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Aliasing of Sampled Signals

This section quantifies aliasing in the general case. This result is then used in the proof of the sampling theorem in the next section.

It is well known that when a continuous-time signal contains energy at a frequency higher than half the sampling rate $ f_s/2$ , sampling at $ f_s$ samples per second causes that energy to alias to a lower frequency. If we write the original frequency as $ f = f_s/2 +
\epsilon$ , then the new aliased frequency is $ f_a = f_s/2 - \epsilon$ , for $ \epsilon\leq f_s/2$ . This phenomenon is also called ``folding'', since $ f_a$ is a ``mirror image'' of $ f$ about $ f_s/2$ . As we will see, however, this is not a complete description of aliasing, as it only applies to real signals. For general (complex) signals, it is better to regard the aliasing due to sampling as a summation over all spectral ``blocks'' of width $ 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