Hello, it seems that every data normalized adaptive filter or with error non-linearity obeys a kind of an energy conservation relation. This relation states that the norm of weight-error vector after the update plus the norm of the apriori estimation error equals the norm of the weight-error vector before the update plus the norm of the aposteriori estimation error. No cross-terms appear! Due to the fact that no cross-terms appear, this relation is the starting point for the transient and steady-state analysis of an adaptive filter. I was wondering: is the existence of this energy conservation relation a convenient coincedence - an outcome of luck, or is it underpinned by some fundamental law of adaptive filtering (data normalized or with error non-linearities)? For example the energy conservation relation in Newton mechanics is a universal law and we expect it to hold everywhere. This is a law of nature. Is the energy conservation relation of adaptive filters also a law of their nature? Is it possible for an adaptive filter (data normalized or with error non-linearity) not to obey an energy conservation relation? Manolis

# Energy Conservation Relation in Adaptive Filtering

Started by ●April 11, 2008

Reply by ●April 11, 20082008-04-11

On 11 Apr, 08:21, "Manolis C. Tsakiris" <el01...@mail.ntua.gr> wrote: ...> I was wondering: is the existence of this energy conservation relation a > convenient coincedence - an outcome of luck, or is it underpinned by some > fundamental law of adaptive filtering (data normalized or with error > non-linearities)?First, the 'energy' of a zero mean discrete time process is the same as the variance of the same sequence of number. The key, then, is to understand how one is allowed to sum variances of different variables while discarding the cross terms. If one computes the variance of the sum of two random variables one must in general account for the cross terms. If the variables are known to be independent, the cross terms disappear. So the energies (or variances) of the data are directly summable because the random variables involved are mutually independent. Rune

Reply by ●April 11, 20082008-04-11

Manolis C. Tsakiris wrote:> it seems that every data normalized adaptive filter or with error > non-linearity obeys a kind of an energy conservation relation. This > relation states that the norm of weight-error vector after the update plus > the norm of the apriori estimation error equals the norm of the > weight-error vector before the update plus the norm of the aposteriori > estimation error. No cross-terms appear! Due to the fact that no > cross-terms appear, this relation is the starting point for the transient > and steady-state analysis of an adaptive filter. > > I was wondering: is the existence of this energy conservation relation a > convenient coincedence - an outcome of luck, or is it underpinned by some > fundamental law of adaptive filtering (data normalized or with error > non-linearities)?As Rune Alnor noted, this has to do with the mutial independence of the variables.> For example the energy conservation relation in Newton mechanics is a > universal law and we expect it to hold everywhere. This is a law of nature.This is a common misconception. There is no such law of nature. The "energy" is the artificially selected functional which is supposed to be invariant in a particular system. The "conservation of energy" is just a technical trick to solve the mechanical problems.> Is the energy conservation relation of adaptive filters also a law of their > nature? Is it possible for an adaptive filter (data normalized or with > error non-linearity) not to obey an energy conservation relation?Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com

Reply by ●April 11, 20082008-04-11

"Vladimir Vassilevsky" <antispam_bogus@hotmail.com> wrote in message news:AaKLj.808$7Z2.404@newssvr12.news.prodigy.net...> > > Manolis C. Tsakiris wrote: > >> it seems that every data normalized adaptive filter or with error >> non-linearity obeys a kind of an energy conservation relation. This >> relation states that the norm of weight-error vector after the update >> plus >> the norm of the apriori estimation error equals the norm of the >> weight-error vector before the update plus the norm of the aposteriori >> estimation error. No cross-terms appear! Due to the fact that no >> cross-terms appear, this relation is the starting point for the transient >> and steady-state analysis of an adaptive filter. >> >> I was wondering: is the existence of this energy conservation relation a >> convenient coincedence - an outcome of luck, or is it underpinned by some >> fundamental law of adaptive filtering (data normalized or with error >> non-linearities)? > > As Rune Alnor noted, this has to do with the mutial independence of the > variables. > > >> For example the energy conservation relation in Newton mechanics is a >> universal law and we expect it to hold everywhere. This is a law of >> nature. > > This is a common misconception. There is no such law of nature. The > "energy" is the artificially selected functional which is supposed to be > invariant in a particular system. The "conservation of energy" is just a > technical trick to solve the mechanical problems. > >> Is the energy conservation relation of adaptive filters also a law of >> their >> nature? Is it possible for an adaptive filter (data normalized or with >> error non-linearity) not to obey an energy conservation relation? > > > Vladimir Vassilevsky > DSP and Mixed Signal Design Consultant > http://www.abvolt.comIn other contexts, we'd refer to "the model" and it would be understood that it may have simplifications for the sake of practicality and ease of analysis. A common model for conservation of energy in mechanical systems is a frictionless model - in an electrical analog, no "resistors", eh? Fred