Noise figure (NF) is a term typically used to help quantify a receiver's sensitivity performance for RF engineers. It can be used when designing the receive system in a variety of applications, such as military software-defined radio (SDR), test and measurement equipment (including spectrum analyzers), and medical equipment. Additionally, it can help to compare the performance of individual signal chain components. In this article, we discuss NF and how it can be calculated in a single-ended-to-differential (SE-DE) fully differential amplifier (FDA), given basic circuit element values.

Many analog engineers are familiar with noise, but not necessarily NF. We begin our discussion with thermal noise, better known as *white noise* . This noise is due to random motion of electrons and is directly proportional to temperature.

K is Boltzman's constant (1.38 x 10^{–23} Joules/°K), T is temperature in K, and B is bandwidth in Hz. This noise is also independent of a given system impedance. For room temperature (290°K), we end up with –174 dBm/Hz. Any time we add an active element to a system, we impact the noise of this system beyond the thermal noise. If there is no impact on the system, the device has a noise factor of 1. This means the signal-to-noise ratio (SNR) at the input of the device is equivalent to the SNR at the device output.

thus creating Fn=1. Noise factor is dimensionless. When converted to dB, it is referred to as *noise figure* . The equation for NF is:

For a noise factor of 1, NF is 0 dBm. This means that there is no impairment on the system SNR. Source impedance has an impact on the system's NF. Noise power is directly proportional to the absolute temperature of the resistor.

We assume capacitors and inductors to be ideally noiseless. Therefore, only the source resistance part is to be referenced for overall noise impact. In order to have no impact on the thermal noise power, or having a system with the noise floor equal to the thermal noise kTB, the load resistance must be equivalent to the source resistance. This is another expression of the SNR ratio equation where, in a system with NF = 0 dBm,

where

Substituting (e_{NO} )^{2} to e_{(NZLOAD) } ^{2} equal load and source resistance minimally impact the system. Using Equation 5, NF is impacted when changing a component's input source resistance. When reviewing the specification of NF in datasheets, always be aware of the Rs used as reference with the NF Equation 5 in mind.

Generally, if a datasheet (for example, for an amplifier) references NF Rs=100 Ohm, while a datasheet for a similar device notes NF Rs=50 Ohms (assuming the same gain), we have an approximate 3 dB difference in NF. From this, we can assume (if input noise is not provided) that the device referencing 100 Ohms actually has lower input noise than one using 200 Ohms — if NF and gains are equal.

In our example, let's calculate NF using the LMH5401, an 8GHz low distortion, voltage feedback (VFB) FDA.

**Figure 1**

In Figure 1, we can calculate NF in a 50 Ohm system. Using Equations 7–14, we can calculate the FDA's NF:

Today we learned that, when using an SE-DE FDA, the NF has a gain of 4 V/V (12 dB) single-ended to differential. We also learned that we have an overall measured noise figure, referred to as 50 Ohms of 9.6 dB, that correlates closely with the calculated value using the provided calculations.

Join us next time, when we will discuss using active filters in clocking devices and synthesizers.

**Editor's note** : In writing this article, the author, Carissa Sipp, used the following as support materials: Calculating Noise Figure in Op Amps, AN-1719 Noise Figure Analysis Fully Differential Amplifier, LMH5401 datasheet, and LMH3401 datasheet.

Noise is everywhere. This is the reason why engineer should deal with it and design circuit to minimize it. All equation could be reused to figure out the total noise in the circuit. I am wondering how noise power induced by these equations is accurate. It could be better to capture the actual noise in the circuit.

Under formula{3} the author wrote that “NF is in dBm” which is wrong.

I am used to work with NF in a 50-Ohm system, so I cannot agree with statements that there is a NF “gain” of 3 db if the impedance changed from 50 to 100 Ohms. NF relates to power ratios, not voltages.

The whole paper contributes rather to a confusion due to these errors. I would like to see a really responsible author to write a correct paper on noise figure in low-frequency symmetrical systems.

