# Time delay estimation question

Started by July 22, 2003
```I have read quite a few papers on teh subject and have a question.

Suppose we have two sensors x1 and x2

x1(t)=s1(t)+n1(t)

and

x2(t)=a.s2(t-D)+n2(t)

where n1(t) and n2(t) are uncorrelated white noise sources also
uncorrelated with the two signals s1 and s2. I need to find the delay
D.The constant a is an attenuation factor.

The Knapp-Carter give an expression for cross-correlation as

Cx1x2(tau) = a.Css(tau) * delta (tau - D) where * is convolution and
delta is the delta function

and in other papers it is given as

Cx1x2(tau) = a.Css(tau-D)

are these the same? I Think they are but the Knapp-Carter one makes
more sense to me than the other one. If I take the Fourier Transform
of both I should get
the cross-spectral density of x1 with x2.

Thanks

Tom
```
```aberdonian_2000@yahoo.com (Tom) wrote in message news:<e1b1658f.0307221459.5a2d5e56@posting.google.com>...
> I have read quite a few papers on teh subject and have a question.
>
> Suppose we have two sensors x1 and x2
>
> x1(t)=s1(t)+n1(t)
>
> and
>
> x2(t)=a.s2(t-D)+n2(t)
>
> where n1(t) and n2(t) are uncorrelated white noise sources also
> uncorrelated with the two signals s1 and s2. I need to find the delay
> D.The constant a is an attenuation factor.
>
> The Knapp-Carter give an expression for cross-correlation as
>
> Cx1x2(tau) = a.Css(tau) * delta (tau - D) where * is convolution and
> delta is the delta function
>
> and in other papers it is given as
>
> Cx1x2(tau) = a.Css(tau-D)
>
> are these the same? I Think they are but the Knapp-Carter one makes
> more sense to me than the other one. If I take the Fourier Transform
> of both I should get
>  the cross-spectral density of x1 with x2.
>
> Thanks
>
> Tom

Hi Tom,

I don't know why it makes you to think so deep.

From fundamental convolution theorem we get,

css(tau-D) = integral [css(tau) deltafunction(tau-D)d(tau)
where, d(tau) is derivative of tau.

or in other words,
css(tau-D) = css * deltafunction(tau-D)
where, * is convolution.

Frequency domain also does not show any differnce.
Conclusion: I don't see any difference ; also no special sense which you
try to hint.

I guess "a" is scalar here.

Regards,
Santosh
```
```aberdonian_2000@yahoo.com (Tom) wrote in message news:<e1b1658f.0307221459.5a2d5e56@posting.google.com>...
> I have read quite a few papers on teh subject and have a question.
>
> Suppose we have two sensors x1 and x2
>
> x1(t)=s1(t)+n1(t)
>
> and
>
> x2(t)=a.s2(t-D)+n2(t)
>
> where n1(t) and n2(t) are uncorrelated white noise sources also
> uncorrelated with the two signals s1 and s2. I need to find the delay
> D.The constant a is an attenuation factor.
>
> The Knapp-Carter give an expression for cross-correlation as
>
> Cx1x2(tau) = a.Css(tau) * delta (tau - D) where * is convolution and
> delta is the delta function
>
> and in other papers it is given as
>
> Cx1x2(tau) = a.Css(tau-D)
>
> are these the same? I Think they are but the Knapp-Carter one makes
> more sense to me than the other one. If I take the Fourier Transform
> of both I should get
>  the cross-spectral density of x1 with x2.
>
> Thanks
>
> Tom

Yes they are the same, convolution with a delta signal will result in
the input signal.
```
```aberdonian_2000@yahoo.com (Tom) wrote in message news:<e1b1658f.0307221459.5a2d5e56@posting.google.com>...
> I have read quite a few papers on teh subject and have a question.
>
> Suppose we have two sensors x1 and x2
>
> x1(t)=s1(t)+n1(t)
>
> and
>
> x2(t)=a.s2(t-D)+n2(t)
>
> where n1(t) and n2(t) are uncorrelated white noise sources also
> uncorrelated with the two signals s1 and s2. I need to find the delay
> D.The constant a is an attenuation factor.
>
> The Knapp-Carter give an expression for cross-correlation as
>
> Cx1x2(tau) = a.Css(tau) * delta (tau - D) where * is convolution and
> delta is the delta function
>
> and in other papers it is given as
>
> Cx1x2(tau) = a.Css(tau-D)
>
> are these the same? I Think they are but the Knapp-Carter one makes
> more sense to me than the other one. If I take the Fourier Transform
> of both I should get
>  the cross-spectral density of x1 with x2.
>
> Thanks
>
> Tom

Ask yourself, wha is the value of delta(tau-D) at any time other than tau-D
```
```aberdonian_2000@yahoo.com (Tom) wrote in message news:<e1b1658f.0307221459.5a2d5e56@posting.google.com>...
> I have read quite a few papers on teh subject and have a question.
>
> Suppose we have two sensors x1 and x2
>
> x1(t)=s1(t)+n1(t)
>
> and
>
> x2(t)=a.s2(t-D)+n2(t)
>
> where n1(t) and n2(t) are uncorrelated white noise sources also
> uncorrelated with the two signals s1 and s2. I need to find the delay
> D.The constant a is an attenuation factor.
>
> The Knapp-Carter give an expression for cross-correlation as
>
> Cx1x2(tau) = a.Css(tau) * delta (tau - D) where * is convolution and
> delta is the delta function
>
> and in other papers it is given as
>
> Cx1x2(tau) = a.Css(tau-D)
>
> are these the same? I Think they are but the Knapp-Carter one makes
> more sense to me than the other one. If I take the Fourier Transform
> of both I should get
>  the cross-spectral density of x1 with x2.
>
> Thanks
>
> Tom

Ask yourself, what is the value of delta(tau-D) at any time other than tau-D
```
```
santosh nath wrote:

> aberdonian_2000@yahoo.com (Tom) wrote in message news:<e1b1658f.0307221459.5a2d5e56@posting.google.com>...
> > I have read quite a few papers on teh subject and have a question.
> >
> > Suppose we have two sensors x1 and x2
> >
> > x1(t)=s1(t)+n1(t)
> >
> > and
> >
> > x2(t)=a.s2(t-D)+n2(t)
> >
> > where n1(t) and n2(t) are uncorrelated white noise sources also
> > uncorrelated with the two signals s1 and s2. I need to find the delay
> > D.The constant a is an attenuation factor.
> >
> > The Knapp-Carter give an expression for cross-correlation as
> >
> > Cx1x2(tau) = a.Css(tau) * delta (tau - D) where * is convolution and
> > delta is the delta function
> >
> > and in other papers it is given as
> >
> > Cx1x2(tau) = a.Css(tau-D)
> >
> > are these the same? I Think they are but the Knapp-Carter one makes
> > more sense to me than the other one. If I take the Fourier Transform
> > of both I should get
> >  the cross-spectral density of x1 with x2.
> >
> > Thanks
> >
> > Tom
>
> Hi Tom,
>
> I don't know why it makes you to think so deep.
>
> From fundamental convolution theorem we get,
>
> css(tau-D) = integral [css(tau) deltafunction(tau-D)d(tau)
> where, d(tau) is derivative of tau.
>
> or in other words,
> css(tau-D) = css * deltafunction(tau-D)
> where, * is convolution.
>
> Frequency domain also does not show any differnce.
> Conclusion: I don't see any difference ; also no special sense which you
> try to hint.
>
> I guess "a" is scalar here.
>
> Regards,
> Santosh

Yes of course it is simple. I only needed to substiitute h =tau-D and take the FT
and they are identical.
Now can somebody anwswer a (vaguely) related question. I have seen books which define sampling as
multiplication by a train of impulses and somebody on this group said it was convolution by impulses - do they
give the same end result?
ie suppose I have an analugue signal f(t) then is the FT of

f(t)*sum delta(t-tau)   * = convolution

the same as

f(t).sum delta (t-tau) where . is multiplication.

We should get the standard sampled spectrum .

Thanks

Tom

