Average sampling of band-limited stochastic processes

We consider the problem of reconstructing a wide sense stationary band-limited process from its local averages taken either at the Nyquist rate or above. As a result, we obtain a sufficient condition under which average sampling expansions hold in mean square and for almost all sample functions. Truncation and aliasing errors of the expansion are also discussed

Band-limited functions and the sampling theorem

The definition of band-limited functions (and random processes) is extended to include functions and processes which do not possess a Fourier integral representation. This definition allows a unified approach to band-limited functions and band-limited (but not necessarily stationary) processes. The sampling theorem for functions and processes which are band-limited under the extended definition is derived

Distortion-Rate Function of Sub-Nyquist Sampled Gaussian Sources

The amount of information lost in sub-Nyquist sampling of a continuous-time Gaussian stationary process is quantified. We consider a combined source coding and sub-Nyquist reconstruction problem in which the input to the encoder is a noisy sub-Nyquist sampled version of the analog source. We first derive an expression for the mean squared error in the reconstruction of the process from a noisy and information rate-limited version of its samples. This expression is a function of the sampling frequency and the average number of bits describing each sample. It is given as the sum of two terms: Minimum mean square error in estimating the source from its noisy but otherwise fully observed sub-Nyquist samples, and a second term obtained by reverse waterfilling over an average of spectral densities associated with the polyphase components of the source. We extend this result to multi-branch uniform sampling, where the samples are available through a set of parallel channels with a uniform sampler and a pre-sampling filter in each branch. Further optimization to reduce distortion is then performed over the pre-sampling filters, and an optimal set of pre-sampling filters associated with the statistics of the input signal and the sampling frequency is found. This results in an expression for the minimal possible distortion achievable under any analog to digital conversion scheme involving uniform sampling and linear filtering. These results thus unify the Shannon-Whittaker-Kotelnikov sampling theorem and Shannon rate-distortion theory for Gaussian sources.Comment: Accepted for publication at the IEEE transactions on information theor

On the multidimensional sampling theorem

The well known sampling theorem is extended to the multidimensional weakly stationary (but not necessarily band-limited) processes. The mean square and almost sure convergence of the sampling expansion sum is derived for full spectrum multidimensional processes

Positive definite functions and Karhunen-Loeve Theorem

Εθνικό Μετσόβιο Πολυτεχνείο--Μεταπτυχιακή Εργασία. Διεπιστημονικό-Διατμηματικό Πρόγραμμα Μεταπτυχιακών Σπουδών (Δ.Π.Μ.Σ.) “Μαθηματική Προτυποποίηση σε Σύγχρονες Τεχνολογίες στην Οικονομία

The sampling formula and A. L. Cauchy

The works of A . L . Cauchy appear in many referencies about band-limited function periodic sampling . The usual Shannon formul a is generally associated with the famous paper of A . L. Cauchy untitled "Mémoire sur diverses formules d'analyse", published in 1841 in the "comptes rendus de l'Académie des Sciences" . This paper shows that the sampling formula may come from anothe r reference by A . L. Cauchy. Moreover, other interpolation formulas (even in the non-periodic case) can be derived from a thir d paper on complex integral calculus .A. L. Cauchy apparaît dans beaucoup de bibliographies concernant l'échantillonnage périodique des fonctions ou des processus à spectre borné. On y associe la formule de Shannon à un article de Cauchy intitulé Mémoire sur diverses formules d'analyse paru en 1841 dans les Comptes rendus de l'Académie des Sciences. Ce qui suit tend à démontrer que c'est dans un autre article de Cauchy que l'on trouve le matériel à l'origine de la formule d'échantillonnage habituelle. On montrera qu'un troisième de ses articles, concernant le calcul des résidus, permet d'envisager d'autres formules d'interpolation, y compris à prises d'échantillons non périodiques

An investigation of sampled data interpolation error

Ph.D.D. L. Fin

Design problems in pulse transmission

"July 28, 1960." Issued also as a thesis, M.I.T. Dept. of Electrical Engineering, May 25, 1960.Bibliography: p.48.Army Signal Corps Contract DA36-039-sc-78108. Dept. of the Army Task 3-99-20-001 and Project 3-99-00-000.Donald Winston Tufts