You guys don't really have a "math" forum, so I wasn't quite sure where to put this question.
I've only taken a passing glance at fractional calculus. I once thought it might help solve a problem of mine. When it didn't, I moved on. Has anyone applied fractional calculus, or does it remain a mathematical curiosity?
I've posted some comments regarding this before and though there aren't a lot of immediate applications, there are still some possibilities and interesting ways that the concepts could be applied to potentially link other areas of mathematics together.
Here are just a couple thoughts regarding possible uses and an insight into how it could be viewed as a link between some fields in mathematics:
1) A potential application that could be useful would be with respect to linear systems. A linear filter can construct various slopes of sensitivities between frequencies that approach limits that are integer powers of the ratio of two frequencies.
For example, a simple lowpass filter, could be used to model motion in a mechanical system or applied to audio processing for effects, though the natural 'fall off' for a single lowpass filter is ~6dB per octave (basically, doubling the frequency reduces the amplitude to 1/2 it's initial amplitude). Multiple such filters can be staged or combined to create quite a variety of characteristics in the transition region, but these all ultimately approach a decay that's simply an integer power of a single stage (so a "two-pole" filter will approach ~12dB per octave drop, or a 1/4th attentuation of the amplitude when the frequency is doubled).
This fall off or gain is also seen when a sine is integrated and the derivative of a sine, where f is proportional to the frequency is:
sin'(f*t)=f*cos(f*t)
So the amplitude increases proportional to the frequency (a single pole "highpass" filter), but replacing this with a process generating an arbitrary fractional dB/octave filter structure would allow for some quite novel filter structures to be created (though this may be possible to do with infinite impulse response filters, and it may only be approximateable within a limited frequency range with finite impulse response techniques or fourier transforms).
Anyway, applying fractional derivatives to a signal allows for intermediate forms of filters to be constructed and this could be used to generate novel audio and graphic algorithms.
2) In physics, a wave can be seen as a rotation in phase of some contrast for a physical measurement occuring at a single location in space. Each measurement is made as a contrast between two states/values, this is similar to recursive differentiation over time as each measurement becomes the new reality that is again resampled and effectively differentiated to become the next moment etc. This creates an evolving wavefront over time similar to this:
1 0 0 0 0 0 0
1 -1 0 0 0 0 0
1 -2 1 0 0 0 0
1 -3 3 -1 0 0 0
1 -4 6 -4 1 0 0
...
This approaches a gaussian function ("bell curve"/normal distribution http://mathworld.wolfram.com/NormalDistribution.html) which propogates in one direction, but recognize that if we use a fractional derivative we can create a virtual "wavepacket" that can proprogate in either direction in time and at fractional velocities. Notice also that this object grows in width in one direction and shrinks in the other, hence can be seen to arise from a particle and become a wave with a wavefunction that matches a gaussian probability distribution.
Sidenote: A normal distribution or gaussian function is quite interesting in that it arises very naturally and includes the irrational sqrt(pi), (notice that a 2-D diffusion makes this area a dependent upon pi) and you have an exponential (which is likely associated with strong binding forces in physics) and a 1/d^2 term which could be seen quite similar to a diffusion that gives 1/d^2 fallout in forces. This form also provides a close link between sinusoidal waves (e^it) and statistical diffusion (e^(-t^2)), also notice that a gaussian retains a gaussian form when converted between time and frequency domain (again a good wave/particle duality). If we square the oscillatory amplitude at each point, we again arrive at a gaussian form and if we then once again recursively differentiate this, we see it shift, expand and becomes an oscillatory gaussian wavepacket that frequency shifts toward higher/bluer frequencies! Notice also that detailed measurements of mass reveal a gaussian probability function. (Also notice that the original recursive differentiation gives us Pascal's Triangle, which once again correlates to many atomic properties http://milan.milanovic.org/math/english/atom/electron.html)
From this perspective, a fractional derivative would allow for something similar to a slower than "lightspeed" motion within this system to occur, so motions resembling masses in this system should arise from such fractional differential computations (which can result in quite complex phenomenon because these can be effectively polynomial roots, this leads to complex structures in number theory).