```
```aberdonian_2000@yahoo.com (Tom) wrote in message news:<e1b1658f.0307221459.5a2d5e56@posting.google.com>...
> I have read quite a few papers on teh subject and have a question.
>
> Suppose we have two sensors x1 and x2
>
> x1(t)=s1(t)+n1(t)
>
> and
>
> x2(t)=a.s2(t-D)+n2(t)
>
> where n1(t) and n2(t) are uncorrelated white noise sources also
> uncorrelated with the two signals s1 and s2. I need to find the delay
> D.The constant a is an attenuation factor.
>
> The Knapp-Carter give an expression for cross-correlation as
>
> Cx1x2(tau) = a.Css(tau) * delta (tau - D) where * is convolution and
> delta is the delta function
>
> and in other papers it is given as
>
> Cx1x2(tau) = a.Css(tau-D)
>
> are these the same? I Think they are but the Knapp-Carter one makes
> more sense to me than the other one. If I take the Fourier Transform
> of both I should get
>  the cross-spectral density of x1 with x2.
>
> Thanks
>
> Tom

Hi Tom,

This can be shown  from fundamental convolution theorem that,
delta(tau - D) * Css(tau) =  Css(tau - D)   ....(1)

I don't know why you suspect and thinking the other makes sense .
There is no difference at all.
In Freq. Domain also there should not be any difference as you hinted.

I guess "a" is scalar.

Regards,
Santosh
```
```Tom wrote:
(snip)
> Now can somebody anwswer a (vaguely) related question. I have seen books which define sampling as
> multiplication by a train of impulses and somebody on this group said it was convolution by impulses - do they
> give the same end result?
> ie suppose I have an analugue signal f(t) then is the FT of
>
> f(t)*sum delta(t-tau)   * = convolution
>
> the same as
>
> f(t).sum delta (t-tau) where . is multiplication.
>
> We should get the standard sampled spectrum .
>
> Thanks
>
> Tom
>
>

Convolution of a continuous function by the delta function would result
in a continuous function, wouldn't it?  There would have to be something
about an infinite sum of shifted continuous functions for the result to
be discrete sequence which doesn't seem obvious to me.

```