The theoretical framework of the analytic wavelet transform used for vibration signals envelope-demodulation was addressed. Under the condition that the Fourier transform of an analytic wavelet is a real-valued function, it has been demonstrated that the analytic wavelet's imaginary part is the Hilbert transform of its real part. Therefore, it has been concisely deduced that combined Morlet wavelets, harmonic wavelets and combined harmonic wavelets belong to the analytic wavelets. These wavelets can also be used in the envelope-demodulation analysis of mechanical faults vibration signals, in a...