So I'm trying to find a good mathematical expression for a unit-length phasor taking a random walk in phase. I imagine there are many ways to do this and was looking for some input. The approach I am taking (if you have a better idea chime in!) is: phasor(t) = exp(j*X*t) where X is a slowly varying, Gaussian random variable. So more of a random frequency... Some things I do know (from my data): * E[phasor] = 0 (in other words--it moves around and is not biased towards any specific phase) * E[||phasor||] = 1 (always unit length) * The analytic signal that this phasor describes has some PSD peaking at DC and decaying fairly rapidly (although not symmetrically at least for finite window measurements) So is there any simple (In the spirit of Occam) model that I might try to fit my observations to? Sorry if this question is ill-posed. Any input would probably help get me on the right direction though, so thanks in advance. -Martin
Random Phase Walk
Started by ●March 29, 2009
Reply by ●March 29, 20092009-03-29
On Mar 30, 3:13�am, "wazerf...@gmail.com" <wazerf...@gmail.com> wrote:> So I'm trying to find a good mathematical expression for a unit-length > phasor taking a random walk in phase. �I imagine there are many ways > to do this and was looking for some input. > > The approach I am taking (if you have a better idea chime in!) is: > phasor(t) = exp(j*X*t) > > where X is a slowly varying, Gaussian random variable. �So more of a > random frequency... > > Some things I do know (from my data): > > * � E[phasor] = 0 (in other words--it moves around and is not biased > towards any specific phase) > * � E[||phasor||] = 1 (always unit length) > * � The analytic signal that this phasor describes has some PSD > peaking at DC and decaying fairly rapidly (although not symmetrically > at least for finite window measurements) > > So is there any simple (In the spirit of Occam) model that I might try > to fit my observations to? > > Sorry if this question is ill-posed. �Any input would probably help > get me on the right direction though, so thanks in advance. > > -MartinNormally we use integrated white noise...
Reply by ●March 29, 20092009-03-29
On Mar 29, 12:54�pm, HardySpicer <gyansor...@gmail.com> wrote:> On Mar 30, 3:13�am, "wazerf...@gmail.com" <wazerf...@gmail.com> wrote: > > > > > So I'm trying to find a good mathematical expression for a unit-length > > phasor taking a random walk in phase. �I imagine there are many ways > > to do this and was looking for some input. > > > The approach I am taking (if you have a better idea chime in!) is: > > phasor(t) = exp(j*X*t) > > > where X is a slowly varying, Gaussian random variable. �So more of a > > random frequency... > > > Some things I do know (from my data): > > > * � E[phasor] = 0 (in other words--it moves around and is not biased > > towards any specific phase) > > * � E[||phasor||] = 1 (always unit length) > > * � The analytic signal that this phasor describes has some PSD > > peaking at DC and decaying fairly rapidly (although not symmetrically > > at least for finite window measurements) > > > So is there any simple (In the spirit of Occam) model that I might try > > to fit my observations to? > > > Sorry if this question is ill-posed. �Any input would probably help > > get me on the right direction though, so thanks in advance. > > > -Martin > > Normally we use integrated white noise...So in this case, X would be the integral of a white gaussian rv?
Reply by ●March 30, 20092009-03-30
On Mar 29, 10:13�am, "wazerf...@gmail.com" <wazerf...@gmail.com> wrote:> So I'm trying to find a good mathematical expression for a unit-length > phasor taking a random walk in phase. �I imagine there are many ways > to do this and was looking for some input. > > The approach I am taking (if you have a better idea chime in!) is: > phasor(t) = exp(j*X*t) > > where X is a slowly varying, Gaussian random variable. �So more of a > random frequency...first it should be integrated somehow as Hardy says. so i think it should be phasor(t) = exp(j*X*t0)*phasor(t-t0) if, in discrete-time, the direction of walk is completely random, then maybe it could be phasor[n] = exp(j*X) * phasor[n-1] but now X is a uniform r.v. with range of [-pi, +pi) r b-j
Reply by ●March 30, 20092009-03-30
On Mar 29, 10:13�pm, robert bristow-johnson <r...@audioimagination.com> wrote:> On Mar 29, 10:13�am, "wazerf...@gmail.com" <wazerf...@gmail.com> > wrote: > > > So I'm trying to find a good mathematical expression for a unit-length > > phasor taking a random walk in phase. �I imagine there are many ways > > to do this and was looking for some input. > > > The approach I am taking (if you have a better idea chime in!) is: > > phasor(t) = exp(j*X*t) > > > where X is a slowly varying, Gaussian random variable. �So more of a > > random frequency... > > first it should be integrated somehow as Hardy says. �so i think it > should be > > � � phasor(t) = exp(j*X*t0)*phasor(t-t0) > > if, in discrete-time, the direction of walk is completely random, then > maybe it could be > > � � phasor[n] = exp(j*X) * phasor[n-1] > > but now X is a uniform r.v. with range of [-pi, +pi) > > r b-jRobert, I like this recursive formulation (although I would still make X a gaussian rather than uniform). How might I should that this has an average of zero? I'll play around with it a bit. Thanks, Martin
