[Sampling, Up--Sampling, Down--Sampling, and Multi--Rate Processing]Sampling, Up--Sampling, Down--Sampling, and Multi--Rate Processing

A very important and fundamental operation in discrete-time signal processing is that of sampling. Discrete-time signals are often obtained from continuous-time signal by simple sampling. This is mathematically modeled as the evaluation of a function of a real variable at discrete values of time [1] . Physically, it is a more complicated and varied process which might be modeled as convolution of the sampled signal by a narrow pulse or an inner product with a basis function or, perhaps, by some nonlinear process.

The sampling of continuous-time signals is reviewed in the recent books by Marks [2] which is a bit casual with mathematical details, but gives a good overview and list of references. He gives a more advances treatment in [3] . Some of these references are [4] [5] [6] [7] [8] [1] [9] . These will discuss the usual sampling theorem but also interpretations and extensions such as sampling the value and one derivative at each point, or of non uniform sampling.

Multirate discrete-time systems use sampling and sub sampling for a variety of reasons [10] [11] . A very general definition of sampling might be any mapping of a signal into a sequence of numbers. It might be the process of calculating coefficients of an expansion using inner products. A powerful tool is the use of periodically time varying theory, particularly the bifrequency map, block formulation, commutators, filter banks, and multidimensional formulations. One current interest follows from the study of wavelet basis functions. What kind of sampling theory can be developed for signals described in terms of wavelets? Some of the literature can be found in [12] [13] [14] [15] [16] .

Another relatively new framework is the idea of tight frames [15] [17] [16] . Here signals are expanded in terms of an over determined set of expansion functions or vectors. If these expansions are what is called a tight frame, the mathematics of calculating the expansion coefficients with inner products works just as if the expansion functions were an orthonormal basis set. The redundancy of tight frames offers interesting possibilities. One example of a tight frame is an over sampled band limited function expansion.