**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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--Pictorial Version

Figure A.1 shows how a sound is reconstructed from its samples. Each sample can be considered as specifying the

scalingandlocationof asinc function. The discrete-time signal being interpolated in the figure is . The sinc functions are drawn with dashed lines, and they sum to produce the solid curve. Note the ''Gibb's overshoot'' near the corners of this continuous rectangular pulse due to band-limiting.Note how each sinc function passes through zero at every sample instant but the one it is centered on, where it passes through 1. An isolated sinc function is shown in Fig. A.2.

The sinc function is the famous ''sine x over x'' curve, defined by

where denotes the sampling rate in samples-per-second, and denotes time in seconds.