Abstract
In discrete methods the significant section of an infinite function is turned into a periodic function, on order to convert a Fourier transform into a Fourier series. The repeated section is sampled at a number of discrete points, and may, of course, be sampled through any number of complete periods. Calculation time is minimised by sampling in the frequency domain at multiples of the lowest frequency, given by the reciprocal of the time period, but it is essential to preserve the cyclic nature of the approximating function. Time waveforms of finite length may be assumed to repeat in an infinite train with spectra recover able, with no aliasing, from regular samples at time separations equal to at least the reciprocal of twice the highest frequency in the waveform. If the first sample is at zero time, there is always an even number of samples in a period, if one sample lies at the centre of the period.
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