## 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 $$\tilde{a}_0(t)e^{jM_0(t)} = \sum\limits_{k=0}^{\infty} \frac{1}{(2k+1)^2}e^{j 2k\omega_0 t}.$$