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From: Randy Yates on 31 May 2010 00:16
"cfy30" <cfy30(a)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(a)ieee.org % but there's a teardrop in his eye..."
http://www.digitalsignallabs.com % 'One Summer Dream', *Face The Music*, ELO
From: dvsarwate on 31 May 2010 08:41
On May 30, 11:16 pm, Randy Yates <ya...(a)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
From: Rune Allnor on 31 May 2010 11:16
On 31 Mai, 14:41, dvsarwate <dvsarw...(a)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
From: Steve Pope on 31 May 2010 13:08
dvsarwate <dvsarwate(a)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
From: glen herrmannsfeldt on 31 May 2010 13:34
dvsarwate <dvsarwate(a)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
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