3) In order to see a link here to other areas of mathematics, notice this:
sin'(f*t)=f*cos(f*t)
sin'(f*t)=f*sin(f*t+pi/2)
A fractional derivative could be generalized to (where d is the fractional derivative):
sin'(d)(f*t)=(f^d)*sin(f*t+d*pi/2)
Notice that every derivative rotates the signal 90 degs and amplifies the signal by a power curve. Integration would then become values of d<0.
Hence we can combine a fourier transform along with a scaling and rotation of the measured amplitudes and then apply an inverse fourier transform to emulate fractional derivatives (notice this could also be done using a finite sample of a function to approximate a fractional derivative that might not be otherwise computable, so for example, there are some functions that aren't particularly amenable to being integrated or differentiated, and this would appear aggrevated further for fractional derivatives, but these can still be approximated to some required finite accuracy if points of these functions can be computed as we can then apply a fourier transform and perform arbitrary filtering equivalent to transformations in calculus upon them).
----------------------------------------------------------------------
After all is said and done, what would allow fractional calculus to be used much more easily and potentially correlate to real physical processes would be to find a mechanism that allows a small finite system to actually computed such a transformation precisely. Off hand, there don't appear to be any classical physical systems capable of performing a fractional derivative or integral etc., but I might be missing something here and also quantum mechanics may allow for such a system to be statistically emulated (I'm just tossing out some ideas).
Here are just a couple thoughts regarding possible uses and an insight into how it could be viewed as a link between some fields in mathematics:
1) A potential application that could be useful would be with respect to linear systems. A linear filter can construct various slopes of sensitivities between frequencies that approach limits that are integer powers of the ratio of two frequencies.
For example, a simple lowpass filter, could be used to model motion in a mechanical system or applied to audio processing for effects, though the natural 'fall off' for a single lowpass filter is ~6dB per octave (basically, doubling the frequency reduces the amplitude to 1/2 it's initial amplitude). Multiple such filters can be staged or combined to create quite a variety of characteristics in the transition region, but these all ultimately approach a decay that's simply an integer power of a single stage (so a "two-pole" filter will approach ~12dB per octave drop, or a 1/4th attentuation of the amplitude when the frequency is doubled).
This fall off or gain is also seen when a sine is integrated and the derivative of a sine, where f is proportional to the frequency is:
sin'(f*t)=f*cos(f*t)
So the amplitude increases proportional to the frequency (a single pole "highpass" filter), but replacing this with a process generating an arbitrary fractional dB/octave filter structure would allow for some quite novel filter structures to be created (though this may be possible to do with infinite impulse response filters, and it may only be approximateable within a limited frequency range with finite impulse response techniques or fourier transforms).
Anyway, applying fractional derivatives to a signal allows for intermediate forms of filters to be constructed and this could be used to generate novel audio and graphic algorithms.
2) In physics, a wave can be seen as a rotation in phase of some contrast for a physical measurement occuring at a single location in space. Each measurement is made as a contrast between two states/values, this is similar to recursive differentiation over time as each measurement becomes the new reality that is again resampled and effectively differentiated to become the next moment etc. This creates an evolving wavefront over time similar to this:
1 0 0 0 0 0 0
1 -1 0 0 0 0 0
1 -2 1 0 0 0 0
1 -3 3 -1 0 0 0
1 -4 6 -4 1 0 0
...
This approaches a gaussian function ("bell curve"/normal distribution http://mathworld.wolfram.com/NormalDistribution.html) which propogates in one direction, but recognize that if we use a fractional derivative we can create a virtual "wavepacket" that can proprogate in either direction in time and at fractional velocities. Notice also that this object grows in width in one direction and shrinks in the other, hence can be seen to arise from a particle and become a wave with a wavefunction that matches a gaussian probability distribution.
Sidenote: A normal distribution or gaussian function is quite interesting in that it arises very naturally and includes the irrational sqrt(pi), (notice that a 2-D diffusion makes this area a dependent upon pi) and you have an exponential (which is likely associated with strong binding forces in physics) and a 1/d^2 term which could be seen quite similar to a diffusion that gives 1/d^2 fallout in forces. This form also provides a close link between sinusoidal waves (e^it) and statistical diffusion (e^(-t^2)), also notice that a gaussian retains a gaussian form when converted between time and frequency domain (again a good wave/particle duality). If we square the oscillatory amplitude at each point, we again arrive at a gaussian form and if we then once again recursively differentiate this, we see it shift, expand and becomes an oscillatory gaussian wavepacket that frequency shifts toward higher/bluer frequencies! Notice also that detailed measurements of mass reveal a gaussian probability function. (Also notice that the original recursive differentiation gives us Pascal's Triangle, which once again correlates to many atomic properties http://milan.milanovic.org/math/english/atom/electron.html)
From this perspective, a fractional derivative would allow for something similar to a slower than "lightspeed" motion within this system to occur, so motions resembling masses in this system should arise from such fractional differential computations (which can result in quite complex phenomenon because these can be effectively polynomial roots, this leads to complex structures in number theory).
3) In order to see a link here to other areas of mathematics, notice this:
sin'(f*t)=f*cos(f*t)
sin'(f*t)=f*sin(f*t+pi/2)
A fractional derivative could be generalized to (where d is the fractional derivative):
sin'(d)(f*t)=(f^d)*sin(f*t+d*pi/2)
Notice that every derivative rotates the signal 90 degs and amplifies the signal by a power curve. Integration would then become values of d<0.
Hence we can combine a fourier transform along with a scaling and rotation of the measured amplitudes and then apply an inverse fourier transform to emulate fractional derivatives (notice this could also be done using a finite sample of a function to approximate a fractional derivative that might not be otherwise computable, so for example, there are some functions that aren't particularly amenable to being integrated or differentiated, and this would appear aggrevated further for fractional derivatives, but these can still be approximated to some required finite accuracy if points of these functions can be computed as we can then apply a fourier transform and perform arbitrary filtering equivalent to transformations in calculus upon them).
----------------------------------------------------------------------
After all is said and done, what would allow fractional calculus to be used much more easily and potentially correlate to real physical processes would be to find a mechanism that allows a small finite system to actually computed such a transformation precisely. Off hand, there don't appear to be any classical physical systems capable of performing a fractional derivative or integral etc., but I might be missing something here and also quantum mechanics may allow for such a system to be statistically emulated (I'm just tossing out some ideas).
1) I get it.
2) You lost me. But wait for me to read it again before you expend any effort. Maybe it will come to me.
3) This sounds close to what I pondered for a time.
1) Noting that a fractional derivative equates to a phase shift for a sine function is what helped make this click for me early on. However, I never connected it to the filtering effects of integration. That's a cool idea.
I can't think how I would apply it, but I've had problems in the past with traditional filters not giving me what I wanted. The next time I have a problem, I might give this a try. Any papers on the subject so I can give proper credit if and when?
3) Basically, I was looking for new ways to integrate functions that can only be done numerically using traditional calculus. I hoped fractional calculus might give me some new insights into those functions. But, like you said, it actually seemed to make things worse.
Even more specific, a simple way to look at the phase plane of the system I was modelling is with x' = sqrt(1-x^4). The idea that fractional derivatives create a phase shift led me to wonder what I could do with this equation. But I never found anything that helped me.
2) You lost me. But wait for me to read it again before you expend any effort. Maybe it will come to me.
3) This sounds close to what I pondered for a time.
1) Noting that a fractional derivative equates to a phase shift for a sine function is what helped make this click for me early on. However, I never connected it to the filtering effects of integration. That's a cool idea.
I can't think how I would apply it, but I've had problems in the past with traditional filters not giving me what I wanted. The next time I have a problem, I might give this a try. Any papers on the subject so I can give proper credit if and when?
3) Basically, I was looking for new ways to integrate functions that can only be done numerically using traditional calculus. I hoped fractional calculus might give me some new insights into those functions. But, like you said, it actually seemed to make things worse.
