Discrete functions - Pure function domain

*To*: mathgroup at smc.vnet.net*Subject*: [mg104403] Discrete functions - Pure function domain*From*: Rui <rui.rojo at gmail.com>*Date*: Fri, 30 Oct 2009 02:20:20 -0500 (EST)

I have been trying to implement a function to compute the continuous (finite) to discrete fourier transform, and I've run into some trouble, I wanna know your opinions as to a nice way to implement this. Ideally, with it you can get a formula, if its possible, and if not it should still be useful to get the values. Ideally also, it also tells the user the frequency domain for which the transformed function is valid (-Infinity,..., -2/P, -1/P, 0, 1/P, ...., Inifitny) Basically, it's just this: CDFT[f_, P_]:= (* P would be the length of the time function domain and f is the function itself*) ( Return[ (Integrate[f[t] * E^(-I 2 \[Pi] # t), {t, 0, P}]//FullSimplify)& ] ) That seemed to work but the function returned is valid for all f, and it doesn't even tell the user where it should be valid. I then added a Print line. Print["frequency \[Element] Z(", TraditionalForm[(P)^-1], ")"]; It seemed to work more or less ok, but I run into trouble in a case where the CCFT[0] gave me indetermination, infinity, blah, where it should give a number. That's when I added a Limit CDFT[f_, P_]:= ( Print["frequency \[Element] Z(", TraditionalForm[(P)^-1], ")"]; Return[ (Limit[Integrate[f[t] * E^(-I 2 \[Pi] r t), {t, 0, P}]//FullSimplify, r->#])& ] ) It's getting uglier and uglier :P. Anyway, CDFT[0] gives a good result now but if CDFT[t] gets evaluated first and then replaced t->0, I run again into the same trouble. And if I want, for example, to plot it, I get a Print message for every time it calls CDFT, which is not good. However, I don't know how else to tell the user of the allowed frequencies (returning a list with the result and the message seems very ugly). I would restrict the domain of the pure function with /; if I knew how. Particularly, I've been working with one period [0, 24] of the function x[t_] := (UnitStep[T + 6] - UnitStep[T])*(T + 6) + (UnitStep[T] - UnitStep[T - 6])*(6 - T) /. T -> Mod[t, 24, -12] ; Rui Rojo