# Correlation Help

Started by October 12, 2011
```I am trying to get my terminology correct on saying that "sinx is not
correlated with cosx", but.....

I know this is not quite true because a correlation function (or in
this case a cross correlation function) conducts zillions of integrals
on the cosine and sine function as it slides one against the other.

When doing the orthogonality test on a sin and cosine , it is
essentially doing one integral (multiplying the two signals together)
within the cross correlation function to determine how "similar" the
cosine function is to the sine function (in this case it is 0 because
they are indeed orthogonal).

What is the correct terminology for one integral  at one particular
time shift within the cross correlation function.  And what is the
interpretation (or name) of the result of that one integral?

This is bugging me because the term "correlation" seems to indicate a
function that includes all the integrals, not just one of them.

Thanks

```
```On Oct 12, 6:06&#4294967295;pm, brent <buleg...@columbus.rr.com> wrote:
> I am trying to get my terminology correct on saying that "sinx is not
> correlated with cosx", but.....
>
> I know this is not quite true because a correlation function (or in
> this case a cross correlation function) conducts zillions of integrals
> on the cosine and sine function as it slides one against the other.
>
> When doing the orthogonality test on a sin and cosine , it is
> essentially doing one integral (multiplying the two signals together)
> within the cross correlation function to determine how "similar" the
> cosine function is to the sine function (in this case it is 0 because
> they are indeed orthogonal).
>
> What is the correct terminology for one integral &#4294967295;at one particular
> time shift within the cross correlation function. &#4294967295;And what is the
> interpretation (or name) of the result of that one integral?
>
> This is bugging me because the term "correlation" seems to indicate a
> function that includes all the integrals, not just one of them.
>
> Thanks

You need to think a little more about what you mean by
correlation.  For example, the cross-correlation (integral)
of sin(x)*cos(x) from 0 to pi/2 is *not* zero, while the
integral from 0 to 2*pi is.  In fact, the integral of sin(x)*cos(x)
over the interval (y, y+2*n*pi) is zero for any choice of
real number y.  In other words, sin(x) and cos(x) are
orthogonal over an interval of length one period (or an
integer multiple of one period) no matter where the
interval starts,  In contrast, sin(x) and cos(x) are orthogonal
over  (0, pi) but not over (y, y+pi).

Turning to your notion of sin(x) and cos(x) sliding relative
to each other, consider that cos(x - pi/2) = sin(x) and
so any hope of getting *all* your zillions of integrals to
equal 0 is shot down right there.  The *cross-correlation*
function can be defined as

R(t) = integral from 0 to 2*pi sin(x)*cos(x-t) dx

and is a *periodic* function of t that has value 0
at t=0, value 0.5 at t = pi/2, and so on.  In fact,
R(t) = 0.5*sin(t).  sin(x) and cos(x) are *not*
uncorrelated signals, but they are orthogonal
over any interval of length a multiple of 2*pi.

Dilip Sarwate
```
```On Oct 12, 8:14&#4294967295;pm, dvsarwate <dvsarw...@yahoo.com> wrote:
> On Oct 12, 6:06&#4294967295;pm, brent <buleg...@columbus.rr.com> wrote:
>
>
>
>
>
>
>
>
>
> > I am trying to get my terminology correct on saying that "sinx is not
> > correlated with cosx", but.....
>
> > I know this is not quite true because a correlation function (or in
> > this case a cross correlation function) conducts zillions of integrals
> > on the cosine and sine function as it slides one against the other.
>
> > When doing the orthogonality test on a sin and cosine , it is
> > essentially doing one integral (multiplying the two signals together)
> > within the cross correlation function to determine how "similar" the
> > cosine function is to the sine function (in this case it is 0 because
> > they are indeed orthogonal).
>
> > What is the correct terminology for one integral &#4294967295;at one particular
> > time shift within the cross correlation function. &#4294967295;And what is the
> > interpretation (or name) of the result of that one integral?
>
> > This is bugging me because the term "correlation" seems to indicate a
> > function that includes all the integrals, not just one of them.
>
> > Thanks
>
> You need to think a little more about what you mean by
> correlation. &#4294967295;For example, the cross-correlation (integral)
> of sin(x)*cos(x) from 0 to pi/2 is *not* zero, while the
> integral from 0 to 2*pi is. &#4294967295;In fact, the integral of sin(x)*cos(x)
> over the interval (y, y+2*n*pi) is zero for any choice of
> real number y. &#4294967295;In other words, sin(x) and cos(x) are
> orthogonal over an interval of length one period (or an
> integer multiple of one period) no matter where the
> interval starts, &#4294967295;In contrast, sin(x) and cos(x) are orthogonal
> over &#4294967295;(0, pi) but not over (y, y+pi).
>
> Turning to your notion of sin(x) and cos(x) sliding relative
> to each other, consider that cos(x - pi/2) = sin(x) and
> so any hope of getting *all* your zillions of integrals to
> equal 0 is shot down right there. &#4294967295;The *cross-correlation*
> function can be defined as
>
> R(t) = integral from 0 to 2*pi sin(x)*cos(x-t) dx
>
> and is a *periodic* function of t that has value 0
> at t=0, value 0.5 at t = pi/2, and so on. &#4294967295;In fact,
> R(t) = 0.5*sin(t). &#4294967295;sin(x) and cos(x) are *not*
> uncorrelated signals, but they are orthogonal
> over any interval of length a multiple of 2*pi.
>
> Dilip Sarwate

Thanks,

From you answer I feel I am close .

but what has me bugged is that the cross correlation integral is the
sliding integral (The sliding integral seems to be the definition of
the cross correlation function) that contains all possible ______
correlations(what is the correct word to put in here??)______ between
the sliding sine function and the cosine function?

