The Exact Frequency Domain Solution for the Periodic Synchronous Averaging Performed in Discrete-Time

This paper deals with the technique known as the periodic synchronous averaging. The exact analytical expression for the fast Fourier transform (FFT) representing the digital spectrum of the signal undergoing periodic synchronous averaging is derived using the general signal and spectral framework. This formula connects the coefficient of Fourier series of the original continuous-time signal with the FFT of the averaged sampled version taking into consideration all the effects such as difference between the true and averaging periods, the attenuation and the leakage. The results of the numerical simulation are presented for the case of periodic signal, which was chosen a train of triangle pulses, the spectrum of which possesses a closed form and whose Fourier series coefficients rapidly decrease with the index. The chosen example allows the authors to illustrate that the waveform of the recovered signal can vary significantly, despite a rather slight difference in values between the true and averaging periods. Another important effect emphasized in the presented paper is that overall distinction between the original and averaged signals measured by means of relative mean square error raises if the total observation length increases while the other parameters remains fixed.