## Thermal Noise in Communication and Optical Systems

By ZM Peterson • Jan 9, 2019Optical systems, communication systems, and other systems that require sensitive electronic and optical measurements can suffer from a variety of noise sources. The challenge in dealing with these noise sources is to identify them from time domain and/or frequency domain measurements.

Extrinsic noise sources are easy to identify as the frequency content of noise in a sensor or receiver will be quite similar to that of the noise source. Intrinsic noise sources like thermal noise, 1/f noise, and shot noise will set a lower limit for noise in your system. Here’s what you need to understand about thermal noise in communication and in optical systems when designing for highly sensitive measurements.

## Extrinsic Noise vs. Thermal Noise in Communication Systems

There are some important noise sources that appear in any communication system. The major noise problem that was overcome in copper wiring for telecom systems was crosstalk. This was solved using twisted pair cable with duplexing, which provides a low loop inductance path for signal transmission that inhibits inductive crosstalk. Because these systems do not operate at high enough frequency for capacitive crosstalk to be noticeable, the issue of crosstalk in copper-wired telephone systems is largely solved with twisted pair cabling. The same applies in Ethernet cabling.

In electronic systems on a PCB, crosstalk is always a signal integrity consideration at high speed and high frequency, and this particular noise problem cannot be solved by simply twisting conductors around each other. Other signal integrity and power integrity problems in PCBs and other systems can be challenging as device speeds and frequencies increase.

Something like ringing due to switching, NEXT/FEXT, and phase noise/jitter are usually easy to identify in the time domain or frequency domain. Some noise sources like phase noise (for analog signals), timing jitter (for digital signals), and intense reflective/transient ringing are seen in the time domain in an oscilloscope trace. Similarly, the stair-step response with ringing on a transmission line can also be easily seen in an oscilloscope trace. However, looking at a noisy oscilloscope trace may hide some noise sources that can only be distinguished in the frequency domain.

*As an example, thermal noise in a resistor produces voltage and current fluctuations. Conceptually, this is tantamount to adding a small voltage source in series with the resistor.*

Extrinsic noise sources, such as radiated/conducted EMI tend to appear as a series of peaks with varying bandwidth. The frequency content of noise induced from EMI will resemble that of the source, unless noise is amplified in a strongly nonlinear portion of the system. Thermal noise is quite different. Thermal noise is a form of intrinsic noise in that it does not arise from some external component or system. In other words, it occurs because all systems have temperature. There are other intrinsic noise sources that are unrelated to thermal noise, but thermal noise in communication systems and in other electronics systems has a unique frequency content and time-dependent behavior.

## What is Thermal Noise?

Thermal noise is also known as white noise. It has the following characteristics:

**Fluctuations are Gaussian distributed in time.**In other words, the size of a fluctuation between two points in time are Gaussian distributed.**Fluctuations in time are uncorrelated.**If you calculate the autocorrelation of thermal noise, you should find that the autocorrelation function is a delta function.**Flat bandwidth.**The frequency content of thermal noise is uniformly distributed throughout the frequency domain. In other words, the Fourier transform of thermal noise is a straight line with slope of 0.

Thermal noise produces fluctuations about some average voltage that are Gaussian distributed in the time domain. These fluctuations in the time domain are shown in the image below.

*Signal fluctuations due to thermal noise in the time domain.*

These fluctuations from thermal noise in communication systems interfere with optical sensors, electro-optical elements, and every other piece of circuitry in a system. These fluctuations arise simply because some part of the system has temperature that is greater than 0 K. Voltage and current produced by thermal noise fluctuations can be derived from statistical mechanicals for an arbitrary impedance or admittance in a particular circuit element. If you know the impedance and admittance as a function of frequency, then you can easily calculate the RMS voltage and current fluctuations from the Bose-Einstein distribution.

The integrals in the these equations can be used to define the thermal noise bandwidth for any system. For a Thevenin equivalent resistor, this reduces to the familiar formula for the bandwidth of thermal noise in communication systems. This tells you everything you would expect to see from thermal noise in an optical sensor, electro-optic element, or other component in a communication system.

*RMS voltage and current fluctuations from thermal noise in electronic systems.*

*RMS voltage fluctuations across a Thevenin equivalent resistor.*

## Other Noise Sources: Shot Noise and 1/f Noise

The other two prominent noise sources are shot noise and 1/f noise. Shot noise arises due to the quantized nature of electrons and photons, which creates discrete fluctuations at low current. At sufficiently high current, these fluctuations occur so rapidly that they can generally not be measured electrically.

1/f noise in any system (electronic, optical, biological, etc.) is a complex topic and encompasses a broad range of possible phenomena. In fact, the exact nature of 1/f noise sources is still up for debate, and there is no universal set of equations that describes the origin of 1/f for all systems. What is generally agreed is that 1/f noise results from stochastic processes, which then produce a noise source that is inversely proportional to some power of frequency. In other words, the power spectral density of 1/f noise is inversely proportional to some power of frequency, and the exact power depends on the nature of the system under consideration.

Each type of noise source can be distinguished in the frequency domain. Eventually, 1/f noise dies out and cannot be seen above thermal noise. An example of the transition from 1/f noise to thermal noise and shot noise in the frequency domain is shown below.

*Transition from 1/f noise to thermal noise and shot noise.*

**Distinguishing between these intrinsic noise sources and other noise sources doesn’t require complex simulation tools. Instead, it requires precision measurements and analysis from an experienced design firm. At NWES, we provide cutting edge PCB design services and technology research services for innovative technology companies. Contact us today to see how we can help you succeed.**