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));