DTFT of aⁿu[n] with spectrum plot

DTFT

$$\begin{aligned} X(e^{j\omega}) &= \sum_{n=-\infty}^{\infty} a^n u[n]\,e^{-j\omega n} = \sum_{n=0}^{\infty}\left(a e^{-j\omega}\right)^n \\ &= \frac{1}{1 - a e^{-j\omega}}, \quad \text{since } |ae^{-j\omega}| = a < 1 \end{aligned}$$

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

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

Spectrum plots

 |X|
 1/(1-a) |                        *
         |                     *     *
         |
         |                  *           *
         |               *                 *
         |         *  *                       *  *
 1/(1+a) |*  *  *                                   *  *  *
         +-------------------------------------------------> w
         -pi                      0                      pi

 phase  (odd; max |phase| = asin(a), reached where cos w = a)
+asin(a) |            *  *  *
         |         *           *
         |      *
         |   *
       0 |*                       *                       *
         |                                             *
         |                                          *
         |                           *           *
-asin(a) |                              *  *  *
         +-------------------------------------------------> w
         -pi                      0                      pi

(Plots drawn for $a = 0.5$; one period $-\pi$ to $\pi$ is shown.)

The magnitude is even and low-pass (peak at $\omega = 0$), the phase is odd, and both are periodic with period $2\pi$.