Nyquist Theorem

The calculation of functions in digital domain from analogue acoustic signals involves a two-step process which includes analog to digital (A/D) conversion and digital calculations performed on digitized acoustical signal. Function V(t) will be represented digitally without any loss of information as long as sampling occurs in accordance with the Nyquist criteria [1-9]. How can we determine the values of digitized function for points between samples when we have only N samples available? The Nyquist formula requires an infinite number of samples to accomplish this task. In situations when digital samples are sufficiently dense, one can approximate many continuous functions with their discrete formulations. Errors generated in these cases will be small, since they depend on spacing between samples. The situation will be different, however, when samples are coarsely spaced. For example, a sinusoidal tone of frequency f = 20,000 Hz sampled at fsampling = 44,100 Hz is represented by only 2.205 samples per period. Calculation of many functions (for example RMS values) may lead in this case to some errors. Finite duration sampling of continuous signal results in errors caused by our limited knowledge of the function for all points in time. It turns out that the more samples we have around the region of interest, the more accurately we are able to reproduce the function there. This paper investigates the error caused by truncation of the Nyquist sampling formula with the aim of quantifying it and establishing ways to minimize its effect.

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