Sine wave with a constant period: what else is constant? [closed]

Question

Which statistic that you can calculate on a window remains constant on a sine wave with a constant period? This is for a window function for a proprietary filter.

Answer 1

The quantities you will find to be constant in a window for a pure sine wave are all derived from

$$ {y''/y} = - \alpha^2$$

where $y(x) = A \sin(\alpha x) $. So if you measure the second derivative in any window of width $W$ around position $x$, for example, by computing

$$ Y''(x) = { Y(x+\epsilon) - 2Y(x) + Y(x-\epsilon) \over \epsilon ^2} $$

for $\epsilon<W$ taken to be small compared to $1/\alpha$ and divide by the value $Y(x)$, no matter which window you take and where you are in the window, the answer will be $-\alpha^2$. You can extract other higher order invariants from this relation, by differentiating. So that

$$ {y'''/y'} = -\alpha^2 $$

and

$$ {y''''/y} = \alpha^4 $$

etc, I hope the others are all obvious. This will not distinguish between cosine and sine, of course, because you can shift one to the other, but it will distinguish between sinusoids and anything else. Different values of $\alpha$ are different periods.

If you are interested in a diagnostic which will give you a sinusoid without knowing $\alpha$, it is given by differentiating the first relation once again:

$$ {d\over dx} ({y''\over y}) = 0 $$

or

$$ {y y''' - y' y'' = 0} $$

At all places where y is nonzero. Using suitable estimators of these quantities in any window will give you a criterion for a function to be sinusoidal.