A bandlimited signal may be reconstructed from its samples with an ideal low-pass filter. But is this ideal low-pass filter still the optimal filter if there is noise. If not how to find out the optimal filter? thanks!
reconstruct a signal from its samples in the presence of noise
Started by ●April 19, 2007
Reply by ●April 20, 20072007-04-20
On Apr 19, 7:30 pm, zqchen <zhiqun.c...@gmail.com> wrote:> A bandlimited signal may be reconstructed from its samples with an > ideal low-pass filter. But is this ideal low-pass filter still the > optimal filter if there is noise. If not how to find out the optimal > filter? thanks!"optimal" with respect to what criteria? Are there known signal characteristics beyond bandlimited? What kind of noise? additive, multiplicative, of a distribution, dropout, jitter, ... More specific questions are more likely to yield useful, interesting or possibly even correct answers. Dale B. Dalrymple http://dbdimages.com
Reply by ●April 20, 20072007-04-20
On 4=D4=C220=C8=D5, =CF=C2=CE=E73=CA=B116=B7=D6, dbd <d...@ieee.org> wrote:> > "optimal" with respect to what criteria? > > Are there known signal characteristics beyond bandlimited? > > What kind of noise? additive, multiplicative, of a distribution, > dropout, jitter, ... > > More specific questions are more likely to yield useful, interesting > or possibly even correct answers. > > Dale B. Dalrymplehttp://dbdimages.comI think since error is the criteria for noiseless case, SINR may be ok for noise case. In fact what I want to know is if there are cases the ideal low pass filter with no reconstruction error in noiseless case will be no more the right choice in the presence of some noise. In such cases I need to choose some other filters according to some criterian, say SINR.
Reply by ●April 20, 20072007-04-20
zqchen wrote:> A bandlimited signal may be reconstructed from its samples with an > ideal low-pass filter. But is this ideal low-pass filter still the > optimal filter if there is noise. If not how to find out the optimal > filter? thanks!You are mixing up two independent concepts. Sampling and reconstruction is concerned with the conditions under which a signal may be sampled and perfectly reproduced from the samples. Noise reduction is concerned with seperating noise and signal. For most real-world noise reduction applications, an LTI filter isn't sufficient. LTI filters only make sense for noise reduction if a) the noise is additive and b) noise and signal are in non-overlapping frequency bands. Regards, Andor
Reply by ●April 20, 20072007-04-20
zqchen wrote:> A bandlimited signal may be reconstructed from its samples with an > ideal low-pass filter. But is this ideal low-pass filter still the > optimal filter if there is noise. If not how to find out the optimal > filter? thanks!Where would the noise come from? Misinterpreted bits? The output of the DAC? Help me to understand the sequence of events you have in mind. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●April 20, 20072007-04-20
On Apr 19, 9:30 pm, zqchen <zhiqun.c...@gmail.com> wrote:> A bandlimited signal may be reconstructed from its samples with an > ideal low-pass filter. But is this ideal low-pass filter still the > optimal filter if there is noise. If not how to find out the optimal > filter? thanks!If you have more specific knowledge of the signal -- such as its power spectral density -- and your noise is AWGN and independent of the signal, and your error measure is the mean- square distortion, then the best is to use a Wiener filter. If the only thing that you know about the signal is that it's bandlimited, and all the other assumptions apply, then the Wiener filter IS a lowpass filter with the same bandwidth. So there are quite a bit of qualifiers before you get to any conclusion. There is no "general" answer, although you asked a good question that is often overlooked. If you want to be concerned with LINEAR reconstruction methods, possibly with non-ideal sampling and reconstruction filters, then read up some of Michael Unser's beautiful papers on the subject. He shows that digital correction is possible, in many cases in form of a digital filter. http://bigwww.epfl.ch/publications/unser0001.html OK, now I am just getting rhapsodic.... Hope that helps, Julius
Reply by ●April 20, 20072007-04-20
zqchen wrote:> A bandlimited signal may be reconstructed from its samples with an > ideal low-pass filter. But is this ideal low-pass filter still the > optimal filter if there is noise. If not how to find out the optimal > filter? thanks!If a signal is corrupt with a noise, there is generally no need to have the reconstruction error better then the noise. Thus the requirements to the lowpass reconstruction filter can be lowered, and the filter can be optimized in that sense. Is it what you are asking about? Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Reply by ●April 21, 20072007-04-21
