Hello, All,
Along with our current work on Touchstone format, I'd like to separately
consider another - analytical, alternative - representation of
frequency-dependent parameters, or at least to plan such consideration
for the nearest future, when we finish with Touchstone specifications.
First of all, let me say that we have two different ways of
frequency-domain representation for S/Y/Z (etc.) matrices: sampled and
analytical.
1. Sampled representation is basically our Touchstone-like formats,
where dependence is given by a collection of its matrix values at
several frequency points. Besides Touchstone, there are some other less
popular ways of sampled representation. In all cases: there dependence
is given on a finite number of points and has lower/upper frequency
bounds. Such dependences may pose problems if we need to evaluate the
function at any given frequency, since then we may need interpolation or
extrapolation, if the desired frequency lies outside given bounds. With
sampled dependences, we may have intrinsic problems with "causality" and
"passivity".
2. Analytical representation however always assumes that our dependence
can be given by some kind of analytical expression or formula. Examples:
pole/zero representation, partial fraction expansion, representation by
means of polynomials, by Laplace elements, and ultimately, by linear
equivalent circuits, that could be unambiguously converted into
analytical expressions, or at least evaluated at any given frequency,
without lower/upper bound and with no interpolation. Such
representations, if consistent, are causal by design. Still, they do not
guarantee 'passivity'. Also, analytical representations almost always
are very concise if compared to sampled ones. (Typically, the file size
ratio is between 1 to 3 orders.) Another advantage: analytical
representations are more suited for frequency and time-domain analysis
because they do not require interpolation and in part of time domain
analysis unambiguously provide time domain responses, that allows either
direct numerical integration (as in case of equivalent circuit) or very
efficient recursive convolution, semi-analytical procedure that is by
orders more efficient than convolution with IFFT'ed sampled dependence
and also is way more accurate.
3. Comparing sampled and analytical representations, we can easily see
their advantages and weaknesses. Sampled dependence is represented by
mere tables of numbers that makes reading/parsing extremely fast and
simple. The tables can be generated from simulation or measurements, no
'tricky' transformations are involved. Analytically defined dependences
sometimes require some extra language constructs of even circuit
description. Some efforts are needed to create 'analytical'
representation from sampled dependence. Because of that, once created,
such analytical model should be re-used (and hence we need to find a way
of storing it).
4. I oppose the idea of combining sampled and analytical dependences
into a single data set. For example, it does not look appealing to me to
combine S11 and S22 given by samples with S12 and S21 defined by any
type of analytical representation. This could be a valid intermediate
step in model creation, but it does not work well as a final standard.
Main reason: by allowing so, we rather add up together disadvantages
from both sampled and analytical world and multiply complexity and
confusion.
5. Hence, it seems reasonable that we need a separate format to
represent analytical dependences. It also seems reasonable that the way
we choose for analytical representation should be identical for all
matrix components (e.g. we should not mix equivalent circuits for
S11/S22 with tables of poles/residues for S12 and S21 or other similar
way). We simply don't need using many ways of analytical representations
because they are easily mutually convertible. I also think that ideally,
we should avoid using linguistic (e.g. HSPICE or others') constructs, to
completely avoid language specifics and their limitations (e.g. on
largest allowed complexity of Laplace construct) and also to avoid
introducing any concept of 'nodes' and 'branches'. We don't need to
introduce the 'structure' or 'topology' to the matter that is purely
numerical: we are dealing with frequency-dependent matrix.
6. I'd like to propose one format we can use for analytical
representation of any causal frequency dependences. (See attached
document). The advantages are:
- all we need is table of numbers, with as many lines (per matrix
component) as we need. Each line always contains 4 numbers sufficient to
represent a summand in partial fraction expansion (see formula 1). If
the pole in the fraction is real, two of four numbers are zero.
- there is an easy and unambiguous way to construct analytical
dependences from those numbers.
- analytical dependence is given as a sum of primitive components,
making it extremely easy to operate in time domain analysis. Or, if
needed, convert it into equivalent circuit or other analytical form, or
simply re-sample in any desirable way.
- no simulator-specific constructs are used. To construct the
dependence, we need only a simple code that reads numbers from such PLS
tables.
- this format was extensively tested during at least 6 years in many
S/Y/Z parameter applications. We found it very convenient, concise and
suitable for representation of matrices with very large number of ports
(e.g. 226x226) where it allowed huge disk space and simulation time
reduction when compared to conventional simulators (compare: 8 hours and
5 minutes single simulation run time for equivalent circuit and this
type of representation, same complexity). The reason: we avoid parsing
time and many extra nodes/ branches needed when using equivalent
circuit. More details in: DesignCon 2006,
http://www.altera.com.cn/literature/cp/cp-ftdsimltn.pdf.
- there were no attempts made to support sparsity with this format, but
this is easy to add. I think it's almost evident how we can add
sparsity. Some of ideas we have for Touchstone format in that respect,
can be completely reused.
- similarly, we can borrow much of our constructs for mixed mode and
combined single and mixed mode representation as introduced in
Touchstone 2.0 format. Interestingly enough, even matrix transformations
needed between mixed-standard mode representations, are fully applicable
to proposed analytical representation (see again expression (1) in
attached document), since all we need is multiplication and summing up
such components. However, note that these transformations are NOT
directly applicable in case of pole-zero, Laplace of equivalent circuit
representation.
Please review the documents and share your thoughts.
Thanks,
Vladimir