I thought this was a great article. Finally a little more information outside of the RF world on how to work with FDAs and figure out how to put their NF in a line up! The noise contributions and equations are very helpful as I have not see such detail before nor an explanation as such. This greatly helps when looking at my link budget and deciphering if I can use FDAs with external configurations or I need to go with fixed gain depending on my gain needs- at times fixed gains do not also meet my specifications.

3dB is a standard “improvement” for the ADC calculation of noise figure- this is due to the full scale power being intregal in the equation and from the analog work Rs matters here—going from 50 to 100 impacts the NF by 3dB simply put. This is what the author (I believe) is trying to help RF engineers understand. Also, a very simplified equation for (not nearly as informative as this explanation) NF for an amplifier shows the improvement the mentioned. This is the case even if it doesn't necessarily make 'sense' to those who work in the world of RF (fixed impedances thus spec in power).

The noise figure of a block or system is generally based on having a matched source and input impedance such that all available noise power (kTB) is absorbed into the input. It is odd to specify a noise figure reference impedance, since if matched the available noise power is always kTB.

@DaeJ: How much of a difference do you think would be between the noise in the equation and the noise in the actual circuit? In other words, are you saying the equation would be impractical to use to calculate noise?

Thank you very much for your feedback. You are correct when referring to the NF as it should always be dB as it is just a ratio.

To calculate the NF of an amplifier calculating the total input-referred noise is necessary. As shown, to do this you use input voltage and current noise specs. The noise contribution of external resistors (including the feedback, gain and source resistors) based on the circuit topology (this will change depending on amplifier configurations, source impedances, etc.)must be calculated. The 'dependancy' on Rs is the nature of amplifiers as they are not 'fixed' to a specific impedance. This source impedance 'delivers' a voltage noise to the amplifier.

For example, if you use a non-inverting amplifier configuration, the Rs generate a noise voltage of sqrt(4kTRs). This is delivered to the amplifier divided by the Rs and Rt resistors. Rt is your termination resistor that is used to 'match' to the Rs. Thus the overall noise figure calculation will be determined using the source impedance and dependant on this value.

The overall goal is to help engineers when looking at an amplifier datasheet. Sometimes NF is referenced to a particular Rs in a particular configuration. If the amplifiers being compared are not referenced to the same Rs and configuration, one amplifier may have a lower overall input noise spec, despite having 'similar' NF. Knowing this helps engineers make a better choice of amplifier.

Moving from a RF world of 50 Ohms and fixed impedances, allowing for specifications in power to an analog world of voltage specifications for such devices as amplifiers and ADCs where load impedances are not typically fixed, the NF calculation changes dependant on configuration. The 3 dB is a helpful rule of thumb used for quick comparison. This could be higher or lower depending on the actual input voltage noise spec of the amplifier and holds true to ADCs due to the nature of the calculation. The main take away is that moving to different Rs for components like amplifiers and ADCs impacts the overall calculations for noise and thus NF. When comparing these types of components, only compare NF referenced to the same Rs.

I would like to thank the author for another clarifcation attempt. I still think there is confusion about NF in low-frequency devices. Due to non-standard impedances and other reasons, “voltage” noise has to be combined with “current noise while the basic NF definition stems from noise POWER.

I think myself as fortunate as had always treated NF in 50-Ohm or similar common impedance, and using noise temperature as the simplest noise-power approach in RF and microwave circuits. I remain confused (and others too according to other responses) when it comes to low-frequency circuits as described.

Add 1/f noise, and I am easily lost.

Just a quick note, using these equations and the configuration shown produces a NF that is just slightly higher than the actual bench measurement.

In a link budget of such these calculations are a helpful tool in a cascaded system. One can see the immediate impact (however estimated due to the nature of calculations themselves) on the SNR (and NF) on the system once the noise contribution of the amplifier's configuration is known.

When investigating with the noise, noise could be defined as white noise or random noise or something else. If engineering tool enables to separate the signal from noise, engineer could figure out what causes the noise source.