solving an equation $\sum_{n=0}^{N-1}h(n)\mathrm{sin}[(n-\alpha)\omega+\beta]=0$

Question

If we have the equation:

$$\sum_{n=0}^{N-1}h(n)\mathrm{sin}[(n-\alpha)\omega+\beta]=0.$$

We want to solve for the combination of $(h,\alpha,\beta)$ that this equation holds for all $\omega\in \mathbb{R}$.

Here, $h:=h(n)$ is a real valued finite sequence, $\alpha,\beta $ are real valued constants to be solved. Of course, we are not interested in the trivial case in which $h(n)\equiv0$.

Can you find the other solutions?

Thanks!

Answer 1

The sinus functions for pairwise different frequencies are linearly independent. There is only the trivial linear combination that combines to zero.