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

# DTFT of aⁿu[n] with spectrum plot

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2073 Magh, 3+2 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-plot
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

Find Fourier transform of signal x[n] = aⁿu[n] (where 0 < a < 1) and plot magnitude and phase spectrum.

## Short answer

For 0 < a < 1, X(e^(jω)) = 1/(1 - ae^(-jω)), with magnitude 1/sqrt(1 - 2a cosω + a²) and phase -arctan(a sinω/(1 - a cosω)). The magnitude is even and low-pass, phase is odd, and both repeat every 2π.

## Answer

#### 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

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

```text
 |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 $2\pi$.

## Check yourself

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

1. What is the DTFT of a^n u[n], 0<a<1?
   A. 1/(1 + a e^(-jω))
   B. a/(1 - e^(-jω))
   C. 1/(1 - a e^(jω))
   D. 1/(1 - a e^(-jω))
   Answer: D. Sum of (a e^(-jω))^n for n ≥ 0 converges since |a e^(-jω)| = a < 1. (Key point: DTFT)

2. What is |X(e^{jω})|?
   A. 1/sqrt(1 - 2a cosω + a^2)
   B. 1/(1 - 2a cosω + a^2)
   C. 1/sqrt(1 + 2a cosω + a^2)
   D. 1/(1 - a)
   Answer: A. |1 - a e^(-jω)|^2 = (1 - a cosω)^2 + a^2 sin^2ω = 1 - 2a cosω + a^2. (Key point: Magnitude)

3. Where are the maximum and minimum of |X| for 0<a<1?
   A. Max at ω = π, min at ω = 0
   B. Max 1/(1-a) at ω = 0, min 1/(1+a) at ω = π
   C. Max 1/(1-a) at ω = π/2
   D. Constant for all ω
   Answer: B. cosω = 1 gives the smallest denominator, cosω = -1 the largest. (Key point: Peak and low)

4. Which statement about the spectra is right?
   A. Magnitude odd, phase even, aperiodic
   B. Both even, period π
   C. Magnitude even, phase odd, both periodic with 2π
   D. Both periodic with period 1
   Answer: C. The DTFT of a real sequence has these symmetries and is always 2π-periodic. (Key point: Symmetry and period)

## Related in Signals and Systems

- [Magnitude and phase of DTFT of aⁿu[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-geometric-magnitude-phase) — Signals and Systems, chapter 3
- [Fourier series vs transform; DTFT of aⁿ](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afs-vs-ft-dtft-geometric) — Signals and Systems, chapter 3
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- [Fourier transform of e^(−a|t|)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-two-sided-exponential) — Signals and Systems, chapter 3, asked 5×
- [CTFT convolution; DTFT of 2ⁿu[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-convolution-dtft-2n) — Signals and Systems, chapter 3
- [H(e^(jω)), magnitude plot and h[n] for y[n] = 0.5y[n−1] + x[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afirst-order-difference-plot) — Signals and Systems, chapter 6, asked 9×

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