On Feb 10, 3:21 pm, Boris Gourévitch <bo...@SUS-AU-SPAMpi314.net> wrote: > Hi, >
Hi, I'm not entirely sure I understand your question, so if my reply doesn't make any sense then please ignore it.
> I'm working in signal processing and I would like to build a function > f(t) (t is time) defined on R+, bounded on [a,b], derivable
I'm wondering if you might mean "differentiable" rather than "derivable"...
> and which > verifies Probability(f(t) is in [c,d])=(d-c), [c,d] being included in > [a,b].
Do you mean f(t) at a uniformly randomly chosen instant in time? I'm not sure it's logically possible to choose such an instant unless you constrain t to lie in a finite time interval -- though you can sidestep this issue by using a periodic function. Also, probability is a number between zero and one, so I'm wondering if you mean the probability is *proportional* to d - c, rather than equal to it.
I'm guessing that your probability requirements are such that a sawtooth wave would work, except that's no good to you because it's not smooth. I don't see that a smooth bounded periodic function defined for all t > 0 is possible: f(t) will inevitably be biased towards the points of maximum and minimum when arbitrarily small intervals [c,d] are considered. For the same reason, I don't see that a smooth function defined over a finite timespan (with t chosen uniformly randomly from that timespan) will be possible either, except for the trivial case of a linear function. The only way I can see is to somehow contrive to have a maximum or minimum at every value within [a,b] (for some t) and make the limiting probability over t = 0 to t = T come right as T -> infinity, if that's possible (I don't know if it is). But the probabilities wouldn't be right with t chosen from any finite time interval.
> in other words, I would like a function that meets each point of > [a,b] with the same probability, If I express it correctly. > > An additional constraint on f would be that its frequential spectrum > (with a given sampling frequency fe) does not contain values above a > given value w. Thus, f is smoothed is a sense: > > The idea I'm currently trying is to get uniform values between a and b > at a rate w, then interpolate between the points to reach a sampling > frequency fe. I'm not completely satisfied because the final curve > reaches extreme values less frequently than central values (its > distribution is a mode, I would like it to be flat). Have you got some > ideas or isn't it clear ? > Thanks in advance, > > Boris