## 3D Hilbert Spectrum Examples

### Sinusoidal FM Waveform

3D Hilbert Spectrum of a Sinusoidal FM Waveform as a Fourier Series. Here the real signal is given by $$x (t) = \Re\left\lbrace \sum\limits_{k=-\infty}^{\infty}J_k(2\pi B/\omega_m)e^{j[(\omega_c+k\omega_m)t+\phi_k]} \right\rbrace$$ where $$J_k(\cdot)$$ denotes the $$k$$th-order Bessel function of the first kind. This yields an infinite number of components with IA given by $$a_k(t)=J_k(2\pi B/\omega_m)$$ and IF given by $$\omega_k(t) =\omega_c+k\omega_m.$$

3D Hilbert Spectrum of a Sinusoidal FM Waveform as a Single FM Component. Here the real signal is given by $$x (t) = \Re\left\lbrace e^{j\left[\omega_c t + B \sin(\omega_m t)\right]}\right\rbrace$$ with IA $$a_0(t) = 1$$ and IF $$\omega_0(t) = \omega_{c} + \frac{d}{dt}B\sin(\omega_m t)$$