Hello everyone!
Can anybody help me with the following statistical signal processing
problem?.
Given 2 independent random variables x and y, where y is uniformly
distributed in [-pi,pi] and x is Rayleigh distributed with parameter sigma,
I have tested empirically that:
E{ (x * cos(y) + 1)/sqrt(x^2 + 2*x*cos(y) + 1) }
can be approximated by
1./sqrt(E{x^2} + 1}
Any ideas of the assumptions that may lead to the above approximation?
Thank you,
Maco
Proof for Expected Value Problem
Started by ●April 18, 2011
Reply by ●April 18, 20112011-04-18
On 04/18/2011 10:03 AM, maco wrote:> Hello everyone! > > Can anybody help me with the following statistical signal processing > problem?. > > Given 2 independent random variables x and y, where y is uniformly > distributed in [-pi,pi] and x is Rayleigh distributed with parameter sigma, > I have tested empirically that: > > E{ (x * cos(y) + 1)/sqrt(x^2 + 2*x*cos(y) + 1) } > > can be approximated by > > 1./sqrt(E{x^2} + 1} > > Any ideas of the assumptions that may lead to the above approximation?I'll bet your prof wants you to think of how the Raleigh distribution relates to the magnitude of a pair of independent Gaussian distributions with identical variances, and how the uniform distribution relates to the angle on the plane of a pair of samples with independent Gaussian distributions. And I'll be that that's _way too much clue_ -- let us know what thoughts get sparked off. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●April 19, 20112011-04-19
>On 04/18/2011 10:03 AM, maco wrote: >> Hello everyone! >> >> Can anybody help me with the following statistical signal processing >> problem?. >> >> Given 2 independent random variables x and y, where y is uniformly >> distributed in [-pi,pi] and x is Rayleigh distributed with parametersigma,>> I have tested empirically that: >> >> E{ (x * cos(y) + 1)/sqrt(x^2 + 2*x*cos(y) + 1) } >> >> can be approximated by >> >> 1./sqrt(E{x^2} + 1} >> >> Any ideas of the assumptions that may lead to the above approximation? > >I'll bet your prof wants you to think of how the Raleigh distribution >relates to the magnitude of a pair of independent Gaussian distributions >with identical variances, and how the uniform distribution relates to >the angle on the plane of a pair of samples with independent Gaussian >distributions. > >And I'll be that that's _way too much clue_ -- let us know what thoughts >get sparked off. > >-- > >Tim Wescott >Wescott Design Services >http://www.wescottdesign.com > >Do you need to implement control loops in software? >"Applied Control Theory for Embedded Systems" was written for you. >See details at http://www.wescottdesign.com/actfes/actfes.html >Thank you Tim, In fact, as you noted, variable x comes from the modulus of a gaussian complex variable and y from its phase. My problem right now to get to the proof is that the numerator and denominator of the expression do not seem to be independent. On the other hand, the expected value operator in the approximation is inside the square-root operator, which is also quite interesting. Here is my Matlab script for checking the validity of the approximated expression: N = 10000; sigma = linspace(0,50,100); for n=1:20 x = random('rayl', sigma(n), [N 1]); y = 2*pi*(rand(N,1)-0.5); f = (x.*cos(y) + 1)./sqrt(x.^2 + 2*x.*cos(y) + 1); est_mean(n) = mean(f); approx_mean(n) = 1/sqrt(mean(x.^2) + 1); end plot(est_mean), hold on, plot(approx_mean,'r') Best regards, maco
Reply by ●April 19, 20112011-04-19
On 04/19/2011 01:31 AM, maco wrote:>> On 04/18/2011 10:03 AM, maco wrote: >>> Hello everyone! >>> >>> Can anybody help me with the following statistical signal processing >>> problem?. >>> >>> Given 2 independent random variables x and y, where y is uniformly >>> distributed in [-pi,pi] and x is Rayleigh distributed with parameter > sigma, >>> I have tested empirically that: >>> >>> E{ (x * cos(y) + 1)/sqrt(x^2 + 2*x*cos(y) + 1) } >>> >>> can be approximated by >>> >>> 1./sqrt(E{x^2} + 1} >>> >>> Any ideas of the assumptions that may lead to the above approximation? >> >> I'll bet your prof wants you to think of how the Raleigh distribution >> relates to the magnitude of a pair of independent Gaussian distributions >> with identical variances, and how the uniform distribution relates to >> the angle on the plane of a pair of