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[Silent]

Reading Skolnik book on radar, I saw this sentence (talking about the probability of false alarm):


The average duration of a threshold crossing by noise is approximately the
reciprocal of the IF bandwidth B

My question is simply: why?


Thank you.

[admin]

Not sure but you can reason your way through to get some clues....

Noise power is proportional to bandwidth: more bandwidth more noise. More bandwidth, faster changes (higher frequency content, more chaotic).

If threshold crossing is determined by noise, and I don't know the context whether signal is there, but the reciprocal of the bandwidth B would be how slow the noise is changing, so it lines up with my intuition

[k3nnw]

I think this paper is a good explanation:

https://www.phys.hawaii.edu/~anita/new/papers/militaryHandbook/rcvr_sen.pdf

It boils down to noise figure (NF) and the previous response, the intuitive approach, is correct. Also, narrower band pass filters pass lower amplitudes of noise power but faster transient signals are not able to passed because their rate of change falls outside of the filter's band width.

Consider a simple single stage LC filter which has two energy storage devices, an inductor and a capacitor. These need to be charged and discharged by a signal whose voltage and current drive charge through the network. Two devices can be quickly filled and emptied of energy. The filter response is very wide and the filter rolloff is very slow.

Now consider a multi-section LC filter comprised of a dozen of these LC stages - that's now a dozen L and a dozen C all cascaded in some form from input to output. This network now takes time to charge and discharge and hence a rapid signal, perhaps a single pulse which rises from 0V to 1V in a microsecond and then immediately decays back down again in a microsecond, won't arrive at the output. There isn't enough signal power in that pulse to cause a response at the output. If there were (say) a million pulses appearing one after the other, some of them would eventually propagate through.

Now consider the amount of noise let through by that first consideration, the wide single stage LC filter - it swamps that single pulse. In the latter case there's a lot less noise but the signal's too weak unless it is a continuous wave. That's why the false alarm rate is described in the way that the paper (and Skolnik) talk about.