[PDF] FREQUENCY MODULATION (FM) SYNTHESIS

27319_304_FM.pdf

FREQUENCY

MODULATION (FM)

SYNTHESIS

+ phase distortion(PD)

Electronic musical instruments

Frequency modulation

(FM) FM ʹfrequency modulation, used since 1920s to transmit radio waves: ͻtransmitted signal (modulator) ʹe.g. radio broadcast ͻcarriersignal ʹhigh frequency sine (e.g. 99.8 MHz)

ͻamplitude of the transmitted signal modulates

instantaneous frequencyof the carrier

ͻmodulated signal is transmitted on air

ͻthe received signal is demodulated

ͻwe obtain the original signal

source: http://slidedeck.io/jsantell/dsp-with-web-audio-presentation FM in sound synthesis

1973 ʹJohn Chowningpublished a paper:

ͣThe Synthesisof ComplexAudio Spectra

by Meansof FrequencyModulation͘͟

ͻIf the two signals have specific frequencies,

a harmonic signal is obtained. ͻChanges in modulator amplitude modify the timbre.

ͻMultiple modulations may be performed.

ͻEasy and cheap method of digital sound synthesis.

ͻPatented in1975-1995by Chowningand Yamaha.

FM in sound synthesis

Ğƚ͛ƐƐŝŵƉůŝĨLJƚŚĞƉƌŽďůĞŵƚŽƚǁŽƐŝŶĞŽƐĐŝůůĂƚŽƌƐ:

ͻcarrier signal (C)

xc(t) = Asin(t)

ͻmodulating signal (M)

xm(t) = Isin(t)

The modulator changes (modulates)

the instantaneous frequency of the carrier signal: x(t) = Asin[t+ xm(t)] x(t) = Asin[t+ Isin(t)]

Frequency modulation in sound

What effect does FM produce?

ͻLow modulating frequency (<1 Hz): slow wobbling of the pitch (just like LFO in the subtractive synthesis).

ͻModulating frequency in 1 Hz ʹ20 Hz range:

an increasing vibrato effect.

ͻFrequency above 20 Hz: an inharmonic sound

is produced, it sounds very rough. ͻIn some configurations, e.g. if both frequencies are the same, we get a nice sounding harmonic signal!

Synthetic spectrum components

Peaks in the spectrum of a modulated sound:

fckfm(k= 0, 1, 2, ...) In FM terms: lower and upper band (below and above fc)

For example,fc= 500 Hz, fm= 100 Hz:

͙͕ϭϬϬ͕ϮϬϬ͕ϯϬϬ͕ϰϬϬ͕500͕ϲϬϬ͕ϳϬϬ͕ϴϬϬ͕ϵϬϬ͕͙

Reflection of spectral components

ͻWhat about components with negative frequencies?

For example, forfc= 400 Hz,fm= 100 Hz,we obtain:

fcʹ5fm= 400 ʹ500 = -100 Hz

ͻWe know that: sin(ʹx) = ʹsin(x)

ͻTherefore:

ʹĂ͞ŶĞŐĂƚŝǀĞ͟ĐŽŵƉŽŶĞŶƚŝƐƌĞĨůĞĐƚĞĚƚŽĂƉŽƐŝƚŝǀĞ

frequency (an absolute value is taken),

ʹphase of the reflected component is inversed,

ʹif 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

͞ŶĞŐĂƚŝǀĞ͟ĐŽŵƉŽŶĞŶƚ

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

Modulation ratiowmʹa ratio ofmodulatingfrequency to thecarrierfrequency. ͻIn order to obtain a harmonic signal, the modulation ratio has to be expressed as a ratio of integers N2andN1. ͻIn practice, low integers are used, e.g.:1:1, 2:1, 3:1, 3:2. 1 2 N N f fw c m m

Modulation ratio

Typical values of the modulation ratio

(spectral frequencies are calculated forfc= 400 Hz):

ͻ1:1ʹall spectral components are present

ϰϬϬ͕ϴϬϬ͕ϭϮϬϬ͕ϭϲϬϬ͕ϮϬϬϬ͕͙ ͻ2:1ʹonly even numbered components(k = 0,2,4,...) ϰϬϬ͕ϭϮϬϬ͕ϮϬϬϬ͕ϮϴϬϬ͕͙

ͻ3:1ʹevery third component is missing

ϰϬϬ͕ϴϬϬ͕ϭϲϬϬ͕ϮϬϬϬ͕ϮϴϬϬ͕͙

Example of an inharmonic spectrum:

ͻwm= 2 : 1

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.

ͻfc= 500 Hz, fm= 500 Hz f0= 500 Hz

(for modulation ratio1:1, both frequencies are the same)

ͻfc= 500 Hz, fm= 100 Hz f0= 100 Hz

(the first peak is at100 Hz: 500 ʹ4 x 100)

ͻfc= 200 Hz, fm= 300 Hz f0= 100 Hz (!!!)

͙͕-700, -400, -ϭϬϬ͕ϮϬϬ͕ϱϬϬ͕ϴϬϬ͕͙;reflection:)

ϭϬϬ͕ϮϬϬ͕ϰϬϬ͕ϱϬϬ͕ϳϬϬ͕ϴϬϬ͕͙

Modulation index

ͻModulation index(I) = modulator amplitude

(do not confuse with the modulation ratio). ͻDetermines the modulated frequency range (f= Ifm).

ͻInfluences the number of important components

in the spectrum. Larger index ʹa richer spectrum.

Carson rule:B= 2(f+ fm) = 2 fm(I+ 1)

ͻAlso influences amplitudesof spectral components and therefore, determines the timbreof the sound!

ͻPractical values: 10 to 100.

Influence of the modulation index

Carrier frequency:220 Hz, modulation:440 Hz

Time signalsSpectra

0.1 1 10 100

Amplitude of spectral components

Amplitudes of spectral components are given by:

}............................................................ ])3sin( )3sin([ )( ])2sin( )2sin([ )( ])sin( )sin([ )( )sin( )({ )( 3 2 1 0