interpolation at data sequence endpoints

In a previous thread, it was mentioned that polynomial spline
interpolation and windowed sinc reconstruction give similar results,
or may even converge under some conditions.

In another recent thread, it was stated the a windowed sinc
reconstruction filter used for upsampling would give "invalid" results
near the endpoints of a data sequence until all the taps of the FIR
filter were filled with actual data.

But there are one-sided polynomial interpolation methods.

Could these polynomial interpolation methods be used to fill in
the missing endpoint subsamples?  Or is there a translation of these
one-sided polynomial interpolaters into the domain of impulse responses
which could be used to make an appropiate adaptive FIR filter for use
near the data endpoints?  Or could the data sequence be padded past its
endpoints with estimated data points produced by the method of finite
differences (to keep the derivatives continuous) to allow a regular
linear phase FIR filter to produce reasonable intepolated results at
(or even past!) the endpoints?


IMHO. YMMV.
-- 
Ron Nicholson   rhn AT nicholson DOT com   http://www.nicholson.com/rhn/ 
#include <canonical.disclaimer>        // only my own opinions, etc.

Dear Ronald,

I think that any interpolation is based on the
hypothesis of the original analog signal nature.
If its nature is exactly described by the very polynomial
then everything is OK.
When restoring the signal at the edges
any method will fit
because nobody knows how the signal behaved in past time and the future.
When I know that that signal =0 in endpoints there then
none interpolation is needed.
When the signal is divided into slices
for example for FFT, then this problem
is usually solved by interleaving and windowing.
Then the polynomial has to fit the used window.
Etc.

A.Ser.


"Ronald H. Nicholson Jr." wrote:

> In a previous thread, it was mentioned that polynomial spline
> interpolation and windowed sinc reconstruction give similar results,
> or may even converge under some conditions.
>
> In another recent thread, it was stated the a windowed sinc
> reconstruction filter used for upsampling would give "invalid" results
> near the endpoints of a data sequence until all the taps of the FIR
> filter were filled with actual data.
>
> But there are one-sided polynomial interpolation methods.
>
> Could these polynomial interpolation methods be used to fill in
> the missing endpoint subsamples?  Or is there a translation of these
> one-sided polynomial interpolaters into the domain of impulse responses
> which could be used to make an appropiate adaptive FIR filter for use
> near the data endpoints?  Or could the data sequence be padded past its
> endpoints with estimated data points produced by the method of finite
> differences (to keep the derivatives continuous) to allow a regular
> linear phase FIR filter to produce reasonable intepolated results at
> (or even past!) the endpoints?
>
> IMHO. YMMV.
> --
> Ron Nicholson   rhn AT nicholson DOT com   http://www.nicholson.com/rhn/
> #include <canonical.disclaimer>        // only my own opinions, etc.

"Ronald H. Nicholson Jr." <rhn@mauve.rahul.net> wrote in message
news:c02f2u$akr$1@blue.rahul.net...
> In another recent thread, it was stated the a windowed sinc
> reconstruction filter used for upsampling would give "invalid" results
> near the endpoints of a data sequence until all the taps of the FIR
> filter were filled with actual data.
>
> But there are one-sided polynomial interpolation methods.
>
> Could these polynomial interpolation methods be used to fill in
> the missing endpoint subsamples?  Or is there a translation of these
> one-sided polynomial interpolaters into the domain of impulse responses
> which could be used to make an appropiate adaptive FIR filter for use
> near the data endpoints?  Or could the data sequence be padded past its
> endpoints with estimated data points produced by the method of finite
> differences (to keep the derivatives continuous) to allow a regular
> linear phase FIR filter to produce reasonable intepolated results at
> (or even past!) the endpoints?
>
Hi Ronald,

 If you want to do an interpolation operation near right-hand edge of a
delay line, you can store the samples from previous time steps that have
"run-off" the right-hand end of the line. So these old samples may not
contribute directly to the system being modeled, but come in handy for
interpolation near the edges, i.e they provide realistic data points for an
FIR filter.

Siddharth

Siddharth Mathur wrote:
> "Ronald H. Nicholson Jr." <rhn@mauve.rahul.net> wrote in message
> news:c02f2u$akr$1@blue.rahul.net...
> > In another recent thread, it was stated the a windowed sinc
> > reconstruction filter used for upsampling would give "invalid" results
> > near the endpoints of a data sequence until all the taps of the FIR
> > filter were filled with actual data.
> >
> > But there are one-sided polynomial interpolation methods.
> >
> > Could these polynomial interpolation methods be used to fill in
> > the missing endpoint subsamples?  Or is there a translation of these
> > one-sided polynomial interpolaters into the domain of impulse responses
> > which could be used to make an appropiate adaptive FIR filter for use
> > near the data endpoints?  Or could the data sequence be padded past its
> > endpoints with estimated data points produced by the method of finite
> > differences (to keep the derivatives continuous) to allow a regular
> > linear phase FIR filter to produce reasonable intepolated results at
> > (or even past!) the endpoints?
> >
> Hi Ronald,
> 
>  If you want to do an interpolation operation near right-hand edge of a
> delay line, you can store the samples from previous time steps that have
> "run-off" the right-hand end of the line. So these old samples may not
> contribute directly to the system being modeled, but come in handy for
> interpolation near the edges, i.e they provide realistic data points for an
> FIR filter.

That would be the overlap-add method for filtering a continuous stream
of data with an FIR. Unless you are doing this in frequency domain
(ie. FFT convolution), there is no need to break down the signal into
chunks (unless you are doing block processing for other reasons, such
as minimizing IO overhead).

If you have a finite vector of length M (signal), and you convolve it
with another vector of length N (FIR filter), you get a new vector
with length M+N-1. There is nothing wrong with that. If the FIR is
linear phase, you have tapering at the start and end of the signal
vector, just as convolution dictates. If the FIR is minimum or maximum
phase, you have tapering at the end or start only, respectively. There
is nothing "invalid" about that.

Regards,
Andor