Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Reflection-Free Port Coefficients

For an $ N$ -port adaptor, with port wave-impedances $ R_i$ , $ i=1,2,\ldots,N$ , let's arbitrarily designate port $ N$ as the reflection-free port (the one on top). It is convenient to define the port conductances $ G_i\isdeftext 1/R_i$ . To suppress reflection on port $ N$ , we need, for a parallel adaptor,

\begin{eqnarray*}
R_N &=& R_1 \;\Vert\; R_2 \;\Vert\; \cdots \;\Vert\; R_{N-1} \;\Leftrightarrow\\ [5pt]
G_N &=& G_1 + G_2 + \cdots + G_{N-1}
\end{eqnarray*}

and, for a series adaptor,

\begin{eqnarray*}
R_N &=& R_1 + R_2 + \cdots + R_{N-1}.
\end{eqnarray*}

Recall the alpha parameters for an $ N$ -port series scattering junction, derived from the physical constraints that the velocities be equal and the forces sum to zero at the (series) junction:

$\displaystyle \alpha_i \isdefs \frac{2R_i}{R_1+R_2+\cdots+R_N} \eqsp \zbox{\frac{R_i}{R_N}}
$

when port $ N$ is reflection free.

Since $ \sum_{i=1}^N \alpha_i=2$ , we have $ \zbox{\alpha_N=1}$ and $ \zbox{\sum_{i=1}^{N-1} \alpha_i=1}$ .


Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Download WaveDigitalFilters.pdf
Download WaveDigitalFilters_2up.pdf
Download WaveDigitalFilters_4up.pdf

``Wave Digital Filters'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2017-06-05 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]