Reply by ●March 30, 20092009-03-30
On Mar 30, 12:09�am, "wazerf...@gmail.com" <wazerf...@gmail.com> wrote:> On Mar 29, 10:13�pm, robert bristow-johnson > > > > <r...@audioimagination.com> wrote: > > On Mar 29, 10:13�am, "wazerf...@gmail.com" <wazerf...@gmail.com> > > wrote: > > > > So I'm trying to find a good mathematical expression for a unit-length > > > phasor taking a random walk in phase. �I imagine there are many ways > > > to do this and was looking for some input. > > > > The approach I am taking (if you have a better idea chime in!) is: > > > phasor(t) = exp(j*X*t) > > > > where X is a slowly varying, Gaussian random variable. �So more of a > > > random frequency... > > > first it should be integrated somehow as Hardy says. �so i think it > > should be > > > � � phasor(t) = exp(j*X*t0) * phasor(t-t0) > > > if, in discrete-time, the direction of walk is completely random, then > > maybe it could be > > > � � phasor[n] = exp(j*X) * phasor[n-1] > > > but now X is a uniform r.v. with range of [-pi, +pi) >> > I like this recursive formulation (although I would still make X a > gaussian rather than uniform).depends if you want the direction to be completely random or to depend a little on the previous direction (then the variance on X is small). what scale (variance) would you make your gaussian X? if it's large, does it matter that it wraps around at +/- pi? if you're picking gaussian because you think it makes your life simpler, that won't be the case if it goes into the argument of a sin() or cos() function.> > How might I should that this has an average of zero?what, the phasor or the directional change? r b-j
Reply by ●March 30, 20092009-03-30
Hi all Sorry to jump in the middle of discussion. If I take complex white noise with variance 1 and low pass filter it. Then measure the angle of the resulting o/p and use this is phase rotation. Is this Ok? Thanks. Chintan
Reply by ●March 30, 20092009-03-30
On Mar 30, 12:03�am, robert bristow-johnson <r...@audioimagination.com> wrote:> On Mar 30, 12:09�am, "wazerf...@gmail.com" <wazerf...@gmail.com> > wrote: > > > > > On Mar 29, 10:13�pm, robert bristow-johnson > > > <r...@audioimagination.com> wrote: > > > On Mar 29, 10:13�am, "wazerf...@gmail.com" <wazerf...@gmail.com> > > > wrote: > > > > > So I'm trying to find a good mathematical expression for a unit-length > > > > phasor taking a random walk in phase. �I imagine there are many ways > > > > to do this and was looking for some input. > > > > > The approach I am taking (if you have a better idea chime in!) is: > > > > phasor(t) = exp(j*X*t) > > > > > where X is a slowly varying, Gaussian random variable. �So more of a > > > > random frequency... > > > > first it should be integrated somehow as Hardy says. �so i think it > > > should be > > > > � � phasor(t) = exp(j*X*t0) * phasor(t-t0) > > > > if, in discrete-time, the direction of walk is completely random, then > > > maybe it could be > > > > � � phasor[n] = exp(j*X) * phasor[n-1] > > > > but now X is a uniform r.v. with range of [-pi, +pi) > > > I like this recursive formulation (although I would still make X a > > gaussian rather than uniform). > > depends if you want the direction to be completely random or to depend > a little on the previous direction (then the variance on X is small). > what scale (variance) would you make your gaussian X? �if it's large, > does it matter that it wraps around at +/- pi? �if you're picking > gaussian because you think it makes your life simpler, that won't be > the case if it goes into the argument of a sin() or cos() function. > > > > > How might I should that this has an average of zero? > > what, the phasor or the directional change? > > r b-jThe phasor itself--how might I show that it has a value of zero? I think I have a method actually... If I assume it is e^(j*X(t)) where X(t) is a Wiener Process, I can find the expected value of the function of the random process using the normal expectation formula-- ie, integrate pdf(x)*e^(jx) for x from -Inf to Inf. Now e^(jx) = cos (x) * j*sin(x) and each part can be done separately. The sin part is easy--by symmetry of the pdf it is zero. The cos part is not so simple. But if you evaluate the integral (using the pdf(x) from http://en.wikipedia.org/wiki/Wiener_process you get e^(-t/2). Pretty nice! So the expected value of a random walking phasor (that starts at zero phase or just = 1) depends on time and is e^(-t/2) So now I want the expected value for all time (ie--an average over time). I assume (by linearity of the expectation operator) I can simply average the "expected value at each t" over t. Sound right? Any nicer way to formulate this?