```
```On Oct 12, 8:33&#4294967295;pm, brent <buleg...@columbus.rr.com> wrote:

>
>  the cross correlation integral is the
> sliding integral

No, the cross-correlation FUNCTION R(t) is the value
of the "sliding integral"

R(t) = integral from 0 to 2*pi sin(x)*cos(x-t) dx

where t denotes the slide.

>(The sliding integral seems to be the definition of
> the cross correlation function) that contains all possible ______
> correlations(what is the correct word to put in here??)______ between
> the sliding sine function and the cosine function?

Not all functions in this world are defined by formulas such
as f(t) = t^2; some are defined as the value of an integral.
Perhaps the following will be familiar to you:  if a linear
time invariant system has impulse response h(t), then
with the input denoted as x(t) and the output as y(t), the
output of the linear system (I do hope that you agree that
the output signal is a *function* of t) is given by

y(t) = integral from -infinity to +infinity h(tau)x(t - tau) d(tau)

meaning that

y(0.576241..) =  integral from -infinity to +infinity h(tau)x(0.576241
- tau) d(tau)

and

y(2.5) =  integral from -infinity to +infinity h(tau)x(2.5 - tau)
d(tau)

and so on for "zillions of values" of real numbers t.
Thus this function y(t) embodies in it all the "zillions
of integrals" corresponding to different values of t.
Similarly, the cross-correlation function includes all
those zillions of integrals that you are worried about.

--Dilip Sarwate