samples with independent Gaussian >> distributions. >> >> And I'll be that that's _way too much clue_ -- let us know what thoughts >> get sparked off. >> >> -- >> >> Tim Wescott >> Wescott Design Services >> http://www.wescottdesign.com >> >> Do you need to implement control loops in software? >> "Applied Control Theory for Embedded Systems" was written for you. >> See details at http://www.wescottdesign.com/actfes/actfes.html >> > > Thank you Tim, > > In fact, as you noted, variable x comes from the modulus of a gaussian > complex variable and y from its phase. > > My problem right now to get to the proof is that the numerator and > denominator of the expression do not seem to be independent. On the other > hand, the expected value operator in the approximation is inside the > square-root operator, which is also quite interesting. > > Here is my Matlab script for checking the validity of the approximated > expression:(scrip snipped) I think you need to stop distracting yourself with Matlab, and see how far you can reduce this with paper and pencil. What is your expectation when you put it into terms of the two rectilinear Gaussians? Do any bells ring when you look at that expression? Does it simplify anything? (Note that I can't guarantee anything -- I just think this is a good path to go down, to see if it looks promising). -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●April 19, 20112011-04-19
On 04/19/2011 01:31 AM, maco wrote:>> On 04/18/2011 10:03 AM, maco wrote: >>> Hello everyone! >>> >>> Can anybody help me with the following statistical signal processing >>> problem?. >>> >>> Given 2 independent random variables x and y, where y is uniformly >>> distributed in [-pi,pi] and x is Rayleigh distributed with parameter > sigma, >>> I have tested empirically that: >>> >>> E{ (x * cos(y) + 1)/sqrt(x^2 + 2*x*cos(y) + 1) } >>> >>> can be approximated by >>> >>> 1./sqrt(E{x^2} + 1} >>> >>> Any ideas of the assumptions that may lead to the above approximation? >> >> I'll bet your prof wants you to think of how the Raleigh distribution >> relates to the magnitude of a pair of independent Gaussian distributions >> with identical variances, and how the uniform distribution relates to >> the angle on the plane of a pair of samples with independent Gaussian >> distributions. >> >> And I'll be that that's _way too much clue_ -- let us know what thoughts >> get sparked off. >> >> -- >> >> Tim Wescott >> Wescott Design Services >> http://www.wescottdesign.com >> >> Do you need to implement control loops in software? >> "Applied Control Theory for Embedded Systems" was written for you. >> See details at http://www.wescottdesign.com/actfes/actfes.html >> > > Thank you Tim, > > In fact, as you noted, variable x comes from the modulus of a gaussian > complex variable and y from its phase. > > My problem right now to get to the proof is that the numerator and > denominator of the expression do not seem to be independent. On the other > hand, the expected value operator in the approximation is inside the > square-root operator, which is also quite interesting.So -- is this homework? -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●April 19, 20112011-04-19
On 04/19/2011 10:19 AM, Tim Wescott wrote:> On 04/19/2011 01:31 AM, maco wrote: >>> On 04/18/2011 10:03 AM, maco wrote: >>>> Hello everyone! >>>> >>>> Can anybody help me with the following statistical signal processing >>>> problem?. >>>> >>>> Given 2 independent random variables x and y, where y is uniformly >>>> distributed in [-pi,pi] and x is Rayleigh distributed with parameter >> sigma, >>>> I have tested empirically that: >>>> >>>> E{ (x * cos(y) + 1)/sqrt(x^2 + 2*x*cos(y) + 1) } >>>> >>>> can be approximated by >>>> >>>> 1./sqrt(E{x^2} + 1} >>>> >>>> Any ideas of the assumptions that may lead to the above approximation? >>> >>> I'll bet your prof wants you to think of how the Raleigh distribution >>> relates to the magnitude of a pair of independent Gaussian distributions >>> with identical variances, and how the uniform distribution relates to >>> the angle on the plane of a pair of samples with independent Gaussian >>> distributions. >>> >>> And I'll be that that's _way too much clue_ -- let us know what thoughts >>> get sparked off. >>> >>> -- >>> >>> Tim Wescott >>> Wescott Design Services >>> http://www.wescottdesign.com >>> >>> Do you need to implement control loops in software? >>> "Applied Control Theory for Embedded Systems" was written for you. >>> See details at http://www.wescottdesign.com/actfes/actfes.html >>> >> >> Thank you Tim, >> >> In fact, as you noted, variable x comes from the modulus of a gaussian >> complex variable and y from its phase. >> >> My problem right now to get to the proof is that the numerator and >> denominator of the expression do not seem to be independent. On the other >> hand, the expected value operator in the approximation is inside the >> square-root operator, which is also quite interesting. > > So -- is this homework? >Hint: I can get this down to an expression in the rectilinear variables along with a bunch of operations against constants. Unfortunately there's a radical and more than one fraction involved -- but hey, the world is not a perfect place. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●April 19, 20112011-04-19
On Apr 19, 12:22=A0pm, Tim Wescott <t...@seemywebsite.com> wrote:> On 04/19/2011 10:19 AM, Tim Wescott wrote: > > > On 04/19/2011 01:31 AM, maco wrote: > >>> On 04/18/2011 10:03 AM, maco wrote: > >>>> Hello everyone! > > >>>> Can anybody help me with the following statistical signal processing > >>>> problem?. > > >>>> Given 2 independent random variables x and y, where y is uniformly > >>>> distributed in [-pi,pi] and x is Rayleigh distributed with parameter > >> sigma, > >>>> I have tested empirically that: > > >>>> E{ (x * cos(y) + 1)/sqrt(x^2 + 2*x*cos(y) + 1) } > > >>>> can be approximated by > > >>>> 1./sqrt(E{x^2} + 1} > > >>>> Any ideas of the assumptions that may lead to the above approximatio=n?> > >>> I'll bet your prof wants you to think of how the Raleigh distribution > >>> relates to the magnitude of a pair of independent Gaussian distributi=ons> >>> with identical variances, and how the uniform distribution relates to > >>> the angle on the plane of a pair of samples with independent Gaussian > >>> distributions. > > >>> And I'll be that that's _way too much clue_ -- let us know what thoug=hts> >>> get sparked off. > > >>> -- > > >>> Tim Wescott > >>> Wescott Design Services > >>>http://www.wescottdesign.com > > >>> Do you need to implement control loops in software? > >>> "Applied Control Theory for Embedded Systems" was written for you. > >>> See details athttp://www.wescottdesign.com/actfes/actfes.html > > >> Thank you Tim, > > >> In fact, as you noted, variable x comes from the modulus of a gaussian > >> complex variable and y from its phase. > > >> My problem right now to get to the proof is that the numerator and > >> denominator of the expression do not seem to be independent. On the ot=her> >> hand, the expected value operator in the approximation is inside the > >> square-root operator, which is also quite interesting. > > > So -- is this homework? > > Hint: I can get this down to an expression in the rectilinear variables > along with a bunch of operations against constants. =A0Unfortunately > there's a radical and more than one fraction involved -- but hey, the > world is not a perfect place. > > -- > > Tim Wescott > Wescott Design Serviceshttp://www.wescottdesign.com > > Do you need to implement control loops in software? > "Applied Control Theory for Embedded Systems" was written for you. > See details athttp://www.wescottdesign.com/actfes/actfes.htmlIf it were not for the horrendous notation chosen by the OP (or his instructor!), the question boils down to the expected value of the ratio of a non-zero mean Gaussian random variable to a Rician random variable, specifically, E[(X+1)/sqrt{(X+1)^2 + Y^2}] where X, Y are independent zero-mean Gaussian random variables, and sqrt{(X+1)^2 + Y^2} is Rician. In fact, a little geometry shows that the question asked is the expected value of cos(theta) where theta is the angle of the radius vector where the polar coordinates of the random point are (R, theta) with R being Rician. The distribution of theta is given in books such as Davenport at Root, Random Signals and Noise, and Papoulis, Probability, Random Variables and Stochastic Processes (at least in the earlier editions; don't know about the latest one) and involves erf. Computing E[cos(theta)] might not be an easy task. As a direct response to the OP's remark>On the other hand, the expected value operator >in the approximation is inside the square-root operator, >which is also quite interesting.I will simply say that sqrt(E{x^2} + 1} is a constant with value sqrt(2*sigma^2 + 1) or sqrt(sigma^2 + 1) depending on your definition of Rayleigh random variable. --Dilip Sarwate
