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Aims and Motivation for the Course

Module by: Nick Kingsbury

Summary: This module describes the aims and motivation for the course.

We aim to:
  • Develop a theory which can characterize the behavior of real-world Random Signals and Processes;
  • Use standard Probability Theory for this.
Random signal theory is important for
  • Analysis of signals;
  • Inference of underlying system parameters from noisy observed data;
  • Design of optimal systems (digital and analogue signal recovery, signal classification, estimation ...);
  • Predicting system performance (error-rates, signal-to-noise ratios, ...).
Example 1: Speech signals 
Use probability theory to characterize that some sequences of vowels and consonants are more likely than others, some waveforms more likely than others for a given vowel or consonant. Please see Figure 1.
Use this to achieve: speech recognition, speech coding, speech enhancement, ...
Figure 1: Four utterances of the vowel sound 'Aah'.
Example 2: Digital communications 
Characterize the properties of the digital data source (mobile phone, digital television transmitter, ...), characterize the noise/distortions present in the transmission channel. Please see Figure 2.
Use this to achieve: accurate regeneration of the digital signal at the receiver, analysis of the channel characteristics ...
Figure 2: Digital data stream from a noisy communications Channel.
Probability theory is used to give a mathematical description of the behavior of real-world systems which involve elements of randomness. Such a system might be as simple as a coin-flipping experiment, in which we are interested in whether 'Heads' or 'Tails' is the outcome, or it might be more complex, as in the study of random errors in a coded digital data stream (e.g. a CD recording or a digital mobile phone).
The basics of probability theory should be familiar from the IB Probability and Statistics course. Here we summarize the main results from that course and develop them into a framework that can encompass random signals and processes.

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