Hi, If x and y and orthogonal, is it true that corr(x,y)=0? How can this be proved? cfy30
orthogonal and correlation
Started by ●May 30, 2010
Reply by ●May 30, 20102010-05-30
cfy30 <cfy30@n_o_s_p_a_m.yahoo.com> wrote:>If x and y and orthogonal, is it true that corr(x,y)=0? How can this be >proved?Um, by definition? S.
Reply by ●May 30, 20102010-05-30
cfy30 wrote:> Hi, > > If x and y and orthogonal, is it true that corr(x,y)=0? How can this be > proved? > > > cfy30Definitionally: http://www-mobile.ecs.soton.ac.uk/bjc97r/pnseq/node5.html "Orthogonal codes have zero cross-correlation." http://www.thefreedictionary.com/orthogonal ... "b. (of a pair of functions) having a defined product equal to zero" -- Les Cargill
Reply by ●May 30, 20102010-05-30
If x = cos(omega*t) and y = sin(omega*t). x and y are orthogonal but it seems to me they are correlated because their difference is only 90degree! What is not right? cfy30>cfy30 wrote: >> Hi, >> >> If x and y and orthogonal, is it true that corr(x,y)=0? How can this be >> proved? >> >> >> cfy30 > >Definitionally: > >http://www-mobile.ecs.soton.ac.uk/bjc97r/pnseq/node5.html > >"Orthogonal codes have zero cross-correlation." > >http://www.thefreedictionary.com/orthogonal >... >"b. (of a pair of functions) having a defined product equal to zero" > >-- >Les Cargill > >
Reply by ●May 30, 20102010-05-30
On 5/30/2010 2:07 PM, cfy30 wrote:> If x = cos(omega*t) and y = sin(omega*t). x and y are orthogonal but it > seems to me they are correlated because their difference is only 90degree! > What is not right? > > > cfy30 > >> cfy30 wrote: >>> Hi, >>> >>> If x and y and orthogonal, is it true that corr(x,y)=0? How can this be >>> proved? >>> >>> >>> cfy30 >> >> Definitionally: >> >> http://www-mobile.ecs.soton.ac.uk/bjc97r/pnseq/node5.html >> >> "Orthogonal codes have zero cross-correlation." >> >> http://www.thefreedictionary.com/orthogonal >> ... >> "b. (of a pair of functions) having a defined product equal to zero"What do you understand the definition of of "correlated" to be? cos(x) = sqrt(1 - sin^2(x)), but they are uncorrelated because T integral(sin(x) * cos(x))dx is zero when T -> infinity or T = k periods. x = 0 k is an integer Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 31, 20102010-05-31
"cfy30" <cfy30@n_o_s_p_a_m.yahoo.com> writes:> If x = cos(omega*t) and y = sin(omega*t). x and y are orthogonal but it > seems to me they are correlated because their difference is only 90degree! > What is not right?Hi cfy30, Good question! Keep in mind there are (at least) two definitions of orthogonal: a probabilistic one, E[XY] = 0, [garcia] and a "functional" one: \int_{A}^{B} f(t) g(t) dt = 0. [spiegel] So as a start to answer your question, ask yourself which definition you're using. --Randy @book{garcia, title = "Probability and Random Processes for Electrical Engineering", author = "{Alberto~Leon-Garcia}", publisher = "Addison-Wesley", year = "1989"} @BOOK{spiegel, title = "{Applied Differential Equations}", author = "{Murray~R.~Spiegel}", publisher = "Prentice Hall", edition = "third", year = "1981"}> > > cfy30 > >>cfy30 wrote: >>> Hi, >>> >>> If x and y and orthogonal, is it true that corr(x,y)=0? How can this be >>> proved? >>> >>> >>> cfy30 >> >>Definitionally: >> >>http://www-mobile.ecs.soton.ac.uk/bjc97r/pnseq/node5.html >> >>"Orthogonal codes have zero cross-correlation." >> >>http://www.thefreedictionary.com/orthogonal >>... >>"b. (of a pair of functions) having a defined product equal to zero" >> >>-- >>Les Cargill >> >>-- Randy Yates % "Bird, on the wing, Digital Signal Labs % goes floating by mailto://yates@ieee.org % but there's a teardrop in his eye..." http://www.digitalsignallabs.com % 'One Summer Dream', *Face The Music*, ELO
