# Fourier transform

1. Sep 25, 2014

### bonildo

Whais is the effect of a multiplication by (-1)^n in the DTFT ??

In other words, whats is this transform : x(n)* (-1)^n ??

2. Sep 25, 2014

### jbunniii

If $X(\omega) = \sum_{n=-\infty}^{\infty} x(n) e^{-i \omega n}$, then
\begin{align} \sum_{n=-\infty}^{\infty} x(n) (-1)^n e^{-i \omega n} &= \sum_{n=-\infty}^{\infty} x(n) e^{-i \pi n} e^{-i \omega n} \\ &= \sum_{n=-\infty}^{\infty} x(n) e^{-i (\omega + \pi) n} \\ &= X(\omega + \pi) \end{align}
Note that this is also equal to $X(\omega - \pi)$ due to the $2\pi$-periodicity of the discrete-time Fourier transform.