# What is the Fourier transform of a phase modulated signal?

I'm studying some Fourier analysis and have for a while been trying to figure out how to apply the Fourier transform to a phase modulated signal.

More rigorously stated, what is $$\mathcal{F}[\,\cos(vt + f_m(t))\,]$$ where $$f_m(t) = A \sin(v_At) + B \cos(v_Bt)$$

After trying several approaches the closest I got was noting that (for example) $\cos(A\sin(v_At))$ obviously has the same periodicity as $\sin(v_At)$, which gives some intuitive information as to what the solution should look like but is far from an actual solution.

When you analyze periodic functions with modulated arguments it can get pretty complex very fast when you want to avoid computational and numeric methods.

You certainly won't always get direct or finite representations !

as a non mathematical example, just look at the spectrum of an amplitude modulated signal (old AM Radio) and a frequency modulated spectrum (FM is only minor different from phase modulation by sin/cos modulation) For AM, you get 3 discrete values (one carrier, 2 sidebands) for FM/PM you get a continuum from $\;-\infty\;$ to $\;+\infty\;$ !

A first clue for you will be the Integral representation of the first kind Bessel functions.

$$J_n(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}Exp(in\theta-ix\;Sin(\theta)\,)\;d\theta=\frac{1}{\pi}\int_0^{\pi}Cos(n\theta-x\;Sin(\theta)\,)\;d\theta$$

As you see, Bessels have the properties you want here.

The second important expression is the Jacobi–Anger identity:

$$Exp(iz\;Cos(\theta))=\sum_{-\infty}^{\infty}\;i^n\,J_n(z)\,Exp(in\theta)=J_n(z)+2\sum_{1}^{\infty}\;i^n\,J_n(z)\,Cos(n\theta)$$ This basically tells you how to expand a trigonometric function in an exponential (or inside another trigonometric function). The coefficients are -what a surprise- the first kind bessels.

I won't post the whole Fourier Transformation here, since one can easily find it on the web and surely here, too.

As a note, you might want to try 'easier' functions first. In ultrafast Optics or femtosecond stuff, you are dealing with chirped pulses, this means wave packets with a (approximately linear) phase modulation:

https://i.stack.imgur.com/LtugI.png

Fresnel Integrals come handy in here.

Hope this could things up.