For a given S-matrix ${\displaystyle \mathbf {S} (p)}$  of degree ${\displaystyle d}$ ;

${\displaystyle \mathbf {S} (p)={\begin{bmatrix}s_{11}&s_{12}\\s_{21}&s_{22}\end{bmatrix}}}$ 

where,

p is the complex frequency variable and may be replaced by ${\displaystyle i\omega }$  in the case of steady state sine wave signals, that is, where only a Fourier analysis is required

d will equate to the number of elements (inductors and capacitors) in the network, if such network exists.

Belevitch's theorem states that, ${\displaystyle \scriptstyle \mathbf {S} (p)}$  represents a lossless rational network if and only if,^[1]

${\displaystyle \mathbf {S} (p)={\frac {1}{g(p)}}{\begin{bmatrix}h(p)&f(p)\\\pm f(-p)&\mp h(-p)\end{bmatrix}}}$ 

where,

${\displaystyle f(p)}$ , ${\displaystyle g(p)}$  and ${\displaystyle h(p)}$  are real polynomials

${\displaystyle g(p)}$  is a strict Hurwitz polynomial of degree not exceeding ${\displaystyle d}$ 

${\displaystyle g(p)g(-p)=f(p)f(-p)+h(p)h(-p)}$  for all ${\displaystyle \scriptstyle p\,\in \,\mathbb {C} }$ .

• Belevitch, Vitold Classical Network Theory, San Francisco: Holden-Day, 1968 OCLC 413916.
• Rockmore, Daniel Nahum; Healy, Dennis M. Modern Signal Processing, Cambridge: Cambridge University Press, 2004 ISBN 0-521-82706-X.