An Introduction to FM

8/8/2019 An Introduction to FM

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An Introduction To FM

Bill Schottstaedt

In frequency modulation we modulate the frequency — "modulation" here is just a

latinate word for "change". Vibrato and glissando are frequency modulation. John

Chowning tells me that he stumbled on FM when he sped up vibrato to the point that itwas creating audible sidebands (perceived as a timbral change) rather than faster

warbling (perceived as a frequency change). We can express this (the vibrato, not the neat

story) as:

where the c subscript stands for "carrier" and f(t) means "some arbitrary function added

to the carrier". Since cos takes an angle as its argument, f(t) modulates (that is, changes)

the angle passed to the cosine, hence the generic name "angle

modulation". We can add that change either to the argument tocos ("phase modulation", cos(angle + change)), or add it to the current phase, then

take cos of that ("frequency modulation", cos(angle += change)), so our formula can

viewed either way. Since the angle is being incremented by the carrier frequency in either

case, the difference is between:

PM: cos((angle += incr) + change)

FM: cos(angle += (incr + change))

FM and phase modulation



To make the difference clear, textbooks put in an integral when they mean frequency

modulation:

In PM we change the phase, in FM we change the phase increment, and to go from FM to

PM, integrate the FM modulating signal. But you can't tell which is in use from theoutput waveform; you have to know what the modulating signal is. In sound synthesis,

where we can do what we want with the modulating signal, there is no essential

difference between frequency and phase modulation.

I would call this issue a dead horse, but it is still causing confusion, even 40 years downthe road. So, here are two CLM instruments, one performing phase modulation, the other

performing frequency modulation. I have tried to make the innards explicit at each step,

and match the indices so that the instruments produce the same results given the same parameters. Also, to lay a different controversy to rest, it should be obvious from these two

functions that there is no difference in run-time computational expense or accuracy.

(define (pm beg end freq amp mc-ratio index) ; "mc-ratio" = modulator to

carrier frequency ratio

(let ((carrier-phase 0.0) ; set to pi/2 if someone tells you PM can't

produce energy at 0Hz

(carrier-phase-incr (hz->radians freq))

(modulator-phase 0.0)

(modulator-phase-incr (hz->radians (* mc-ratio freq))))

(do ((i beg (+ 1 i)))

((= i end))(let* ((modulation (* index (sin modulator-phase)))

(pm-val (* amp (sin (+ carrier-phase modulation)))))

;; no integration in phase modulation

(set! carrier-phase (+ carrier-phase carrier-phase-incr))

(set! modulator-phase (+ modulator-phase modulator-phase-incr))

(outa i pm-val)))))

(define (fm beg end freq amp mc-ratio index)

(let* ((carrier-phase 0.0)

(carrier-phase-incr (hz->radians freq))

(modulator-phase-incr (hz->radians (* mc-ratio freq)))

(modulator-phase (* 0.5 (+ pi modulator-phase-incr)))

;; (pi+incr)/2 to get (centered) sin after integration, to match

pm case above

(fm-index (hz->radians (* mc-ratio freq index))))

;; fix up fm index (it's a frequency change)

(do ((i beg (+ 1 i)))

((= i end))

(let ((modulation (* fm-index (sin modulator-phase)))

(fm-val (* amp (sin carrier-phase))))

(set! carrier-phase (+ carrier-phase modulation carrier-phase-



incr))

(set! modulator-phase (+ modulator-phase modulator-phase-incr))

(outb i fm-val)))))

(with-sound (:channels 2)

(pm 0 10000 1000 .5 0.25 4)

(fm 0 10000 1000 .5 0.25 4))

(with-sound (:channels 2)

(pm 0 10000 1000 .5 0.5 10)

(fm 0 10000 1000 .5 0.5 10))

Given our formula for FM, let's assume, for starters, that f(t) isa sinusoid:

where the "m" stands for "modulator" and the "B" factor is usually called the modulation

index. The corresponding CLM code is:

(oscil carrier (* B (oscil modulator)))

where oscil is (essentially):

(define* (oscil oscillator :optional (fm-input 0.0) (pm-input 0.0))

(let ((result (sin (+ oscillator-phase pm-input))))

(set! oscillator-phase (+ oscillator-phase (+ oscillator-phase-

increment fm-input)))result))

Since it is generally believed that the ear performs some sort of projection of the timedomain waveform into the frequency domain (a Fourier Transform), and that timbre is at

least partly a matter of the mix of frequencies present (the spectrum), our main interest in

the FM formula is in the spectrum it produces. To determine that spectrum, we have toendure some tedious mathematics. By the trigonometric identity:

we can substitute for "a" and for "b" and get:

If we can get a Fourier transform of the two inner portions: and

, we can use:

simple FM: sin(sin)



to get the final results. "A" here is in the earlier formulas, and "B" is either

or . The Fourier transform we want is not obvious to

us (not to me, certainly!), so we go to Abramowitz and Stegun, "Handbook of Mathematical Functions" and find (formulas 9.1.42 and 9.1.43):

Here the J's refer to the Bessel functions which we will return to later.

