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Vibration signal analysis is the most common technique used for mechanical vibration monitoring. By using vibration sensors, the fault prognosis of rotating machinery provides a way to detect possible machine damage at an early stage and prevent property losses by taking appropriate measures. We first propose a digital integrator in frequency domain by combining fast Fourier transform with digital filtering. The velocity and displacement signals are, respectively, obtained from an acceleration signal by means of two digital integrators. We then propose a fast method for the calculation of the envelope spectra and instantaneous frequency by using the spectral properties of the signals. Cepstrum is also introduced in order to detect the unidentifiable periodic signal in the power spectrum. Further, a fault prognosis algorithm is presented by exploiting these spectral analyses. Finally, we design and implement a visualized real-time vibration analyzer on a Raspberry Pi embedded system, where our fault prognosis algorithm is the core algorithm. The real-time signals of acceleration, velocity, displacement of vibration, as well as their corresponding spectra and statistics, are visualized. The developed fault prognosis system has been successfully deployed in a water company.