Quick question for you guys...

Is there any relationship between constraining a functions derivitives and
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 bandwith
of the process)


x_lp[n] 'smoothness' should also be able to be described by the functions
derivitives either existing or also being continuous.  Is there a straight
forward relationship?

thanks in advance

On Wed, 31 Oct 2012 15:05:10 -0500, westocl wrote:

> Quick question for you guys...
> 
> Is there any relationship between constraining a functions derivitives
> and 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
> bandwith of the process)
> 
> 
> x_lp[n] 'smoothness' should also be able to be described by the
> functions derivitives either existing or also being continuous.  Is
> there a straight forward relationship?
> 
> thanks in advance

I suspect a straightforward relationship, but probably only in a limiting 
sense, i.e. you could make some sort of a statement that said "if the 
spectral content is no more than X at f, then the n'th derivative can be 
no more than y = (X, f).

But it would take me some work to prove that (or disprove it).

-- 
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com

>On Wed, 31 Oct 2012 15:05:10 -0500, westocl wrote:
>
>> Quick question for you guys...
>> 
>> Is there any relationship between constraining a functions derivitives
>> and 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
>> bandwith of the process)
>> 
>> 
>> x_lp[n] 'smoothness' should also be able to be described by the
>> functions derivitives either existing or also being continuous.  Is
>> there a straight forward relationship?
>> 
>> thanks in advance
>
>I suspect a straightforward relationship, but probably only in a limiting

>sense, i.e. you could make some sort of a statement that said "if the 
>spectral content is no more than X at f, then the n'th derivative can be 
>no more than y = (X, f).
>
>But it would take me some work to prove that (or disprove it).
>
>-- 
>My liberal friends think I'm a conservative kook.
>My conservative friends think I'm a liberal kook.
>Why am I not happy that they have found common ground?
>
>Tim Wescott, Communications, Control, Circuits & Software
>http://www.wescottdesign.com
>

Hmmm. well I guess quickly we can say in the limiting case where the
bandwidth goes to zero, the resulting sequence approaches a sinusoid and
any of its derivitives are well known to also be continuous sinusoids.

Not sure what to say in the limiting sense going towards infinity..

>On Wed, 31 Oct 2012 15:05:10 -0500, westocl wrote:
>
>> Quick question for you guys...
>> 
>> Is there any relationship between constraining a functions derivitives
>> and 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
>> bandwith of the process)
>> 
>> 
>> x_lp[n] 'smoothness' should also be able to be described by the
>> functions derivitives either existing or also being continuous.  Is
>> there a straight forward relationship?
>> 
>> thanks in advance
>
>I suspect a straightforward relationship, but probably only in a limiting

>sense, i.e. you could make some sort of a statement that said "if the 
>spectral content is no more than X at f, then the n'th derivative can be 
>no more than y = (X, f).
>
>But it would take me some work to prove that (or disprove it).
>
>-- 
>My liberal friends think I'm a conservative kook.
>My conservative friends think I'm a liberal kook.
>Why am I not happy that they have found common ground?
>
>Tim Wescott, Communications, Control, Circuits & Software
>http://www.wescottdesign.com
>

Hmmm. well I guess quickly we can say in the limiting case where the
bandwidth goes to zero, the resulting sequence approaches a sinusoid and
any of its derivitives are well known to also be continuous sinusoids.

Not sure what to say in the limiting sense going towards infinity..

On Wed, 31 Oct 2012 17:07:47 -0500, westocl wrote:

>>On Wed, 31 Oct 2012 15:05:10 -0500, westocl wrote:
>>
>>> Quick question for you guys...
>>> 
>>> Is there any relationship between constraining a functions derivitives
>>> and 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
>>> bandwith of the process)
>>> 
>>> 
>>> x_lp[n] 'smoothness' should also be able to be described by the
>>> functions derivitives either existing or also being continuous.  Is
>>> there a straight forward relationship?
>>> 
>>> thanks in advance
>>
>>I suspect a straightforward relationship, but probably only in a
>>limiting
> 
>>sense, i.e. you could make some sort of a statement that said "if the
>>spectral content is no more than X at f, then the n'th derivative can be
>>no more than y = (X, f).
>>
>>But it would take me some work to prove that (or disprove it).
>>
>>--
>>My liberal friends think I'm a conservative kook. My conservative
>>friends think I'm a liberal kook. Why am I not happy that they have
>>found common ground?
>>
>>Tim Wescott, Communications, Control, Circuits & Software
>>http://www.wescottdesign.com
>>
>>
> Hmmm. well I guess quickly we can say in the limiting case where the
> bandwidth goes to zero, the resulting sequence approaches a sinusoid and
> any of its derivitives are well known to also be continuous sinusoids.
> 
> Not sure what to say in the limiting sense going towards infinity..

