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The Sinc Function

Figure: The sinc function $ \protect$sinc$ (x) \isdef \sin (\pi x)/(\pi x)$ .

The sinc function, or cardinal sine function, is the famous ``sine x over x'' curve, and is illustrated in Fig.D.2. For bandlimited interpolation of discrete-time signals, the ideal interpolation kernel is proportional to the sinc function

   sinc$\displaystyle (f_st) \isdef \frac{\sin(\pi f_st)}{\pi f_st}.

where $ f_s$ denotes the sampling rate in samples-per-second (Hz), and $ t$ denotes time in seconds. Note that the sinc function has zeros at all the integers except 0, where it is 1. For precise scaling, the desired interpolation kernel is $ f_s$sinc$ (f_st)$ , which has a algebraic area (time integral) that is independent of the sampling rate $ f_s$ .

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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
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University