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Reconstruction from Samples--Pictorial Version

Figure D.1 shows how a sound is reconstructed from its samples. Each sample can be considered as specifying the scaling and location of a sinc function. The discrete-time signal being interpolated in the figure is a digital rectangular pulse:

$\displaystyle x = [\dots, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, \dots]
$

The sinc functions are drawn with dashed lines, and they sum to produce the solid curve. An isolated sinc function is shown in Fig.D.2. Note the ``Gibb's overshoot'' near the corners of the continuous rectangular pulse in Fig.D.1 due to bandlimiting. (A true continuous rectangular pulse has infinite bandwidth.)

Figure D.1: Summation of weighted sinc functions to create a continuous waveform from discrete-time samples.
\includegraphics[width=\twidth]{eps/SincSum}

Notice that each sinc function passes through zero at every sample instant but the one it is centered on, where it passes through 1.


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``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 © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA