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
Relationship between a functions derivitives and BW?
Started by ●October 31, 2012
Reply by ●October 31, 20122012-10-31
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 advanceI 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
Reply by ●October 31, 20122012-10-31
>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..
Reply by ●October 31, 20122012-10-31
>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..
Reply by ●October 31, 20122012-10-31
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
Reply by ●October 31, 20122012-10-31
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 advanceHave 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.
Reply by ●October 31, 20122012-10-31
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
Reply by ●October 31, 20122012-10-31
>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 functionsderivitives>>>>> 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, whichisn'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.
Reply by ●October 31, 20122012-10-31
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
Reply by ●November 1, 20122012-11-01
>> 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).