Even more specific, a simple way to look at the phase plane of the system I was modelling is with x' = sqrt(1-x^4). The idea that fractional derivatives create a phase shift led me to wonder what I could do with this equation. But I never found anything that helped me.
QUOTE (Resha Caner+)
1) I get it.
Great to hear, from your post it sounds like you have some familiarity with filters and possibly digital signal processing. Nice work
Oh, I made a mistake in one comment, I meant to that that I don't believe a linear infinite impulse response filter can approach anything other than the ~6dB/octave per pole asymptotic limit and neither with a finite impulse response filter (though again, you can generate many transition band approximations to these), and even though a windowed fourier can approximate many of these, it's still limited to generating nothing more than a finite impulse response filter can, so as far as I know (though I admit this is just from my familiarity with the subject and I haven't delved into this too deeply ... I do believe that if you're working with a single frequency though, or potentially a narrow band signal, there may be some adaptive ways of emulating such a filter and then filtering out the non-linear harmonics generated - basically you could construct a narrow frequency sensitive compressor and have the compression be a function of the center frequency - not idea, but at least it's something).
Great to hear, from your post it sounds like you have some familiarity with filters and possibly digital signal processing. Nice work
Oh, I made a mistake in one comment, I meant to that that I don't believe a linear infinite impulse response filter can approach anything other than the ~6dB/octave per pole asymptotic limit and neither with a finite impulse response filter (though again, you can generate many transition band approximations to these), and even though a windowed fourier can approximate many of these, it's still limited to generating nothing more than a finite impulse response filter can, so as far as I know (though I admit this is just from my familiarity with the subject and I haven't delved into this too deeply ... I do believe that if you're working with a single frequency though, or potentially a narrow band signal, there may be some adaptive ways of emulating such a filter and then filtering out the non-linear harmonics generated - basically you could construct a narrow frequency sensitive compressor and have the compression be a function of the center frequency - not idea, but at least it's something).
QUOTE (Resha Caner+)
2) You lost me. But wait for me to read it again before you expend any effort. Maybe it will come to me.
Don't worry about it too much. I was messing around with some of these ideas for quite a while and truly there's a lot of background material embedded within these comments, so I wouldn't expect these ideas to click into place immediately, but it's just that a fractional derivative could potentially be applied to a system similar to a discrete quantum one, with waves being composed of discrete states, and then using fractional derivatives, we could possibly emulate how these appear to be continuous waves with intermediate states.
Something interesting here is that the gain term, f in f*cos(f*t) would become a power function for intermediate derivatives and we'd have (f^d)*sin(f*t+pi*d/2) and that the f^d term would be irrational for most values of f and d, though it's interesting to consider that not all fractional powers would create an f^d that's irrational and those rational points, such as (4^1/2)=2 could possibly represent discrete physical processes.
Something else quite interesting here is that the additional attentuation or amplification of various frequencies relative to each other gives a manner to estimate the fine scale properties of matter by looking at the amplitudes in a spectrum. For example, black body radiation has a central peak with 2 different rates of asymptotic fallout outside that. You can tell by the approximation "rolloff" how many processes are integrating information and how many are differentiating it (you can apply the same concept to audio or mechnical signals as well and estimate how many and what types of processes are occuring by looking at the rate or slope of attentuation of the filter, so, for example a signal contains few high or low frequencies and grow by 6dB per decade at low frequencies, then decays at a high frequency by ~12dB per decade requires that a single differentiating stage exist between your sample and a white noise signal and that two integrating stages exist (and you can determine relative rates at which each process occurs by looking where the transitions occur, so there's quite a bit of possible detailed information available regarding a mechanical system simply by listening to the effect "timber"/spectrum it generates.
Don't worry about it too much. I was messing around with some of these ideas for quite a while and truly there's a lot of background material embedded within these comments, so I wouldn't expect these ideas to click into place immediately, but it's just that a fractional derivative could potentially be applied to a system similar to a discrete quantum one, with waves being composed of discrete states, and then using fractional derivatives, we could possibly emulate how these appear to be continuous waves with intermediate states.
Something interesting here is that the gain term, f in f*cos(f*t) would become a power function for intermediate derivatives and we'd have (f^d)*sin(f*t+pi*d/2) and that the f^d term would be irrational for most values of f and d, though it's interesting to consider that not all fractional powers would create an f^d that's irrational and those rational points, such as (4^1/2)=2 could possibly represent discrete physical processes.
Something else quite interesting here is that the additional attentuation or amplification of various frequencies relative to each other gives a manner to estimate the fine scale properties of matter by looking at the amplitudes in a spectrum. For example, black body radiation has a central peak with 2 different rates of asymptotic fallout outside that. You can tell by the approximation "rolloff" how many processes are integrating information and how many are differentiating it (you can apply the same concept to audio or mechnical signals as well and estimate how many and what types of processes are occuring by looking at the rate or slope of attentuation of the filter, so, for example a signal contains few high or low frequencies and grow by 6dB per decade at low frequencies, then decays at a high frequency by ~12dB per decade requires that a single differentiating stage exist between your sample and a white noise signal and that two integrating stages exist (and you can determine relative rates at which each process occurs by looking where the transitions occur, so there's quite a bit of possible detailed information available regarding a mechanical system simply by listening to the effect "timber"/spectrum it generates.
QUOTE (ReshaCaner+)
3) This sounds close to what I pondered for a time.
1) Noting that a fractional derivative equates to a phase shift for a sine function is what helped make this click for me early on. However, I never connected it to the filtering effects of integration. That's a cool idea.
Yes, if you plot the gain logarithmically then integration is a straight line decay in amplitude versus frequency (45 deg, if you have the same scaling for both freq and amp) and differentiation grows at a 45 deg angle, fractional derivatives would be intermediate slopes.
1) Noting that a fractional derivative equates to a phase shift for a sine function is what helped make this click for me early on. However, I never connected it to the filtering effects of integration. That's a cool idea.
Yes, if you plot the gain logarithmically then integration is a straight line decay in amplitude versus frequency (45 deg, if you have the same scaling for both freq and amp) and differentiation grows at a 45 deg angle, fractional derivatives would be intermediate slopes.
QUOTE (ReshaCaner+)
I can't think how I would apply it, but I've had problems in the past with traditional filters not giving me what I wanted. The next time I have a problem, I might give this a try.
The most general purpose method would be to use a finite impulse response filter with coefficients generated by an inverse fourier approximation to the response of a fractional derivative (you can even cancel the phase shift components by using this technique, if desired. Most professional audio applications prefer finite impulse response filters because of this ability to control the phase, though you tend to get additional delays, because you can only delay the faster components and not speed up slower phases, simply because low frequencies have a longer wavelength and require a longer time to process, so again you're limited in bandwidth over which you can approximate a fractional filter structure)
The most general purpose method would be to use a finite impulse response filter with coefficients generated by an inverse fourier approximation to the response of a fractional derivative (you can even cancel the phase shift components by using this technique, if desired. Most professional audio applications prefer finite impulse response filters because of this ability to control the phase, though you tend to get additional delays, because you can only delay the faster components and not speed up slower phases, simply because low frequencies have a longer wavelength and require a longer time to process, so again you're limited in bandwidth over which you can approximate a fractional filter structure)
QUOTE (ReshaCaner+)
Any papers on the subject so I can give proper credit if and when?
I don't have any papers written (this stuff is all in my head now) but you can google "FIR filter design" and should find information related to constructing FIR (finite impulse response) filter coefficients to generate semi-arbitrary filter responses (there should be programs around to enter an idealized response, along generally with a preferred window function and it should be able to iterate an approximate set of coefficients for you).
I don't have any papers written (this stuff is all in my head now) but you can google "FIR filter design" and should find information related to constructing FIR (finite impulse response) filter coefficients to generate semi-arbitrary filter responses (there should be programs around to enter an idealized response, along generally with a preferred window function and it should be able to iterate an approximate set of coefficients for you).
QUOTE (ReshaCaner+)
3) Basically, I was looking for new ways to integrate functions that can only be done numerically using traditional calculus. I hoped fractional calculus might give me some new insights into those functions. But, like you said, it actually seemed to make things worse.
There might be some "sweet spot" for a fractional derivative that simplifies some computation. Recognize that you're effective working with a polynomial root -
Let me give an example here. A single pole high pass filter could be described as s(t)-s(t-1), where you're computing a result as the delta/change in amplitude of a signal a some time, t.
An iteratively generated higher dimensional derivative of this would appear as a binomial expansion of coefficents for samples over time. We can use a 'z transform' (http://www.ling.upenn.edu/courses/ling525/z.html) to represent this simpler as a polynomial and then you could potential find roots to this that would allow for some fractional derivative to be computed (though it seems highly unlikely that a general solution would exist but instead some specific rational derivative might exist).
Something to consider here is that there are also differences between a discrete sampled version of a derivative (similar to finite calculus) and a continuous system with infinitesimal calculus (you get a cosine "rolloff" in attentuation, versus an exponential decay for example with a single pole)
There might be some "sweet spot" for a fractional derivative that simplifies some computation. Recognize that you're effective working with a polynomial root -
Let me give an example here. A single pole high pass filter could be described as s(t)-s(t-1), where you're computing a result as the delta/change in amplitude of a signal a some time, t.
An iteratively generated higher dimensional derivative of this would appear as a binomial expansion of coefficents for samples over time. We can use a 'z transform' (http://www.ling.upenn.edu/courses/ling525/z.html) to represent this simpler as a polynomial and then you could potential find roots to this that would allow for some fractional derivative to be computed (though it seems highly unlikely that a general solution would exist but instead some specific rational derivative might exist).
Something to consider here is that there are also differences between a discrete sampled version of a derivative (similar to finite calculus) and a continuous system with infinitesimal calculus (you get a cosine "rolloff" in attentuation, versus an exponential decay for example with a single pole)
QUOTE (ReshaCaner+)
Even more specific, a simple way to look at the phase plane of the system I was modelling is with x' = sqrt(1-x^4).
Actually a discrete sampled computation, with a cosine rolloff appears more likely to match this.
Notice that if we square both sides, we get:
(x')^2=1-x^4
And then we can see this as:
(x')^2=(1)^2-(x^2)^2
(x')^2=(x^0)^2-(x^2)^2
And then recognize that if x^0 and x^2 are sampled at 90 degs to each other, this should give you an x' centered at a 45 deg phase shift between both of these (there are ways of altering the phase of the result too, if necessary).
So basically you'd just be computing the difference between two samples, and you could interpolate half a sample, if desired to delay the result (assuming you're working with a preferably low delay or small computational requirement, realtime system).
Very interesting. Yes, this definitely gives me something to play with. I'm off to give it a try.
But before I go, let me ask this: did you just come up with that now, or is it a standard trick? If you just came up with it, I'm impressed.
The linear case (or a simple circle), would be x' = sqrt(1-x^2). So, I could do the same thing.
(x')^2 = (x^0)^2 - (x^1)^2
I've got some other issues rattling around in my head, so after I chew through this, I'll be back again.
P.S. Is there a way to code in equations rather than texting them?
Very interesting. Yes, this definitely gives me something to play with. I'm off to give it a try.
But before I go, let me ask this: did you just come up with that now, or is it a standard trick? If you just came up with it, I'm impressed.
The linear case (or a simple circle), would be x' = sqrt(1-x^2). So, I could do the same thing.
(x')^2 = (x^0)^2 - (x^1)^2
I've got some other issues rattling around in my head, so after I chew through this, I'll be back again.
I came up with it off the top of my head
(but I have worked with digital signal processing and IIR and FIR filters before in some control and audio applications, so the basic tools to create it I'm already familiar with).
There are a couple other functions you could be interested in as well that could relate here.
