I'm recently encountering such a problem, which I think is very intuitive but I have no background knowledge on this field:


Given a signal with certain frequency distribution, e.g. we know that the signal has only high frequency part, we can expect that the number of zero-crossings of the signal has to be large. Is there a way to quantify this statement? For example, Can we give a (deterministic or stochastic) lower bound on the zero-crossings?


Thanks a lot for any hint or reference.



Estimate on zero-crossings of band passing signals
In the discrete frequency setting (periodic signal) everything is nice and clean: since we can find a real-valued trigonometric polynomial $P$ of degree $\le n$ with given $2n$ or fewer roots on the circle, the real-valued signal $f=\sum\_k a\_k z^k$ that changes sign at most than $2n$ times, should have some non-zero $a\_k$ with $|k|\le n$ (just take the polynomial $P$ with the same sign changes and observe that $fP$ preserves sign, so $\int fP\ne 0$) and that is sharp ($z^{n+1}+z^{-(n+1)}$ has $2n+2$ zeroes and the number of sign changes is always even).


The continuous case is much messier. What exactly do you need? 


