User:KYN/Sampling and Reconstruction

In signal processing, sampling and reconstruction are two dual or complementary processes which are frequently used in many types of applications. What follows is an overview of the technical and mathematical framework which applies to sampling and reconstruction processes. It starts with a basic model of the two processes which, however, has wide range of applications. It then presents various types of generalizations which follows more or less natural from this simple model.

Introduction
The basic model of sampling and reconstruction considers a continuous time signal, a function of time as a real-valued variable. In what follows, this signal is refered to as an analog signal, and as an example you may think of an audio signal which is a funtion of time. By means of a sampling process, the analog signal is converted to a discrete time signal, a function over the set of integers. This representation of the signal is here refered to as a digital signal, and you may think of it as a long list of numbers. The digital signal is then transmitted or stored, possibly also processed or compressed and uncompressed. Finally, the digital representation of the signal is converted back to an analog signal. In some cases, the formation of the analog signal implies that it shold be as close or similar to the original signal as possible. In order cases, the processing of the digital signal implies that an altered version of the original signal should be reconstructed.

The analog signal
Before going in the details of the sampling process we need a thoroug understanding of the analog signal. Not only is this a function of a real-valued variable, e.g., time, but it maps time to a real number. This is not just a matter of mathematics or a suble technicality, it is a fundamental property of analog signals which make them fundamentally different from digital signals, which are described below. In pratice, the analog signal normally assumes values within a limited and well-defined range. This limitation is only related to the practical implementation of how the signals are processed and not something which needs to be taken into account in the following presentation.

The digital signal
In particular, it raises the question about what is the difference between an analog signal and its digital representation. One part of the answer is that the analog signal varies over a real-valued variable when the digital signal varies over the set of integers. The other part is that the analog signal can assume an arbitrary value, at least within a well-defined range. The digital signal, on the other hand, can only assume discrete values, also within a well-defined range. Both these differences make analog and digital signals qualitatively different.

Simple model of the sampling process
Let us return to the original and analog signal.