> I had to resample a signal with requirement of minimal processing delay.
> So I made a Lagrange polynomial extrapolator to predict the signal on
> the duration of +1 sample into the future. That is straightforward and
> it works good enough for the job.
> However what could be the other options for extrapolation of the Nyquist
> bandlimited signal? What is an optimal solution for this case?
Wouldn't be that _equivalent_ to
- not extrapolating at all
- using a _different_ lowpass filter
?!
(assuming linear extrapolation)
It seems to me that you'd be better off _directly_ designing the
appropriate lowpass filter that suits your needs w.r.t. a low group
delay for your band of interest.
SG
Reply by Steve Pope●October 27, 20112011-10-27
Greg Berchin <gjberchin@charter.net> wrote:
>Since you have the past signal of infinite duration, I suspect that
>the least-squares optimal solution would be an ideal lowpass filter
>applied to the digital signal interpreted as an analog impulse train.
>Tough to implement, though, and not exactly low-delay.
Doesn't this usually come down to being identical to the Lagrangian
interpolator (which is the first approach I would have used).
Steve
Reply by Tim Wescott●October 26, 20112011-10-26
On Wed, 26 Oct 2011 17:07:56 -0700, Greg Berchin wrote:
> On Oct 26, 6:15 pm, Tim Wescott <t...@seemywebsite.com> wrote:
>
>> The Wiener filter optimizes for least squares. It's very much like the
>> asymptotic solution to a Kalman filter.
>
> Got it. It's been a long time since I studied them. I have this vague
> memory that a least-squares solution with a Gaussian distribution *is*
> the maximum likelihood solution. I could be mistaken.
>
>> > PS Tim, are you the same Tim Wescott who occasionally shows up in the
>> > Control Line Stunt Forumhttp://www.stunthanger.com/smf/?
>>
>> Yes -- I don't recall seeing you on there: do you have a nym, or are
>> you just a lurker?
>
> I visit the site occasionally, but have never registered. In my younger
> days I was capable of the entire AMA Precision Aerobatics pattern
> (though I never competed), but it's been 35 years since I did a
> wingover. To put it into perspective, the last stunt plane that I flew
> was powered by a Fox .35!
I have yet to do the complete pattern, ever. I was working up to it in
my late teens, but then college happened, and marriage, and kids, and I
never got the time until recently to pick up the reins again. I started
competing last October, flying the beginner's pattern. I can do every
maneuver of the real thing, but haven't put them all together in one
flight yet.
I might have done one today, but for breaking the plane. It's a good
thing I was flying over soft ground.
(To complete my day I was flying a friend's 1/2A Ringmaster on a very
sick engine and poorly adjusted controls. The flight involved a lot of
quick yanks backward on the handle; it ended when I went over backward
onto my head. I've never gotten whacked hard enough to see a white flash
before; I don't think it's necessary to my future happiness to do it ever
again. At any rate, when I opened my eyes and looked around my buddies
were gathered around the plane, making sure that it was in good shape.
It's a _really good_ thing I was flying over soft ground today.)
--
www.wescottdesign.com
Reply by Greg Berchin●October 26, 20112011-10-26
On Oct 26, 6:15�pm, Tim Wescott <t...@seemywebsite.com> wrote:
> The Wiener filter optimizes for least squares. �It's very much like the
> asymptotic solution to a Kalman filter.
Got it. It's been a long time since I studied them. I have this vague
memory that a least-squares solution with a Gaussian distribution *is*
the maximum likelihood solution. I could be mistaken.
> > PS Tim, are you the same Tim Wescott who occasionally shows up in the
> > Control Line Stunt Forumhttp://www.stunthanger.com/smf/?
>
> Yes -- I don't recall seeing you on there: do you have a nym, or are you
> just a lurker?
I visit the site occasionally, but have never registered. In my
younger days I was capable of the entire AMA Precision Aerobatics
pattern (though I never competed), but it's been 35 years since I did
a wingover. To put it into perspective, the last stunt plane that I
flew was powered by a Fox .35!
Greg
Reply by Tim Wescott●October 26, 20112011-10-26
On Wed, 26 Oct 2011 14:11:46 -0700, Greg Berchin wrote:
> On Oct 26, 3:05 pm, Tim Wescott <t...@seemywebsite.com> wrote:
>
>> If you could express the spectrum of the signal with a rational
>> polynomial I would expect that a Wiener filter would be the best -- I'm
>> not sure that's not just another name for your Lagrangian thing,
>> though.
>
> Tim, that jogged a memory. I think that the Weiner Filter is related to
> the Maximum Likelihood estimate. Give that Vladimir knows the statistics
> of the signal, a Maximum Likelihood estimate might be the right way to
> go.
