I am reading a paper and am having difficult reconstructing an equation. The paper begins with the following equations where $\omega_{n} = \omega_{o} + n\Delta\omega$ and:

(1) $A_{in}(t) = \sum_{n}a_{n}(t)e^{-i\omega_{n}t}$

(2) $B_{out}(t) = \sum_{n}b_{n}(t)e^{-i\omega_{n}t}$

(3) $B_{out}(t) = e^{i\phi(t)}A_{in}(t)$

From the above equations ($\phi(t)$ is $\frac{2\pi}{\Delta\omega}$ periodic), and using Fourier series coefficients of $e^{i\phi(t)}$, they state:

(4) $b_{m}(t) = \sum_{n=-\infty}^{\infty}c_{m-n}a_{n}(t)$

(5) $c_{k} = \frac{\Delta\omega}{2\pi}\int_{0}^{\frac{\Delta\omega}{2\pi}}dte^{i\phi(t)}e^{ik\Delta\omega t}$

I have been trying to confirm equations (4) and (5) without success. Specifically, I am not sure how they reduce the sum on the left side to the mth element. I appreciate any and all help! Thanks


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