3D Hilbert Spectrum Examples

Triangle Waveform

3D Hilbert Spectrum of a Triangle Waveform as a Fourier Series. The real signal is given by \( x(t) = \Re\left\lbrace\sum\limits_{k=0}^{\infty} \frac{8A}{\pi^2}\frac{1}{(2k+1)^2} e^{j (2k+1) \omega_0 t}\right\rbrace \) where the IA is given by \( a_k(t) = \frac{8A}{\pi^2}\frac{1}{(2k+1)^2}\) the IF is given by \( \omega_k(t) = (2k+1) \omega_0.\)

3D Hilbert Spectrum of a Triangle Waveform as a Single AM Component. The real signal is given by \(x(t)= \Re\left\lbrace a_0(t)e^{j\omega_0 t}\right\rbrace\) where the IA is given by \({a_0}(t) = x(t)/\cos(\omega_0 t).\)

3D Hilbert Spectrum of a Triangle Waveform as a Single FM Component. The real signal is given by \( x ( t) =\Re\left\lbrace A e^{j \left[\omega_0 t+M_0(t)\right]} \right\rbrace) \) here the IF is \({\omega_0}(t) = \omega_0+\frac{d}{dt}M_0(t)\) where \(M_0(t) = \arccos\left[ x(t)/A \right]-\omega_0 t.\)

3D Hilbert Spectrum of a Triangle Waveform as a Single AM-FM Component with Harmonic Correspondence. The real signal is given by \(x (t) = \Re\left\lbrace a_0(t)e^{j[\omega_0 t+M_0(t)]} \right\rbrace\) where the IA is given by \( a_0(t) = \frac{8A}{\pi^2} \tilde{a}_0(t)\) and the IF is \(\omega_0(t) = \omega_0+\frac{d}{dt}M_0(t)\) where \(\tilde{a}_0(t)\) and \(M_0(t)\) are related through \(\tilde{a}_0(t)e^{jM_0(t)} = \sum\limits_{k=0}^{\infty} \frac{1}{(2k+1)^2}e^{j 2k\omega_0 t}.\)