DTFT of aⁿu[n]

The signal $x[n] = a^n u[n]$ with $0 < a < 1$ is a right-sided decaying exponential: $x[n] = 1, a, a^2, \dots$ for $n = 0, 1, 2, \dots$

DTFT

$$\begin{aligned} X(e^{j\omega}) &= \sum_{n=-\infty}^{\infty}a^n u[n]e^{-j\omega n} = \sum_{n=0}^{\infty}\left(ae^{-j\omega}\right)^n \end{aligned}$$

This is a geometric series with ratio $r = ae^{-j\omega}$. Since $|r| = a < 1$, it converges to $\frac{1}{1-r}$:

$$\boxed{X(e^{j\omega}) = \frac{1}{1 - ae^{-j\omega}}}$$

Magnitude and phase

Write $1 - ae^{-j\omega} = (1 - a\cos\omega) + ja\sin\omega$:

$$|X(e^{j\omega})| = \frac{1}{\sqrt{1 - 2a\cos\omega + a^2}}, \qquad \angle X(e^{j\omega}) = -\tan^{-1}\!\left(\frac{a\sin\omega}{1 - a\cos\omega}\right)$$

For example, with $a = 0.5$: $|X| = 2$ at $\omega = 0$, $0.894$ at $\pi/2$, and $0.667$ at $\pi$.

 |X(e^jw)|  (a = 0.5)
 2.0 |              *
     |            *   *
     |          *       *
0.89 |        *           *
0.67 |  * * *               * * *
     +--+-----------+-----------+--> w
       -pi          0           pi
 (repeats every 2pi)

Observations

• The magnitude is even and the phase is odd in $\omega$, because $x[n]$ is real.
• The spectrum is periodic with period $2\pi$.
• With $0 < a < 1$ the signal is low-pass: most of its energy is near $\omega = 0$. For $-1 < a < 0$ the peak moves to $\omega = \pi$ (high-pass).
• If $a \ge 1$ the sum diverges and the DTFT does not exist.