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Cheung–Marks theorem
In information theory, the Cheung–Marks theorem,〔J.L. Brown and S.D.Cabrera, "On well-posedness of the Papoulis generalized sampling expansion," IEEE Transactions on Circuits and Systems, May 1991 Volume: 38 , Issue 5, pp. 554–556〕 named after K. F. Cheung and Robert J. Marks II, specifies conditions〔K.F. Cheung and R. J. Marks II, "Ill-posed sampling theorems", IEEE Transactions on Circuits and Systems, vol. CAS-32, pp.829–835 (1985).〕 where restoration of a signal by the sampling theorem can become ill-posed. It offers conditions whereby "reconstruction error with unbounded variance () when a bounded variance noise is added to the samples."〔D. Seidner, "Vector sampling expansion," IEEE Transactions on Signal Processing. v. 48. no. 5. 2000. p. 1401–1416.〕
==Background==
In the sampling theorem, the uncertainty of the interpolation as measured by noise variance is the same as the uncertainty of the sample data when the noise is i.i.d.〔R.C. Bracewell, ''The Fourier Transform and Its Applications,'' McGraw Hill (1968)〕 In his classic 1948 paper founding information theory, Claude Shannon offered the following generalization of the sampling theorem:〔Claude E. Shannon, "Communication in the presence of noise", Proc. Institute of Radio Engineers, vol. 37, no.1, pp. 10–21, Jan. 1949. (Reprint as classic paper in: ''Proc. IEEE'', Vol. 86, No. 2, (Feb 1998) )〕
Although true in the absence of noise, many of the expansions proposed by Shannon become ill-posed. An arbitrarily small amount of noise on the data renders restoration unstable. Such sampling expansions are not useful in practice since sampling noise, such as quantization noise, rules out stable interpolation and therefore any practical use.
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