102
D. Rocchesso: Sound Processing
side components
sound bandwidth
Figure 8: pd patch for phase modulation. Adapted from a help patch of the pd
distribution.
k=1
J
k
(I) sin ((
c
+ k
m
)n) + (-1)
k
sin ((
c
- k
m
)n)
side frequencies
,
where J
k
(I) is the k-th order Bessel function of the first kind. These Bessel
functions are plotted in figure 9 for several values of k (number of side frequency)
and I (modulation index).
Therefore, the effect of phase modulation is to introduce side components
that are shifted in frequency from the fundamental by multiples of
m
and whose
amplitude is governed by J
k
(I). Generally speaking, the larger the modulation
index, the wider is the sound bandwidth. Since the number of side components
that are stronger than one hundredth of the carrier magnitude is approximately
M = I + 0.24I
0.27
,
(20)
the bandwidth is approximately
BW = 2 I + 0.24I
0.27
m
2I
m
.
(21)
If the ratio
c
/
m
is rational the resulting spectrum is harmonic, and the
partials are multiple of the fundamental frequency
0
=
c
N
1
=
m
N
2
,
(22)
where
N
1
N
2
=
c
m
, with N
1
, N
2
irreducible couple .
(23)
For instance, if N
2
= 1, all the harmonics are present, and if N
2
= 2 only the
odd harmonics are present.
When calculating the spectral components, some of the partials on the left
of the carrier may assume a negative frequency. Since sin (-) = - sin =