Dear Jaoua Amna,

Forgive me before you in the interpretation of the curve "ROC" I propose to review the file whose link is as follows and will be discussed after.

-Detection Theory Notes - Personal.psu.edu

www.personal.psu.edu/faculty/m/x/mxm14/sonar/detection.pdf

Best regards

Hello Sir,

I have read the course that you gave me.

Dear Jaoua Amna,

Here are the notes:

It seems to me that there is a problem in the attachment of the ROC curve.

- The probability of false alarm is a threshold which is set beforehand (well understood that the probability of detection is superimposed on the probability of false alarm)

- The probability of detection and probability of false alrme should be on the same axis

Best regards

Dear Jaoua Amna,

I followed the dialog between you and Dr. Mohamed and detected that there is some study relation between you. Any how. You asked about the physical explanation of your results concerning the relation between the probability of detection Pd, and the probability of false alarm Pfa.

In the energy detection problem you have a receiver in a certain location whose function is to detect the the existence of an active  primary user. So, the sensing receiver will receive the the primary user signal x if it exists and the receiver itself produces noise n.

The problem of the detection is not deterministic since the signal it self may vary in  a large range because for example the channel fading due to multipath effect.

In addition the noise is a random variable.  So, the problem is that we have to distinguish two signals which are the noise only if there is no activity of the primary transmitter n, and the signal x+n. Having the probability distribution of the two signals, one can define a decision threshold where the probability density  function of both signals will be equal. One can then find the probability of the false alarm Pfa which is the total probability that the noise is equal or greater than the threshold level. This means that the primary transmitter is not active in spite of that we decided it is active. The probability of detection is Pd is the total probability that x+n will be greater than the threshold level.

It is so that the two distributions may be Gaussian with noise distribution has a zero mean value and variance sigman^2. While the signal and noise has an average value equals the signal average value at the receiver site and a standard deviation sigman due to the additive white noise. The signal it self may change randomly due to fading then one has to take into consideration by the proper probability density function according to the type pf fading.

The mathematical computation is just to evaluate the areas under the probability density curves in the intended range which is above the threshold for the x+n and above the threshold for the nose n for the noise n.

In short one get the given expression above for the Pd.

There is also an expression for the Pfa, you have to give it as Dr. Mohamed said.

As the two probabilities are required to describe the detection process. i would like to advice you to calculate both as a function of the signal to noise ratio and plot them. 

Physically increasing the probability of the false alarm increases the detection probability for any given signal to noise ratio.

I shed some light on the detection process and would like to say you explain your results based on your physical model of your system.

wish you success

Dear Jaoua Amna,

I totally agree with Prof. Abdelhalim speech and for Jaoua he needed to put his thoughts in order.

Best regards

Thank you Sirs for your help.