Sobhy’s Method

Sobhy’s algorithm also converts the rational impedance \(Z(s)\) (Equation (81)) to the \(H\)–\(h\) continued fraction of Equation (82), but it bypasses Markov parameters, which are calculated in Khatwani’s method entirely. Instead it builds two interlaced triangular rows \(A\) and \(B\) whose first columns yield \(H_i , h_i\) directly.

Sobhy’s method is faster than Khatwani’s, but requires the same arbitrary-precision as polynomial long division.

Initialisation

Write again

(81)\[Z(s)= \frac{\alpha_0+\alpha_1s+\dots+\alpha_{N-1}s^{N-1}} {\beta_0+\beta_1s+\dots+\beta_{N-1}s^{N-1}+\beta_N s^{N}} .\]

The \(H\)–\(h\) continued fraction for this is:

(82)\[Z(s)= \cfrac{1}{ H_1 s+h_1+ \cfrac{1}{ H_2 s+h_2+\ddots+ \cfrac{1}{H_N s+h_N}}}\]

Create the calculation table (un-labelled positions are zero)

(83)\[\begin{split}\begin{pmatrix} A_{0,1}&A_{0,2}&A_{0,3}&\dots&\dots&A_{0,N+1}\\ B_{0,1}&B_{0,2}&B_{0,3}&\dots&\dots&B_{0,N+1}\\[2pt] A_{1,1}&A_{1,2}&A_{1,3}&\dots&A_{1,N}&\\ B_{1,1}&B_{1,2}&B_{1,3}&\dots&B_{1,N}&\\ \vdots &\vdots & & & &\\ A_{N,1}\\ B_{N,1} \end{pmatrix}\!.\end{split}\]

Seed rows

(84)\[A_{0,j}=B_{0,j}= \beta_{N+1-j},\qquad A_{1,j}= \alpha_{N-j}.\]

Iterative filling rule

For \(i = 1,2,\dots, N\) and \(j = 1,2,\dots\)

(85)\[A_{i,j}=B_{\,i-1,j+1}-\frac{B_{\,i-1,1}}{A_{\,i-1,1}}\;A_{\,i-1,j+1},\]

(86)\[B_{i,j}=A_{\,i-1,j+1}-\frac{A_{\,i-1,1}}{A_{\,i,1}}\;A_{\,i,j+1}.\]

Proceed until every element needed for the first columns \(A_{i,1},\,B_{i,1}\) (i = 1…N) has been computed.

Extracting H and h

Simply

(87)\[H_i=\frac{B_{i-1,1}}{A_{i,1}},\qquad h_i=\frac{B_{i,1}} {A_{i,1}},\qquad i=1,\dots,N.\]

These coefficients give the \(H\)–\(h\) continued fraction (Equation (82)); convert to J- and S-fractions with the formulas collected on the J-fraction Route to the Cauer Ladder page to obtain the Cauer ladder.