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Resampling with minimum delay

Started by Vladimir Vassilevsky October 26, 2011
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?


Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com


Upsampling or downsampling? I ask because downsampling requires an appropriate
anti-aliasing lowpass operation, whereas upsampling does not.

What is your definition of "optimal"? Optimal from a computational standpoint
might be a zero-order hold.

Greg
>> 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? Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
> 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

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
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
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
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/?
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
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