Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Normed vector spaces
- 2 Analytic tools
- 3 Fourier series
- 4 The Fourier transform
- 5 Compressive sampling
- 6 Discrete transforms
- 7 Linear filters
- 8 Windowed Fourier and continuous wavelet transforms. Frames
- 9 Multiresolution analysis
- 10 Discrete wavelet theory
- 11 Biorthogonal filters and wavelets
- 12 Parsimonious representation of data
- References
- Index
5 - Compressive sampling
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Normed vector spaces
- 2 Analytic tools
- 3 Fourier series
- 4 The Fourier transform
- 5 Compressive sampling
- 6 Discrete transforms
- 7 Linear filters
- 8 Windowed Fourier and continuous wavelet transforms. Frames
- 9 Multiresolution analysis
- 10 Discrete wavelet theory
- 11 Biorthogonal filters and wavelets
- 12 Parsimonious representation of data
- References
- Index
Summary
Introduction
Although it is convenient for conceptual and theoretical purposes to think of signals as general functions of time, in practice they are usually acquired, processed, stored and transmitted as discrete and finite time samples. We need to study this sampling process carefully to determine to what extent a sampling or discretization allows us to reconstruct the original information in the signal. Furthermore, real signals such as speech or images are not arbitrary functions. Depending on the type of signal, they have special structure. No one would confuse the output of a random number generator with human speech. It is also important to understand the extent to which we can compress the basic information in the signal to minimize storage space and maximize transmission time.
Shannon sampling is one approach to these issues. In that approach we model real signals as functions f(t) in L2(ℝ) that are bandlimited. Thus if the frequency support of f(ω) is contained in the interval [-Ω, Ω] and we sample the signal at discrete time intervals with equal spacing less than 1/2πΩ, i.e., faster than the Nyquist rate, we can reconstruct the original signal exactly from the discrete samples. This method will work provided hardware exists to sample the signal at the required rate. Increasingly this is a problem because modern technologies can generate signals of higher bandwidth than existing hardware can sample.
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- Chapter
- Information
- The Mathematics of Signal Processing , pp. 164 - 207Publisher: Cambridge University PressPrint publication year: 2011
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