Multiple Aliasing of Windowed Real-Valued Signal as a Cause of Accuracy Limitation of DFT Methods

DFT spectral signal analysis is a widely used method for various types of signal processing due to its several advantages. In the case of incoherently sampled real-valued signals, spectral leakage and aliasing occur, causing errors in the signal spectrum estimate. Standard techniques consider that negative spectral leakage components influence the positive part of the spectrum by aliasing. Consequently, these methods are trying to suppress these spectral components. The paper shows that this way may not be satisfactory, because windowing of the signal, necessarily associated with the use of DFT, causes the appearance of repetitive bands in addition to the considered negative mirror components. These components cause additional aliasing, albeit with a decreasing but non-negligible influence, causing further errors in the calculated spectral components. This effect of multiple aliasing, which has not been considered so far, explains the limited maximum achievable accuracy of commonly used DFT methods. The paper further explains that the effect of the random part of such multiple aliasing can be fully compensated converting the real-valued signal to its analytic counterpart. Then the permanent part of the multiple aliasing can be simply eliminated from the obtained analytic signal. Furthermore, the paper introduces a simple procedure for converting real-valued sinusoidal signal to analytic signal in time domain. This procedure is used to develop a new frequency estimation method with minimal estimation error even for very small numbers of signal samples. By using a rectangular window, it minimizes the effect of noise practically to the Cramer-Rao bound.