Short note: Frequency shifting of DTFT

Frequency shifting property: Multiplying a sequence by a complex exponential $e^{j\omega_0 n}$ shifts its DTFT by $\omega_0$:

$$x[n] \leftrightarrow X(e^{j\omega}) \;\Rightarrow\; e^{j\omega_0 n}x[n] \leftrightarrow X(e^{j(\omega-\omega_0)})$$

Proof:

$$\begin{aligned} \sum_{n=-\infty}^{\infty} e^{j\omega_0 n}x[n]\,e^{-j\omega n} &= \sum_{n=-\infty}^{\infty} x[n]\,e^{-j(\omega-\omega_0)n} \\ &= X(e^{j(\omega-\omega_0)}) \end{aligned}$$

Points to note:

• The whole spectrum moves right by $\omega_0$; the shape does not change. The shifted spectrum is still periodic with period $2\pi$.
• It is the dual of the time-shifting property ($x[n-n_0] \leftrightarrow e^{-j\omega n_0}X(e^{j\omega})$).
• Modulation: Since $\cos\omega_0 n = \frac{1}{2}(e^{j\omega_0 n} + e^{-j\omega_0 n})$, $x[n]\cos\omega_0 n \leftrightarrow \frac{1}{2}\left[X(e^{j(\omega-\omega_0)}) + X(e^{j(\omega+\omega_0)})\right]$.
• Special case $\omega_0 = \pi$: $(-1)^n x[n] \leftrightarrow X(e^{j(\omega-\pi)})$, which turns a low-pass spectrum into a high-pass one.

Example: $a^n u[n] \leftrightarrow \frac{1}{1-ae^{-j\omega}}$, so $e^{j\pi n/4}a^n u[n] \leftrightarrow \frac{1}{1-ae^{-j(\omega-\pi/4)}}$; the peak moves from $\omega = 0$ to $\omega = \pi/4$.