Consider the following signal:
\tilde{u}(t) = A_c e^{j\beta \sin(2\pi f_m t)}
where $\beta$ is a constant and $f_m$ is a fixed frequency tone.
(a) Note that $\tilde{u}(t)$ is a periodic function in time. Find its fundamental period or fundamental
frequency.
(b) It has been known from your earlier classes that any periodic function $s(t)$ with a fundamental
frequency of $f_0$ can be represented in a Fourier series as follows:
$s(t) = \sum_{n=-\infty}^{\infty} c_n e^{j2\pi nf_0 t}$
where $c_n$ represents the Fourier coefficients. Show that the Fourier coefficients are computed
as
$c_n = \frac{1}{T} \int_0^T s(t)e^{-j2\pi nf_0 t} dt$ where $T = 1/f_0$.
(c) Using the Fourier series in (b), find the Fourier series of $\tilde{u}(t)$ in terms of the following
function, known as the $n$-th order Bessel function of the first kind:
$J_n(\beta) = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{j(\beta \sin x - nx)} dx$