If you want a frequency dependent phase shift, you can use a Hilbert transform http://en.wikipedia.org/wiki/Hilbert_transform. This is similar to 90 deg phase shift without the frequency dependent scaling that would occur with an integration (you can use this to compensate for the phase shift induced by time delays). (There are some very novel uses for this function)
If you want to construct interpolations of intermediate points (for example, sampling between two points in a stream), a "sinc function", sin(x)/x, http://en.wikipedia.org/wiki/Sinc_function gives the idealized weighted transform (for an infinite impulse response filter). Notice also that this allows for oversampling a single to a higher rate (and you could utilize this to create the equivalent of x^2 terms from a sample rate constructing x^4 terms - you just interleave real samples with virtual interpolated ones, but you can actually resample a signal to any rational ratio of rate you want).
So between these two functions you can create linear time delays that are even less than a single sample (though the precision of such an interpolation will require the entire group be delayed) and linear phases delays (relative to frequency), as well as rescale time steps and create rational ratios of frequencies between the two representations. Between these two function then, you can both shift a signal in time to any rational/fractional location and ratio you desire and then compensation for the induced linear phase delay that this shifting incurs, using a Hilbert transform.
I admit I don't know the specific application you're working on, but I assume it's a multipole low pass filter in a discrete/time sampled system, that could, for example, be the response of the output stage of a digital to analog convert.
If you can give me more detailed information regarding the specific application realm you're working in, I could probably give you additional thoughts to consider or construct a more detailed example of a solution. (I'm still assuming your x' term is refering to an amplitude response).
Actually a discrete sampled computation, with a cosine rolloff appears more likely to match this.
Notice that if we square both sides, we get:
(x')^2=1-x^4
And then we can see this as:
(x')^2=(1)^2-(x^2)^2
(x')^2=(x^0)^2-(x^2)^2
And then recognize that if x^0 and x^2 are sampled at 90 degs to each other, this should give you an x' centered at a 45 deg phase shift between both of these (there are ways of altering the phase of the result too, if necessary).
So basically you'd just be computing the difference between two samples, and you could interpolate half a sample, if desired to delay the result (assuming you're working with a preferably low delay or small computational requirement, realtime system).
QUOTE (ReshaCaner+)
The idea that fractional derivatives create a phase shift led me to wonder what I could do with this equation. But I never found anything that helped me.
I don't know if any of this helps you, but hopefully there's something there for you to dig into.
I don't know if any of this helps you, but hopefully there's something there for you to dig into.
QUOTE
Actually a discrete sampled computation, with a cosine rolloff appears more likely to match this.
Notice that if we square both sides, we get:
(x')^2=1-x^4
And then we can see this as:
(x')^2=(1)^2-(x^2)^2
(x')^2=(x^0)^2-(x^2)^2
And then recognize that if x^0 and x^2 are sampled at 90 degs to each other, this should give you an x' centered at a 45 deg phase shift between both of these (there are ways of altering the phase of the result too, if necessary).
So basically you'd just be computing the difference between two samples, and you could interpolate half a sample, if desired to delay the result (assuming you're working with a preferably low delay or small computational requirement, realtime system).
Notice that if we square both sides, we get:
(x')^2=1-x^4
And then we can see this as:
(x')^2=(1)^2-(x^2)^2
(x')^2=(x^0)^2-(x^2)^2
And then recognize that if x^0 and x^2 are sampled at 90 degs to each other, this should give you an x' centered at a 45 deg phase shift between both of these (there are ways of altering the phase of the result too, if necessary).
So basically you'd just be computing the difference between two samples, and you could interpolate half a sample, if desired to delay the result (assuming you're working with a preferably low delay or small computational requirement, realtime system).
Very interesting. Yes, this definitely gives me something to play with. I'm off to give it a try.
But before I go, let me ask this: did you just come up with that now, or is it a standard trick? If you just came up with it, I'm impressed.
The linear case (or a simple circle), would be x' = sqrt(1-x^2). So, I could do the same thing.
(x')^2 = (x^0)^2 - (x^1)^2
I've got some other issues rattling around in my head, so after I chew through this, I'll be back again.
P.S. Is there a way to code in equations rather than texting them?
QUOTE (Resha Caner+Mar 5 2008, 02:19 PM)
Very interesting. Yes, this definitely gives me something to play with. I'm off to give it a try.
