104

D. Rocchesso: Sound Processing

spectral envelope

feedback modulation index

sawtooth wave

amplitude modulation

Complex modulator

The modulating waveform can be non-sinusoidal. In this case the analysis can

be quite complicated. For instance, a modulator with two partials

1

and

2

,

acting on a sinusoidal carrier, gives rise to the expansion

x(n) = A

k

m

J

k

(I

1

)J

m

(I

2

) sin ((

c

+ k

1

+ m

2

) n) .

(24)

Partials are found at the positions |

c

± k

1

± m

2

| . If

M

= MCD(

1

,

2

),

the spectrum has partials at |

c

± k

M

|. For instance, a carrier f

c

= 700Hz

and a modulator with partials at f

1

= 200Hz and f

1

= 300Hz, produce a

harmonic spectrum with fundamental at 100Hz. The advantage of using complex

modulators in this case is that the spectral envelope can be controlled with more

degrees of freedom.

Feedback FM

A sinusoidal oscillator can be used to phase-modulate itself. This is a feedback

mechanism that, with a unit-sample feedback delay, can be expressed as

x(n) = sin (

c

n + x(n - 1)) ,

(25)

where is the feedback modulation index. The trigonometric expansion

x(n) =

k

2

k

J

k

(k) sin (k

c

n)

(26)

holds for the output signal. By a gradual increase of we can gradually trans-

form a pure sinusoidal tone into a sawtooth wave [78]. If the feedback delay is

longer than one sample we can easily produce routes to chaotic behaviors as

is increased [12, 15].
FM with Amplitude Modulation

By introducing a certain degree of amplitude modulation we can achieve a more

compact distribution of partials around the modulating frequency. In particular,

we can use the expansion
e

I cos (

m

n)

sin (

c

n + I sin (

m

n)) = sin (

c

n) +

k=1

I

k

k

sin ((

c

+ k

m

) n) ,

(27)

to produce a sequence of partials that fade out as 1/k in frequency, starting from

the carrier. Figure 10 shows the magnitude spectrum of the sound produced

by the mixed amplitude/frequency modulation (27) with carrier frequency at

3000Hz, modulator at 1500Hz, modulation index I = 0.2, and sample rate

F
s

= 22100Hz.

5

The reader is invited to verify the expansion (27) using an octave script with wm = 100; wc
= 200; I = 0.2; n = [1:4096]; y1 = exp(I*cos(wm*n)) .* sin(wc*n + I*sin(wm*n));