Magnitude and phase of DTFT of aⁿu[n]

DTFT:

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

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

Magnitude:

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

Phase:

$$\angle X(e^{j\omega}) = -\tan^{-1}\left(\frac{a\sin\omega}{1 - a\cos\omega}\right)$$

Values for $0<a<1$ (taking $a = 0.5$ for the sketch):

The phase reaches its most negative value, $-\sin^{-1}a$ ($-30^\circ$ for $a=0.5$), where $\cos\omega = a$ ($\omega = \pi/3$ for $a = 0.5$). By odd symmetry it reaches $+30^\circ$ at $\omega = -\pi/3$.

 |X(e^jw)|  (a = 0.5, low-pass shape)
 2.0 |              *
     |            *   *
     |          *       *
0.89 |        *           *
0.67 |  * * *               * * *
     +--+-----------+-----------+--> w
       -pi          0           pi

 phase (a = 0.5, odd function)
 +30 |        * * *
 +15 |    * *
   0 +--*-----------*-----------*--> w
 -15 |                    * *
 -30 |              * * *
       -pi          0           pi

 (both plots repeat every 2pi)

Observations:

• The spectrum is continuous in $\omega$ and periodic with period $2\pi$.
• The magnitude is even and the phase is odd (real signal).
• For $0<a<1$ the signal is low-pass (peak at $\omega = 0$). For $-1<a<0$ the peak moves to $\omega=\pi$ (high-pass).