Dayum.  I think I was wrong.

Man, there's got to be some statement that you can make -- but it 
probably involves the ratio of various orders of derivatives, which isn't 
going to be "simple".

OTOH, it sure seems like there's a relationship in there, to be teased 
out and pondered on.

-- 
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com

On Wednesday, October 31, 2012 1:05:11 PM UTC-7, westocl wrote:
> Quick question for you guys... Is there any relationship between constraining a functions
derivitives and 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 bandwith of the process) x_lp[n]
'smoothness' should also be able to be described by the functions derivitives either existing or
also being continuous. Is there a straight forward relationship? thanks in advance

Have you looked at the time derivative property of Fourier transforms?:

F[dx(t)/dt] = jwX(jw)

where x(t) is your time domain sequencem and X(jw) is your Fourier transform of the time domain
sequence. I understand the angle you're considering, and I'm not sure if a simple relationship
exists, but perhaps looking at it this way may shed some light.

Am 31.10.2012 23:23, schrieb Tim Wescott:
>>> On Wed, 31 Oct 2012 15:05:10 -0500, westocl wrote:
>>>
>>>> Quick question for you guys...
>>>>
>>>> Is there any relationship between constraining a functions derivitives
>>>> and constraining its bandwith?

[snip]

> Man, there's got to be some statement that you can make -- but it 
> probably involves the ratio of various orders of derivatives, which isn't 
> going to be "simple".
> 
> OTOH, it sure seems like there's a relationship in there, to be teased 
> out and pondered on.

I hope you all are aware of the fact that the derivative can be seen as
a filtered signal where low frequencies jave been attenuated, high
frequencies have been amplified and the phase has been shifted by 90°
over all frequencies.

  d a*sin(f*t) / dt = a*f*cos(f*t)

sin->cos: 90° phase shift
a->a*f: frequency dependent amplitude response

Cheers!
SG

>Am 31.10.2012 23:23, schrieb Tim Wescott:
>>>> On Wed, 31 Oct 2012 15:05:10 -0500, westocl wrote:
>>>>
>>>>> Quick question for you guys...
>>>>>
>>>>> Is there any relationship between constraining a functions
derivitives
>>>>> and constraining its bandwith?
>
>[snip]
>
>> Man, there's got to be some statement that you can make -- but it 
>> probably involves the ratio of various orders of derivatives, which
isn't 
>> going to be "simple".
>> 
>> OTOH, it sure seems like there's a relationship in there, to be teased 
>> out and pondered on.
>
>I hope you all are aware of the fact that the derivative can be seen as
>a filtered signal where low frequencies jave been attenuated, high
>frequencies have been amplified and the phase has been shifted by 90°
>over all frequencies.
>
>  d a*sin(f*t) / dt = a*f*cos(f*t)
>
>sin->cos: 90° phase shift
>a->a*f: frequency dependent amplitude response
>
>Cheers!
>SG
>


Ok, Ok, we're getting there... but somebody needs to make that statement
that pulls this all together..

Lets say we are looking at a lowpass sequence that is looking nice and
smooth and then it suddenly jumps to a strangely high or low value. From a
bandwidth prospective, i can easily say... 'That looks wrong because that
signal doesnt have that kind of bandwidth to jump like that'.

I want to be able to say, 'that looks wrong because that signal has
continuous second and third derivitives' (or something like that).  

There has to be a connection between the two.

Just thinking out loud here .....

If you have a filter with a given impulse response h(n) and an input which is bounded between
plus and minus 1 then the highest output value possible is reached with an input sequence of +/-
1's equal to sgn(h(n)).  This max output will occur when the input sequence aligns with the
impulse response at the point where all the products of coefficient times data gives a positive
number.

So suppose you takes the derivative of h(n) using 1- Z^-1 and then take the sgn() of this. Then
you will get the highest value possible at the output of the derivative filter, which MIGHT mean
that if you apply this same sequence to the filter h(n) you will get the highest derivative
possible at one pair of output points.

Again I haven't really thought this through very well so I'm probably wrong. 

Bob

>> Just thinking out loud here .....

me too...
From the filter argument, it seems like also the derivatives aren't allowed
to contain any energy beyond the cutoff frequency for a band-limited
signal. 
Never looked at it this way.

There are many methods that match derivatives of a waveform to suppress
high-frequency energy (Lagrange interpolation or N-continuous OFDM come to
mind).