Discrete-time Fourier transform (DTFT) Slecture by Jacob Holtman - Rhea

Definition

$ X(\omega) := \sum_{k=-\infty}^{\infty}x[n]e^{-j\omega k} $

$ X(\omega) $ is seen to be periodic with a period of $ 2\pi $ to see this $ \omega $ is replaced with $ \omega + 2k\pi $ where k is an integer

$ X(\omega + 2\pi) = \sum_{n=-\infty}^{\infty}x[n]e^{-j(\omega + 2k\pi)n} $

Using the multiplicative rule of exponential the $ \omega $ and $ 2k\pi $ are split into two different exponential

$ X(\omega + 2\pi) = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}e^{2k\pi n} $

given that n and k are integers k and so $ e^{-j2k\pi n} = 1 $ from Euler's identity and so

$ X(\omega + 2\pi) = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n} $

so $ X(\omega + 2\pi) = X(\omega) $ for all $ \omega $

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