First, let's finish this expansion; we take these two sums and and plug

them into our first expansion of the FM formula, and out pops:

or in a slightly more compact form:

Here we are using the fact that . We can changeour point of view on the first part of the expansion given above, and ask

for the amplitude of a given sideband:

We end up with a spectrum made up of a "carrier" at and symmetrically

C G J Jacobi,

GesammelteWerke, VI

101



placed sidebands separated by . The amplitudes follow the Bessel

functions. I put carrier in quotes because in computer music we listen to

the result of the modulation (this was Chowning's idea — see "TheSynthesis of Complex Audio Spectra by Means of Frequency

Modulation"). The Bessel functions are nearly 0 until the index (B) equalsthe order (n). Then they have a bump and tail off as a sort of dampedsinusoid:

As the index sweeps upward, energy is swept gradually outward into higher order side

bands; this is the originally exciting, now extremely annoying "FM sweep". The

important thing to get from these Bessel functions is that the higher the index, the more

dispersed the spectral energy — normally a brighter sound.

carrier=1000, mod=100,

index=1.0


index=2.0


index=3.0



J0(1.0) = 0.765 ->

1.0 (*)

J1(1.0) = 0.440 ->

0.575

J2(1.0) = 0.115 ->

0.150

J3(1.0) = 0.019 ->

0.025

J4(1.0) = 0.002 ->

0.003

(* Jn values normalized

to match

the fft peak values

given above)

J0(2.0) = 0.224 ->

0.388 (*)

J1(2.0) = 0.577 ->

1.0

J2(2.0) = 0.353 ->

0.611

J3(2.0) = 0.129 ->

0.223

J4(2.0) = 0.034 ->

0.058

J5(2.0) = 0.007 ->

0.012J6(2.0) = 0.001 ->

0.002

(A larger FFT reduces the

mismatch)

J0(3.0) = -0.260 ->

-0.534 (*)

J1(3.0) = 0.339 ->

0.697

J2(3.0) = 0.486 ->

1.0

J3(3.0) = 0.309 ->

0.635

J4(3.0) = 0.132 ->

0.271

J5(3.0) = 0.043 ->0.088

J6(3.0) = 0.011 ->

0.023

There is a rule of thumb, Mr Carson's rule, about the overall bandwidth of the resultant

spectrum (it follows from our description of the Bessel functions): Roughly speaking,there are fm-index+1 significant sidebands on each side of the carrier, so our total

bandwidth is more or less

2 * modulator-frequency * (fm-index + 1)

This is a good approximation — 99% of the signal power is within its limits. To turn that

around, we can reduce the danger of aliasing by limiting the fm index to approximately

(srate/2 - carrier_frequency) / modulator_frequency; use srate/4 to be safer. (Mr Carson'sopinion of FM: "this method of modulation inherently distorts without any compensating

advantages whatsoever").



One hidden aspect of the FM expansion is that it produces a time domain waveform that

is not "spikey". If we add cosines at the amplitudes given by the Bessel functions (using

additive synthesis to produce the same magnitude spectrum as FM produces), we get avery different waveform. Doesn't the FM version sound richer and, far more importantly,

louder? (The FM RMS value here is 0.707, whereas the sum of cosines RMS value is

0.243 because we scaled its original peak amplitude of 2.89 to 1.0):

FM waveform (index: 3.0) vs sum of cosines with the same (relative) component

amplitudes

I think FM gives the minimum peak amplitude that its components could have produced,

given any set of initial phases. From one point of view (looking at FM as changing the

phase passed to the sin function), it's obvious that the output waveform should be thiswell behaved, but looking at it from its components, it strikes me as a minor miracle that

there is a set of amplitudes (courtesy of the Bessel functions) that fits together so

perfectly. Here is an attempt to graph the 15 main components, with their sum in black:



Then there's the perennial question "why Bessel functions?". Most explanations start with

: obscurum per obscurius! A different tack might be to start with

, a definition of sine, and call the "e^(ix)" terms "t", then cos(sin) involves terms like

, which is one (convoluted) way to define Bessel functions. Or perhaps most forthright, start withthe formula for Jn(B) given above (the integral), and say "we want cos(sin) expanded as a sum of cosines,

and we simply define Jn to be the nth coefficient in that sum". This was the approach of Bessel and other 19th century mathematicians, but it is not very satisfying for some reason. Perhaps history can help? These

functions were studied by Daniel Bernoulli (the vibrations of a heavy chain, 1738), Euler (the vibrations of a

membrane, 1764), Lagrange (planetary motion, 1770), and Fourier (the motion of heat in a cylinder, 1822);

Bessel studied them in the context of Kepler's equation, and wrote a monograph about them in 1824. For an

explanation of the connection between planetary motion and FM, see Benson, "Music: A Mathematical

Offering". Just for completeness, here's a derivation following Gray and Mathews, "A Treatise on Bessel

Functions":

Here's a simple FM instrument:

(define* (fm beg dur freq amp mc-ratio index :optional (index-env '(0 1

100 1)))

(let* ((start (seconds->samples beg))

(end (+ start (seconds->samples dur)))

(cr (make-oscil freq))

(md (make-oscil (* freq mc-ratio)))

(fm-index (hz->radians (* index mc-ratio freq)))

(ampf (make-env index-env :scaler amp :duration dur))

(indf (make-env index-env :scaler fm-index :duration dur)))

(run

(lambda ()

(do ((i start (+ 1 i)))

((= i end))

(outa i (* (env ampf) ; amplitude env

(oscil cr (* (env indf) ; carrier +

modulation env

(oscil md)))))))))) ; modulation

I put an envelope on the fm-index ("indf" above) to try out dynamic spectra ("dynamic"

means "changing" here). For now, don't worry too much about the actual side band

amplitudes. They will not always match Chowning's description, but we'll get around toan explanation eventually.

