Spectra of FMSpectra of FM• FM spectra contains the carrier FM spectra contains the carrier

frequency plus sideband components frequency plus sideband components whose amplitudes depend on the whose amplitudes depend on the Bessel functions (of the first kind).Bessel functions (of the first kind).

• I I is the modulation index, is the modulation index, ffcc the carrier the carrier frequency, frequency, ffmm the modulator frequency the modulator frequency

Bessel function of the first Bessel function of the first kind of orders 0 ~ 3kind of orders 0 ~ 3

JJ00(I)(I) corresponds to order 0, corresponds to order 0, JJ11(I)(I) corresponds to order 1, corresponds to order 1,

……

Spectra of FMSpectra of FM

• Bessel functions look like damped Bessel functions look like damped sine waves, where the order of the sine waves, where the order of the function is given by the subscriptfunction is given by the subscript

• A property of Bessel functions:A property of Bessel functions:JJ-i-i(I) = J(I) = Jii(I) * (-1)(I) * (-1)ii

• C library for Bessel functions: C library for Bessel functions: jn(order, I)jn(order, I)

Properties of Formant FM SpectraProperties of Formant FM Spectra• Negative frequencies fold up to Negative frequencies fold up to

corresponding positive harmonic corresponding positive harmonic frequencies.frequencies.

FM SpectraFM Spectra• May get negative frequency components: May get negative frequency components:

• these fold up with change of sign:these fold up with change of sign:

FM SpectraFM Spectra• With larger modulation index (With larger modulation index (II), we get more ), we get more

sidebands with larger amplitudes (i.e., sidebands with larger amplitudes (i.e., spectrum gets brighter). spectrum gets brighter).

• May get negative amplitude partials: May get negative amplitude partials: • from negative Bessel values from negative Bessel values JJnn(I) (I) • from odd left sidebands from odd left sidebands

JJ-i-i(I) = J(I) = Jii(I) * (-1)(I) * (-1)ii

FM SpectraFM Spectra• May get components above the Nyquist May get components above the Nyquist

frequency (causing aliasing)frequency (causing aliasing)

• To avoid aliasing with FM: To avoid aliasing with FM: • use low carrier frequency use low carrier frequency ffcarcar

0 <= f0 <= fcarcar <= 10*f <= 10*fmodmod

(0 <= n(0 <= ncc <= 10) <= 10)

• use low modulation indices use low modulation indices II 0 <= I <= 100 <= I <= 10

Generating Harmonic FM Generating Harmonic FM SpectraSpectra

• Formant FMFormant FMA special case of FM with:A special case of FM with:ffmm = f = f11

ffcc = n = nccffmm = n = nccff11

where where nncc is an integer representing the is an integer representing the carrier carrier frequency ratiofrequency ratio in the range: in the range:0 ≤ n0 ≤ ncc ≤ 10 ≤ 10..

Formant FMFormant FM• ““formant” means resonanceformant” means resonance• ffcc acts like a resonance with acts like a resonance with

sidebands falling off at harmonics sidebands falling off at harmonics around it.around it.

amplitude

fm=f1=100

fc=500 (nc=5)

100 200 300 400 500

fc

600 700

fc+2fm

800 900fc+fm

frequency

Properties of Formant FM Properties of Formant FM SpectraSpectra

• 1) Negative frequencies fold up to 1) Negative frequencies fold up to corresponding positive harmonic corresponding positive harmonic frequencies.frequencies.

amplitude

100 200 300 400 500 600 700 800 900

frequency

1000

1100

1200

-100 0

Properties of Formant FM Properties of Formant FM SpectraSpectra

• 2) Amplitude of each harmonic 2) Amplitude of each harmonic kk is is given by:given by:aakk = J = J(k-n(k-ncc))(I) – J(I) – J-(k+n-(k+ncc))(I)(I)

Example: Example: nncc = 5 = 5aa11 = J = J(1-5)(1-5)(I) – J(I) – J-(1+5)-(1+5)(I) = J(I) = J-4-4(I) – J(I) – J-6-6(I)(I)aa66 = J = J(6-5)(6-5)(I) – J(I) – J-(6+5)-(6+5)(I) = J(I) = J11(I) – J(I) – J-11-11(I)(I)

amplitude

100 200 300 400 500 600 700 800 900

frequency

1000

1100

1200

-100 0fc

fc=f1=100

fc=500 (nc=5)

Dynamic (Time-Varying) Dynamic (Time-Varying) Modulation IndicesModulation Indices

• Time-varying indices produce a dynamic spectrumTime-varying indices produce a dynamic spectrum• Spectral harmonics fade in and out as the Spectral harmonics fade in and out as the

modulation index modulation index II varies (unlike acoustic varies (unlike acoustic instruments)instruments)

• Fixed modulation index Fixed modulation index II used in modeling used in modeling acoustic instrumentsacoustic instruments [iii:27] FM trumpet

[iii:28] real trumpet [iii:7] FM sound

Dynamic Spectra withDynamic Spectra withMultiple Carrier FMMultiple Carrier FM

• Problem:Problem:• Single carrier-modulator pair with fixed Single carrier-modulator pair with fixed

modulation index produces a fixed modulation index produces a fixed spectrum (not dynamic).spectrum (not dynamic).

• Solution:Solution:• Multiple Carrier FMMultiple Carrier FM

Multiple Carrier FMMultiple Carrier FM• uses multiple carriers, each with its own uses multiple carriers, each with its own

modulation index, amplitude envelope and modulation index, amplitude envelope and carrier frequency ratio carrier frequency ratio

Multiple Carrier FMMultiple Carrier FM• carriers may add or partially cancel one carriers may add or partially cancel one

another (complex interactions) another (complex interactions)

[iii:29] 3-carrier FM trumpet parametersmod is the fundamental andnc is the carrier/mod ratio

negative amplitude is a (180°) phase shift

[iii:28][iii:28] real trumpet real trumpet

Multiple Carrier FMMultiple Carrier FM

• [iii:30][iii:30] 5-carrier fm soprano 5-carrier fm soprano