System analysis is only as good as the model used. For non-linear systems, the Volterra
model is arguably the most complete model we have. However, we need to know the kernels of the
model. For linear systems where the x'h model is used (product of x-transpose and h), the
kernel,h, is found by solving the normal equation, and many adaptive algorithms exist.

However, this does not seem to be true for the Volterra model. In 1983, Stephen Boyd came up
with a way to measure the Volterra kernels. In 2006, Niclas Bjorsell showed how to measure
Volterra kernels for A-to-D converters. But, I have not been able to find anything on adaptive
systems using the Volterra model where the kernels are derived from the input/output
relationship.

Does anyone have information on this?


Many thanks,
Maurice

On 24.07.2012 19:48, maury wrote:
> System analysis is only as good as the model used. For non-linear systems, the Volterra
model is arguably the most complete model we have. However, we need to know the kernels of the
model. For linear systems where the x'h model is used (product of x-transpose and h), the
kernel,h, is found by solving the normal equation, and many adaptive algorithms exist.
>
> However, this does not seem to be true for the Volterra model. In 1983, Stephen Boyd came
up with a way to measure the Volterra kernels. In 2006, Niclas Bjorsell showed how to measure
Volterra kernels for A-to-D converters. But, I have not been able to find anything on adaptive
systems using the Volterra model where the kernels are derived from the input/output
relationship.
>
> Does anyone have information on this?
>
>
> Many thanks,
> Maurice
>

Hi Maurice,
What do you mean by deriving kernels? Truncated Volterra series have 
been widely used for power amplifier linearization and in that case 
coefficients of the Volterra model in a predistorter can be obtained, 
for example, by means of adaptive minimization of the error between the 
predistorter input and the downconverted/scaled power amplifier output 
(so-called model referenced adaptive system (MRAS) predistorter). But 
the system operates with a fixed order of the Volterra model and fixed 
memory depths, so you have to make some reasonable assumption about 
these parameters beforehand.

-- 

Alexander

On Tuesday, July 31, 2012 3:08:46 AM UTC-5, Alexander Sotnikov wrote:
> On 24.07.2012 19:48, maury wrote: > System analysis is only as good as the model used.
For non-linear systems, the Volterra model is arguably the most complete model we have. However,
we need to know the kernels of the model. For linear systems where the x'h model is used
(product of x-transpose and h), the kernel,h, is found by solving the normal equation, and many
adaptive algorithms exist. > > However, this does not seem to be true for the Volterra
model. In 1983, Stephen Boyd came up with a way to measure the Volterra kernels. In 2006, Niclas
Bjorsell showed how to measure Volterra kernels for A-to-D converters. But, I have not been able
to find anything on adaptive systems using the Volterra model where the kernels are derived from
the input/output relationship. > > Does anyone have information on this? > > >
Many thanks, > Maurice > Hi Maurice, What do you mean by deriving kernels? Truncated
Volterra series have been widely used for power amplifier linearization and in that case
coefficients of the Volterra model in a predistorter can be obtained, for example, by means of
adaptive minimization of the error between the predistorter input and the downconverted/scaled
power amplifier output (so-called model referenced adaptive system (MRAS) predistorter). But the
system operates with a fixed order of the Volterra model and fixed memory depths, so you have to
make some reasonable assumption about these parameters beforehand. -- Alexander

Hello Alexander,
What I'm looking at is non-linear adaptive system identification. Several models exist (e.g.,
Hammerstein, etc.), but the Volterra is probably the most complete (at least that I know about).
I am familiar with, and have used, truncated Volterra, but have not seen any work on adaptively
deriving an unknown non-linear system impulse response from the system input/output
information.

The closest I've seen is the measurement/determination of the kernels using a frequency-domain
version of the Volterra series, but not an adaptive system identification treatment.

Maurice

>> For non-linear systems, the Volterra model is arguably the most complete
model we have

that could be argued :-) 
For -weakly- nonlinear systems, maybe. But if things get heavily nonlinear
(clipping), a run-of-the-mill piecewise linear least-squares approximation
- maybe with some memory as in Wiener / Hammerstein memory might turn out
to be a better choice overall.

>What I'm looking at is non-linear adaptive system identification.


Look at this book

http://www.amazon.com/Adaptive-Nonlinear-System-Identification-Communication/dp/0387263284/ref=s
r_1_1/002-6131040-9646467?ie=UTF8&s=books&qid89708740&sr=1-1

On Wednesday, August 1, 2012 4:32:42 AM UTC-5, Alexander Petrov wrote:
> >What I'm looking at is non-linear adaptive system identification. Look at this book
http://www.amazon.com/Adaptive-Nonlinear-System-Identification-Communication/dp/0387263284/ref=s
r_1_1/002-6131040-9646467?ie=UTF8&s=books&qid89708740&sr=1-1

Thanks Alexander, I'll take a gander.