Sampling-50 years after Shannon
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Citations
Method and apparatus for multiplexed fabry-perot spectroscopy
Stable Phaseless Sampling and Reconstruction of Real-Valued Signals with Finite Rate of Innovation
Joint reconstruction of misaligned images from incomplete measurements for cardiac MRI
Behavior of Shannon’s Sampling Series for Hardy Spaces
Sampling Theorem Associated with Multiple-parameter Fractional Fourier Transform
References
A theory for multiresolution signal decomposition: the wavelet representation
A wavelet tour of signal processing
Ten Lectures on Wavelets
A practical guide to splines
Orthonormal bases of compactly supported wavelets
Related Papers (5)
Frequently Asked Questions (14)
Q2. What future works have the authors mentioned in the paper "Sampling—50 years after shannon" ?
The subject is far from being closed and its importance is most likely to grow in the future with the ever-increasing trend of replacing analog systems by digital ones ; typical application areas are 582 PROCEEDINGS OF THE IEEE, VOL. Many of the results reviewed in this paper have a potential for being useful in practice because they allow for a realistic modeling of the acquisition process and offer much more flexibility than the traditional band-limited framework. Last, the authors believe that the general unifying view of sampling that has emerged during the past decade is beneficial because it offers a common framework for understanding—and hopefully improving—many techniques that have been traditionally studied by separate communities. Areas that may benefit from these developments are analog-to-digital conversion, signal and image processing, interpolation, computer graphics, imaging, finite elements, wavelets, and approximation theory.
Q3. What are some recent applications of generalized sampling?
Recent applications of generalized sampling include motion-compensated deinterlacing of televison images [11], [121], and super-resolution [107], [138].
Q4. What is the way to obtain the expansion coefficients of a signal?
Given the equidistant samples (or measurements) of a signal , the expansion coefficients areusually obtained through an appropriate digital prefiltering procedure (analysis filterbank) [54], [140], [146].
Q5. What is the interesting generalization of (9)?
An interesting generalization of (9) is to consider generating functions instead of a single one; this corresponds to the finite element—or multiwavelet—framework.
Q6. What is the computational solution to the special case?
The computational solution takes the form of a multivariate filterbank and is compatible with Papoulis’ theory in the special case .
Q7. What is the advantage of the projection interpretation of the sampling process?
The projection interpretation of the sampling process that has just been presented has one big advantage: it does not require the band-limited hypothesis and is applicable for any function .
Q8. What is the way to suppress aliasing?
When this is not the case, the standard signal-processing practice is to apply a low-pass filter prior to sampling in order to suppress aliasing.
Q9. What is the graph of the error kernels for the least squares spline app?
8. This graph clearly shows that, for signals that are predominantly low-pass (i.e., with a frequency content within the Nyquist band), the error tends to be smaller for higher order splines.
Q10. What is the logical approach to the sampling theorem?
Having reinterpreted the sampling theorem from the more abstract perspective of Hilbert spaces and of projection operators, the authors can take the next logical step and generalize the approach to other classes of functions.
Q11. What is the main reason for the study of the error estimation technique?
This has led researchers in signal processing, who wanted a simple way to determine the critical sampling step, to develop an accurate error estimation technique which is entirely Fourier-based [20], [21].
Q12. What is the common symbol used in the theory of the wavelet transform?
Note that the orthogonalized version plays a special role in the theory of the wavelet transform [81]; it is commonly represented by the symbol .
Q13. What is the way to predict the loss of performance when an approximation algorithm is?
It is especially interesting to predict the loss of performance when an approximation algorithm such as (30) and (24) is used instead of the optimal least squares procedure (18).
Q14. How much decay does a polynomial spline provide?
The authors observe that a polynomial spline approximation of degree provides an asymptotic decay of 1 20 dB per decade, which is consistent with (45).