General Problems of the Sampling-Reconstruction Procedure of Random Process Realizations

Absract: The Sampling-Reconstruction Procedure (SRP) of random process and field realizations is a very popular problem during a lot of decades. Unfortunately, this problem is completely not solved until present time. There is a well-known Balakrishnan ́s theorem. This theorem describes SRP of stationary random process realizations. This theorem is characterized by some principal drawbacks: the probability density function (PDF) is not used; the mathematical model of sampled process is not realizable; the number of samples is equal to infinity; the reconstruction procedure is linear and the same for all types of random processes; the reconstruction error is equal to zero for all types of the processes. In order to overcome these drawbacks, we suggest using the conditional mean rule (CMR) for the statistical description of the SRP of random processes and fields. We take into account the PDF of sampled processes and fields. We can analyze the SRP of the following types: the process can be stationary or no stationary, the number of samples is arbitrary and limit, the intervals between neighbor samples can be arbitrary or periodical, etc. Generally, one can declare: any random process must have its own optimal reconstruction algorithm and reconstruction error function. In the case of Gaussian processes the reconstruction function is linear function and the reconstruction error function does not depend on the values of samples. If the sampled process is non Gaussian, then the reconstruction function is a non linear function of samples and the reconstruction error function depends on the samples. We illustrate these points by a lot of examples, including the SRP with jitter. Majority of our results are related with the SRP of continuous processes. But, additionally, we give same new results with respect of the SRP of processes with jumps.