The function $f(x,y)$ has a periodicity of $a$ in the $x$-direction and $b$ in the $y$-direction,
\[ \begin{equation} f(x,y)=3\sin \left(\frac{2\pi x}{a}\right)\cos \left(\frac{2\pi y}{b}-1\right). \end{equation} \]This function can be expressed as a sum of complex exponentials,
\[ \begin{equation} f(x,y)=\sum\limits_{\vec{G}}f_{\vec{G}}e^{i(G_xx+G_yy)}. \end{equation} \]Calculate Fourier coefficients, $f_{\vec{G}}$.