On 2013-05-01, Fons Adriaensen wrote:
As a technical nitpick, they are too: if you just shift a signal in
time, the output will be a shifted version of the original.
That corresponds to a very weak definition of 'time-invariant'...
I didn't think there could be but one: either it varies in time or it
does not. Of course you can always talk about "time-invariance" in
intuitive terms, and that's fine for most kinds of technical work. But
when you get down to it, eventually you have to operationalize your
concept and know which formal, mathematical properties it has.
Preferably those properties will also be somehow orthogonal to the rest
of the concepts you use, such as linearity (or, actually,
homogeneity+additivity, which linearity means).
My definition simply says an operator F from some suitable closed class
of operators over functions over an additive group is shift invariant,
if for all f belonging to the domain of F, f(x+a)==F(f)(x+a). That is a
useful definition which lets you do general, often useful math with the
property, and it's nicely several from most other properties you might
then use at the same time. In most cases, where you can somehow formally
measure the relative sizes of the sets of operators with the property
against those without, there are vanishingly few operators with the
property, so that the property is pretty restrictive. In the fully
discrete world, I seem to remember there are many more additive
homomorphisms around, for example, than shift invariant ones in this
sense. So it's rather restrictive at least in that sense.
Still, do give me your definition of time-invariant. Perhaps there are
stronger definitions I haven't heard of yet, and which can be useful in
e.g. more fine grained analysis.
--
Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front
+358-50-5756111, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
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