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Chapter 9:
Discrete Transforms
Published: 1999
DOI: 10.1117/3.339204.ch9
In many engineering applications the function (signal) under consideration is a continuous function of time that needs to be processed by a digital computer. The only way this can be accomplished is to sample the continuous function at discrete intervals of time. The sampled signal x ∗ (t) is then processed as an approximation to the true signal x(t). The relation between a continuous function x(t) and its sample values x(kT), k=0,±1,±2,…, where T is a fixed interval of time, is one of prime importance in digital processing techniques. If the Fourier transform of x(t) is nonzero only over a finite range of the transform variable, it turns out that the continuous function x(t) can always be recovered (theoretically) from knowledge of only its sample values x(kT), provided that the sampling rate is fast enough. This remarkable result is known as the sampling theorem and plays a central role in digital processing techniques. Functions whose transform is zero everywhere except for a finite interval are known as band-limited waveforms in signal analysis. Such functions do not actually exist in the real world, but theoretical considerations of band-limited waveforms is fundamental to the digital field. If the function under consideration is closely approximated by a band-limited waveform, then the sampled version of the function gives a reasonably accurate description of that function, provided the sample values are taken at a rate that is at least twice the highest frequency that is significant in the continuous waveform.
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Digital signal processing

Signal processing

Fourier transforms

Signal analysis

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