How to calculate the Fourier transform of absolute-valued sinusoid?
For $0 &lt; a \le 1$, the Fourier Transform is straightforward:

$$\begin{align*}\mathscr{F}\left\{|1+a\sin(2\pi s_0t)|\right\} &amp;= \mathscr{F}\left\{1+a\sin(2\pi s_0t)\right\} \\
&amp;= \int_{-\infty}^{\infty}\left[1+a\sin(2\pi s_0t)\right]e^{-2\pi i st} \mathrm{d}t\\
&amp; = \delta(s) +\dfrac{a}{2i}\left[\delta(s-s_0)-\delta(s+s_0)\right] \\
\end{align*}$$

For $ a &gt; 1$, we have to rewrite the function to get rid of the absolute value bars:

$$|1+a\sin(2\pi s_0t)| = |s_0|\mathrm{III}(s_0t) *\left(\left[1+a\sin(2\pi s_0t)\right]\cdot\left[\mathrm{\Pi}\left(\dfrac{\pi s_0}{\frac{\pi}{2}-\mathrm{Arcsin}\left[-\frac1{a}\right]}t-\dfrac{\frac{\pi}{2}}{2\pi s_0}\right)-\mathrm{\Pi}\left(\dfrac{\pi s_0}{\frac{\pi}{2}+\mathrm{Arcsin}\left[-\frac1{a}\right]}t+\dfrac{\frac{\pi}{2}}{2\pi s_0}\right)\right]\right)$$

Where $*$ denotes convolution, $\mathrm{Arcsin}()$ denotes the principal arcsine,

$$\mathrm{III}(at) = \dfrac1{|a|}\sum_{n=-\infty}^{\infty}\delta\left(t-\dfrac{n}{a}\right)$$
and
$$\begin{equation}\mathrm{\Pi}(at) = \begin{cases}
1 &amp; |at| &lt;\frac1{2}\\
0 &amp; |at| &gt;\frac1{2}\end{cases}\end{equation}$$

Note that the $\mathrm{\Pi}()$ functions are two rectangles snipping out the two different segments of one cycle, and the convolution with the $\mathrm{III}()$ function creates all the periodic replicas of those two segments of that one cycle.

Now using tables of Fourier Transforms, and various, basic Fourier Transform theorems we get, for $a &gt;1$,

$$\mathscr{F}\left\{|1+a\sin(2\pi s_0t)|\right\} =\mathrm{III}\left(\dfrac{s}{s_0}\right)\cdot\left(\left[\delta(s)+\dfrac{a}{2i}\left(\delta(s-s_0)-\delta(s+s_0)\right)\right]*\left[\dfrac{\frac{\pi}{2}-\mathrm{Arcsin}\left(-\frac1{a}\right)}{\pi|s_0|}e^{2\pi i\left(\dfrac{-\frac{\pi}{2}}{2\pi s_0}\right)\left(\dfrac{\frac{\pi}{2}-\mathrm{Arcsin}\left(-\frac1{a}\right)}{\pi s_0}\right)s}\mathrm{sinc}\left(\dfrac{\frac{\pi}{2}-\mathrm{Arcsin}\left(-\frac1{a}\right)}{\pi s_0}s\right)-\dfrac{\frac{\pi}{2}+\mathrm{Arcsin}\left(-\frac1{a}\right)}{\pi|s_0|}e^{2\pi i\left(\dfrac{\frac{\pi}{2}}{2\pi s_0}\right)\left(\dfrac{\frac{\pi}{2}+\mathrm{Arcsin}\left(-\frac1{a}\right)}{\pi s_0}\right)s}\mathrm{sinc}\left(\dfrac{\frac{\pi}{2}+\mathrm{Arcsin}\left(-\frac1{a}\right)}{\pi s_0}s\right)\right]\right)$$

So you have the difference of two modulated $\mathrm{sinc}()$ functions, with 1 real replica at DC and 2 imaginary replicas at $\pm s_0$ (due to convolution with the $\delta()$ functions), sampled at integer multiples of $s_0$ (by the $\mathrm{III}()$ function).