(with-sound () (fm 0 1.0 100 .5 1.0 4.0))

is Chowning's first example. Sure enough, it's a complex spectrum (that is, it has lots of components; try an index of 0 to hear a sine wave, if you're suspicious). Since our

simple FM examples



modulating frequency to carrier frequency ratio (mc-ratio above) is 1.0, we get sidebands

at harmonics of the carrier. If we use an mc-ratio of .25 and a carrier of 400:

(with-sound () (fm 0 1.0 400 .5 0.25 4.0))

we end up with the same perceived pitch because the sidebands are still at multiples of

100 Hz.

(with-sound () (fm 0 1.0 400 .5 1.1414 4.0))

has inharmonic sidebands. Most real sounds seem to change over the course of a note,

and it was at one time thought that most of this change was spectral. To get a changingspectrum, we need only put an envelope on the fm-index:

(with-sound () (fm 0 0.5 400 .5 1.0 5.0 '(0 0 20 1 40 .6 90 .5 100

0)))

making a brass-like sound. Similarly, Chowning suggests that

(with-sound () (fm 0 1.0 900 .5 1/3 2.0 '(0 0 6 .5 10 1 90 1 100

0)))

is a woodwind-like tone,

(with-sound () (fm 0 1.0 500 .5 .2 1.5 '(0 0 6 .5 10 1 90 1 100 0)))

is bassoon-like, and finally

(with-sound () (fm 0 1.0 900 .5 2/3 2 '(0 0 25 1 75 1 100 0)))

is clarinet-like. Now start at 2000 Hz, set the mc-ratio to .1, and sweep the FM index

from 0 to 10, and the spectrogram looks like this:



There is a lot of music in simple FM. You get a full spectrum at little computational

expense, and the index gives you a simple and intuitive way to change that spectrum.Since the output peak amplitude is not affected by the modulating signal (cos(x) is

between -1 and 1 no matter what x is, as long as it is real), we can wrench the index

around with wild abandon. And since the number of significant components in thespectrum is nearly proportional to the index (Carson's rule), we can usually predict more

or less what index we want for a given spectral result.

A slightly bizarre sidelight: there's no law

against a modulating signal made up of complex

numbers. In this case, cos is no longer bounded,so the output can peak at anything, but we still

get FM-like spectra. , where "I"

is the modified Bessel function, so if our index

is purely imaginary, we can expand cos(wc + bi

sin wm)t as

If our index is a + bi, we get

index: 6+3i



This looks similar to normal FM, but with

normalization headaches. Perhaps we can take

advantage of the split betweeen the real andimaginary parts — unexplored territory!

I am getting carried away — we need to back up a bit and clear

up one source of confusion. If you looked at the spectrum of our first example, and compared it to the spectrum Chowning

works out, you may wonder what's gone awry. We have to return to our initial set of

formulas. If we consider that:

and using our previous formulas for the expansion of the cos(sin) and sin(sin) terms, with

the identity:

we see that we still have a spectrum symmetric around the carrier, and the amplitude and

frequencies are just as they were before, but the initial phases of the side bands have

changed. Our result is now

This is Chowning's version of the expansion. In general:

Our first reaction is, "well so what if one's a sine and the other's a cosine — they'll soundthe same", but we are being hasty. What if (for example), the modulator has the same

frequency as the carrier, and its index (B) is high enough that some significant energy

appears at ? Where does energy at a negative frequency go? We once

again fall back on trigonometry: , but , so the negativefrequency component adds to the positive frequency component if it's a cosine, but

subtracts if it's a sine. We get a different pattern of cancellations depending on the initial phases of the carrier and modulator. Take the CLM instrument:

phase quibbles: cos(sin)



(define (pm

beg dur freq

amp fm-index

mod-phase)

(let*

((start

(seconds->samples

beg))

(end (+

start

(seconds-

>samples

dur)))

(cr

(make-oscil

freq))

(md

(make-oscil

freq mod-

phase)))

(run

(lambda

()

(do

((i start (+

1 i)))

((= i end))

(outa i (*

amp (oscil

cr 0.0

(* fm-index

(oscil

md))))))))))

(with-sound

() (pm 0 1.0

100 .5 8 0))

(with-sound

() (pm 0 1.0

100 .5 8

(* .5 pi)))

mod phase = 0.0

mod phase = pi/2

There is a slight difference! We're using phase-modulation for simplicity (the integration

in FM changes the effective initial phase). By varying the relative phases, we can get a

changing spectrum from these cancellations. Here is a CLM instrument that shows this(subtle) effect:

(define (fm beg dur freq amp mc-ratio index)






(md (make-oscil (* freq mc-ratio)))

(skewf (make-env (list 0.0 0.0 1.0 pi) :duration dur)))

(run

(lambda ()

(do ((i start (+ 1 i)))

((= i end))

(outa i (* amp (oscil cr 0.0 (* index (oscil md 0.0 (env

skewf)))))))))))

(with-sound () (fm 0 2 100 0.5 1.0 30.0))

The next question is "if we can get cancellations, can we fiddle

with the phases and get asymmetric FM spectra?". There areseveral approaches; an obvious one uses the fact that:

If we have a spectrum B made up entirely of sines (or entirely cosines), we can multiply

it by sin A (or cos A), add the two resulting spectra, and the (A + B) parts cancel.