Reply by ●May 31, 20102010-05-31
On May 30, 11:16�pm, Randy Yates <ya...@ieee.org> wrote:> Keep in mind there are (at least) two definitions of orthogonal: a > probabilistic one, > > � E[XY] = 0, �[garcia] > > and a "functional" one: > > � \int_{A}^{B} f(t) g(t) dt = 0. �[spiegel] > > So as a start to answer your question, ask yourself which definition > you're using.and there are at least three definitions of uncorrelated: For random variables X and Y, E[XY} must equal E[X]E[Y] while for signals, some say x(t) and y(t) are uncorrelated if \int_{-oo}^{oo} x(t) y(t) dt = 0 or more precisely, lim_{T --> oo} (1/T) \int_{-T}^{T} x(t) y(t) dt = 0 and others insist on the stronger condition that \int_{-oo}^{oo} x(t) y(t + tau) dt = 0 for all choices of tau. Note that the stronger condition implies that X(f)Y(f) = 0, that is, the two signals occupy non-overlapping frequency bands. Thus, sin(wt) and cos(wt) are correlated in this sense, but sinusoids at two different frequencies are not. Just another contribution to the confusion.... --Dilip Sarwate
Reply by ●May 31, 20102010-05-31
On 31 Mai, 14:41, dvsarwate <dvsarw...@gmail.com> wrote:> and others insist on the stronger condition that > > \int_{-oo}^{oo} x(t) y(t + tau) dt = 0 for all choices of tau. > > Note that the stronger condition implies that X(f)Y(f) = 0, > that is, the two signals occupy non-overlapping frequency > bands. �Thus, sin(wt) and cos(wt) are correlated in this > sense, but sinusoids at two different frequencies are not.I know of the two first definitions, but I can't remember to have seen this last one before. In what contexts does this definition / claim / requirement occur? Rune
Reply by ●May 31, 20102010-05-31
dvsarwate <dvsarwate@gmail.com> wrote:>and there are at least three definitions of uncorrelated: > >For random variables X and Y, E[XY} must equal E[X]E[Y] > >while for signals, some say x(t) and y(t) are uncorrelated if >\int_{-oo}^{oo} x(t) y(t) dt = 0 or more precisely, > >lim_{T --> oo} (1/T) \int_{-T}^{T} x(t) y(t) dt = 0 > >and others insist on the stronger condition that > >\int_{-oo}^{oo} x(t) y(t + tau) dt = 0 for all choices of tau. > >Note that the stronger condition implies that X(f)Y(f) = 0, >that is, the two signals occupy non-overlapping frequency >bands. Thus, sin(wt) and cos(wt) are correlated in this >sense, but sinusoids at two different frequencies are not.>Just another contribution to the confusion....Thanks for posting this, because it may relate to a question I have often heard argued: in 2-MSK, are the two tones orthogonal or not? (Actually it may not quite hit that case, but it's similar.) Steve
Reply by ●May 31, 20102010-05-31
dvsarwate <dvsarwate@gmail.com> wrote: (snip)> Note that the stronger condition implies that X(f)Y(f) = 0, > that is, the two signals occupy non-overlapping frequency > bands. Thus, sin(wt) and cos(wt) are correlated in this > sense, but sinusoids at two different frequencies are not.I don't know about correlation, but for coherence sin(wt) and cos(wt) should be coherent. At different frequencies, the coherence time or length depends on the frequency difference. -- glen
