Definition

In microwave circuits described by scattering parameters, it is advantageous to regard noise as noise waves [5]. The noise characteristics of an n-port is then defined completely by one outgoing noise wave $\underline{b}_{noise,n}$ at each port (see 2-port example in fig. 2.1) and the correlation between these noise sources. Therefore, mathematically, you can characterize a noisy n-port by its $n\times n$ scattering matrix $(\underline{S})$ and its $n\times n$ noise wave correlation matrix $(\underline{C})$.

$$\begin{split}(\underline{C}) = \begin{pmatrix}\overline{\unde... ..._{n2} & \ldots & \underline{c}_{nn}\\ \end{pmatrix} \end{split}$$ (2.1)

Where $\overline{x}$ is the time average of $x$ and $\underline{x}^*$ is the conjugate complex of $\underline{x}$. Noise correlation matrices are hermitian matrices because the following equations hold.

$$\left(\underline{c}_{nn}\right) = \left(\overline{\left\vert b_{noise,n}\right\vert^2}\right) = 0$$ (2.2)

$$\underline{c}_{nm} = \underline{c}_{mn}^*$$ (2.3)

Where Im$(\underline{x})$ is the imaginary part of $\underline{x}$ and $\vert\underline{x}\vert$ is the magnitude of $\underline{x}$.

Figure 2.1: signal flow graph of a noisy 2-port