Editing Johnson–Nyquist noise (section)

=== Multiport electrical networks ===
[[Richard Q. Twiss]] extended Nyquist's formulas to multi-[[Port (circuit theory)|port]] passive electrical networks, including non-reciprocal devices such as [[circulator]]s and [[Isolator (microwave)|isolator]]s.<ref>{{Cite journal | doi = 10.1063/1.1722048| title = Nyquist's and Thevenin's Theorems Generalized for Nonreciprocal Linear Networks| journal = Journal of Applied Physics| volume = 26| issue = 5| pages = 599–602| year = 1955| last1 = Twiss | first1 = R. Q.| bibcode = 1955JAP....26..599T}}</ref> 
Thermal noise appears at every port, and can be described as random series voltage sources in series with each port. The random voltages at different ports may be correlated, and their amplitudes and correlations are fully described by a set of [[cross-spectral density]] functions relating the different noise voltages,
: <math>S_{v_m v_n}(f) = 2 k_\text{B} T \eta(f) (Z_{mn}(f) + Z_{nm}(f)^*)</math>
where the <math>Z_{mn}</math> are the elements of the [[impedance matrix]] <math>\mathbf{Z}</math>.
Again, an alternative description of the noise is instead in terms of parallel current sources applied at each port. Their cross-spectral density is given by
: <math>S_{i_m i_n}(f) = 2 k_\text{B} T \eta(f) (Y_{mn}(f) + Y_{nm}(f)^*)</math>
where <math>\mathbf{Y} = \mathbf{Z}^{-1}</math> is the [[Admittance parameters|admittance matrix]].