Problem:
In certain signal detection problems (e.g. radar or sonar) the probability of false alarm (FA) (i.e., of saying that a certain signal is present in the data when it actually is not) is given by:
∞
pFA = ∫ _______1_______ x^p/2-1 e-x/2 dx
η Γ(p/2) 2^p/2
Eq. 1.1
where η is called the detection threshold. If p is an even number, it can be shown that the Eq. 1.1 reduces to the finite series
(p/2)-1
pFA = e^(-1/2 η) Σ (1/k!)(η/2)^k
k=0
The detection threshold η is a very important design parameter in signal detectors. Often it is desired to specify an acceptable value for pFA (where 0 < pFA < 1), and then it is necessary to solve nonlinear equation (Eq 1.2) for η. Let p = 6. Use the bisection method to find η for pFA = 0.001. Use a tolerance of 0.00001.
Comment:
I just need to verify if the equation I'm going to use in bisection method is
0.001=(e^(1/2(n)))(1/0)(n/2)^0+(1/1)(n/2)^1+(1/2)(n/2)^2
Equations didn't turn out right
......∞
pFA = ∫ _______1_______ x^p/2-1 e-x/2 dx
......η .Γ(p/2) 2^p/2
Eq. 1.1
................(p/2)-1
pFA = e^(-1/2 η) Σ (1/k!)(η/2)^k
.................k=0