Coherent averaging
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From: vale on 29 Aug 2006 11:45 Hi all, we are dealing with N synchronous measures of a spatial signal corrupted with a non-zero mean noise. Those measures are mutually uncorrelated. Is it feasible to extract the signal using coherent averaging? We have a reasonable model of the signal pattern (but not of the signal peak to peak). Can we use this model to improve the signal extraction? We are now using the correlation within model and the measure as weight in the averaging procedure. Thanks, Valentina Camomilla ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Valentina Camomilla, PhD Biomechanics Laboratory Department of Human Movement and Sport Science IUSM - Istituto Universitario di Scienze Motorie Piazza Lauro de Bosis, 6 00195 ROMA - ITALY Phone +39 06 36733522 Fax +39 06 36733517 e-mail: camomilla(a)iusm.it ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
From: Andor on 30 Aug 2006 09:20 vale wrote: > Hi all, > > we are dealing with N synchronous measures of a spatial signal > corrupted with a non-zero mean noise. Those measures are mutually > uncorrelated. Is it feasible to extract the signal using coherent > averaging? Is this your setup: there exists a (real) vector x = [x1,x2, ... x_N] of length N, and you have, say, M measurements y_k = x + e_k, 1 <= k <= M, where the M vectors e_k are non-correlated and non-zero mean noise? Do you know the mean value E[ e_{k,l} ], where e_{k,l} is the l-th coordinate in the k-th error vector? > > We have a reasonable model of the signal pattern (but not of the signal > peak to peak). Can we use this model to improve the signal extraction? One should think so. What is the model? > We are now using the correlation within model and the measure as weight > in the averaging procedure. > > Thanks, > > Valentina Camomilla Regards, Andor
From: Scott Seidman on 30 Aug 2006 09:41 "vale" <smirtariata(a)gmail.com> wrote in news:1156866310.581560.148830 @p79g2000cwp.googlegroups.com: > Hi all, > > we are dealing with N synchronous measures of a spatial signal > corrupted with a non-zero mean noise. Those measures are mutually > uncorrelated. Is it feasible to extract the signal using coherent > averaging? > > In my field, we usually refer to this as "ensemble averaging". For zero- mean noise, this improves your signal to noise ratio by sqrt(N), where N is the number of epochs in your ensemble. The derivation to demonstrate this sqrt(N) improvement depends upon E(noise)=0. You specify that this doesn't exist. I suspect that things will be much the same for your case if you subtract out the mean of the noise. > We have a reasonable model of the signal pattern (but not of the signal > peak to peak). Can we use this model to improve the signal extraction? > We are now using the correlation within model and the measure as weight > in the averaging procedure. > I suspect that any model-based technique would also depend upon you subtracting out the mean of the noise. -- Scott Reverse name to reply
From: vale on 30 Aug 2006 11:39 Andor wrote: > vale wrote: > > Hi all, > > > > we are dealing with N synchronous measures of a spatial signal > > corrupted with a non-zero mean noise. Those measures are mutually > > uncorrelated. Is it feasible to extract the signal using coherent > > averaging? > > Is this your setup: there exists a (real) vector > > x = [x1,x2, ... x_N] > > of length N, and you have, say, M measurements > > y_k = x + e_k, 1 <= k <= M, > > where the M vectors e_k are non-correlated and non-zero mean noise? Yes, this is the case. Do you know the mean value E[ e_{k,l} ], > where e_{k,l} is the l-th coordinate in the k-th error vector? Generally speaking, we do not know E[ e_{k,l} ]. However, we do have a validation data set for which we can evaluate the mean value. > > We have a reasonable model of the signal pattern (but not of the signal > > peak to peak). Can we use this model to improve the signal extraction? > > One should think so. What is the model? The model is a time history correlated with the pattern of the signal (I'm not sure I understood your question...). > > We are now using the correlation within model and the measure as weight > > in the averaging procedure. > > > Regards, > Andor Thanks, Valentina
From: Andor on 30 Aug 2006 17:01
vale wrote: > Andor wrote: > > vale wrote: > > > Hi all, > > > > > > we are dealing with N synchronous measures of a spatial signal > > > corrupted with a non-zero mean noise. Those measures are mutually > > > uncorrelated. Is it feasible to extract the signal using coherent > > > averaging? > > > > Is this your setup: there exists a (real) vector > > > > x = [x1,x2, ... x_N] > > > > of length N, and you have, say, M measurements > > > > y_k = x + e_k, 1 <= k <= M, > > > > where the M vectors e_k are non-correlated and non-zero mean noise? > > Yes, this is the case. > > Do you know the mean value E[ e_{k,l} ], > > where e_{k,l} is the l-th coordinate in the k-th error vector? > > Generally speaking, we do not know E[ e_{k,l} ]. > However, we do have a validation data set for which we can evaluate the > mean value. > > > > > We have a reasonable model of the signal pattern (but not of the signal > > > peak to peak). Can we use this model to improve the signal extraction? > > > > One should think so. What is the model? > > The model is a time history correlated with the pattern of the signal > (I'm not sure I understood your question...). I didn't understand your original statement, so that makes two of us. What do mean by "signal pattern" and not having a model peak-to-peak ? A reasonable first step is to subtract the mean (or at least your available estimate thereof). Now depending on the distribution of the noise, you are essentially faced with estimating N locations (the x_l, 1 <= l <= N). Averaging is the maximum likelihood estimator if the residual zero-mean noise is Gaussian (and, of course, uncorrelated in the k-direction, but you said that this is the case). If the noise follows another distribution, then you might want to use another estimator. For example, in case of uniform noise, the maximum likelihod estimator would be the average of the maximum and the minimum, along the k-direction. Etc. Perhaps the signal vector x has some internal correlation that could be used for a further estimation step (Kalman smoothing, for example). Just thinking out loud. Regards, Andor |