Noise Parameters

Having the noise wave correlation matrix, one can easily compute the noise parameters [5]. The following equations calculate them with regard to port 1 (input) and port 2 (output). (If one uses an n-port and want to calculate the noise parameters regarding to other ports, one has to replace the index numbers of S- and c-parameters accordingly. I.e. replace "1" with the number of the input port and "2" with the number of the output port.)

Noise figure:

$$F = 1 + \frac{\underline{c}_{22}}{k\cdot T_0\cdot \vert\underline{S}_{21}\vert^2}$$ (2.4)

$$NF\,[ ]\, = 10\cdot\lg F$$ (2.5)

Optimal source reflection coefficient (normalized according to the input port impedance):

$$\Gamma_{opt} = \eta_2\cdot\left( 1-\sqrt{1-\frac{1}{\vert\eta_2\vert^2}} \right)$$ (2.6)

With

$$\eta_1 = \underline{c}_{11}\cdot \vert\underline{S}_{21}\vert^2 - 2\cdot \left(\underline{c}_{12}\cdot\underline{S}_{21}\cdot\underline{S}_{11}^*\right) + \underline{c}_{22}\cdot\vert\underline{S}_{11}\vert^2$$ (2.7)

$$\eta_2 = \frac{1}{2}\cdot\frac{\underline{c}_{22} + \eta_1} {\underline{c}_{22}\cdot\underline{S}_{11} - \underline{c}_{12}\cdot\underline{S}_{21}}$$ (2.8)

Minimum noise figure:

$$F_{min} = 1 + \frac{\underline{c}_{22} - \eta_1\cdot \vert\Gamma_... ...cdot T_0\cdot \vert\underline{S}_{21}\vert^2\cdot (1+\vert\Gamma_{opt}\vert^2)}$$ (2.9)

$$NF_{min} = 10\cdot \lg F_{min}$$ (2.10)

Equivalent noise resistance:

$$R_n = \frac{Z_{port,in}}{4\cdot k\cdot T_0}\cdot \left( \underlin... ...left\vert \frac{1+\underline{S}_{11}}{\underline{S}_{21}} \right\vert^2 \right)$$ (2.11)

With	$Z_{port,in}$ internal impedance of input port
	Boltzmann constant $k = 1.380658\cdot 10^{-23}$ J/K
	standard temperature $T_0 = 290$K

Calculating the noise wave correlation coefficients from the noise parameters is straightforward as well.

$$\underline{c}_{11} = k\cdot T_{min}\cdot (\vert S_{11}\vert^2-1) + K_x\cdot \vert 1-S_{11}\cdot\Gamma_{opt}\vert^2$$ (2.12)

$$\underline{c}_{22} = \vert S_{21}\vert^2\cdot\left( k\cdot T_{min} + K_x\cdot\vert\Gamma_{opt}\vert^2 \right)$$ (2.13)

$$\underline{c}_{12} = \underline{c}_{21}^* = -S_{21}^*\cdot\Gamma_{opt}^*\cdot K_x + \frac{S_{11}}{S_{21}}\cdot\underline{c}_{22}$$ (2.14)

with

$$K_x = \frac{4\cdot k\cdot T_0\cdot R_n}{Z_0\cdot\vert 1+\Gamma_{opt}\vert^2}$$ (2.15)

Once having the noise parameters, one can calculate the noise figure for every source admittance $Y_S=G_S+j\cdot B_s$, source impedance $Z_S=R_S+j\cdot X_s$, or source reflection coefficient $r_S$.

$$F = \frac{SNR_{in}}{SNR_{out}} = \frac{T_{equi}}{T_0} + 1$$ (2.16)

$$= F_{min} + \frac{G_n}{R_S}\cdot\left( (R_S-R_{opt})^2 + (X_S-X_{opt})^2 \right)$$ (2.17)

$$= F_{min} + \frac{G_n}{R_S}\cdot\left\vert \underline{Z}_S - \underline{Z}_{opt} \right\vert ^2$$ (2.18)

$$= F_{min} + \frac{R_n}{G_S}\cdot\left( (G_S-G_{opt})^2 + (B_S-B_{opt})^2 \right)$$ (2.19)

$$= F_{min} + \frac{R_n}{G_S}\cdot\left\vert \underline{Y}_S - \underline{Y}_{opt} \right\vert ^2$$ (2.20)

$$= F_{min} + 4\cdot\frac{R_n}{Z_0}\cdot\frac{\left\vert \underline... ...ine{r}_S\vert^2\right)\cdot\left\vert 1+\underline{\Gamma}_{opt}\right\vert ^2}$$ (2.21)

Where $SNR_{in}$ and $SNR_{out}$ are the signal to noise ratios at the input and output, respectively, $T_{equi}$ is the equivalent (input) noise temperature. Note that $G_n$ does not equal $1/R_n$.

All curves with constant noise figures are circles (in all planes, i.e. impedance, admittance and reflection coefficient). A circle in the reflection coefficient plane has the following parameters.

center point:

$$\underline{r}_{center} = \frac{\Gamma_{opt}}{1+N}$$ (2.22)

radius:

$$R = \frac{\sqrt{N^2 + N\cdot(1-\vert\Gamma_{opt}\vert^2)}}{1+N}$$ (2.23)

with

$$N = \frac{Z_0}{4\cdot R_n}\cdot (F-F_{min})\cdot \vert 1+\Gamma_{opt}\vert^2$$ (2.24)