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"The Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain." fr

"The Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain." from https://www.tutorialspoint.com/differentiation-in-z-domain-property-of-z-transform https://www.tutorialspoint.com/differentiation-in-z-domain-p.... I was taught in engineering math that z-domain is the Laplacian discrete equivelent of the s domain, which is continuous and used by EE as well in analog. z^-1 (or z(-1) means last sample. It is commonly used in digital signal processing, FFT etc. s domain is used for (from memory) the operater e^-jw, which is for electrical engineers the transform for use with sine waves, such that impedance is 1/sC for capacitance and sL for inductance. z domain has useful properties like "The differentiation in z-domain property of Z-transform states that the multiplication by n in time domain corresponds to the differentiation in zdomain." I am sure I have made some technical mistakes in the above, but it is how I remember it and they don't impact my ability to apply it for my limited EE needs. As to the question about the valid approach, intuitively I see this, but wondering if there was a formal proof of some kind, or is it taken as given?

Sniffnoy 4y ago
Oh, I see! So it's the Fourier transform (in the abstract sense) from the integers to the circle group, except that then you extend to all of C (well, possibly minus the origin). Interesting! Yeah as someone previously unfamiliar with it, it would have been substantially clearer had you referred explcitly to applying the Z-transform, rather than switching domains, which is the sort of thing that will only make sense to someone who already knows about this. So wait does that mean you weren't actually sure if this approach to that step would work? I assumed you were saying you had a proof, not just outlining an approach you thought would work. (I mean, obviously you didn't have a proof of the whole thing as the overall statement is false, but individual steps might have worked.) But I have to note -- even if the proof works, then if you're applying the Z-transform, then you are once again not sticking to the realm of the discrete! Complex numbers are a continuum matter. So that approach still doesn't yield an integers-only proof! > As to the question about the valid approach, intuitively I see this, but wondering if there was a formal proof of some kind, or is it taken as given? I'm not really sure how to answer this -- what would a formal proof here even consist of? It's easy enough to do it in any instance, but the problem is, how would you even formally state the general principle? Like you could do large classes of statements, certainly. So for instance, if what we're doing is purely algebraic, then you could say, if A and B are algebraic structures, and i:A->B is an injective homomorphism, and S is a set of algebraic equations all of which are always in the image of i, then all of them are always true in A; but of course there's way more types of statements one can make than algebraic equations. So, uh, yeah, one can write down any number of statements like that, but I don't know how you'd formally abstract it into a general principle...?
- Sniffnoy 4y ago
  Or, to say a bit more about this -- I guess the basic principle here is "it's OK to use auxiliaries?" Introducing a larger number system is no different from drawing an auxiliary line in geometry (for instance). There's no rule that the proof of a statement may only contain the entities appearing in the statement! You can introduce whatever auxiliaries you like.

Sniffnoy 4y ago

Oh, I see! So it's the Fourier transform (in the abstract sense) from the integers to the circle group, except that then you extend to all of C (well, possibly minus the origin). Interesting! Yeah as someone previously unfamiliar with it, it would have been substantially clearer had you referred explcitly to applying the Z-transform, rather than switching domains, which is the sort of thing that will only make sense to someone who already knows about this. So wait does that mean you weren't actually sure if this approach to that step would work? I assumed you were saying you had a proof, not just outlining an approach you thought would work. (I mean, obviously you didn't have a proof of the whole thing as the overall statement is false, but individual steps might have worked.) But I have to note -- even if the proof works, then if you're applying the Z-transform, then you are once again not sticking to the realm of the discrete! Complex numbers are a continuum matter. So that approach still doesn't yield an integers-only proof! > As to the question about the valid approach, intuitively I see this, but wondering if there was a formal proof of some kind, or is it taken as given? I'm not really sure how to answer this -- what would a formal proof here even consist of? It's easy enough to do it in any instance, but the problem is, how would you even formally state the general principle? Like you could do large classes of statements, certainly. So for instance, if what we're doing is purely algebraic, then you could say, if A and B are algebraic structures, and i:A->B is an injective homomorphism, and S is a set of algebraic equations all of which are always in the image of i, then all of them are always true in A; but of course there's way more types of statements one can make than algebraic equations. So, uh, yeah, one can write down any number of statements like that, but I don't know how you'd formally abstract it into a general principle...?

Sniffnoy 4y ago
Or, to say a bit more about this -- I guess the basic principle here is "it's OK to use auxiliaries?" Introducing a larger number system is no different from drawing an auxiliary line in geometry (for instance). There's no rule that the proof of a statement may only contain the entities appearing in the statement! You can introduce whatever auxiliaries you like.