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Math Help - Frequency modulation synthesis in series

  1. #1
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    Frequency modulation synthesis in series

    Hello there, I'm currently working on a project about the mathematics of FM synthesis (for a general overview see the relevant chapter in ☮ Music: a Mathematical Offering ☮).

    I'm trying to do some expansion and simplification of using multiple modulating waves and the maths is getting a bit trying. I want to rearrange an equation of trignometric functions so they are in bessel function form. So far I've done it for parallel modulating waves, by showing if we have something of the form

    sin(c_2 + I_2 sin(\theta_2) + I_1 sin(\theta_2)) then it can be rearranged to

    \sum_{k_1} \sum_{k_2} J_{k_1} (I_1) J_{k_2} (I_2) sin (c_2 + k_1 \theta_1 + k_2 \theta_2))

    using standard addition formulae for trignometric equations and the Bessel function expansion of sin(z sin\theta)) = \sum_{n=0}^{oo} J_{2n+1}(z)sin((2n+1) \theta)

    (I can provide a full proof if needed, but I hope the sketch will give an idea of what I'm aiming for)

    So I'm trying to do the same for series:

    I'm starting with sin(\alpha_1 + I_1 sin (\alpha_2 + I_2 sin \theta_2)

    Using addition formula for sin and then applying the Bessel function formula again, I arrive at

    sin (c_1 + I_1 \sum_{k_1}J_{k_1} (I_2) sin (c_1 + k_1 \theta_1)))

    Now here's where I get stuck. Can anyone suggest how I might expand/simplify this last equation? Can it even be done?
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  2. #2
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    I've actually found the solution to the problem but not the proof, so if someone could help me understand it that would be great.

    Rewriting in the authors own terms, he starts with the equation

    s(t) =  sin(2 \pi f_c t + I sin[2 \pi f_1 t + I_2 sin\{2 \pi f_2 t\}])

    and ends with s(t) = \sum_k J_k(I_1) \times J_n (k I_2) sin(2\pi [f_c + k_1 f_1 + n f_2]t)

    Can anyone explain the inbetween steps for me?
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