How do we know that CTFT of the autocorrelation function is the PSD?
$$\begin{align}
R_x(\tau) &amp;=\int_{-\infty}^\infty x(t)x^*(t-\tau)\,\mathrm dt\\
&amp;= \int_{-\infty}^\infty \left[\int_{-\infty}^\infty X(f)e^{j2\pi ft}\,\mathrm df\right]x^*(t-\tau)\,\mathrm dt\\
&amp;= \int_{-\infty}^\infty X(f) \left[\int_{-\infty}^\infty x^*(t-\tau) e^{j2\pi ft}\,\mathrm dt\right]\,\mathrm df\\
&amp;= \int_{-\infty}^\infty X(f) \left[\int_{-\infty}^\infty x^*(t-\tau) e^{j2\pi f(t-\tau)}\,\mathrm dt\right]e^{j2\pi f\tau}\,\mathrm df\\
&amp;= \int_{-\infty}^\infty X(f) \left[\int_{-\infty}^\infty x^*(\lambda) e^{j2\pi f\lambda}\,\mathrm d\lambda\right]e^{j2\pi f\tau}\,\mathrm df\\
&amp;= \int_{-\infty}^\infty X(f) \left[\int_{-\infty}^\infty x(\lambda) e^{-j2\pi f\lambda}\,\mathrm d\lambda\right]^*e^{j2\pi f\tau}\,\mathrm df\\
&amp;= \int_{-\infty}^\infty X(f)X^*(f) e^{j2\pi f\tau}\,\mathrm df\\
&amp;= \int_{-\infty}^\infty |X(f)|^2 e^{j2\pi f\tau}\,\mathrm df
\end{align}$$