# Where does $\frac{N}{2}$ came from in approximating an N-point DFT?

I've came across the author saying that

... for a real cosine input having k cycles in the N-point input time sequence, the amplitude response of an N-point DFT bin in terms of the bin index m is approximated by the sinc function $$X(m)=\frac{A_{0}\color{red}{N}}{\color{red}{2}}\cdot\frac{\sin[\pi(k-m)]}{\pi(k-m)}$$ where $A_{0}$ is the peak value of the DFT’s input sinusiod.

I have no idea where $\frac{N}{2}$ came from. I'm wondering if you can give me some hints.

• Does this help at all? – Peter K. Apr 24 '16 at 17:28
• @PeterK. That seems what I've been looking for. Thank you very much. – bae Apr 24 '16 at 17:45