““Normal” Single-Modulator FMNormal” Single-Modulator FM““Normal” Single-Modulator FMNormal” Single-Modulator FM

• Frequency ModulationFrequency Modulation

MModulation in frequency.odulation in frequency.

x(t) = w(t)sin[2x(t) = w(t)sin[2(f(fcc + If + Ifmmsin(2sin(2ffmmt))t]t))t]

Phase ModulationPhase ModulationPhase ModulationPhase Modulation

• Phase ModulationPhase Modulation

Equivalent to FM – implements the Equivalent to FM – implements the modulation in the phase instead of modulation in the phase instead of frequency.frequency.

x(t) = w(t)sin[2x(t) = w(t)sin[2ffcct + Isin(2t + Isin(2ffmmt)]t)]

Double Modulator FMDouble Modulator FMDouble Modulator FMDouble Modulator FM

• Double Modulator FMDouble Modulator FM

Uses two parallel modulators.Uses two parallel modulators.

x(t) = w(t) sin[2x(t) = w(t) sin[2ffcctt

+ I+ Im1m1sin(2sin(2ffm1m1t)t)

+ I+ Im2m2sin(2sin(2ffm2m2t)]t)]

Double Modulator FMDouble Modulator FMDouble Modulator FMDouble Modulator FM

• Double FM-produced harmonic Double FM-produced harmonic amplitudes depend on a sum of Bessel amplitudes depend on a sum of Bessel function differences and productsfunction differences and products

aakk(I(Im1m1, I, Im2m2)=)=

[J[J(k-n(k-ncc-hn-hnmm))(I(Im1m1) - J) - J- (k+n- (k+ncc+hn+hnmm))

(I(Im1m1)] J)] Jhh(I(Im2m2))

h=-

Double Modulator FMDouble Modulator FMDouble Modulator FMDouble Modulator FM

• This is a more complicated relationship This is a more complicated relationship than single modulator FM, where each than single modulator FM, where each carrier’s harmonic amplitudes depend carrier’s harmonic amplitudes depend on a single Bessel function difference.on a single Bessel function difference.

• This complexity makes double FM This complexity makes double FM parameter optimization a more difficult parameter optimization a more difficult task than formant FM parameter task than formant FM parameter optimization.optimization.

Double Modulator FMDouble Modulator FMDouble Modulator FMDouble Modulator FM

• Double FM modulation indices and Double FM modulation indices and frequency ratios are usually smaller than frequency ratios are usually smaller than those of formant FM.those of formant FM.

• Example: Spectrum of double modulator Example: Spectrum of double modulator FM with FM with nncc=10, n=10, nmm=5, I=5, Im1m1=6=6, and , and IIm2m2=2=2::

Double versus Double versus single Modulator FMsingle Modulator FM

Double versus Double versus single Modulator FMsingle Modulator FM

• Convergence of error for different numbers of Convergence of error for different numbers of carriers using double modulator FM and formant carriers using double modulator FM and formant FM to model the trumpet.FM to model the trumpet.

• Double FM can always do better than single Double FM can always do better than single modulator FM for the same number of carriers.modulator FM for the same number of carriers.

Double versus Double versus Single Modulator FMSingle Modulator FM

Double versus Double versus Single Modulator FMSingle Modulator FM

• Convergence of error vs computation (number of Convergence of error vs computation (number of table lookups) using double modulator FM and table lookups) using double modulator FM and formant FM to model the trumpet.formant FM to model the trumpet.

• Double modulator FM is only cost-effective when using 1 carrierDouble modulator FM is only cost-effective when using 1 carrier• Otherwise it is better to just add more single modulated carriers.Otherwise it is better to just add more single modulated carriers.

Nested Modulator FMNested Modulator FMNested Modulator FMNested Modulator FM

• Nested Modulator FMNested Modulator FM

Uses nested (serial) modulators.Uses nested (serial) modulators.x(t) = w(t)sin[2x(t) = w(t)sin[2ffcct+t+

IIm1m1sin(2sin(2ffm1m1t + t + I Im2m2sin(2sin(2ffm2m2t))]t))]

Nested FM is more nonlinear than Nested FM is more nonlinear than double FM, making optimization double FM, making optimization more difficult. more difficult.

Feedback FMFeedback FMFeedback FMFeedback FM

• Feedback FMFeedback FM

A discrete formula for feedback FM is the A discrete formula for feedback FM is the following:following:

xxnn = w = wnnsin[(2sin[(2ff11n/SR) + (Bxn/SR) + (Bxn-1n-1/w/wn-1n-1)])]

with with wwnn the discrete carrier amplitude the discrete carrier amplitude

envelope, and envelope, and ff11 the desired fundamental the desired fundamental

frequency.frequency.

Feedback FMFeedback FMFeedback FMFeedback FM• The output of the carrier is used to The output of the carrier is used to

modulate the following sample, scaled by modulate the following sample, scaled by the modulation index the modulation index BB. . • When When BB is less than about 1.5, a monotonically is less than about 1.5, a monotonically

decreasing spectrum results.decreasing spectrum results.• Because of this, feedback FM is potentially Because of this, feedback FM is potentially

more easily controlled than the other forms more easily controlled than the other forms of FM (where the harmonics oscillate as the of FM (where the harmonics oscillate as the modulation index changes).modulation index changes).

• Another advantage of feedback FM over Another advantage of feedback FM over other forms of FM is that its spectral other forms of FM is that its spectral components are strictly positive for components are strictly positive for BB < 1.5. < 1.5. • This avoids phase cancellation when multiple This avoids phase cancellation when multiple

carriers are added together.carriers are added together.

FM Types ComparedFM Types ComparedFM Types ComparedFM Types Compared• Convergence of error for different Convergence of error for different

numbers of carriers using various types of numbers of carriers using various types of FM to model the trumpet.FM to model the trumpet.

FM with 3 FM with 3 nested nested

modulatorsmodulators is best for a is best for a fixed number fixed number of carriers.of carriers.

FM Types ComparedFM Types ComparedFM Types ComparedFM Types Compared• Convergence of error vs computation Convergence of error vs computation

(number of table lookups) using various (number of table lookups) using various types of FM to model the trumpet.types of FM to model the trumpet.

FeedbackFeedbackFM is bestFM is bestfor a fixed for a fixed number of number of

table lookups table lookups ((computations)computations)