Reconstruction from Samples--The Math

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Reconstruction from Samples--The Math

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Reconstruction from Samples–The Math

Let $x_d(n) \ denote the $n$th sample of the original sound $x(t)$, where $t$ is time in seconds. Thus, $n$ ranges over the integers, and $T_s$ is the sampling period in seconds. Thesampling rate in Hertz (Hz) is just the reciprocal of the sampling period,i.e.,


To avoid losing any information as a result of sampling, we must assume $x(t)$ is band-limited to less than half the sampling rate. This means there can be no energy in $x(t)$ at frequency$F_s/2$ or above. We will prove this mathematically when we proveShannon’s Sampling Theorem in §A.3 below.

Let $X(\ denote the Fourier transform of $x(t)$, i.e.,


Then we can say $x$ is band-limited to less than half the sampling rate if and only if $X(\ for all $\. In this case, Shannon’s sampling theorem gives us that $x(t)$ can be uniquely reconstructed from the samples $x(nT_s)$ by summing up shifted, scaled, sinc functions:


The sinc function is the impulse response of the ideal lowpassfilter. This means its Fourier transform is a rectangular window in the frequency domain. The particular sinc function used here corresponds to the ideal lowpass filter which cuts off at half the sampling rate. In other words, it has a gain of 1 between frequencies 0 and $F_s/2$, and a gain of zero at all higher frequencies.

The reconstruction of a sound from its samples can thus be interpreted as follows: convert the sample stream into a weighted impulsetrain, and pass that signal through an ideal lowpass filter which cuts off at half the sampling rate. These are the fundamental steps ofdigital to analog conversion (DAC). In practice, neither the impulses nor the lowpass filter are ideal, but they are usually close enough to ideal that you cannot hear any difference. Practical lowpass-filter design is discussed in the context ofband-limited interpolation.

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