complex body: acoustic signal (sound) Elements: parameters and/or basic signals
Motivations: Reproduce existing sounds Reproduce the physical process of sound generation Generate new pleasant sounds Control/explore timbre
Model / representation Sound
Synthesis
How can I generate new sounds?
FilterOscillator
Envelope
vibrato
Pitch
Trigger
Cutoff freq
GainSound
Networks of basic elements synthesis techniques
Two main types: linear and non-linear
Additive Synthesis It is based on the idea that complex waveforms can be created by
the addition of simpler ones. It is a linear technique, i.e. do not create frequency components
that were not explicitly contained in the original waveforms Commonly, these simpler signals are sinusoids (sines or cosines)
with time-varying parameters, according to Fouriers theory:
( )!=
+=0
2sin)(i
iii tfAts "#
Amp1(t)Freq1(t)
Amp2(t)Freq2(t)
AmpN(t)FreqN(t)
Additive Synthesis:
A Pipe Organ
Additive Synthesis Square wave: only odd harmonics. Amplitude of the nth harmonic= 1/n
Time-varying sounds
AmpFreq
Amp1(t)
Freq1(t)
AmpFreq
Amp2(t)
Freq2(t)
AmpFreq
AmpN(t)
FreqN(t)
According to Fourier, all sounds can be described and reproducedwith additive synthesis.
Even impulse-like components can be represented by using ashort-lived sinusoid with infinite amplitude.
Additive synthesis is very general (perhaps the most versatile). Control data hungry: large number of parameters are required to
reproduce realistic sounds
Examples
water
guitar
flute
transSMSstochdetorig
Is another linear technique based on the idea that sounds can begenerated from subtracting (filtering out) components from a veryrich signal (e.g. noise, square wave).
Its simplicity made it very popular for the designof analog synthesisers (e.g. Moog)
Subtractive Synthesis
fA Sound
GainParameters
ComplexWaveform Filter Amplifier
The human speech system The vocal chords act as an oscillator, the mouth/nose cavities,
tongue and throat as filters We can shape a tonal sound (oooh vs aaah), we can whiten the
signal (sssshhh), we can produce pink noise by removing highfrequencies
Source-Filter model Subtractive synthesis can be seen as a excitation-resonator or
source-filter model The resonator or filter shapes the spectrum, i.e. defines the spectral
envelope
Source-Filter model
Envelope estimation
Whitening of the signal Transformations
Analysis Processing Synthesis
Amplitude modulation Non-linear technique, i.e. results on the creation of frequencies
which are not produced by the oscillators. In AM the amplitude of the carrier wave is varied in direct
proportion to that of a modulating signal.
Ampm(t)
Freqm(t)
Ampc(t)
Freqc(t)
modulator carrier
bipolar unipolar
Bipolar -> Ring modulationUnipolar -> Amplitudemodulation
Let us define the carrier signal as:
And the (bipolar) modulator signal as:
The Ring modulated signal can be expressed as:
Which can be re-written as:
s(t) presents two sidebands at frequencies: c - m and c + m
Ring Modulation
)cos()( tAtccc
!=
)cos()( tAtmmm
!=
( ) ( )tAtAtsmmcc
!! coscos)( "=
!
s(t) =AcAm
2cos "
c#"
m[ ]t( ) + cos "c +"m[ ]t( )[ ]
Ring Modulation
freq
amp
fc
fc - fm fc + fm
Let us define the carrier signal as:
And the (unipolar) modulator signal as:
The amplitude modulated signal can be expressed as:
Which can be re-written as:
s(t) presents components at frequencies: c , c - m and c + m
Amplitude Modulation
)cos()( ttcc
!=
)cos()( tAAtmmmc
!+=
( )[ ] ( )ttAAtscmmc
!! coscos)( +=
!
s(t) = Accos "
ct( ) +
Am
2cos "
c#"
m[ ]t( ) + cos "c +"m[ ]t( )[ ]
Modulation index In modulation techniques a modulation index is usually defined such
that it indicates how much the modulated variable varies around itsoriginal value.
For AM this quantity is also known as modulation depth:
c
m
A
A=!
If = 0.5 then thecarriers amplitudevaries by 50% aroundits unmodulated level.
For = 1 it varies by100%.
> 1 causes distortionand is usually avoided
C/M frequency ratio
Lets define the carrier to modulator frequency ratio c/m (= c /m) for a pitched signal m(t)
If c/m is an integer n, then c, and all present frequencies, aremultiples of m (which will become the fundamental)
If c/m = 1/n, then c will be the fundamental
When c/m deviates from n or 1/n (or more generally, from aratio of integers), then the output frequencies becomes moreinharmonic
Example of C/M frequency variation
Frequency Modulation Frequency modulation (FM) is a form of modulation in which the
frequency of a carrier wave is varied in direct proportion to theamplitude variation of a modulating signal.
When the frequency modulation produces a variation of lessthan 20Hz this results on a vibrato.
Ampm(t)
Freqm(t)
Ampc(t)
Freqc(t)
modulator carrier
Let us define the carrier signal as:
And the modulator signal as:
The Frequency modulated signal can be expressed as:
This can be re-written as
Frequency Modulation
!
c(t) = cos("ct)
!
m(t) = " sin(#mt)
( )( )tttsmc
!"! sincos)( +=
!
s(t) = Jk(")cos #
c+ k#
m( )t[ ]k=$%
%
&
If 0 then the FM spectrum contains infinite sidebands atpositions c km.
Frequency Modulation
k
Jk
The amplitudes of each pair ofsidebands are given by the Jkcoefficients which are functionsof
Modulation index
As in AM we define a FM modulation index that controls themodulation depth.
In FM synthesis this index is equal to , the amplitude of themodulator and is directly proportional to f.
As we have seen the value of determines the amplitude of thesidebands of the FM spectrum
Furthermore the amplitude decreases with the order k. Thus, although theoretically the number of sidebands is infinite, in
practice their amplitude makes them inaudible for higher orders. The number of audible sidebands is a function of , and is
approximated by 2+1 Thus the bandwidth increases with the amplitude of m(t), like in
some real instruments
C/M frequency ratio The ratio between the carrier and modulator frequencies c/m
is relevant to define the (in)harmonic characteristic of s(t).
The sound is pitched (harmonic) if c/m is a ratio of positiveintegers: c / m = Nc / Nm
E.g. for fc = 800 Hz and fm = 200 Hz, we have sidebands at600Hz and 1kHz, 400Hz and 1.2kHz, 200Hz and 1.4kHz, etc
Thus the fundamental frequency of the harmonic spectrumresponds to: f0 = fc / Nc = fm / Nm
If c/m is not rational an inharmonic spectrum is produced
If f0 is below the auditory range, the sound will not beperceived as having definitive pitch.
FM examples