Time-domain aliasing and anti-aliasing effects in differentiating a band-unlimited signal

In this paper, we investigate time-domain errors occurring in the two extreme discrete-time differentiation modes of band-unlimited signals: in the full-band processing mode and in the processing mode with ideal anti-aliasing filtering (AAF) with a cut-off at the Nyquist frequency. We disclosed that regardless sampling frequency the error from AAF is greater than the aliasing error. It is found that type IV differentiators designed by three commonly known digital filter design methods approximately equally process the high frequency portion of the signal above the Nyquist frequency with nearly equal aliasing errors having a weak dependence on differentiator length. In contrast, the differentiators very differently compute the derivatives of the low frequency portion bellow the Nyquist frequency providing rather dissimilar the common differentiation accuracy. The results show that the differentiators derived by using maximal linearity constraints are more accurate than those designed by the Parks-McClellan algorithm and the impulse response truncation method.