The Y-parameters of a multi-port component defined by its S-parameters required for a small signal AC analysis can be obtained by converting the S-parameters into Y-parameters.
In order to extend a -port to have a S-parameter device with ports assuming that the original reference port had a reflection coefficient the new S-parameters are according to T. O. Grosch and L. A. Carpenter :
If the reference port has been ground potential, then simply folds to -1. The reverse transformation by connecting a termination with a reflection coefficient of to the th port writes as follows.
With the S-parameter transformation done the -port noise wave correlation matrix is
whence denotes the equivalent noise temperature of the original reference port and the operator indicates the transposed conjugate matrix (also called adjoint or adjugate).
The reverse transformation can be written as