DTFT of x(n)*(-1)^n: Effect & Transform Explained

[bonildo]

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

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

 

[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.