Introduction to Random Signals and Noise

8

Poisson Processes and Shot Noise

8.1 INTRODUCTION

Random point processes described in this chapter deal with a sequences of events, where both the time and the amplitude of the random variable is of a discrete nature. However, in contrast to the discrete-time processes dealt with so far, the samples are not equidistant in time, but the randomness is in the arrival times. In addition, the sample values may also be random. In this chapter we will not deal with the general description of random point processes but will confine the discussion to a special case, namely those processes where the number of events k, in fixed time intervals of length T, is described by a Poisson probability distribution

In this equation we do not use the notation for the probability density function, since it is a discrete function and thus indeed a probability (denoted by P(·)) rather than a density, which is denoted by f_X(x).

An integer-valued stochastic process is called a Poisson process if the following properties hold [1,5]:

1. The probability that k events occur in any arbitrary interval of length T is given by Equation (8.1).
2. The number of events that occur in any arbitrary time interval is independent of the number of events that occur in any other arbitrary non-overlapping time interval.

Many physical phenomena are accurately modelled by a Poisson process, such as the emission of photons from ...