To answer to your question, I would like to illustrate my explanation with the figure attached below. It corresponds to the probability density functions associated with both hypothesis (H1: signal present, H0: signal abscent). I am assuming that the signal of interest is a baseband signal (e.g. pulses, PWM, PPM, etc.), and that the noise is Gaussian and zero-mean.

Let gamma be the threshold for discriminating between H1 and H0. Then the probability of detection (Pd) is the area under f1(x) at the right-side of gamma. On the other hand, the probability of false alarm (Pfa) is equal to the area under f0(x) at the right-side of gamma.

The ROC curves are plots of Pd vs. Pfa. There will be different ROC curves for different SNRs (signal to noise ratio). If the noise variance is constant, and a change in SNR is only due to a change in the pulses amplitudes, then an increase of SNR means that f1(x) will "move" to the right (the mean will be increased). Thus, the gray area (Pfa) in the figure will remain the same, but the area under f1(x) at the right-side of gamma (Pd) will be increased. And viceversa: if the SNR is decreased, then the Pd will be decreased.

Anyway, the Pfa doesn't change, and the Pd is strictly related with the SNR.

That is why two different ROC curves (associated to differents SNRs) will never cross each other, because for every Pfa, different Pd values will be obtained for a different SNR values.

I hope I have explained it well.

Best regards,

LM Gato.

Using Laplace Transform, is it possible to transfer function with out knowing input signals to the system.

Mr. Kaliannan, 

With ROC, in this case we are not referring to "Region of Convergence" but "Receiver Operating Characteristics".

Best regards,

LM Gato.