Hello DSP folks,

This question has been bothering me for a long time since I took my
Microwave Engineering course:

Suppose we have a linear system H(z) we can easily find its poles and
zeros and perform stability analyis of the system. Does such a thing
exist for general network analysis using the S-parameters ? I know
that stability analysis can be done at every frequency, especially if
one is designing amplifiers. There are stability criteria that are
valid only at specific frequencies.

A lot of times we need to do DSP work with discretized S-paramters
{S(f_1),...,S(f_N)}, and hence the question arises ... does the
Z-transform mean anything for these S-parameters ? S_21(f) of a
network is same as the H(f) of the system. Does a general treatment of
the S-parameters using Z-transform exist ?

Thanks in advance,
LL.
---------------------------------------------------------------
PS: I searched Google and came up with nothing.

l...@yahoo.com (Lord Labakudas) wrote in message
news:<d...@posting.google.com>...
> Hello DSP folks,
> 
> This question has been bothering me for a long time since I took my
> Microwave Engineering course:
> 
> Suppose we have a linear system H(z) we can easily find its poles and
> zeros and perform stability analyis of the system. Does such a thing
> exist for general network analysis using the S-parameters ? I know
> that stability analysis can be done at every frequency, especially if
> one is designing amplifiers. There are stability criteria that are
> valid only at specific frequencies.
> 
> A lot of times we need to do DSP work with discretized S-paramters
> {S(f_1),...,S(f_N)}, and hence the question arises ... does the
> Z-transform mean anything for these S-parameters ? S_21(f) of a
> network is same as the H(f) of the system. Does a general treatment of
> the S-parameters using Z-transform exist ?

My understanding is that S-parameters(S-matrix) represent analog(RF)
network where S_22 represents  return loss and indirectly talk about
network stabilty through VSWR(since VSWR can be related with
reflection coeff.).

I see a possibility of Laplace transform rather than Z-transform since
we are interested in analog domain!!

Regards,
Santosh
> 
> Thanks in advance,
> LL.
> ---------------------------------------------------------------
> PS: I searched Google and came up with nothing.

"Lord Labakudas" <l...@yahoo.com> wrote in message
news:d...@posting.google.com...
> Hello DSP folks,
>
> This question has been bothering me for a long time since I took my
> Microwave Engineering course:
>
> Suppose we have a linear system H(z) we can easily find its poles and
> zeros and perform stability analyis of the system. Does such a thing
> exist for general network analysis using the S-parameters ? I know
> that stability analysis can be done at every frequency, especially if
> one is designing amplifiers. There are stability criteria that are
> valid only at specific frequencies.
>
> A lot of times we need to do DSP work with discretized S-paramters
> {S(f_1),...,S(f_N)}, and hence the question arises ... does the
> Z-transform mean anything for these S-parameters ? S_21(f) of a
> network is same as the H(f) of the system. Does a general treatment of
> the S-parameters using Z-transform exist ?

LL,

I have no idea about a "general treatment" and I think I understand why.
As a student I was always amused that we had to do "network synthesis" from
rational functions but the orgin of those rational functions was always just
"assumed".  How one arrived at them was obscured by the curriculum/approach.
The answer is that they come from applications of approximation theory or
optimization algorithms and that many common cases have been treated - (and
many have not).

Let's see if I understand your context:

You have an analog system so S parameters can be measured.
The S parameters are being measured at a sequence of frequencies S(f_1) ...
S(F_N).
The S parameters, or Y parameters represent a system model for each of these
frequencies - often a single-frequency approach.
You're using them over a range of frequencies - so their values vary.

So, it appears that you should be able to model the S or Y parameters as a
function of frequency using rational functions to represent them by applying
approximation methods.  Doing this needs to be comprehensive enough that
zero and infinite frequency is accounted for in the model.  Once this is
done, you should have a structure for which ordinary linear differential
equations with constant coefficients can be written.  Once this is done, I
believe the Laplace transform can be used.

By extension, one should be able to extend this to using linear difference
equations with constant coefficients and use the z-transform.

I've never done it.  But this seems like a likely approach.
None of this makes sense if the "network" is really a digital system of
course - because things like "reflection" don't apply.

I hope this helps move you in a productive direction.

Fred

Fred,

Thanks a bunch for your thoughful insight into my question !! 

-LL.

> So, it appears that you should be able to model the S or Y parameters as a
> function of frequency using rational functions to represent them by applying
> approximation methods.  Doing this needs to be comprehensive enough that
> zero and infinite frequency is accounted for in the model.  Once this is
> done, you should have a structure for which ordinary linear differential
> equations with constant coefficients can be written.  Once this is done, I
> believe the Laplace transform can be used.
>