The Role of Probability in Signal Processing
Let’s begin by defining probability.
Probability is a branch of mathematics concerned with the analysis of random phenomena. We understand probability as the simple concept of “chance” as experienced in everyday life. Probability applies to the weather forecast in the chance of rain today, to the daily commute in your possibility of being delayed by a traffic snafu, and to your lottery purchase and the odds that you will not be a winner.
In fact, probability theory was actually developed in response to a question about gambling, first asked of Pascal in 1654 by the Chevalier de Mere. The question was regarding who should be the winner if a game is interrupted. Should it be the person with the most points at the time, or the person who had the best chance of winning if the game had gone to completion?
The resulting computation was called Pascal’s triangle. With this, probability theory was formed and today provides electrical engineers with a way to analyze and compensate for random events when designing and developing technology. In terms of signal processing those random events would typically come in the form of noise or missing bits of data.
When radar is picking up a signal the objective is to determine if the signal is clean or if it is also reflecting noise. We’ve all heard of a radar echo which gives false returns. Probability theory, through a series of complex equations, helps us estimate how much noise should be expected as a random fluctuation of the signal. Radar waves can be put through a series of filters to clean up the signal and eliminate the probability of noise corruption.
Another example comes in wireless communications, where the signal goes through a number of amplifiers to enhance signal strength and reach. However, quality is degraded by the thermal noise picked up at each amplification point. Again, knowing the probability of such occurrences helps electrical engineers design technology that anticipates noise and can take the randomness out of the process.
Regardless of the intricacies of the circuit manufacturing process, there will be some random variations between individual components. Defects that are outside of accepted parameters will be fairly easy to spot during the quality assurance phase, and thus, will be removed from the batch. However, there is still the probability that slight deviations, while within accepted parameters, may still impact the performance of the signal processing circuitry. It all depends on what random circumstances encountered further on down the line.
During your study of probability in signal processing, you will come to understand the importance of varying types of random variables, confidence intervals, regression, curve fitting, filters, vectors, matrices, and so much more. All thanks to Pascal and his nobleman friend who enjoyed a game of chance.
Sourceshttp://www.britannica.com/EBchecked/topic/477530/probability-theory http://www.worldtracker.org/media/library/College%20Books/Cambridge%20University%20Press/0521864704.Cambridge.University.Press.Probability.and.Random.Processes.for.Electrical.and.Computer.Engineers.Jun.2006.pdf http://books.google.com/books?id=7NH5yNWMCFEC&pg=PA67&lpg=PA67&dq=Chevalier+dc+Merc&source=bl&ots=4y7soGrmsG&sig=efuBNcIoKTS8YYjoZTg8cRkb0DI&hl=en&sa=X&ei=Knj3U9P3HZPgsATpxoHgCg&ved=0CC0Q6AEwAg#v=onepage&q=Chevalier%20dc%20Merc&f=false