Introduction

In this and the following modules the basic concepts of information theory will be introduced. For simplicity we assume that the signalsare time discrete. Time discrete signals often arise from sampling a time continous signal. The assumption of time discrete signal is valid becausewe will only be looking at bandlimited signals . (Which can, as we know , be perfectly reconstructed).

In treating time discrete signal and their information content we have to distinguish between two types of signals:

• signals have amplitude levels belonging to a finite set
• signals that have amplitudes taken from the real line

Examples of information sources

The signals treated are mainly of a stochastic nature, i.e. the signal is unknown to us.Since the signal is not known to the destination (because of it's stochastic nature), it is then bestmodeled as a random process, discrete-time or continuous time. Examples of information sources that we model as random processes are:

• Digital data source (e.g. a text) can be modeled as a random process.
• Video signals can be modeled as a random process. Such signals are mainly bandlimited toaround 5 MHz (the value depends on the standards used to raster the frames of image).
• Audio signals can be modeled as a random process. Speech is typically between 300 Hz and3400 Hz, see .

Video and speech are analog information signals are bandlimited. Therefore, if sampled faster than two times the highest fequency component, they can be reconstructedfrom their sample values.

The core of information theory

The key observation from the discussion above is that for a reveiver the signals are unknown . It is exact this uncertainty that enables the signalto transmit information. This is the core of information theory:

Some statistics

Here we present some statistics with the intent of reviewing a few basic concepts and to introduce the notation.

Let X be a stochastic variable. Let $X=x_i$ and $X=x_j$ denote two outcomes of X.

• Dependent outcomes implies: $(X=x_i, X=x_j)=(X=x_i)(x_i, X=x_j)=(X=x_j)(x_j, X=x_i)$
• Independent outcomes implies $(X=x_i, X=x_j)=(X=x_i)(X=x_j)$
• Bayes' rule: $(x_i, X=x_j)=\frac{(x_j, X=x_i)(X=x_j)}{(X=x_i)}$