Message from discussion Relationship between a functions derivitives and BW?
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Subject: Re: Relationship between a functions derivitives and BW?
Date: Tue, 06 Nov 2012 14:21:36 -0600
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>>>>Quick question for you guys...
>>>>Is there any relationship between constraining a functions derivitives
>>>>constraining its bandwith?
>>>>For example, if we were to consider the output of a lowpass random
>>>>process... call it x_lp[n]. One could argue that that sequence must be
>>>>'smooth' by some account. (how smooth has to do with the lowpass
>>>>of the process)
>>>>x_lp[n] 'smoothness' should also be able to be described by the
>>>>derivitives either existing or also being continuous. Is there a
>>>>thanks in advance
>>>There is a fundamental inequality, called Bernstein's inequality, that
>>>seems to be what you are looking for. If a signal s(t) has two-sided
>>>bandwidth B and is bounded on the time domain by a constant A>0, then
>>> |s'(t)|<= pi * A B/2.
>>>The signal attaining the bound is A*sin(pi*B*t+phi), where phi is any
>>>phase. For the n-th order derivative, it is
>>> |s^(n)(t)|<= (pi*B/2)^(n-1) A.
>>>I hope this helps.
>>Ups, the last inequality is
>>|s^(n)(t)|<= (pi*B/2)^n A.
>Sorry, another correction. The inequalities are
>|s'(t)|<= pi * A B/2
>|s^(n)(t)|<= (pi*B)^n A.
I will explore this.
There was a little talk as to whether this result only holds for the
continuous case or not. Does this result hold for the discrete case also
where i can conclude x[n+1] - x[n] is bounded by (pi * A B/2)??
If so, this is a pretty neat result.