1.3 Sampling, up--sampling, down--sampling, and multi--rate

A very important and fundamental operation in discrete-time signal processing is that of sampling. Discrete-time signals are often obtainedfrom continuous-time signal by simple sampling. This is mathematically modeled as the evaluation of a function of a real variable at discretevalues of time [link] . Physically, it is a more complicated and varied process which might be modeled as convolution of the sampled signalby 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 [link] which is a bit casual with mathematical details, but gives a good overview and list of references. He gives a more advancedtreatment in [link] . Some of these references are [link] , [link] , [link] , [link] , [link] , [link] , [link] . These will discuss the usual sampling theorem but also interpretations and extensions such assampling 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 [link] , [link] . A very general definition of sampling might be any mapping of a signal into a sequence of numbers. It might bethe 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 followsfrom the study of wavelet basis functions. What kind of sampling theory can be developed for signals described in terms of wavelets? Some of theliterature can be found in [link] , [link] , [link] , [link] , [link] .

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

Fourier techniques

We first start with the most basic sampling ideas based on various forms of Fourier transforms [link] , [link] , [link] .

The spectrum of a continuous-time signal and the fourier transform

Although in many cases digital signal processing views the signal as simple sequence of numbers, here we are going to pose the problem asoriginating with a function of continuous time. The fundamental tool is the classical Fourier transform defined by

and its inverse

where $j=\sqrt{-1}$ . The Fourier transform of a signal is called its spectrum and it is complex valued with a magnitude and phase.

If the signal is periodic with period $f\left(t\right)=f(t+P)$ , the Fourier transform does not exist as a function (it may as a distribution)therefore the spectrum is defined as the set of Fourier series coefficients

with the expansion having the form

The functions $g_k\left(t\right)=e^{j2\pi kt/P}$ form an orthogonal basis for periodic functions and [link] is the inner product $C\left(k\right)=⟨f\left(t\right),g_k\left(t\right)⟩$ .