Unfortunately, in this case there are some pesky -1's floating around, so we getasymmetric or gapped spectra, but not anything we'd claim was single side-band.

(define (pm-cancellation beg dur

carfreq modfreq amp

index)

(let* ((cx 0.0)

(mx 0.0)

(car-incr

(hz->radians

carfreq))

(mod-incr

(hz->radians

modfreq))

(start

(seconds->samplesbeg))

(stop (+

start (seconds-

>samples dur))))

(do ((i start (+

1 i)))

((= i stop))

(outa i (* amp

cos side by itself

both sides (showing cancellations)

asymmetric spectra



(- (* (cos cx) ;

cos * sum-of-cos

(sin (*

index (cos mx))))

(* (sin cx); sin * sum-of-sin

(* (sin

(* index (sin

mx))))))))

(set! cx (+ cx

car-incr))

(set! mx (+ mx

mod-incr)))))

(with-sound () (pm-

cancellation 0 1

1000.0 100.0 0.3

9.0))

I really like the sounds you get from this cancellation; I can't resist adding the followingexamples which come from a collection of "imaginary machines":

(definstrument (machine1 beg dur cfreq mfreq amp index gliss)

(let* ((gen (make-fmssb cfreq mfreq :index 1.0)) ; defined in

generators.scm

(start (seconds->samples beg))

(stop (+ start (seconds->samples dur)))

(ampf (make-env '(0 0 1 .75 2 1 3 .1 4 .7 5 1 6 .8 100 0) :base

32 :scaler amp :duration dur))(indf (make-env '(0 0 1 1 3 0) :duration dur :base 32 :scaler

index))

(frqf (make-env (if (> gliss 0.0) '(0 0 1 1) '(0 1 1 0))

:duration dur :scaler (hz->radians (abs gliss)))))

(run

(lambda ()

(do ((i start (+ 1 i)))

((= i stop))

(set! (fmssb-index gen) (env indf))

(outa i (* (env ampf) (fmssb gen (env frqf)))))))))

(with-sound (:play #t)

(do ((i 0.0 (+ i .2)))

((>= i 2.0))

(machine1 i .3 100 540 0.5 4.0 0.0)

(machine1 (+ i .1) .3 200 540 0.5 3.0 0.0))

(do ((i 0.0 (+ i .6)))

((>= i 2.0))

(machine1 i .3 1000 540 0.5 6.0 0.0)

(machine1 (+ i .1) .1 2000 540 0.5 1.0 0.0)))

(with-sound (:scaled-to .5 :play #t)



(let ((gen (make-rkoddssb 1000.0 2000.0 0.875)) ; defined in

generators.scm

(noi (make-rand 15000 .04))

(gen1 (make-rkoddssb 100.0 10.0 0.9))

(ampf (make-env '(0 0 1 1 11 1 12 0) :duration 11.0 :scaler .5))

(frqf (make-env '(0 0 1 1 2 0 10 0 11 1 12 0 20 0) :duration 11.0

:scaler (hz->radians 1.0))))(run

(lambda ()

(do ((i 0 (+ 1 i)))

((= i (* 12 44100)))

(outa i (* (env ampf)

(+ (rkoddssb gen1 (env frqf))

(* .2 (sin (rkoddssb gen (rand noi)))))))))))

(do ((i 0.0 (+ i 2)))

((>= i 10.0))

(machine1 i 3 100 700 0.5 4.0 0.0)

(machine1 (+ i 1) 3 200 700 0.5 3.0 0.0))

(do ((i 0.0 (+ i 6)))

((>= i 10.0))

(machine1 i 3 1000 540 0.5 6.0 0.0)

(machine1 (+ i 1) 1 2000 540 0.5 1.0 0.0)))

A different approach, also using a form of amplitude modulation, is mentioned by

Moorer in "Signal Processing Aspects of

Computer Music":

This is the rxyk!cos generator ingenerators.scm. It produces beautifulsingle-sided spectra. We might grumble

that the sideband amplitudes don't leave us

much room for maneuver, but the factorialin the denominator overwhelms any

exponential in the numerator, so we can

get many interesting effects: moving

formants, for example.

a: 2, x:1000, y: 100

Palamin et al in "A Method of Generating and Controlling Musical Asymmetrical

Spectra" came up with a slightly more complicated version:

But the peak amplitude of this formula is hard to predict; we'd rather have a sum of

cosines:



so we can use

to normalize the output to -1.0 to 1.0. The spectrum produced for a given "r" is mirrored

by -1/r (remembering that ).

