# Here Comes The Noise!

July 10, 20241 comment

Noise. That wiggle or apparent fuzziness you see riding on top your pristine signal which you automatically turn a blind eye to until you enter the world of precision where that wiggle now causes supreme headaches/stress for your company at large.

Imagine you've found yourself embedded within a company (working as an embedded systems engineer if you'll pardon the terrible pun) working on a project where the noise observed on an ADC input is starting to make your manager sweat bullets (Those lower 2 bits are now considered garbage) where the accumulation of sweat appears to be almost rising by the hour and just to exasipate the issue you've been tasked to tame the noise but you have no idea what noise is other than some random natured thing which we just make random swipes at to make go away. Sometimes we can get lucky but more than likely we will enter a world of pain, so how can we stop the pain? What is this noise thing exactly?

# What is Electrical Noise?

By electrical noise we are referring to fluctuations in voltage or current that are injected into our circuit which often appear random in nature that are almost always not exactly welcome (especially in precision circuits). In other words we can consider them to be undesired signals which are coupled into our electronic circuits. It is common to hear the noise level referred to as the noise floor for the system.

# Types of Noise

There are unfortunately many different types of noise which can manifest in electronic systems, but let's start off with the simplest kind which almost every single electronic circuit will have, even the simplest circuit with a battery and resistor, which is Johnson Noise.

## Johnson Noise

Johnson Noise refers to the thermal noise present in resistors (and any resistance actually) which arises from the electrons random motion in a conductor, it is a physical phenomenon which we have to live with. However some good noise is that we can reduce the noise voltage.

Due to the random motion component of Johnson Noise, which we will see is a common feature of most forms of noise, we commonly speak of RMS noise voltages and currents to describe their intensity.

$$V_{rms} = \sqrt{4k_bTR\Delta f}$$

From the above we can see that the noise is directly proportional to a constant, kb (Boltzmann's constant), temperature T (In Kelvin), the resistance R and the bandwidth delta f.

While we can't do much about that constant kb and usually we don't have too much say about the temperature environment our device is in (although depending on industry we may need to support a certain range), we often can control the resistance value and bandwidth. Hence to reduce Johnson Noise we can reduce the resistor value and/or reduce the bandwidth.

## Pink or 1/f Noise

Pink noise arises whenever current is flowing. It is commonly called 1/f noise as the noise exhibits a Power Spectral Density function with a 1/f profile as below:

What's particularly interesting about pink noise is that its intensity is highest at low frequencies and as such it is this noise that is often heard in audio systems as that annoying random noise sound.

# How can we treat noise systematically?

A very powerful systematic approach to analysing noise is to consider each type of noise present in your system as a noise source and then sum the effects together. Superposition applied to noise if you will.

As an example, let's consider this elementary circuit:

We can replace R1 and R2 with noiseless resistors connected in series to noise voltage sources. When doing this we express the noise voltage sources as Power Spectral Densities (Mean Square Voltage/Hz of bandwidth). We also typically denote Power Spectral Density with an S. A related term, Voltage Spectral Density is given by taking the square root of the power spectral density and has units of V/sqrt(Hz). Note: This quantity is used frequently to express noise metrics in datasheets such as the input voltage noise of a BJT for example and often in units of nV/sqrt(Hz) as this happens to be convenient for calculations.

We can therefore apply our Johnson Noise Equation with both sides squared and divide by sqrt(Hz) to end up with a much simpler equation of:

$$V(rms)^2/\sqrt{\Delta f} = 4k_bTR$$

If we assume room temperature of 300 Kelvin (T) and Boltzmann's constant is 1.38*10^-23 (kb) we can then calculate the noise voltages for each resistor and redraw the circuit as:

The voltage spectral densities for each resistor are then given by taking the square root of each which is 5.772nV/sqrt(Hz) and 7.048nV/sqrt(Hz) respectively.

Note: We cannot sum the voltage spectral densities together with straight addition to determine the Total Voltage Spectral Density as the noises are uncorrelated, instead what we can do is sum the power spectral densities for each resistor together using Superposition and then take the square root of the result. Or we can add them in quadrature (sum the square of each voltage spectral density together and then take the square root at the end.)

Hence for the above we have:

$$S = 3.3312*10^{-17} + 4.968*10^{-17}$$

$$= 8.2992*10^{-17} V^2/Hz$$

$$V = \sqrt{S}$$

$$= 9.110nV/\sqrt{Hz}$$

Datasheets for IC's will often quote various noise metrics which we can then use in our circuits to more accurately model our circuits with noise considered from the get go.

# How can we limit the noise observed at our ADC input?

Returning to our cliffhanging opener, how can we improve our unfortunate ADC situation? Well, from our brief noise tour above we have learnt that the noise level is proportional to both resistance and bandwidth, hence we can reduce the resistance values we present at the ADC input and also perform filtering (low pass filter for example) to reduce the bandwidth down to the signal frequencies we actually care about.

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Comment by July 11, 2024
The voltage spectral densities for each resistor are then given by taking the square root which is 5.772nV/sqrt(Hz) and 7.048nV/sqrt(Hz). From here we can sum each voltage noise source contribution using Superposition to work out the Total Voltage Spectral Density for the circuit of 12.82nV/sqrt(Hz)
This is wrong; you cannot sum noise voltages this way. They are uncorrelated and therefore sum in quadrature: (5.772)² + (7.048)² = (9.110)².

This is also why Johnson noise is proportional to the square root of resistance --- you can treat each resistor equivalently as a bunch of resistors in series, each of which has its own noise that adds in quadrature. One 100 KΩ resistor has the same noise voltage as one hundred 1 KΩ resistors in series, both of which have an equivalent noise voltage as ten times the noise voltage of a single 1 KΩ resistor.

As a side note, I´d recommend writing V_{rms} = \sqrt{4 k_b T R \Delta f}; no need to show asterisks for multiplication in MathJax.

$$V_{rms} = \sqrt{4 k_b T R \Delta f}$$

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