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Just as one can write $$ \left(\frac{e^{jw} + e^{-jw} }{2}\right) = cos(w) $$ where $ j = \sqrt -1 $

Can one represent sum of two exponential functions, which contain summation in the power in a compact way? As an example consider

$$ e^{j\sum_{k=0}^m a_k} + e^{j\sum_{k=0}^m a_{k-1}} $$ where $ a_k $ can be samples of signal implying that $ a_{k-1} $ are delayed versions of $ a_k $.

One possible solution that comes to my mind is to somehow multiply one of the exponential functions with an exponential that will remove that delay and then the two exponential functions will essentially be the same and perhaps lead to a simplification, but I am not sure if that will work.

Any suggestions on how to simplify the given example i.e. write it compactly using some identity will be much appreciated.

Thank you.

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With $\sigma=\sum_{k=0}^ma_k$ and $\theta=a_m-a_{-1}$, these are essentially

$$e^{j\sigma}+e^{j(\sigma-\theta)}.$$

In other words, they essentially differ by a phase $\theta$.
I guess you can simplify it to $e^{j\sigma}\left(1+e^{-j\theta}\right)$?

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