Reply by Sebastian October 29, 20112011-10-29
On 26 Okt., 18:06, 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?
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&nbsp;pm, Tim Wescott <t...@seemywebsite.com> wrote: > >> The Wiener filter optimizes for least squares. &nbsp;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&#2013266080;pm, Tim Wescott <t...@seemywebsite.com> wrote:

> The Wiener filter optimizes for least squares. &#2013266080;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&nbsp;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&#2013266080;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&#2013266080;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