Reply by ●March 31, 20092009-03-31
On Mar 30, 5:19�pm, "wazerf...@gmail.com" <wazerf...@gmail.com> wrote:> On Mar 30, 12:03�am, robert bristow-johnson > > > > <r...@audioimagination.com> wrote: > > On Mar 30, 12:09�am, "wazerf...@gmail.com" <wazerf...@gmail.com> > > wrote: > > > > On Mar 29, 10:13�pm, robert bristow-johnson > > > > <r...@audioimagination.com> wrote: > > > > On Mar 29, 10:13�am, "wazerf...@gmail.com" <wazerf...@gmail.com> > > > > wrote: > > > > > > So I'm trying to find a good mathematical expression for a unit-length > > > > > phasor taking a random walk in phase. �I imagine there are many ways > > > > > to do this and was looking for some input. > > > > > > The approach I am taking (if you have a better idea chime in!) is: > > > > > phasor(t) = exp(j*X*t) > > > > > > where X is a slowly varying, Gaussian random variable. �So more of a > > > > > random frequency... > > > > > first it should be integrated somehow as Hardy says. �so i think it > > > > should be > > > > > � � phasor(t) = exp(j*X*t0) * phasor(t-t0) > > > > > if, in discrete-time, the direction of walk is completely random, then > > > > maybe it could be > > > > > � � phasor[n] = exp(j*X) * phasor[n-1] > > > > > but now X is a uniform r.v. with range of [-pi, +pi) > > > > I like this recursive formulation (although I would still make X a > > > gaussian rather than uniform). > > > depends if you want the direction to be completely random or to depend > > a little on the previous direction (then the variance on X is small). > > what scale (variance) would you make your gaussian X? �if it's large, > > does it matter that it wraps around at +/- pi? �if you're picking > > gaussian because you think it makes your life simpler, that won't be > > the case if it goes into the argument of a sin() or cos() function. > > > > How might I should that this has an average of zero? > > > what, the phasor or the directional change? > > > The phasor itself--how might I show that it has a value of zero? >you mean a *mean* value of zero, right? it never has a value of zero. you have to show that after passing X(t) through a "wrapping function" (essentially modulo 2*pi), that it is uniformly distributed from 0 to 2*pi. then the phasor will point in any direction equally likely and will have a zero mean value. this would have been easy with an explicit uniform pdf. r b-j
Reply by ●March 31, 20092009-03-31
On Mar 30, 11:18�pm, robert bristow-johnson <r...@audioimagination.com> wrote:> On Mar 30, 5:19�pm, "wazerf...@gmail.com" <wazerf...@gmail.com> wrote: > > > > > On Mar 30, 12:03�am, robert bristow-johnson > > > <r...@audioimagination.com> wrote: > > > On Mar 30, 12:09�am, "wazerf...@gmail.com" <wazerf...@gmail.com> > > > wrote: > > > > > On Mar 29, 10:13�pm, robert bristow-johnson > > > > > <r...@audioimagination.com> wrote: > > > > > On Mar 29, 10:13�am, "wazerf...@gmail.com" <wazerf...@gmail.com> > > > > > wrote: > > > > > > > So I'm trying to find a good mathematical expression for a unit-length > > > > > > phasor taking a random walk in phase. �I imagine there are many ways > > > > > > to do this and was looking for some input. > > > > > > > The approach I am taking (if you have a better idea chime in!) is: > > > > > > phasor(t) = exp(j*X*t) > > > > > > > where X is a slowly varying, Gaussian random variable. �So more of a > > > > > > random frequency... > > > > > > first it should be integrated somehow as Hardy says. �so i think it > > > > > should be > > > > > > � � phasor(t) = exp(j*X*t0) * phasor(t-t0) > > > > > > if, in discrete-time, the direction of walk is completely random, then > > > > > maybe it could be > > > > > > � � phasor[n] = exp(j*X) * phasor[n-1] > > > > > > but now X is a uniform r.v. with range of [-pi, +pi) > > > > > I like this recursive formulation (although I would still make X a > > > > gaussian rather than uniform). > > > > depends if you want the direction to be completely random or to depend > > > a little on the previous direction (then the variance on X is small). > > > what scale (variance) would you make your gaussian X? �if it's large, > > > does it matter that it wraps around at +/- pi? �if you're picking > > > gaussian because you think it makes your life simpler, that won't be > > > the case if it goes into the argument of a sin() or cos() function. > > > > > How might I should that this has an average of zero? > > > > what, the phasor or the directional change? > > > The phasor itself--how might I show that it has a value of zero? > > you mean a *mean* value of zero, right? �it never has a value of zero. > > you have to show that after passing X(t) through a "wrapping > function" (essentially modulo 2*pi), that it is uniformly distributed > from 0 to 2*pi. �then the phasor will point in any direction equally > likely and will have a zero mean value. �this would have been easy > with an explicit uniform pdf. > > r b-jYes, mean. I'm going to go with the approach I mentioned above. This has been interesting and actually got me into a few topics including the Hilbert-Huang Transformation... -Martin