Asymmetric FM with

index=2.0, r=0.5:

(using the

asymmetric-fm

generator in CLM)

(with-sound ()

(let ((gen (make-

asymmetric-fm 2000.0

:ratio .2 :r 0.5)))

(do ((i 0 (+ 1

i)))

((= i 20000))

(outa i

(asymmetric-fm gen

2.0)))))

We can put an envelope on either the index or "r"; the index affects how broad thespectrum is, and "r" affects its placement relative to the carrier (giving the effect of a

moving formant). Here we sweep "r" from -1.0 to -20.0, with an index of 3, m/c ratio of .

2, and carrier at 1000 Hz:



So far we have been using just a sinusoid for the modulator;

what if we make it a more complicated signal? Here again

trigonometry can be used to expand

The modulating signal is now made up of two sinusoids (don't despair; this is a

terminating sequence). Since sine is not linear (it is ), this is not the samething as

In the second case we just add together the two simple FM spectra, but in the first case

we get a more complex mixture involving all the sums and differences of the modulatingfrequencies. These sum and difference tones ("intermodulation products") are not limited

to FM. Any nonlinear synthesis technique produces them. Being non-linear, it must have

something that involves a power of its input other than 0 or 1; if we feed in sin a + sin b,for example, that term will produce not just (sin a)^n and (sin b)^n, but all sorts of stuff

involving sin a * sin b (in various powers), and this produces things like cos(a+b) and

cos(a-b). For a less impressionistic derivation of the spectrum, see Le Brun, "A

complex FM: sin(sin+sin)



Derivation of the Spectrum of FM with a Complex Modulating Wave". The result can be

expressed:

You can chew up any amount of free time calculating the resulting side band amplitudes

— see the immortal classic: Schottstaedt, "The Simulation of Natural Instrument Tones

Using Frequency Modulation with a Complex Modulating Wave". (There's a function todo it for you in dsp.scm: fm-parallel-component). In simple cases, the extra modulating

components flatten and spread out the spectrum somewhat (see ncos for a discussion of a

very different non-simple case). In general:

A CLM instrument to produce this is:

(define (fm beg dur freq amp mc-ratios indexes carrier-phase mod-phases)



(cr (make-oscil freq carrier-phase))

(n (length mc-ratios))

(modulators (make-vector n))

(fm-indices (make-vct n)))

(do ((i 0 (+ 1 i)))

((= i n))

(vector-set! modulators i (make-oscil (* freq (list-ref mc-ratios

i)) (list-ref mod-phases i)))(vct-set! fm-indices i (hz->radians (* freq (list-ref indexes i)

(list-ref mc-ratios i)))))

(run

(lambda ()

(do ((i start (+ 1 i)))

((= i end))

(let ((sum 0.0))

(do ((k 0 (+ 1 k)))

((= k n))

(set! sum (+ sum (* (vct-ref fm-indices k) (oscil (vector-

ref modulators k))))))

(outa i (* amp (oscil cr sum)))))))))

(with-sound () (fm 0 2.0 440 .3 '(1 3 4) '(1.0 0.5 0.1) 0.0 '(0.0 0.0

0.0)))

http://ccrma.stanford.edu/software/snd/snd/sndclm.html#ncosdoc





200Hz is

-0.106 (i =

-1, k = -1)

-0.106 (i =

-1, k = 1)

-0.213 ->

0.306

normalized

2000Hz:

-0.023 (i =

-2, k = 0)

0.718 (i =

0, k = 0)

0.695 ->

1.0

normalized

1800Hz:

-0.013 (i =

-2, k = 1)

-0.413 (i =

0, k = -1)

-0.426 ->

0.614

normalized

i is the

2000 Hzpart, k the

200 Hz,

red dots

mark pure

sum/differe

nce tones

(with-sound () (fm 0 2.0 2000 .5 '(1 .1) '(0.5 1.0) 0.0 '(1.855 1.599)))

My favorite computer instrument, the FM violin, uses three sinusoidal components in themodulating wave; for more complex spectra these violins are then ganged together (see

fmviolin.clm for many examples). By using a few sines in the modulator, you get awayfrom the simple FM index sweep that has become tiresome, and the broader, flatter spectrum is somewhat closer to that of a real violin. A pared down version of the fm-

violin is:

(define (violin beg dur frequency amplitude fm-index)



(frq-scl (hz->radians frequency))



(maxdev (* frq-scl fm-index))

(index1 (* maxdev (/ 5.0 (log frequency))))

(index2 (* maxdev 3.0 (/ (- 8.5 (log frequency)) (+ 3.0 (/

frequency 1000)))))

(index3 (* maxdev (/ 4.0 (sqrt frequency))))

(carrier (make-oscil frequency))

(fmosc1 (make-oscil frequency))

(fmosc2 (make-oscil (* 3 frequency)))

(fmosc3 (make-oscil (* 4 frequency)))

(ampf (make-env '(0 0 25 1 75 1 100 0) :scaler amplitude

:duration dur))

(indf1 (make-env '(0 1 25 .4 75 .6 100 0) :scaler index1

:duration dur))


:duration dur))


:duration dur))

(pervib (make-triangle-wave 5 :amplitude (* .0025 frq-scl)))

(ranvib (make-rand-interp 16 :amplitude (* .005 frq-scl))))

(run

(lambda ()(do ((i start (+ 1 i)))

