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'''Nonuniform sampling''' is a branch of [[Nyquist–Shannon_sampling_theorem|Nyquist–Shannon_sampling theorem]]. Nonuniform sampling is based on Lagrange Interpolation and the relationship between itself and (uniform) sampling theorem. Nonuniform sampling is a generalisation of the Whittaker-Shannon-Kotel (WSK) sampling theorem. |
'''Nonuniform sampling''' is a branch of [[Nyquist–Shannon_sampling_theorem|Nyquist–Shannon_sampling theorem]]. Nonuniform sampling is based on Lagrange Interpolation and the relationship between itself and (uniform) sampling theorem. Nonuniform sampling is a generalisation of the Whittaker-Shannon-Kotel (WSK) sampling theorem. |
Revision as of 21:18, 4 July 2012
Nonuniform sampling is a branch of Nyquist–Shannon_sampling theorem. Nonuniform sampling is based on Lagrange Interpolation and the relationship between itself and (uniform) sampling theorem. Nonuniform sampling is a generalisation of the Whittaker-Shannon-Kotel (WSK) sampling theorem.
Lagrange (Polynomial) Interpolation
For a given function, it is possible to construct a polynomial of degree n which has the same value with the function at n+1 points.[1]
Let the n+1 points to be , and the n+1 values to be .
In this way, there exists a unique polynomial such that
Furthermore, it is possible to simplify the representation of using the interpolating polynomials of Lagrange interpolation:
From the above equation:
As a result,
To make the polynomial form more useful:
In that way, the Lagrange Interpolation Formula appears:
Note that if , then the above formula becomes:
Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem
Whittaker tried to extend the Lagrange Interpolation from polynomials to entire functions. He showed that it is possible to construct the entire function[5]
which has the same value with at the points
Moreover, can be written in a similar form of the last equation in previous section:
When a=0 and W=1, then the above equation becomes almost the same as WSK theorem:[6]
If a function f can be represented in the form
then f can be reconstructed from its samples as following:
Nonuniform Sampling
For a sequence satisfying[7]
then
- and is Bernstein space
- is uniformly convergent on compact sets.[8]
The above is called Paley-Wiener-Levinson Theorem which generalize WSK sampling theorem from uniform samples to non uniform samples. Both of them can reconstruct a band-limited signal from those samples, respectively.
References
- F. Marvasti, Nonuniform sampling: Theory and Practice. Plenum Publishers Co., 2001, pp. 123-140.