Reply by ●April 20, 20112011-04-20
> >If it were not for the horrendous notation chosen >by the OP (or his instructor!), the question boils >down to the expected value of the ratio of a non-zero >mean Gaussian random variable to a Rician random >variable, specifically, E[(X+1)/sqrt{(X+1)^2 + Y^2}] >where X, Y are independent zero-mean Gaussian >random variables, and sqrt{(X+1)^2 + Y^2} is Rician. >In fact, a little geometry shows that the question asked >is the expected value of cos(theta) where theta is >the angle of the radius vector where the polar coordinates >of the random point are (R, theta) with R being Rician. >The distribution of theta is given in books such as >Davenport at Root, Random Signals and Noise, and >Papoulis, Probability, Random Variables and Stochastic >Processes (at least in the earlier editions; don't know >about the latest one) and involves erf. Computing >E[cos(theta)] might not be an easy task. > >As a direct response to the OP's remark > >>On the other hand, the expected value operator >>in the approximation is inside the square-root operator, >>which is also quite interesting. > >I will simply say that sqrt(E{x^2} + 1} is a constant >with value sqrt(2*sigma^2 + 1) or sqrt(sigma^2 + 1) >depending on your definition of Rayleigh random variable. > >--Dilip Sarwate >Thank you for your answers, I think they have brought some light to the issue. Certainly, the question becomes easier by considering the expression as E[(X+1)/sqrt{(X+1)^2 + Y^2}], where X and Y are zero-mean independent Gaussian variables. As noted by Dilip, this leads to study the ratio distribution of a non-zero mean Gaussian and a Rician random variable. Considering the E[cos(theta)] form, it seems that theta is given by: theta = atan ( Y/(X + 1) , where Y and X are again zero-mean independent Gaussians. If the denominator had zero mean, i.e. (X + 0), then the ratio Y/X would have the standard Cauchy distribution and applying the arctangent would result in an uniform distribution. However, as said by Tim, life is not a perfect place :D. Apparently (with numerical experiments), depending on the variance of X and Y, the resulting arctangent distribution may change from a bounded Gaussian distribution in [-pi, pi] to a uniform distribution in the same range when the variance becomes much greater than 1. Any ideas to follow? Maco
Reply by ●April 20, 20112011-04-20
On 04/19/2011 06:01 PM, dvsarwate wrote:> On Apr 19, 12:22 pm, Tim Wescott<t...@seemywebsite.com> wrote: >> On 04/19/2011 10:19 AM, Tim Wescott wrote: >> >>> On 04/19/2011 01:31 AM, maco wrote: >>>>> On 04/18/2011 10:03 AM, maco wrote: >>>>>> Hello everyone! >> >>>>>> Can anybody help me with the following statistical signal processing >>>>>> problem?. >> >>>>>> Given 2 independent random variables x and y, where y is uniformly >>>>>> distributed in [-pi,pi] and x is Rayleigh distributed with parameter >>>> sigma, >>>>>> I have tested empirically that: >> >>>>>> E{ (x * cos(y) + 1)/sqrt(x^2 + 2*x*cos(y) + 1) } >> >>>>>> can be approximated by >> >>>>>> 1./sqrt(E{x^2} + 1} >> >>>>>> Any ideas of the assumptions that may lead to the above approximation? >> >>>>> I'll bet your prof wants you to think of how the Raleigh distribution >>>>> relates to the magnitude of a pair of independent Gaussian distributions >>>>> with identical variances, and how the uniform distribution relates to >>>>> the angle on the plane of a pair of samples with independent Gaussian >>>>> distributions. >> >>>>> And I'll be that that's _way too much clue_ -- let us know what thoughts >>>>> get sparked off. >> >>>>> -- >> >>>>> Tim Wescott >>>>> Wescott Design Services >>>>> http://www.wescottdesign.com >> >>>>> Do you need to implement control loops in software? >>>>> "Applied Control Theory for Embedded Systems" was written for you. >>>>> See details athttp://www.wescottdesign.com/actfes/actfes.html >> >>>> Thank you Tim, >> >>>> In fact, as you noted, variable x comes from the modulus of a gaussian >>>> complex