((= i end))

(let ((vib (+ (triangle-wave pervib) (rand-interp ranvib))))

(outa i (* (env ampf)

(oscil carrier

(+ vib

(* (env indf1) (oscil fmosc1 vib))

(* (env indf2) (oscil fmosc2 (* 3.0 vib)))

(* (env indf3) (oscil fmosc3 (* 4.0

vib)))))))))))))

(with-sound () (violin 0 1.0 440 .1 1.0))

We can, of course, use FM (or anything) to produce the

modulating signal. When FM is used, it is sometimes called"cascade FM":

In CLM:

(* A (oscil carrier (* B (oscil modulator (* C (oscil cascade))))))

Each component of the lower pair of oscillators is surrounded by the spectrum produced

by the upper pair, sort of like a set of formant regions.

cascade FM: sin(sin(sin))



osc A: 2000 Hz, osc B: 500 Hz, index 1.5, osc C: 50 Hz, index 1.0

The resemblance of cascade FM to parallel FM is not an accident:

Unfortunately, FM and PM can produce energy at 0Hz (when, for example, the carrier

frequency equals the modulating frequency), and in FM that 0Hz component becomes aconstant offset in the phase increment (the "instantaneous frequency") of the outer or lowermost carrier. Our fundamental frequency no longer has any obvious relation to !

That is, we can expand our cascade formula (in the sin(x + cos(sin)) case) into:

but now whenever the , we get , and the carrier is offset

by (radians->hz (bes-jn B)), that is, (Jn(B) * srate / (2 * pi)). For example, if we

have (oscil gen 0.05), where we've omitted everything except the constant (DC) term(0.05 in this case), this oscil produces a sine wave at its nominal frequency + (radians-

>hz 0.05), an offset of about 351 Hz at a 44100 Hz sampling rate. This extra offset could

be a disaster, because in most cases where we care about the perceived fundamental, weare trying to create harmonic spectra, and that is harder if our modulator/carrier ratio

depends on the current FM index. If you are using low indices and the top pair's mc-ratios

are below 1.0 (in vibrato, for example), you have a good chance of getting usable results.

If you want cascade FM to work in other situations, make sure the top oscil has an initial phase of (pi + mod-incr)/2. The middle FM spectrum will then have only sines (not

cosines), so the DC component will be thoroughly discouraged. Or use phase modulation

instead; in that case, we have effectively (oscil gen 0.0 0.05), which has no effect on

the pitch, but offsets the phase by a constant (0.05), usually not a big deal.

The irascible reader may be grumbling about angels and pins, so here's an example of

cascade FM to show how strong this effect is:

(define (cascade beg dur freq amp modrat modind casrat casind caspha)






(md (make-oscil (* freq modrat)))

(ca (make-oscil (* freq casrat) caspha))

(fm-ind0 (hz->radians (* modind modrat freq)))

(fm-ind1 (hz->radians (* casind casrat freq))))

(run

(lambda ()(do ((i start (+ 1 i)))

((= i end))

(outa i (* amp

(oscil cr (* fm-ind0

(oscil md (* fm-ind1

(oscil ca))))))))))))

(with-sound ()

(cascade 0 1.0 400 .25 1.0 1.0 1.0 1.0 0)

(cascade 1.5 1.0 400 .25 1.0 1.0 1.0 1.0 (* .5 pi)))

;;; clean it up by using the no-DC initial phase:

(with-sound ()

(cascade 0 1.0 400 .25 1 1.0 1 1.0 (* 0.5 (+ pi (hz->radians 400)))))

Why stop at three sins? Here's an experiment that calls sin(sin(sin...)) k times; it seems to

be approaching a square wave as k heads into the stratosphere:

k=3 k=30 k=300

If we use "cos" here instead of "sin", we get a constant, as Bill Gosper has shown:

As z increases above 1.27, we get a square wave, then period doubling, and finally (ca. 1.97) chaos.

A similar trick comes up in feedback FM used in somesynthesizers. Here the output of the modulator is fed back into

its input: sin(y <= w + B sin y). This is expanded by Tomisawa as:

feedback FM: sin(x=sin(x))



As Tomisawa points out, this is very close to the other FM formulas, except that the

argument to the Bessel function depends on the order, we have only multiples of the

carrier frequency in the expansion, and the elements of the sequence are multiplied by2/nB. The result is a much broader, flatter spectrum than you normally get from FM. If

you just push the index up in normal FM, the energy is pushed outward in a lumpy sort of

fashion, not evenly spread across the spectrum. In effect we've turned the axis of theBessel functions so that the higher order functions start at nearly the same time as the

lower order functions. The new function Jn(nB) decreases (very!) gradually. For example

if the index (B) is 1:

Since the other part of the equation goes down as 1/n, we get essentially a sawtooth waveout of this equation (its harmonics go down as 1/n). Tomisawa suggests that B should be

between 0 and 1.5. Since we are dividing by B in the equation, we might worry that as B

heads toward 0, all hell breaks loose, but luckily and for all the other

components so, just as in normal FM, if the index is 0, we get a pure

sine wave.

