# 12

# RANDOM PROCESSES AND NOISE

## 12-1 INTRODUCTION

The free electrons in a resistor at a given temperature—we are thinking in terms of the Drude-Lorentz model—are continually moving about and colliding with the lattice ions. Given the statistical nature of this process, we should expect that at any instant there are more electrons in one half of a resistor than in the other half, and that there is therefore a *thermal noise voltage* ν_{R}(*t*) across an open-circuited resistor. This is in fact so; as shown in Fig. 12-1, the noise voltage adds to whatever signal voltage ν_{R}(*t*) is applied in series with a resistor *R*, and being highly irregular, it can mask a weak signal.

The thermal noise of a resistor is an example of a *random process* in which the value of a variable at any given time is unpredictable, and it is only one of many random processes that generate noise in electronic circuits. We will see that such processes can be described in terms of their statistical measures—that is, in terms of certain averages—and that despite its random nature, we will be able to estimate the degree to which noise is a nuisance.

For the large class of *stationary* random processes, whose properties do not change in time, we will introduce the *autocorrelation function R*(τ), the average product of values taken at times separated by an interval τ. The Fourier transform of the autocorrelation function, the *power spectrum*. (*f*), will then give us a description of the frequency content of a random process, and we ...

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