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Complex, or analytic wavelets enable: Instantaneous frequency, amplitude, and phase extraction - detailed post. Robust feature extraction for classification, stable against time-warping deformations (and, if coupled with time averaging, robust to time shifts), and averaged information recovery via higher-order transforms -- paper, lecture Exact analyticity ...


A complex-valued analytic signal: $$\begin{align} x_\mathrm{a}(t) &= \sum\limits_{k=1}^{K} g_k \ e^{-\alpha_k t} \ e^{j(2\pi f_k t + \phi_k)} \ u(t) \\ &= \sum\limits_{k=1}^{K} \underbrace{g_k e^{j\phi_k}}_{c_k} \ e^{(-\alpha_k+j2\pi f_k)t} \ u(t) \\ &= \sum\limits_{k=1}^{K} c_k \ e^{(-\alpha_k+j2\pi f_k)t} \ u(t) \\ ...


just generate two independent gaussian random numbers (with a good generator like the Twister) having zero mean and identical variance, call one the real part and call the other the imaginary part.

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