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 =