But before I go, let me ask this: did you just come up with that now, or is it a standard trick? If you just came up with it, I'm impressed.
The linear case (or a simple circle), would be x' = sqrt(1-x^2). So, I could do the same thing.
(x')^2 = (x^0)^2 - (x^1)^2
I've got some other issues rattling around in my head, so after I chew through this, I'll be back again.
I came up with it off the top of my head
(but I have worked with digital signal processing and IIR and FIR filters before in some control and audio applications, so the basic tools to create it I'm already familiar with).There are a couple other functions you could be interested in as well that could relate here.
If you want a frequency dependent phase shift, you can use a Hilbert transform http://en.wikipedia.org/wiki/Hilbert_transform. This is similar to 90 deg phase shift without the frequency dependent scaling that would occur with an integration (you can use this to compensate for the phase shift induced by time delays). (There are some very novel uses for this function)
If you want to construct interpolations of intermediate points (for example, sampling between two points in a stream), a "sinc function", sin(x)/x, http://en.wikipedia.org/wiki/Sinc_function gives the idealized weighted transform (for an infinite impulse response filter). Notice also that this allows for oversampling a single to a higher rate (and you could utilize this to create the equivalent of x^2 terms from a sample rate constructing x^4 terms - you just interleave real samples with virtual interpolated ones, but you can actually resample a signal to any rational ratio of rate you want).
So between these two functions you can create linear time delays that are even less than a single sample (though the precision of such an interpolation will require the entire group be delayed) and linear phases delays (relative to frequency), as well as rescale time steps and create rational ratios of frequencies between the two representations. Between these two function then, you can both shift a signal in time to any rational/fractional location and ratio you desire and then compensation for the induced linear phase delay that this shifting incurs, using a Hilbert transform.
I admit I don't know the specific application you're working on, but I assume it's a multipole low pass filter in a discrete/time sampled system, that could, for example, be the response of the output stage of a digital to analog convert.
If you can give me more detailed information regarding the specific application realm you're working in, I could probably give you additional thoughts to consider or construct a more detailed example of a solution. (I'm still assuming your x' term is refering to an amplitude response).
QUOTE (ReshaCaner+)
P.S. Is there a way to code in equations rather than texting them?
I've seen some people do it, but I don't personally know how. (probably some forum help link has the info)
I've seen some people do it, but I don't personally know how. (probably some forum help link has the info)
Yeah, all this makes me sound like a EE, doesn't it? The EE's have done most of the good signal processing work. Actually, I'm an ME, so I'm working on a machine. My specialty is rotating dynamics (engines and transmissions).
I frequently work with NVH (noise, vibration, and harshness), which means I spend a lot of time processing the signals off microphones and accelerometers. That's where I cross over into EE territory.
I have used the Hilbert transform before, so I'm familiar with that. I've also tried applying some wavelets. Most of the guys I work with use the FFT with some averaging and standard bandpass filters to pick out the events they're looking for.
And that works a good portion of the time.
But, sometimes what they throw away as "noise" isn't necessarily noise. So, I'm looking for other techniques.
I'm using x as the state variable and x' as it's derivative. So, in the case of accelerometers we might use acceleration and velocity.
Or, we've used the pip off a proximity probe to get a displacement signal (or velocity by transforming it with Hilbert).
We engineers can get very creative
. We tend to break a lot of rules.
I frequently work with NVH (noise, vibration, and harshness), which means I spend a lot of time processing the signals off microphones and accelerometers. That's where I cross over into EE territory.
I have used the Hilbert transform before, so I'm familiar with that. I've also tried applying some wavelets. Most of the guys I work with use the FFT with some averaging and standard bandpass filters to pick out the events they're looking for.
And that works a good portion of the time.
But, sometimes what they throw away as "noise" isn't necessarily noise. So, I'm looking for other techniques.
I'm using x as the state variable and x' as it's derivative. So, in the case of accelerometers we might use acceleration and velocity.
Or, we've used the pip off a proximity probe to get a displacement signal (or velocity by transforming it with Hilbert).
We engineers can get very creative
. We tend to break a lot of rules.
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