question on ratio of trigonometric series

Consider following ratio of trigonometric series

$\displaystyle H_{special}(\omega) = \frac{\sum_{n=1}^N a_n\sin(n\omega)}{1+\sum_{n=1}^N a_n\cos(n\omega)}$

where the coefficients $\displaystyle a_n$ in the numerator and denominator are equal. $\displaystyle N$ is any positive integer. This is a special case of the general form

$\displaystyle H_{generic}(\omega) = \frac{\sum_{n=0}^N c_n\sin(n\omega)}{\sum_{n=0}^N d_n\cos(n\omega)}$

Does anyone know if it is possible to quantify in anyway what 'shapes' $\displaystyle H_{special}(\omega)$ is constrained to in comparison to $\displaystyle H_{generic}(\omega)$. Any pointers to relevant literature is very much appreciated. I have read about finite Fourier series, but so far that has not provided much insight.

niaren