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 \(\omega_0(t) = \omega_{c} + \frac{d}{dt}B\sin(\omega_m t)\)