I would like to know how to solve or approximately express the summation below $$\sum_{n=0}^{N-1}\frac{e^{i2\pi\alpha n}-e^{i2\pi(\beta n-\gamma)}}{i2\pi((\alpha-\beta)n+\gamma)} $$ where $\alpha$, $\beta$, and $\gamma$ are all reals, $i = \sqrt{-1}$. In my practical problem $$\alpha = \nu T_r+2\Delta f T-\Delta f|\tau|$$ $$\beta = \nu T_r+\Delta f |\tau|$$ $$\gamma = ((1-N)\Delta f +\nu)(T-|\tau|)$$ where, $\tau$ and $\nu$ are variables, and vary from $[-T,T]$ and $[-1/T_r, 1/T_r]$, respectively. $T<T_r<<\Delta f$ and the three are constants.
I have done some simulations about this summation, as shown in Fig. 1. It is a clear evident that the most amplitude concentrates in the diamond areas. I have found that the boundary of the diamond is $\alpha = 0$ and $\beta = 0$, and the summation have high amplitude, i.e. the diamond area, when $\alpha>0$ and $\beta<0$. Fig. 1 A case of $N=32$
Based on the integral of a sinc function, $$\int_0^{\infty}\frac{sin(ax)}{x}dx = \frac{\pi}{2}\mathrm{sign}(a)$$ I supose that the first term and the second term in the numerator are cancled when $\alpha$ and $\beta$ have the same sign, i.e. $\alpha\cdot\beta>0$. I belive the results are regularity, but I do not konw how to trakle this problem. I even do not konw which kind of book or paper should I refer. Welcome and thank all the suggestions!!
