Date: Feb 10, 2009 8:52 PM Author: matt271829-news@yahoo.co.uk Subject: Re: function of uniform probability 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