Where Sin,(Nin) and Sout (Nout) are the signal (noise) levels at the input and output of the device, respectively, and Na is the noise added by the device itself, G is the gain of the device, B is the system bandwidth and k is Boltzman's constant (1.38 x 10-23 joule/kelvin). Noise figure is of particular interest to receiver designers as the the minimum detectable signal is dependant on the Noise Figure as follows:
This approximation assumes the background or ambient (= kTB ) noise is that due to a passive device held at T = 290K, and the Required SNRdB represents the minimum signal to noise ratio for acceptable system performance.
Noise parameters describe how the noise figure varies with source reflection coefficient Γs. Consider Figure 1. There are several formulations for noise figure in terms of noise parameters. One of the popular forms is summarized in (Eq 4).
4 Noise Parameters:
Rn= Equivalent Noise Resistance
FMin= Minimum Noise Figure
|Γopt| : magnitude of optimum noise reflection coefficient
∠Γopt : phase of optimum noise reflection and
Zo = Characteristic impedance (50 Ω)
The above equation provides a description of noise figure in terms of four (4) noise parameters and the source impedance. One common parameter is the minimum noise figure Fmin which will be achieved at some specific optimum (complex) reflection coefficient (Γopt). So in addition to Fmin, two of the other parameters magnitude and angle Γopt, with the fourth parameter being the equivalent noise resistance Rn It should also be noted that there are other noise parameter formulations in addition to those listed in (Eq 4).
One of the common application for noise parameters is for low noise amplifier (LNA) design. Typically a LNA is used at the front end of a receiver to improve the noise figure of the receiver, or essentially boost the signal while adding a low amount of noise to the signal. In addition to its noise figure the gain of the LNA (and correspondingly the transistors used to make up the LNA) is also important. To better understand this, the following equation can be used to calculate the total noise figure of a cascade connection of three different two port devices with gains Gi and noise figure Fi (i= 1, 2, 3).
If an LNA with high Gain G1 and low noise figure F1 is the first device then the system noise figure FTOT can remain low even if the second and third devices have a much higher noise figure. The gain used in most noise calculations, including the cascade noise figure equation above is the available gain which can be expressed in terms of S-parameters as follows:
The associated gain assumes that the load port of Fig. 1 is terminated in a conjugate match for a given source reflection coefficient Γs . The associated gain is often tabulated along with noise parameters and is simply the available gain from (Eq 7) for the particular case of Γs = Gsopt.
Another set of parameters often plotted are the maximum stable gain, MSG, and maximum available gain, MAG. Often amplifiers or transistors are unconditionally stable over a certain frequency range and conditionally stable at other frequencies.
For frequencies where the device is potentially unstable (with stability factor K < 1), the maximum stable gain is defined as the highest realizable gain with passive terminations, after the device is stabilized with cascaded resistance to border line stability; that is, to bring about the condition K = 1. MSG is given by:
Maximum available gain at frequencies where K>1 (unconditionally stable) is given by:
Hence, MSG and MAG numbers in decibels give the amplifier designer a measure of the maximum gain realizable through impedance matching of the amplifier or transistor. Of course the conditions for matching the input for maximum gain and minimum noise figure may be conflicting and a trade-off between these two may be required. While outside the scope of this note, the plotting of noise figure circles and available gain circles can often be used to aid the designer in choosing the best compromise in matching impedance taking noise and gain into consideration .
A basic test configuration used to perform combined noise parameter and S-parameter testing is shown in Figure 2. The network analyzer is needed for performing S-parameters of the DUT, which are required for design analysis along with the noise parameters. The network analyzer is also needed to make measurements that are required for calibration of the noise parameter test system.
The test system of Figure 2 is referred to as an "NP5" system and uses a hardware setup and measurement method originally developed by Adamian  and commercialized by ATN Microwave. The system consists of a noise figure measurement system (e.g. HP 8971C/HP8970B combination or alternative), a network analyzer (e.g. HP8510C or alternative), a mismatched noise source (MNS), and a remote receiver module (RRM). The MNS and RRM each contain a switch that is used to select either the S-parameter measurement mode or the noise parameter measurement mode. They each also contain a bias Tee for applying bias to the input and output of the device under test (DUT). The MNS is generally a solid-state tuner capable of presenting multiple different values of Γs (see Fig. 1) to the DUT, along with the ability to have a "50 ohm" thru-state that allows the noise source to be connected to the DUT through essentially a transmission line. With the RRM switch in the noise measurement position, the RRM includes a low-noise amplifier in the path to the noise receiver to improve the measurement receiver noise figure.
In the S-parameter measurement mode the system is calibrated using thru-reflect-line (TRL) or alternative high accuracy calibration approaches. To calibrate and operate the noise parameter measurement system, the Maury ATS software is used to perform a series of steps that may be summarized as follows:
In the following we will use data taken on samples from the Mini-Circuits SAV amplifier series. Figure 3 shows pictures of a device sample as mounted in a coplanar waveguide test fixture setup for RF wafer probe testing. TRL standards, fabricated with the same ground-signalground test interface were used along with the NIST Multical method  to establish measurement reference planes at the locations indicated in Fig. 3. These same reference planes were established for noise parameter measurements using the Maury ATS software. Fig. 4 and Fig. 5 show example S-parameter and noise parameter measurement results made on 3 test samples.