(define

(feedbk beg

dur freq ampindex)

(let*

((start

(seconds-

>samples

beg))

(end

(+ start

(seconds-

>samples

dur)))

(y

0.0)(x-

incr (hz-

>radians

freq)))

(do ((i

start (+ 1

i))

(x

2/1 J1(1) = 0.880 -> 1.000 (normalized to match fft)

2/2 J2(2) = 0.353 -> 0.401

2/3 J3(3) = 0.206 -> 0.234

2/4 J4(4) = 0.141 -> 0.159

2/5 J5(5) = 0.104 -> 0.118

2/6 J6(6) = 0.082 -> 0.093

2/7 J7(7) = 0.066 -> 0.076



0.0 (+ x x-

incr)))

((= i

end))

(set! y

(+ x (* index

(sin y))))(outa i

(* amp (sin

y))))))

(with-sound

() (feedbk 0

1 100.0 1.0

1.0))

Why does the FFT show a 0 Hz component? Increasing the sampling rate, or decreasing

the carrier frequency reduces this component without affecting the others, but low-pass

filtering the output does not affect it (so it's unlikely to be an artifact of aliasing which isa real problem in feedback FM). Change the sine to cosine in (* amp (sin y)) and

suddenly there's a ton of DC. Fiddle with the initial phase in that line, and there's always

some choice that reduces it to 0.0. Groan — it appears to be another "centering" problem, but I haven't found the magic formula yet (a reasonable stab at it is: -(phase-incr^(1-

(B/3)))).

Why does an index over 1.0 create bursts of noise? Each burst happens as the modulator

phase goes through an odd multiple of pi (where sine is going negative as the phaseincreases). Since the index (B) is high enough, the change between successive samples in

(B * sin(y)) is eventually greater in magnitude than the phase increment. When that

happens on the downslope of the sine curve, B * sin(y) + phase-increment (our overall phase increment) is so much more negative on the current sample than the previous one

that the phase actually backs up. (This is confusing to analyze because at this point in the

curve, the feedback is already holding the phase back,so we need to reach a point where the increase in the

backup overwhelms the increment on that sample,

thereby backing up the overall phase beyond its previous held back value). So the modulator phase

backs into the less negative part of the sine curve: our

next y value is less negative (it can even be positive)!

But now B * sin(y) is also less negative, so the phase increment lurches us forward, and y

is now even more negative. We've started to zig-zag down the sine curve. Depending onthe index, this bouncing can reach any amplitude, and start anywhere after the high point

of the curve. Eventually, the sine slope lessens (as it reaches its bottom), the overall phasecatches up, and the bouncing stops for that cycle. The noise is not chaos (in the sense of

period doubling), or an error in the computation. Our largest safe index is

increment/sin(increment) which is just over 1.0. If we change the code to make sure thecarrier phase doesn't back up, the bursts go away until the index reaches about 1.4, then

Tomisawa's picture of the noise



we start to zigzag at the zero crossing. The take-home message is: "keep the index below

1.0!".

One way to make noise (deliberately) with FM is to increase the index until massivealiasing is taking place. A more controllable approach is to use a random number

generator as our modulator. In this case, the power spectraldensity of the output has the same form as the value

distribution function (amplitude distribution as opposed tofrequency) of the modulating noise, centered around the carrier. The bandwidth of the

result is about 4 times the peak deviation (the random number frequency times its index

— is this just Mr Carson again?):

(with-sound ()

(let ((gen (make-oscil 5000))

(noise (make-rand 1000 :envelope '(-1 1 0 0 1 1))) ; "eared"

(index (hz->radians 1000))) ; index=1.0 so bandwith=4 Khz (2 Khz

on each side)

(do ((i 0 (+ 1 i)))((= i 50000))

(outa i (oscil gen (* index (rand noise)))))))

flat gaussian (bell curve) eared

Simple FM with noise gives both whooshing sounds (high index) and hissing or whistling

sounds (low index), useful for Oceanic Music, but more subtle kinds of noise can be hardto reach. Heinrich Taube had the inspired idea of feeding the noise (as a sort of cascade

FM) into the parallel modulators of an fm-flute, but not into the carrier. The modulating

signal becomes a sum of two or three narrow band noises (narrow because normally the

amplitude of the noise is low), and these modulate the carrier. In CLM:

(oscil carrier (* fm-index (oscil fm (* noise-index (rand noise)))))

You may have noticed that this is one case where phase modulation is different from FM.

Previously, we could fix up each modulating sinusoid (both in amplitude and initial

phase), but here we have no such handles on the components of the incoming signal. If

someone insists, we can still match outputs by integrating the modulating signal:

FM and noise:

sin(sin(rand))



FM(white-noise) = PM(brownian-noise). Similarly, FM(square-wave) = PM(triangle-

wave), FM(nxy1sin) = PM(square-wave), and FM(e^x) = PM(e^x). FM(square-wave) is

.

In the realm of "anything" as the modulating signal, consider

(sin (+ sound-file (* index (sin (* 2 pi sound-file)))))

where "sound file" is any recorded sound. I call this "contrast-enhancement" in the CLM

package. It makes a sound crisper; "Wait for Me!" uses it whenever a sound needs to cutthrough a huge mix.

