FREQUENCY MODULATION (FM) SYNTHESIS, Schemes and Mind Maps of Voice

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FREQUENCY

MODULATION (FM)

SYNTHESIS

+ phase distortion (PD)

Electronic musical instruments

Frequency modulation (FM)

FM – frequency modulation, used since 1920s to transmit radio waves:

• transmitted signal (modulator) – e.g. radio broadcast
• carrier signal – high frequency sine (e.g. 99.8 MHz)
• amplitude of the transmitted signal modulates instantaneous frequency of the carrier
• modulated signal is transmitted on air
• the received signal is demodulated
• we obtain the original signal source: http://slidedeck.io/jsantell/dsp-with-web-audio-presentation

FM in sound synthesis

Let’s simplify the problem to two sine oscillators:

• carrier signal (C)

x

c ( t ) = A sin(  t )

• modulating signal (M)

x

m ( t ) = I sin(  t ) The modulator changes (modulates) the instantaneous frequency of the carrier signal: x ( t ) = A sin[  t + x m

( t )]

x ( t ) = A sin[  t + I sin(  t )]

Frequency modulation in sound

What effect does FM produce?

• Low modulating frequency (<1 Hz): slow wobbling of the pitch (just like LFO in the subtractive synthesis).
• Modulating frequency in 1 Hz – 20 Hz range: an increasing vibrato effect.
• Frequency above 20 Hz: an inharmonic sound is produced, it sounds very rough.
• In some configurations, e.g. if both frequencies are the same, we get a nice sounding harmonic signal!

Reflection of spectral components

• What about components with negative frequencies? For example, for f c = 400 Hz, f m = 100 Hz, we obtain: f c - 5 f m = 400 – 500 = - 100 Hz
• We know that: sin(– x ) = – sin( x )
• Therefore:
  • a “negative” component is reflected to a positive frequency (an absolute value is taken),
  • phase of the reflected component is inversed,
  • if another component is present at this frequency, amplitudes are summed up (with phase).

Reflection of spectral components

amplituda f f f f c Spectrum with a “negative” component The component is reflected, its sign changes. The components are summed, taking their phase into account Absolute values of the amplitude are taken.

Modulation ratio

Typical values of the modulation ratio (spectral frequencies are calculated for f c = 400 Hz):

• 1:1 – all spectral components are present 400, 800, 1200, 1600, 2000, …
• 2:1 – only even numbered components (k = 0,2,4,...) 400, 1200, 2000, 2800, …
• 3:1 – every third component is missing 400, 800, 1600, 2000, 2800, … Example of an inharmonic spectrum:
• w m

Modulation ratio and fundamental frequency

Warning: this is a common mistake. Carrier frequency does not have to be equal to the fundamental frequency! The latter is determined by the first peak in the harmonic series.

• f c = 500 Hz, f m = 500 Hz → f 0 = 500 Hz (for modulation ratio 1:1, both frequencies are the same)
• f c = 500 Hz, f m = 100 Hz → f 0 = 100 Hz (the first peak is at 100 Hz: 500 – 4 x 100)
• f c = 200 Hz, f m = 300 Hz → f 0 = 100 Hz (!!!) …, - 700, - 400, - 100, 200, 500, 800, … (reflection:) 100, 200, 400, 500, 700, 800, …

Influence of the modulation index

Carrier frequency: 220 Hz, modulation: 440 Hz Time signals Spectra

1 10 100

Amplitude of spectral components

Amplitudes of spectral components are given by: ............................................................} ( ) [sin( 3 ) sin( 3 ) ] ( ) [sin( 2 ) sin( 2 ) ] ( ) [sin( ) sin( ) ] ( ) { ( ) sin( ) 3 2 1 0

•  + − − 
•  + + − 
•  + − −  = J I nT nT J I nT nT J I nT nT x n A J I nT c m c m c m c m c m c m c              Minus dla nieparzystych prążków wstęgi dolnej! Note: odd numbered components in the lower band have inversed phase – negative sign. J n ( I ): n - th order Bessel functions, argument: modulation idx.

Influence of modulation index on spectrum

f amplituda f f f f c c c c c c-3m c-m c+m c+3m c-2m c+2m c-4m c+4m c-5m c+5m I= I= I= c-4m c+4m I= I=

Calculating a synthetic spectrum

Parameters: carrier frequency f c , modulating frequency f m

modulation index I. How to compute the spectrum of a FM-modulated signal:

• calculate frequencies of components ( f c  k f m

• compute amplitudes of components [ J k ( I )], remember that some lower band components have negative phase.
• reflect components at negative freqs., invert their phase,
• sum up amplitudes of overlapping components,
• take absolute values of amplitudes. Note: it is not possible to reverse this process and compute parameters that yield a desired spectrum.

Operator

Operator is a basic building block of FM synthesis. It consists of:

• sine oscillator (OSC)
• amplifier (VCA)
• envelope generator (EG) freq – fixed frequency mod – modulating frequency OSC generates a sine with instantaneous frequency = freq + mod OSC VCA EG freq mod amp

FM algorithm

A connection of two or more operators creates a FM synthesis algorithm.

• Two operators ( Simple FM , 2 - op FM ): one carrier and one modulator. The simplest algorithm possible, not sufficient to obtain useful effects.
• In practice, more operators (usually 6) are used, many algorithms are possible.
• The same operators with the same settings, but connected in a different algorithm, produce completely different sound!