The Wiener filter optimizes for least squares. It's very much like the
asymptotic solution to a Kalman filter.
> PS Tim, are you the same Tim Wescott who occasionally shows up in the
> Control Line Stunt Forum http://www.stunthanger.com/smf/?
Yes -- I don't recall seeing you on there: do you have a nym, or are you
just a lurker?
I'm currently down to 0 control line planes -- I had the controls freeze
up on my Skyray this morning on the downward leg of a reverse wingover.
Oh, it's a good thing the rains have started and the grass is soft!
--
www.wescottdesign.com
Reply by Greg Berchin●October 26, 20112011-10-26
On Oct 26, 3:05�pm, Tim Wescott <t...@seemywebsite.com> wrote:
> If you could express the spectrum of the signal with a rational
> polynomial I would expect that a Wiener filter would be the best -- I'm
> not sure that's not just another name for your Lagrangian thing, though.
Tim, that jogged a memory. I think that the Weiner Filter is related
to the Maximum Likelihood estimate. Give that Vladimir knows the
statistics of the signal, a Maximum Likelihood estimate might be the
right way to go.
Greg
PS Tim, are you the same Tim Wescott who occasionally shows up in the
Control Line Stunt Forum http://www.stunthanger.com/smf/?
Reply by Tim Wescott●October 26, 20112011-10-26
On Wed, 26 Oct 2011 12:39:32 -0500, Vladimir Vassilevsky wrote:
>>> I had to resample a signal with requirement of minimal processing
>>> delay. So I made a Lagrange polynomial extrapolator to predict the
>>> signal on the duration of +1 sample into the future. That is
>>> straightforward and it works good enough for the job. However what
>>> could be the other options for extrapolation of the Nyquist
>>> bandlimited signal? What is an optimal solution for this case?
>
> Greg Berchin wrote:
>
>> Upsampling or downsampling? I ask because downsampling requires an
>> appropriate anti-aliasing lowpass operation, whereas upsampling does
>> not.
>
> Upsampling.
>
>> What is your definition of "optimal"? Optimal from a computational
>> standpoint might be a zero-order hold.
>
> Let the future known for a finite number of N samples (N could be zero),
> and the past is known for indefinite duration. The signal is Gaussian
> amplitude distribution, flat specrum, Nyquist bandlimited. What would be
> the most accurate interpolator (or extrapolator if N = 0) algorithm in
> the least square error sense?
If you could express the spectrum of the signal with a rational
polynomial I would expect that a Wiener filter would be the best -- I'm
not sure that's not just another name for your Lagrangian thing, though.
--
www.wescottdesign.com
Reply by Greg Berchin●October 26, 20112011-10-26
On Oct 26, 1:15�pm, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
> That is for an infinite case. I am not sure about semi-infinite.
Yes; I'm having the same problem with "semi-infinite" (negative
infinity to present sample) and finite cases. I suspect that a finite
duration ideal lowpass filter (truncated sinc interpolator in time
domain) might provide the minimum least-squares solution, but if it
does then perhaps least-squares is not the best figure of merit -- by
way of analogy, recall that a rectangular time-domain window applied
to a long impulse response is the least-squares optimal shorter-
duration approximation, proving only that least-squares is not always
a good optimization metric.
Greg
Reply by Vladimir Vassilevsky●October 26, 20112011-10-26
Greg Berchin wrote:
>>Let the future known for a finite number of N samples (N could be zero),
>>and the past is known for indefinite duration. The signal is Gaussian
>>amplitude distribution, flat specrum, Nyquist bandlimited. What would be
>>the most accurate interpolator (or extrapolator if N = 0) algorithm in
>>the least square error sense?
>
>
> Since you have the past signal of infinite duration, I suspect that
> the least-squares optimal solution would be an ideal lowpass filter
> applied to the digital signal interpreted as an analog impulse train.
That is for an infinite case. I am not sure about semi-infinite.
VLV
Reply by Greg Berchin●October 26, 20112011-10-26
> Let the future known for a finite number of N samples (N could be zero),
> and the past is known for indefinite duration. The signal is Gaussian
> amplitude distribution, flat specrum, Nyquist bandlimited. What would be
> the most accurate interpolator (or extrapolator if N = 0) algorithm in
> the least square error sense?
Since you have the past signal of infinite duration, I suspect that
the least-squares optimal solution would be an ideal lowpass filter
applied to the digital signal interpreted as an analog impulse train.
Tough to implement, though, and not exactly low-delay.
I'll have to ponder the problem a bit more for a finite-duration
solution. I recall that RB-J and Duane Wise published some work on
optimal interpolators. Robert, care to comment?
Greg