Finding a DTFT of a signal


I'm trying to figure out what's the DTFT of \$ (-1)^nx[n]\$. (I'm given the DTFT of \$x[n]\$)

So I tried this, but I can't figure out how to proceed from here, if this is even correct.
Any help and advice would be appreciated!

\$F_{DTFT}\{ (-1)^n x[n] \} = \sum\limits_{n=-\infty}^\infty (-1)^n x[n] e^{-i\theta n}\$

\$=\sum\limits_{even \space n's}x[n]e^{-i\theta n}-\sum\limits_{odd \space n's}x[n]e^{-i\theta n} =\$

\$=\sum\limits_{k=-\infty}^{\infty}x[2k]e^{-i\theta 2k}-\sum\limits_{m=-\infty}^{\infty}x[2m+1]e^{-i\theta (2m+1)} = ?\$


Best Answer

Note that \$(-1)^n=e^{i\pi n}\$, so you get

$$\sum_{n=-\infty}^{\infty}x[n](-1)^ne^{-in\theta}= \sum_{n=-\infty}^{\infty}x[n]e^{i\pi n}e^{-in\theta}= \sum_{n=-\infty}^{\infty}x[n]e^{-in(\theta-\pi)}=X(\theta-\pi)$$

The spectrum is just shifted by \$\pi\$. This is basically a consequence of the modulation property.

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