False Alarm Integral

**CaptainBlack** · March 3rd 2006, 06:50 AM

This is a problem asked on the usenet group: sci.engr.radar+sonar

Show:

$ \int_0^{\infty} e^{-(T+N+1-k)y} (1-e^{-y})^{k-1} dy= \frac{(k-1)!(T+N-k)!}{(T+N)!} $

RonL

**ThePerfectHacker** · March 3rd 2006, 09:07 AM

I wanted to call the term $T+N+1$ as $M$ but I am afraid to because maybe you mean that all three need to be positive integers in that case $M\geq 3$ . Can you provide the domain for all the integers?

Maybe you can induct on $k$ ? That what I am trying now.

**ThePerfectHacker** · March 3rd 2006, 09:31 AM

Originally Posted by CaptainBlack

This is a problem asked on the usenet group: sci.engr.radar+sonar

Show:

$ \int_0^{\infty} e^{-(T+N+1-k)y} (1-e^{-y})^{k-1} dy= \frac{(k-1)!(T+N-k)!}{(T+N)!} $

RonL

I was trying to simplify the integral,
$ \int e^{-(T+N+1-k)y} (1-e^{-y})^{k-1} dy $
into an easier form. Notice it can be expressed as,
$ \int e^{-y}(1-e^{-y})^{k-1}(e^{-y})^{T+N-k} $
Use the substitution, $u=1-e^{-y}$ then we have that $u'=e^{-y}$ and $(e^{-y})^{T+N-k}=(1-u)^{T+N-k}$ . Thus, we have that,
$\int u'u^{k-1}(1-u)^{T+N-k}dx$ by the substitution rule we have that,
$\int u^{k-1}(1-u)^{T+N-k}du$ . But I do not know if this form helps?

**CaptainBlack** · March 3rd 2006, 10:46 AM

Originally Posted by CaptainBlack

This is a problem asked on the usenet group: sci.engr.radar+sonar

Show:

$ \int_0^{\infty} e^{-(T+N+1-k)y} (1-e^{-y})^{k-1} dy= \frac{(k-1)!(T+N-k)!}{(T+N)!} $

RonL

Observe that the nature of the problem implies $1 \le k<=T+N$
and $T+N>1$ amd these are all integers.

The first thing that I will do is simplify this a little by putting:

$M=T+N$

Then what we have to show becomes:

$ \int_0^{\infty} e^{-(M+1-k)y} (1-e^{-y})^{k-1} dy= \frac{(k-1)!(M-k)!}{M!} $

Let:

$ I_{k,M}=\int_0^{\infty} e^{-(M+1-k)y} (1-e^{-y})^{k-1} dy $ ,

and proceed by integrating by parts:

$ I_{k,M}=\left[-\frac{(1-e^{-y})^{k-1}e^{-(M+1-k)y} }{M+1-k}\right]_0^\infty$ $-\int_0^{\infty}\frac{e^{-(M+1-k)y}}{M+1-k}(k-1)e^{-y}(1-e^{-y})^{k-2}dy $

Tiding this up:

$ I_{k,M}=\frac{k-1}{M+1-k}I_{k-1,M} $

Then repeatedly using this recurrence and observing the end condition:

$ I_{1,M}=\frac{1}{M} $

gives:

$ I_{k,M}=\frac{(k-1)!(M-k)!}{M!} $

RonL

**topsquark** · March 3rd 2006, 11:54 AM

Hmmm...again I did it the hard way.

I expanded the integrand in powers of $e^{-y}$ and integrated. The integral is easy that way, but the form is nasty. However, I found a way to simplify using, of all things, Lagrange's Interpolation formula.

Just when you thought it was useless...

(Actually, to be honest I'm not quite there...I'm still off by a constant. But that ought to go away once I go through it again. I've got a method, that's what I was after.)

-Dan

BTW Thanks for the integral. I haven't had that much fun integrating for a while!

**topsquark** · March 3rd 2006, 11:59 AM

Since this came from an engineering usenet group, I presume this integral represents something. I'm curious to know if you had found that out.

-Dan

**ThePerfectHacker** · March 3rd 2006, 12:05 PM

I am curious has anyone tried to bring this to the Gamma Integral Form?

**CaptainBlack** · March 3rd 2006, 12:05 PM

Originally Posted by topsquark

Since this came from an engineering usenet group, I presume this integral represents something. I'm curious to know if you had found that out.