We can use more than one sinusoidal component in our carrier, or multiple banks of

carriers and modulators, and depend upon vibrato and "spectral fusion" to make the result

sound like one voice. In this cross between additive synthesis (the multiple carriers) andFM (the formant centered on each carrier), we get around many of the limitations of the

Bessel functions. There are numerous examples in

fmviolin.clm. One of the raspier versions of the fm-violin useda sawtooth wave as the carrier, and some sci-fi sound effects use triangle waves as both

carrier and modulator. See generators.scm for many other FM-inspired synthesis

techniques, including J0(B sin x): "Bessel FM". An elaborate multi-carrier FMinstrument is the voice instrument written by Marc Le Brun, used in "Colony" and other

pieces:

(define* (vox beg dur freq amp :optional (indexes '(.005 .01 .02))

(formant-amps '(.86 .13 .01)))


(end (+ start (seconds->samples dur)))(car-os (make-oscil 0))

(evens (make-vector 3))

(odds (make-vector 3))

(amps (apply vct formant-amps))

(ampf (make-env '(0 0 25 1 75 1 100 0) :scaler amp :duration

dur))

(frmfs (make-vector 3))

(indices (apply vct indexes))

(per-vib (make-triangle-wave 6 :amplitude (* freq .03)))

(ran-vib (make-rand-interp 20 :amplitude (* freq .5 .02))))

(do ((i 0 (+ 1 i)))

((= i 3))

(vector-set! evens i (make-oscil 0))

(vector-set! odds i (make-oscil 0)))

(vector-set! frmfs 0 (make-env '(0 520 100 490) :duration dur))



(run

(lambda ()

(do ((i start (+ 1 i)))

FM voice



((= i end))

(let* ((frq (+ freq (triangle-wave per-vib) (rand-interp ran-

vib)))

(car (oscil car-os (hz->radians frq)))

(sum 0.0))

(do ((k 0 (+ 1 k)))

((= k 3))

(let* ((frm (env (vector-ref frmfs k)))

(frm0 (/ frm frq))

(frm-int (inexact->exact (floor frm0)))

(even-amp 0.0) (odd-amp 0.0)

(even-freq 0.0) (odd-freq 0.0))

(if (even? frm-int)

(begin

(set! even-freq (hz->radians (* frm-int frq)))

(set! odd-freq (hz->radians (* (+ frm-int 1) frq)))

(set! odd-amp (- frm0 frm-int))

(set! even-amp (- 1.0 odd-amp)))

(begin

(set! odd-freq (hz->radians (* frm-int frq)))

(set! even-freq (hz->radians (* (+ frm-int 1) frq)))(set! even-amp (- frm0 frm-int))

(set! odd-amp (- 1.0 even-amp))))

(set! sum (+ sum (+ (* (vct-ref amps k)

(+ (* even-amp

(oscil (vector-ref evens k)

(+ even-freq (* (vct-

ref indices k) car))))

(* odd-amp

(oscil (vector-ref odds k)

(+ odd-freq (* (vct-ref

indices k) car)))))))))))

(outa i (* (env ampf) sum))))))))

(with-sound ()

(vox 0 1.0 220.0 0.5)

(vox 1.5 1.0 110 .5 '(0.02 0.01 0.02) '(.9 .09 .01)))

which produces this spectrogram:



References

Abramowitz and Stegun, "Handbook of Mathematical Functions", Dover 1965.

Benson, "Music: A Mathematical Offering", Cambridge University Press,

Nov 2006. Also available

on-line: http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf. If

the math side of myarticle is of any interest, you might like Benson's discussion of

FM.

Chowning, "The Synthesis of Complex Audio Spectra by Means of Frequency

Modulation", JAES 21:526-534, 1973

Gagliardi, "Introduction to Communications Engineering", Wiley

Interscience, 1978.

Gray and Mathews, "A Treatise on Bessel Functions and Their

Applications to Physics", MacMillan and Co, 1895.

Klapper, "Selected Papers on Frequency Modulation", Dover 1970. (Out ofprint, but available

via used book markets such as abebooks or amazon — usually about

$25).

The Bessel function graph is from Corrington, "Variation of

Bandwidth with Modulation Index in FM",

The picture below of an early radio is from Armstrong, "A Method

of Reducing Disturbances in Radio

Signaling by a System of FM".



LeBrun, "A Derivation of the Spectrum of FM with a Complex Modulating

Wave", CMJ vol1, no 4 1977 p51-52

Moorer, "Signal Processing Aspects of Computer Music: A Survey" Proc

IEEE vol 65 1977.

Palamin, Palamin, Ronveaux "A Method of Generating and Controlling

Asymmetrical Spectra", JAES vol 36,

no 9, Sept 88, p671-685.

Schottstaedt, "The Simulation of Natural Instrument Tones Using

Frequency Modulation

with a Complex Modulating Wave", CMJ vol 1 no 4 1977 p46-50

Taub and Schilling, "Principles of Communications Systems", McGraw-

Hill, 1986.

Tomisawa, "Tone Production Method for an Electronic Musical Instrument"

US Patent 4,249,447, 1981.

Watson, "A Treatise on the Theory of Bessel Functions", Cambridge, 1922.

FM (and AM) radio ca 1933

Music 5 FM?

An Introduction to FM

Documents

Transcript of An Introduction to FM