Probability in Electronics

Probability and Noise in Electronics

Random variables are used to model noise in electronic circuits. By nature, the thermal noise generated by heat in circuits can and often is modeled as so called white noise. To be able to deal with and understand this noise, it is important to have a fundamental way of thinking about probability and statistics.

Thermal noise in circuits

One of the main sources of noise in circuits is what is called thermal noise. This arises due to the random motion of electrons in the material (metal usually). When the heat increases, the electrons increase their energy, and their motion increases. The electrons can be seen as bouncing around randomly, and hitting structures within the metal. This random movement will generate electromagnetic waves and induce voltages.

The thermal noise is often given as

$latex V = sqrt(4 k_b T R W) $

Where V is the RMS (root mean square) voltage of the noise. Where $latex k_b $ is Boltzmanns constant , T is the temperature (in Kelvin), R is the resistance of the metal, and W is the bandwidth in Hertz. Hence, what we can observe is that the noise increases with the square root of the temperature. The Kelvin scale is defined as 0 Kelvin when all molecules are completely not moving, and 0 degree Celsius is achieved at approximately 273 K.

We also see that the noise is dependent on the resistance of the line. This makes intuitive sense as it can be thought that when the resistance increases, there are more things for the electrons to bump into whilst randomly moving around. The bandwidth is simply what range of frequencies we are looking at. For example, we might have a filter at the output, only accepting a certain range of frequencies. In this case, we only see noise contributions from these frequencies. Hence a smaller bandwidth gives less noise, whilst a larger one gives more noise.

The important thing which we assumed is that the noise is white. That means that if you look in the frequency domain, the signal power of the noise is spread out evenly over the whole frequency spectrum. In practice, we know that this is not the case, however, since we always have some kind of filtering of the output (even cables act as filters for high frequencies), this model still works well.

Actually the thermal noise is the most used and important noise to fully grasp and understand. It is used when designing electronic circuits and systems. However, there exists a wide array of other noise types which also can be important in some applications.

Flicker noise

The flicker noise is sometimes called the (1/f) noise. This is because the power of the noise decreases as the inverse of the frequency. Hence, as the frequency increases, the power of the flicker noise decreases. Flicker noise can arise for several reasons, however one of them being impurities in the conductor (metal) used.

Noise correlation in circuits

Correlation between two or more random variables describes how “connected” variables. For example, imagine that we have two random variables, which are very correlated. If we know that the first variable has a certain value, then we can make a better guess what value the other random variable has, without looking at it.

There are two extreme cases, in the first, the noise from different sources are completely correlated, and in the second case the noise sources are completely uncorrelated. If the noise is correlated, then the voltages add up, since we know that the noise power is proportional to the voltage squared, we obtain:

$latex P_{total} = \frac{(V_1 + V_2)^2 }{R} = \frac{ {V_1}^2 + 2 V_1 V_2 + {V_2}^2}{R}$

Here we see that the voltages are added before they are squared, this gives a rather high noise. Hence we can not simply add the noises. In fact, if we add two noise sources with the same RMS voltage, their new noise power will be four times that of one of the sources!

If then noise sources are uncorrelated, we obtain

$latex P_{total} = \frac{(V_1^2 + V_2^2) }{R} $

Which is just an addition of the squared voltages. Hence the powers are simply added. To conclude, if the sources are correlated, the total noise will be higher, if they are uncorrelated, the noises will cancel each other out a bit, and the noise powers are only added. However if they are added the noise will be significantly higher.