I've been trying to Fourier expand

$$\psi(t)=\sin\left(2\pi at\ +\ \sin\left(2\pi bt\right)\right)$$

Please help me evaluate the integral:

$$\nu(x)=\frac{1}{\sqrt{ab}}\int_0^{ab}\sin\left(2\pi at+\sin\left(2\pi bt\right)\right)\cdot e^{-\frac{2\pi ixt}{\sqrt{ab}}}dt$$

(a,b being constants - frequency with my case)

(please consider the fact that I'm not a complex analysis guy, actually a High Schooler, and that I cannot fathom your answer if it has all advanced terminologies [I know what Lebesgue Space is though] )

Edit/Bump: I got till here:

$$\nu(x)=\int_0^{\frac 12}\cos (\omega_at+m\sin \omega_bt)\cdot \cos (xt)dt$$

I'm able to numerically get the Spectrum using Desmos and also taking reference from a Video on MIT Opencourseware on Signals Processing but not mathematically perform this integral.

My attempt

I think the methods I've tried all along evolved with time. I'm facing a problem with the integration part. I first tried u-sub, tried trigonometric simplification and all of them didn't work. Back then I did not realize that it was periodic over $ab$ and that I shouldn't have attempted a Fourier Transform. Then I realized that it was periodic, applied Integration by parts. Didn't work. I tried doing the "Feynmann trick of partial inside the integral sign", I got till a differential equation to which I got an exponential function as a non-constant solution. I found it to be a very weird solution and I realized that the Fourier Coefficients are time-dependent and my differential equation doesn't work.

Using Transformation Formulae gives 2 terms that are 3 trig-functions multiplied. Applying more Transformation helps, I think it leads us back to the original form?

  • 2
    $\begingroup$ I won't write an answer since I'm not a big fan of integrals, but it's worth knowing that this thing has a name: "frequency modulation" - and you can find a lot of both quantitative and qualitative knowledge about its Fourier spectrum by looking up FM synthesis, which is a technique in sound synthesis that uses functions like this - and people interested in sound generally care a lot about harmonic spectrum. It looks like Wikipedia gives an explicit expression of your function as a sum of sine waves. $\endgroup$ Commented Aug 20, 2019 at 12:55
  • 2
    $\begingroup$ $\psi(t) = \Im(e^{2i \pi at} \exp(\frac12 e^{2i \pi bt})\exp(-\frac12 e^{-2i \pi bt}))$ whose Fourier series is found from expanding the two $\exp$ in power series, obtaining the Bessel function as the coefficients $\endgroup$
    – reuns
    Commented Aug 20, 2019 at 18:49
  • 2
    $\begingroup$ The inner sine function has period $1/b$ while the outer sine function has period $1/a$. As such, the overall function will only have period $T$ if $T$ is an integer multiple of both $1/b$ and $1/a$---that is, if $T=n/b=m/a$ for integers $m,n$. But this implies $b/a=n/m$ is rational. In that case, the frequencies $a,b$ are said to be commensurate. If $a,b$ are incommensurate, then the function is said to be quasiperiodic since no period actually exists. (Try plotting the case of $a=1,b=\sqrt{2}$ for a demonstration.) $\endgroup$ Commented Aug 20, 2019 at 20:56
  • $\begingroup$ @MiloBrandt I pretty much arrived at this thing from Frequency Mod. I am working on an FM synthesized sound and I really wanna know that the actual Fourier Analysis looks like. $\endgroup$ Commented Aug 21, 2019 at 13:34
  • $\begingroup$ @reuns Can you please explain how you got the expression? Is that Gamma function? I can see that you wrote the entire expression in terms of $e$ and made it into a product of 3 exponentials. Is that $I$ the Amplitude? $\endgroup$ Commented Aug 21, 2019 at 13:41


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