variable and y from its phase. >> >>>> My problem right now to get to the proof is that the numerator and >>>> denominator of the expression do not seem to be independent. On the other >>>> hand, the expected value operator in the approximation is inside the >>>> square-root operator, which is also quite interesting. >> >>> So -- is this homework? >> >> Hint: I can get this down to an expression in the rectilinear variables >> along with a bunch of operations against constants. Unfortunately >> there's a radical and more than one fraction involved -- but hey, the >> world is not a perfect place. >> >> -- >> >> Tim Wescott >> Wescott Design Serviceshttp://www.wescottdesign.com >> >> Do you need to implement control loops in software? >> "Applied Control Theory for Embedded Systems" was written for you. >> See details athttp://www.wescottdesign.com/actfes/actfes.html > > If it were not for the horrendous notation chosen > by the OP (or his instructor!), the question boils > down to the expected value of the ratio of a non-zero > mean Gaussian random variable to a Rician random > variable, specifically, E[(X+1)/sqrt{(X+1)^2 + Y^2}] > where X, Y are independent zero-mean Gaussian > random variables, and sqrt{(X+1)^2 + Y^2} is Rician. > In fact, a little geometry shows that the question asked > is the expected value of cos(theta) where theta is > the angle of the radius vector where the polar coordinates > of the random point are (R, theta) with R being Rician. > The distribution of theta is given in books such as > Davenport at Root, Random Signals and Noise, and > Papoulis, Probability, Random Variables and Stochastic > Processes (at least in the earlier editions; don't know > about the latest one) and involves erf. Computing > E[cos(theta)] might not be an easy task. > > As a direct response to the OP's remark > >> On the other hand, the expected value operator >> in the approximation is inside the square-root operator, >> which is also quite interesting. > > I will simply say that sqrt(E{x^2} + 1} is a constant > with value sqrt(2*sigma^2 + 1) or sqrt(sigma^2 + 1) > depending on your definition of Rayleigh random variable.That "Gaussian/Rician" statement is not as simple as it sounds -- the Rician in question is not independent of the Gaussian, so you have to be careful about what equations you pull out of manuals. I'm also not sure that it's wise to pull trigonometry back into the problem once you've gotten rid of it, unless the aim is just to better visualize the problem to help you along to a solution. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●April 20, 20112011-04-20
On Apr 20, 11:44=A0am, Tim Wescott <t...@seemywebsite.com> wrote:> > That "Gaussian/Rician" statement is not as simple as it sounds -- the > Rician in question is not independent of the Gaussian, so you have to be > careful about what equations you pull out of manuals.I didn't claim that the Gaussian RV and Rician RV were independent, and indeed the expression that I gave (X+1)/sqrt{(X+1)^2 + Y^2} makes it reasonably obvious that they are not. *I* do not need to be careful what equations I pull out of manuals, and the references that I gave for the distribution for theta are the ones that need to be used: they take into account the non-independence of the two RVs. For the specific case brought up by the OP, the probability density function of theta is exp(-1/(2*sigma^2)) + cos(theta)/(2*sigma*sqrt{2 pi}*[1 + erf{cos(theta)/ (sigma*sqrt{2})}]*exp(-sin^2(theta)/(2*sigma^2) for theta between -pi and +pi, and the OP needs to compute E[cos(theta)] via LOTUS from this pdf. Numerical evaluation of the integral is likely to be easier than trying to evaluate it analytically, though giving it to Mathematica or Maple is probably worthwhile. I am sure that someone has computed an analytical expression already or a good approximation thereof.> I'm also not sure that it's wise to pull trigonometry back into the > problem once you've gotten rid of it, unless the aim is just to better > visualize the problem to help you along to a solution.Different strokes for different folks.... Best of luck trying to simplify your>expression in the rectilinear variables >along with a bunch of operations against constants. >Unfortunately there's a radical and more than one >fraction involved--Dilip Sarwate