-Dan

I don't know exactly what the details are, but its a detector of some kind.
I expect its something like an M from N detector, where a detection is flagged
if M threshold crossings are made out of N opportunities. Except of course this
has three parameters so presumably is more complicated than that.

Also somewhere in there the distribution of the noise is hidden (since this
is related to the calculation of the probability of false alarm)

RonL

**CaptainBlack** · March 3rd 2006, 12:14 PM

Originally Posted by CaptainBlack

This is a problem asked on the usenet group: sci.engr.radar+sonar

Show:

$ \int_0^{\infty} e^{-(T+N+1-k)y} (1-e^{-y})^{k-1} dy= \frac{(k-1)!(T+N-k)!}{(T+N)!} $

RonL

I must admit that I knew (or at least thought I knew) how
to do this before I posted the question.

I posted it here as I wanted the solution on line and in mathematical
notation (it would be impossible to follow in plain ASCII), so I could post a
link to it in its original forum.

RonL

**John Mullane** · March 5th 2006, 06:10 PM

Hey Guys,

I posted that integral in the engineering forum, thanks alot for your help. Indeed it is a detector, for a radar, in my work, an FMCW radar. It is an equation to establish the detection threshold multiplier (T) to obtain a constant probablity of false alarm (Constant false alarm rate (CFAR) detector). In this case N, is the number of 'cells' used in the moving window function, and k is the kth cell, which is used to estimate the mean of the (Assumed exponentiallly distirubuted) noise. This appears in the Ordered-Statistics CFAR detector. The interested reader may look at :
ieeexplore.ieee.org/iel3/ 4347/12454/00573784.pdf?arnumber=573784

or, if you don't have access :
www.nato-asi.org/sensors2005/papers/rohling.pdf

I too believe it can be solved through the gamma function and its factorial properties.

Thanks again,

John.

**John Mullane** · March 5th 2006, 08:39 PM

If you are guys are interested, I'd also like to post the following integral which I'm also banging my head against. It is generally the second step in detector analysis - evaluating the probability of detection.

Show:
$ P_{D} = \int_{0}^{\infty}P[x\geq Tz|H_{1}]f_{Z}(z) dz = \bigg(1 + \displaystyle\frac{Tz}{1+S} \bigg)^{-N} $
where
$ P[x\geq Tz|H_{1}] = \int_{Tz}^{\infty}f_{X}(x|H_{1})dx. $

and,
$ f_{X}(x|H_{1}) = \displaystyle\frac{1}{\mu}e^{(\frac{-x}{\mu} + S)}.I_{0}\bigg(2\sqrt{\displaystyle\frac{S x}{\mu}} \bigg) $
where $I_{0}$ is a modified Bessel function of order zero. (This is a Ricean probability distribution function). S in this case, represents the average signal to noise ratio from the target.

As before , for an exponentially distributed Z, the probability density function of the kth element of N ordered (according to amplitude) independent samples of Z is given by:
$ f_{Z}(z) = \frac{k}{\mu}\binom{N}{k}(e^{\displaystyle{-z/\mu}})^{N-k+1}(1-e^{\displaystyle{-z/\mu}})^{k-1}. $

As the cdf of a Ricean ( $P[x\geq Tz|H_{1}]$ ) is generally evaluated in closed form using the Gaussian error (or complimentary error) function, I fail to see how $P_{D}$ can be obtained in the above form.

Any comments/ideas/suggestions are greatly appreciated.
Regards,
John.

**CaptainBlack** · March 5th 2006, 11:01 PM

Originally Posted by John Mullane

Hey Guys,

I posted that integral in the engineering forum, thanks alot for your help. Indeed it is a detector, for a radar, in my work, an FMCW radar. It is an equation to establish the detection threshold multiplier (T) to obtain a constant probablity of false alarm (Constant false alarm rate (CFAR) detector). In this case N, is the number of 'cells' used in the moving window function, and k is the kth cell, which is used to estimate the mean of the (Assumed exponentiallly distirubuted) noise. This appears in the Ordered-Statistics CFAR detector. The interested reader may look at :
ieeexplore.ieee.org/iel3/ 4347/12454/00573784.pdf?arnumber=573784

or, if you don't have access :
www.nato-asi.org/sensors2005/papers/rohling.pdf

I too believe it can be solved through the gamma function and its factorial properties.

Thanks again,

John.

Thanks for the reference, it looks interesting.

RonL

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