# Reconstructing a Sampled Signal Using Interpolation

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If a continuous signal is sampled with a sampling period then how can you approximately reconstruct the original signal? Three methods that are in use are zero-order hold interpolation, first-order hold interpolation, and band-limited interpolation. The reconstructed signal is calculated using , where is the value of the sample taken at time and is one of the three interpolation functions.

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Contributed by: David Bremner and Bruce Atwood (April 2011)

(Beloit College)

Open content licensed under CC BY-NC-SA

## Details

For zero-order hold interpolation , where is the rectangle (or unit box) function, equal to 1 for and 0 otherwise; for first-order hold interpolation , where is the unit triangle function on the interval ; and for band-limited interpolation , where . For these examples band-limited (sinc) interpolation gives the best results. This is not a surprise since the sampling theorem says a band-limited signal sampled at a frequency at least twice the highest frequency in the signal can be perfectly reconstructed from its samples using sinc interpolation if the summation is taken from to .

For additional information see the lecture on interpolation by Prof. Alan V. Oppenheim at http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-17-interpolation.

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