2
$\begingroup$

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.

$\endgroup$

1 Answer 1

2
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ Thanks Fedja. Discrete result is enough for me. I was actually expecting some condition on the distribution on Fourier coefficients of the function, e.g. if we know \sum ka_k^2 is large instead of a_k=0 for |k|<=N, can we say something on the zero crossings, but the current statement is useful as well. Thanks again. $\endgroup$
    – David
    Jan 17, 2011 at 0:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.