```
```On Oct 12, 9:10&#4294967295;pm, dvsarwate <dvsarw...@yahoo.com> wrote:
> On Oct 12, 8:33&#4294967295;pm, brent <buleg...@columbus.rr.com> wrote:
>
>
>
> > &#4294967295;the cross correlation integral is the
> > sliding integral
>
> No, the cross-correlation FUNCTION R(t) is the value
> of the "sliding integral"
>
> R(t) = integral from 0 to 2*pi sin(x)*cos(x-t) dx
>
> where t denotes the slide.
>
> >(The sliding integral seems to be the definition of
> > the cross correlation function) that contains all possible ______
> > correlations(what is the correct word to put in here??)______ between
> > the sliding sine function and the cosine function?
>
> Not all functions in this world are defined by formulas such
> as f(t) = t^2; some are defined as the value of an integral.
> Perhaps the following will be familiar to you: &#4294967295;if a linear
> time invariant system has impulse response h(t), then
> with the input denoted as x(t) and the output as y(t), the
> output of the linear system (I do hope that you agree that
> the output signal is a *function* of t) is given by
>
> y(t) = integral from -infinity to +infinity h(tau)x(t - tau) d(tau)
>
> meaning that
>
> y(0.576241..) = &#4294967295;integral from -infinity to +infinity h(tau)x(0.576241
> - tau) d(tau)
>
> and
>
> y(2.5) = &#4294967295;integral from -infinity to +infinity h(tau)x(2.5 - tau)
> d(tau)
>
> and so on for "zillions of values" of real numbers t.
> Thus this function y(t) embodies in it all the "zillions
> of integrals" corresponding to different values of t.
> Similarly, the cross-correlation function includes all
> those zillions of integrals that you are worried about.
>
> --Dilip Sarwate

Is one of those zillions of integrals called a correlation?

I cannot seem to find any definition of correlation that has just a
single expression of the form:

Intgral of cosx times sinx dx
```
```On Oct 12, 8:33&#4294967295;pm, brent <buleg...@columbus.rr.com> wrote:
> On Oct 12, 8:14&#4294967295;pm, dvsarwate <dvsarw...@yahoo.com> wrote:
>
>
>
>
>
> > On Oct 12, 6:06&#4294967295;pm, brent <buleg...@columbus.rr.com> wrote:
>
> > > I am trying to get my terminology correct on saying that "sinx is not
> > > correlated with cosx", but.....
>
> > > I know this is not quite true because a correlation function (or in
> > > this case a cross correlation function) conducts zillions of integrals
> > > on the cosine and sine function as it slides one against the other.
>
> > > When doing the orthogonality test on a sin and cosine , it is
> > > essentially doing one integral (multiplying the two signals together)
> > > within the cross correlation function to determine how "similar" the
> > > cosine function is to the sine function (in this case it is 0 because
> > > they are indeed orthogonal).
>
> > > What is the correct terminology for one integral &#4294967295;at one particular
> > > time shift within the cross correlation function. &#4294967295;And what is the
> > > interpretation (or name) of the result of that one integral?
>
> > > This is bugging me because the term "correlation" seems to indicate a
> > > function that includes all the integrals, not just one of them.
>
> > > Thanks
>
> > You need to think a little more about what you mean by
> > correlation. &#4294967295;For example, the cross-correlation (integral)
> > of sin(x)*cos(x) from 0 to pi/2 is *not* zero, while the
> > integral from 0 to 2*pi is. &#4294967295;In fact, the integral of sin(x)*cos(x)
> > over the interval (y, y+2*n*pi) is zero for any choice of
> > real number y. &#4294967295;In other words, sin(x) and cos(x) are
> > orthogonal over an interval of length one period (or an
> > integer multiple of one period) no matter where the
> > interval starts, &#4294967295;In contrast, sin(x) and cos(x) are orthogonal
> > over &#4294967295;(0, pi) but not over (y, y+pi).
>
> > Turning to your notion of sin(x) and cos(x) sliding relative
> > to each other, consider that cos(x - pi/2) = sin(x) and
> > so any hope of getting *all* your zillions of integrals to
> > equal 0 is shot down right there. &#4294967295;The *cross-correlation*
> > function can be defined as
>
> > R(t) = integral from 0 to 2*pi sin(x)*cos(x-t) dx
>
> > and is a *periodic* function of t that has value 0
> > at t=0, value 0.5 at t = pi/2, and so on. &#4294967295;In fact,
> > R(t) = 0.5*sin(t). &#4294967295;sin(x) and cos(x) are *not*
> > uncorrelated signals, but they are orthogonal
> > over any interval of length a multiple of 2*pi.
>
> > Dilip Sarwate
>
> Thanks,
>
> From you answer I feel I am close .
>
> but what has me bugged is that the cross correlation integral is the
> sliding integral (The sliding integral seems to be the definition of
> the cross correlation function) that contains all possible ______
> correlations(what is the correct word to put in here??)______ between
> the sliding sine function and the cosine function?- Hide quoted text -
>
> - Show quoted text -

I think I have found the correct way of stating this.

The missing word should be "correlation values"

or "measures of similarity"

On looking over several articles on the web, they sometimes refer to
the single integral  as " the measure of similarity" between the two
functions.

It was bugging me that a  single value on the cross-correlation
function should not itself be refered to as a correlation.

I think I have it straight :-)

```