# Source/Load-Pull Noise Measurements at K\({}_{\text{a}}\) [PERSON]
## 2 Materials and Methods  
### Formalism of Forward and Reverse Noise Measurements  
Consider a generic linear noisy 2-port terminated in a source admittance \(Y_{S}=G_{S}+JB_{S}\) (associated with a noise temperature \(T_{S}\)) and thus presenting an output admittance \(Y_{out}=G_{out}+JB_{out}\) (its noise temperature being \(T_{out}\)). It is well known that the small-signal and noise behavior at a given frequency \(f_{0}\) of such a 2-port can be modeled as the cascade of:  
* a noisy thru (fully characterized by a noise correlation matrix), and
* a noiseless 2-port (fully characterized by a small-signal matrix),  
as depicted on the left side of Figure 1. Here the expression 'noisy thru' denotes an ideal 2-port network made up of a series noise voltage generator and a shunt noise current generator, such as the one enclosed between sections \(s\)-\(F1\) and \(s\)-\(F\) in the figure. In general, the two generators are partially correlated, which is quantified by way of a correlation admittance \(Y_{\gamma}\):  
\[\dot{i_{n}} = \gamma_{\gamma}e_{n}+\dot{i_{u}} \tag{1}\] \[\overline{e_{n}\dot{i_{u}}^{s}} = 0. \tag{2}\]  
In this framework, it is convenient to express the small-signal behavior of the noisy 2-port in the transmission (ABCD) representation. The associated form of the noise correlation matrix is as follows:  
\[\mathbf{C}=\overline{\left[ \begin{array}{c}e_{n}\\ i_{n}\end{array} \right]\cdot\left[ \begin{array}{c}e_{n}\\ i_{n}\end{array} \right]^{H}}=\overline{\left[ \begin{array}{cc}e_{n}e_{n}^ {*}&e_{n}\dot{i_{n}^{*}}\\ i_{n}e_{n}^{*}&i_{n}\dot{i_{n}^{*}}\end{array} \right]}=\left[ \begin{array}[ ]{cc}c_{11}&c_{12}\\ c_{21}&c_{22}\end{array} \right]=\left[ \begin{array}{cc}c_{11}&c_{12}\\ c_{12}^{*}&c_{22}\end{array} \right]\, \tag{3}\]  
where the \(H\) superscript denotes the Hermitian operator (transpose and conjugate). It proves useful to normalize \(\mathbf{C}\) to \(4k_{B}T_{0}B\), where \(k_{B}=1.3806488\cdot 10^{-23}\) m\({}^{2}\) kg s\({}^{-2}\) K\({}^{-1}\) is Boltzmann's constant, \(T_{0}=290\) K is the standard noise temperature and \(B\) is the equivalent noise bandwidth:  
\[\mathbf{\hat{C}}=\frac{1}{4k_{B}T_{0}B}\mathbf{C}\,\ \dot{e}_{ij}=\frac{1}{4k_{B}T_{0}B }c_{ij}. \tag{4}\]Now focus for a moment on the noisy thru (_NT_) alone. The relation between the equivalent noise temperature of the noisy thru and the elements of its (normalized) correlation matrix can be expressed as follows:  
\[\frac{T_{eq,NT}}{T_{0}}\cdot G_{S,NT}=\left|Y_{S,NT}\right|^{2}\hat{c}_{11}+2G_{S,NT}\text{Re}[\hat{c}_{12}]-2B_{S,NT}\text{Im}[\hat{c}_{12}]+\hat{c}_{22}. \tag{5}\]  
Reversing the noisy thru yields another noisy thru, differing in the orientation of the noise generator and, therefore, with opposite correlation admittance:  
\[e_{n}^{\prime} = -e_{n} \tag{6}\] \[i_{n}^{\prime} = i_{n}\] (7) \[Y_{\gamma}^{\prime} = -Y_{\gamma}\, \tag{8}\]  
where primes denote in this work the reverse mode. Thus, the correlation matrix of the reversed noisy thru is analogous to that of the original one, but with opposite terms in the secondary diagonal. Consequently, the equivalent noise temperature of the reversed noisy thru, \(T_{eq,NN}^{\prime}\), can be expressed as:  
\[\frac{T_{eq,NT}^{\prime}}{T_{0}}\cdot G_{S,NT}=\left|Y_{S,NT}\right|^{2}\hat{ c}_{11}-2G_{S,NT}\text{Re}[\hat{c}_{12}]+2B_{S,NT}\text{Im}[\hat{c}_{12}]+ \hat{c}_{22}\, \tag{9}\]  
where the elements of the correlation matrix are those of the original (i.e., forward) noisy thru, so they are not primed. From these premises, it is easily shown that a consistent set of equations can be written down for the original 2-port, associated both with forward and reverse noise measurements. In particular, in the forward case, equations in the same form as (5) can be written down for various choices of the source admittance, since cascading the noiseless 2-port behind the noisy thru does not alter the equivalent noise temperature.