

Ask HN: Does a discrete time signal have a formula or one definite equation? - sgy

Suppose we had a heart rate signal, can we describe it as one linear function; y = f(x) ? Or extract the function via MATLAB, for instance?
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graeham
Any formula of real-world data would have to be an approximation of some sort.
It is always possible to make some model (function), but the actual question
is how accurate and useful is it.

It isn't possible to answer your question without knowing what the data you
have looks like and what you want to get out of it.

edit: there isn't one (and only one) correct function for _anything_ from real
data. All models are approximations. 'All models are wrong, but some are
useful'.

~~~
sgy
Douglas, let's say we had a sequence of real or integer numbers (y-axis)
against seconds [time] (x-axis). At second 0, I have a corresponding value 29
on the y-axis. At second 1, I have 52 on the y-axis.

I'm not interested in a model that would describe what might be in between the
points, but rather in a function that can return an exact y for a given x.
That is, if we passed 52, we get the 1.

Somehow close to linear interpolation, but this is not what I'm looking for
because, with interpolation, with 100 points the function will be of degree
99.

~~~
ggchappell
> I'm not interested in a model that would describe what might be in between
> the points, but rather in a function that can return an exact y for a given
> x.

Okay, so you want to reconstruct the original data, exactly as received.
Nothing else.

> Somehow close to linear interpolation, but this is not what I'm looking for
> because, with interpolation, with 100 points the function will be of degree
> 99.

It sounds like you're looking for _compression_ : describing the original
signal using less information than in the signal itself. And you want the
compression to be _lossless_ : it should reconstruct the original exactly.

In that case, the answer is "no": lossless compression of an arbitrary signal
is known to be impossible. (It's pretty easy to show, actually.)

But maybe that isn't what you really want? Lossless compression of a signal
like a heartbeat is not something anyone really cares about doing. So, what is
your application? What is it that you really need from this?

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P.S. What you describe as "linear interpolation" is really "fitting a
polynomial".

