## Course Edit

This page is for the course ELEC3106 - Electronics, run by Torsten Lehman, senior lecturer at the University of New South Wales. It is based on the lecture on the 17/03/09.

## Overview Edit

Noise is a completely random, non uniform signal that can be superimposed on a signal in a circuit, thus distorting its natural function. All real life circuits have some element of noise, learning how to control this is vital to designing an effective circuit.

## Prerequisites Edit

The reader is assumed to have some basic electrical knowledge, such as that covered by the courses ELEC2133 - Analogue Electronics and ELEC1111 - Electrical and Telecommunications Engineering. Also basic some mathematical knowledge, such as that covered by MATH2069 - Mathematics 2B would be useful.

## Noise Edit

### Types of Noise Edit

**White noise** is the most general type of noise. The name comes from the fact that as a signal it covers all frequencies (sort of like white light). This noise can come from surrounding circuitry, and from thermal noise to name a few.

**Thermal noise** is the random movement of electrons due to thermal energy, whilst the electrons do generally move in one direction to generate current they also do move around, which generates noise, as shown in figure 1.

**Pink noise** is noise where lower frequencies have more effect then higher frequencies. Figure 2 has a comparison of white and pink noise.

**Shot noise** comes from the quantum effects of semiconductor devices. As only single electrons can cross the junctions of the devices the current is quantised, instead of just being a nice smooth current. We can model this noise as shown in figure 3, where the current source has a value $ 2qI $ (where q is the charge on a single electron and I is the conventional current through the device).

### Electrical noise model Edit

Noise in circuits can be modelled by a variable current source or voltage source as shown in figure 4. Note that the sources can be in either direction as noise is random.

### Quantifying noise Edit

There are many different ways of quantifying noise. In general noise is quantified by it's variance (as it is a random variable). This is shown in the following equation:

$ \sigma^2 = V_N^2 = \frac{1}{t}\int_{0}^{t}V_N^2\, dt $

where t is the time, $ V_N^2 $ is the noise power. Note that the noise power is simply averaged over time. It should also be noted that $ \sqrt{V_N^2} $ is the rms noise.

Another method is the **signal to noise ratio** **(SNR)** this is defined as:

$ SNR = 20\log_{10}\left (\frac{V_{rms}}{\sqrt{V_N^2}}\right) $

where $ \frac{V_{rms}}{\sqrt{V_N^2}} $ is essentially $ \frac{V_{in}}{V_N} $

Another method is **dynamic range (DR)**, this is virtually the same as SNR but is concerned with the maximum signal the amplifier can take. It is defined as:

$ DR = 20\log_{10}\left (\frac{V_{rmsmax}}{V_{rmsmin}}\right) $

where the fraction this time is the maximum input signal over the noise power.

### Total noise Edit

We are usually concerned with more than one source of noise, as most circuit elements generate noise. Because of this we are interested in the total noise of the circuit.

For example consider figure 5. Instantaneously we can see from KVL on the circuit that $ V_{Ntot} = V_{N1} + V_{N2} $. Now

$ V_{Ntot}^2 = \frac{1}{t}\int_{0}^{t}V_N^2\, dt = V_{N1}^2 +V_{N2}^2 + \frac{1}{t}\int_{0}^{t}2V_{N1}V_{N2}\, dt $

The third term is the covariance of the noise. When we consider an instantanous time the covariance disappears (as the integral has both limits at zero). However this term is vital in averaging the noise over time.

Calculating the noise in the frequency domain is usually prefered because it is better defined (not as random). In this case take the fourier transform of the noise and the formula for the noise becomes: