| Are you familiar with Oppenheim and Shaffer? - this DSP textbook
explains what you need to know to do basic deconvolution.
If you can measure the step response or swept frequency response
(via a network analyzer) of your system, then it is possible to
de-imbed a device under test fairly easily - but you will want a
low reflection impedance match at the test equipment end, or reflections
will add to the complexity of the situation. A TDR would be very
useful as well.
The important thing is that it is not a simple matter of just
multiplying the measured transfer function of a device with the
inverse of the transfer function of the test equipment - you have to
do network analysis as well (in terms of a signal flow diagram
involving scattering parameters.)
The standard DSP books concentrate on deconvolving "echos" (such as
arise in geologic and acoustic measurements) - but this is not an
accurate model of a microwave or wideband pulse response system because
there is a lot of smearing of the impulses taking place.
The IEEE test and measurement, Microwave Theory and Technique, and
other journals will have information that may useful, but you'll have
to ferret it out. I've been away from microwave work for a while and
am somewhat rusty - and don't know what's been happening lately in the
field in any detail.
- Jim
|
| OK thanks. Do I have to do it in the time domain or the frequency
domain. I want the answer in the time domain, but the frequency domain
might be faster.
I have TDRd all my equipment several times, but now I should
store the wave and analyze it more thoroughly. The impedance matching
is pretty good on the bench but slightly flawed on the automatic
tester.
Some National Bureau of Standards (now the National Institute of
Standards and Technology) papers I found follow; I can't order them
from NTIS (national tech info service); if you know where the
publications mentioned are available, please let me know. Thanks.
I'd order them from the authors, but the first one retired.
PB88-121967 Software Correction of Measured Pulse Data
Final reprint.
N. S. Nahman 1986 67p; Pub. in Fast Electrical and Optical Measurements
V1, NATO ASI Series E, n108 p351-417 1986
"The fundamental concern in the software correction of measured
pulse waveform data is the solution of an ill-posed deconvolution
problem which arises when one (or both) of the known waveforms is
(are) corrupted by erros due to interference, noise, instrumentation
dirft, etc. The variables concerned are related to each other by
the convolution integral. When one of the integrand functions is
unknown while the other two function are known, the convolution
equation becomes an integral equation for the nknwon waveform.
Solution of an illposed deconvolution problem is obtained by signal
processing or filtering and at most yields an estimate for the unknown
waveform. The objective of the discussion is to bring out the ideas
of ill-posedness and to give examples of applications to pulse
measurement problems which require deconvolution, i.e., the removal
(correction) of pulse source effects and/or measurement system effects
as encountered in signal pulse waveform measurements and system
impluse response measurements."
Also, in the math section of the NIST catalog:
PB89-233524; Infinitely Divisible Pulses, Continuous Deconvolution,
and the Characterizaton of Linear Time Invariant systems.
Final rept. A. S. Carasso Aug 89 36p Contract ARO-63-82. Pub in
SIAM (Society for Industrial and Applied Math) Jnl. on Applied Math
47, n4 p892-927 Aug 98.
"The paper addresses the problem of determing the impulse response
of a linear time invariant system, by probing the system with a
causal, C infinity approximation to the Dirac delta-function. The
authors analyze the ill-posed deconvolution problem which results
from a wide choice of possibly multimodal, infinitely divisible,
proble pulses. The notion of inite divisibility is shown to play
a key role when the systems' response is suspect of having
nondifferentiable singularities. The authors reformulate the Volterra
inegral equation as a Cauchy problem for a linear partial differential
equation in two indipendent variables, and introduce the concepts
of partial and continuous deconvolution. The authors then show
that partial deconvolution of the output waveform results in infinity
error bounds for the regularized solution and its derivatives under
L2 a prioir bounds on the data noise and the unknonw system response.
Using the Poisson summation formula and FFT algorithms [this implies
they use the frequency domain - TEJ], the authors
construct an efficient computational algorithm for performing continuous
deconvolution, given sufficiently long but finite records of the
probe pulse, and the output waveform. The theory is illustrated
with several examples of computational reconstructions of singular
elastic Green's functions, from smooth synthetic noisy data."
Tom
|
| Doing it in the frequency domain requires a pointwise operaton with
a pair of transforms and will probably win for any reasonable
number of points. The time to do these transforms should not be
a problem, I feel. If your data set was so short that an optimized
convolution would be faster than a pair of FFT's then the time to do
the processing will be so little it won't matter.
You can have the library order the papers - it costs your cost center
a nominal charge, but the DLN (at least the Maynard library) seems
very good at ferreting out papers, and the cost is probably comparable
to ordering it yourself.
The first paper you mention is probably going to be the most useful of
the two, and it looks pretty recent too. The second does not sound
as practical as the first, though it may be interesting for its own
sake.
My own knowledge of this area comes from microwave measurements in the
frequency domain, where TDR measurements (on a discontinuity) are used
to make up the effective scattering parameters of the discontinuity
so this can be included in further circuit modeling. This is probably
a cleaner problem than you're facing. Some of the older papers (early
70's) on this included swept measurements of waveguides that were
sensitive enough that a waveguide joint could be easily spotted in the
data.
- Jim
|