---
title: "Magnitude and phase of DTFT of aⁿu[n]"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:dtft-geometric-magnitude-phase"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/dtft-geometric-magnitude-phase
site: BCT Engineering
---

# Magnitude and phase of DTFT of aⁿu[n]

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2077 Chaitra, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/dtft-geometric-magnitude-phase
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

Given discrete time signal is x[n] = aⁿu[n], |a| < 1. Given that Fourier transform of x[n] is continuous in nature, plot magnitude and phase response of the transformed signal.

## Short answer

The DTFT is X(e^{jω}) = 1/(1 - a e^{-jω}). Magnitude: |X| = 1/√(1+a^2-2a cos ω), peaking at ω=0 with value 1/(1-a) for 0<a<1 (low‑pass). Phase: ∠X = -tan^{-1}(a sin ω/(1-a cos ω)), odd, zero at ω=0 and ±π.

## Answer

**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):**

| $\omega$ | $\lvert X\rvert$ (general) | $\lvert X\rvert$, $a=0.5$ | Phase, $a=0.5$ |
|---|---|---|---|
| $0$ | $\frac{1}{1-a}$ | 2 | $0^\circ$ |
| $\pi/2$ | $\frac{1}{\sqrt{1+a^2}}$ | 0.894 | $-26.57^\circ$ |
| $\pi$ | $\frac{1}{1+a}$ | 0.667 | $0^\circ$ |

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 $\omega = -\pi/3$.

```text
 |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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The DTFT of a^n u[n], |a|<1, is:
   A. 1/(1 - a e^(-jω))
   B. 1/(1 + a e^(-jω))
   C. a/(1 - e^(-jω))
   D. 1/(1 - a e^(jω))
   Answer: A. Geometric series Σ(a e^(-jω))^n converges to 1/(1 - a e^(-jω)) when |a|<1. (Key point: DTFT of a^n u[n])

2. The magnitude |X(e^(jω))| is:
   A. 1/sqrt(1 + a^2 - 2a cos ω)
   B. 1/sqrt(1 + a^2 + 2a cos ω)
   C. 1/(1 - a cos ω)
   D. 1/sqrt(1 - a^2)
   Answer: A. |1 - a e^(-jω)|^2 = (1 - a cos ω)^2 + a^2 sin^2 ω = 1 + a^2 - 2a cos ω. (Key point: Magnitude)

3. For a = 0.5, |X| at ω = 0 and ω = π are:
   A. 2 and 0.667
   B. 0.667 and 2
   C. 1 and 1
   D. 2 and 2
   Answer: A. 1/(1-a) = 2 at ω=0 and 1/(1+a) = 0.667 at ω=π, a low-pass shape. (Key point: Values for a = 0.5)

4. Which describes the spectrum for real a^n u[n]?
   A. Magnitude odd, phase even, period π
   B. Magnitude even, phase odd, period 2π
   C. Magnitude even, phase even, aperiodic
   D. Magnitude odd, phase odd, period 2π
   Answer: B. A real signal gives even magnitude and odd phase, and every DTFT is 2π-periodic. (Key point: Symmetry and periodicity)

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