On Apr 20, 4:14 am, zqchen <zhiqun.c...@gmail.com> wrote:> I think since error is the criteria for noiseless case, SINR may be ok > for noise case. > In fact what I want to know is if there are cases the ideal low pass > filter with no reconstruction error in noiseless case will be no more > the right choice in the presence of some noise.Well yes. :) I am guessing that you are thinking about the difference between convolving the image of [signal] with a frequency-domain impulse train, then low-pass filtering the result; versus convolving the image of [signal + noise] with the same impulse train, then low-pass filtering that result. Naturally, if the noise is not band-limited as is the signal, then the bandwidth of filter will have to be increased until it accommodates the noise, in which case you will reconstitute the noise with the signal. If the noise is band-limited along with the signal, then obviously the same filter would be used, again, reconstituting the noise with the signal. So you see, the whole point of choosing the correct sampling rate and/or the correct width of the pre-sampling filter or reconstruction filter is to prevent [signal] or [signal+noise] sub-images from overlapping themselves during reconstitution, and has not much to do with whether noise is present. So the answer is, yes, if there is noise in your signal, and its bandwidth is great for the chosen sampling rate/filter, an ideal filter is not going to help you do anything special.> In such cases I need to choose some other filters according to some > criterian, say SINR.Depends on the characteristics of the noise (PSD, stationarity, etc.) For example, if the "noise" consists of a DC-offset, then the low-pass filter would still be optimal. You would just use it, then subtract whatever is being added (cheating). -Le Chaud Lapin-
Reply by ●April 21, 20072007-04-21
Le Chaud Lapin wrote:> On Apr 20, 4:14 am, zqchen <zhiqun.c...@gmail.com> wrote: >> I think since error is the criteria for noiseless case, SINR may be ok >> for noise case. >> In fact what I want to know is if there are cases the ideal low pass >> filter with no reconstruction error in noiseless case will be no more >> the right choice in the presence of some noise. > > Well yes. :) > > I am guessing that you are thinking about the difference between > convolving the image of [signal] with a frequency-domain impulse > train, then low-pass filtering the result; versus convolving the image > of [signal + noise] with the same impulse train, then low-pass > filtering that result. Naturally, if the noise is not band-limited as > is the signal, then the bandwidth of filter will have to be increased > until it accommodates the noise, in which case you will reconstitute > the noise with the signal. If the noise is band-limited along with > the signal, then obviously the same filter would be used, again, > reconstituting the noise with the signal.The noise, once sampled, is bandlimited "along with the signal." If it wasn't bandlimited before being sampled, then it aliased. Is there a test to distinguish aliased randomness from bandlimited randomness?> So you see, the whole point > of choosing the correct sampling rate and/or the correct width of the > pre-sampling filter or reconstruction filter is to prevent [signal] or > [signal+noise] sub-images from overlapping themselves during > reconstitution, and has not much to do with whether noise is present. > So the answer is, yes, if there is noise in your signal, and its > bandwidth is great for the chosen sampling rate/filter, an ideal > filter is not going to help you do anything special.The reconstruction filter that removes images without noise removes images with noise, whether that noise has aliased or not. The premise of the question is unfounded. There is no good answer to a bad question.>> In such cases I need to choose some other filters according to some >> criterian, say SINR. > > Depends on the characteristics of the noise (PSD, stationarity, etc.) > For example, if the "noise" consists of a DC-offset, then the low-pass > filter would still be optimal. You would just use it, then subtract > whatever is being added (cheating).Would you consider DC to be random? It may be interference if it's not random, but is it noise? Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●April 21, 20072007-04-21
On Apr 20, 5:47 am, Andor <andor.bari...@gmail.com> wrote:> zqchen wrote: > > A bandlimited signal may be reconstructed from its samples with an > > ideal low-pass filter. But is this ideal low-pass filter still the > > optimal filter if there is noise. If not how to find out the optimal > > filter? thanks! > > You are mixing up two independent concepts. Sampling and > reconstruction is concerned with the conditions under which a signal > may be sampled and perfectly reproduced from the samples. > > Noise reduction is concerned with seperating noise and signal. For > most real-world noise reduction applications, an LTI filter isn't > sufficient. LTI filters only make sense for noise reduction if > a) the noise is additive and > b) noise and signal are in non-overlapping frequency bands.dunno if i completely agree, Andor. but i must admit that this takes me back to grad school and is not stuff that i think about or work on currently. as Dale said, "'optimal' with respect to what criteria? " if reconstructing the signal where scaling was important (and you measure error with a Euclidian norm) then a Weiner filter is optimal (if scaling isn't important, then it's a Matched filter), and assuming it's been turned on a long time and the statastics of the noise do not change, it's LTI. and although you get better performance if the noise and signal are disjoint in either time or frequency, that is not the requirement in general. r b-j
