---
title: "Fourier series vs transform; DTFT of aⁿ"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:fs-vs-ft-dtft-geometric"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/fs-vs-ft-dtft-geometric
site: BCT Engineering
---

# Fourier series vs transform; DTFT of aⁿ

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2073 Bhadra, 2+6 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/fs-vs-ft-dtft-geometric
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

What are the differences between Fourier series and Fourier Transform? Find the Fourier transform of the discrete time signal x[n] = aⁿ, |a| < 1 (printed with the two conditions |a| < 1 and 0 < a < 1).

## Short answer

Fourier series represents periodic signals as a sum of harmonics with discrete spectrum; Fourier transform represents aperiodic signals as an integral with continuous spectrum. The DTFT of the causal signal x[n] = a^n u[n], |a|<1, is X(e^{jω}) = 1/(1 − a e^{-jω}).

## Answer

#### Fourier series vs Fourier transform

| Point | Fourier series | Fourier transform |
|---|---|---|
| Signal type | Periodic signals | Aperiodic (and, with impulses, periodic) signals |
| Spectrum | Discrete (lines at $k\omega_0$) | Continuous in $\omega$ |
| Representation | Sum $\sum a_k e^{jk\omega_0 t}$ | Integral $\frac{1}{2\pi}\int X(j\omega)e^{j\omega t}d\omega$ |
| Coefficients | $a_k = \frac{1}{T_0}\int_{T_0} x(t)e^{-jk\omega_0 t}dt$ | $X(j\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt$ |
| Integration range | One period | Whole time axis |
| Physical meaning | Amplitude of each harmonic | Spectral density (amplitude per unit frequency) |
| Relation | Samples of FT of one period, scaled by $1/T_0$ | Limit of FS as $T_0 \to \infty$ |

#### DTFT of $x[n] = a^n$, $|a| < 1$

Taken literally for all $n$, $a^n$ grows without bound as $n \to -\infty$ (for $|a|<1$), so it is not absolutely summable and its DTFT does not exist. The intended (textbook) signal is the causal one, $x[n] = a^n u[n]$.

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

**Magnitude and phase:**

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

For $0 < a < 1$ the magnitude is largest at $\omega = 0$ ($\frac{1}{1-a}$) and smallest at $\omega = \pi$ ($\frac{1}{1+a}$), so the sequence is low-pass. For $-1<a<0$ it is the other way round (high-pass).

**If the two-sided signal $x[n] = a^{|n|}$ is meant:**

$$
\begin{aligned}
X(e^{j\omega}) &= \sum_{n=0}^{\infty} a^n e^{-j\omega n} + \sum_{n=1}^{\infty} a^n e^{j\omega n} \\
&= \frac{1}{1-ae^{-j\omega}} + \frac{ae^{j\omega}}{1-ae^{j\omega}} = \frac{1 - a^2}{1 - 2a\cos\omega + a^2}
\end{aligned}
$$

which is real and even, as expected for a real even sequence.

## Check yourself

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

1. How does the spectrum of a Fourier series differ from that of a Fourier transform?
   A. FS is continuous, FT is discrete
   B. Both are discrete
   C. Both are continuous
   D. FS is discrete (lines at kω0), FT is continuous
   Answer: D. Periodic signals give line spectra; aperiodic signals give continuous spectra. (Key point: FS vs FT spectrum)

2. Which signals does the Fourier series handle?
   A. Periodic signals, integrating over one period
   B. Aperiodic signals over all time
   C. Only discrete signals
   D. Only even signals
   Answer: A. FS coefficients are found from one period; FT integrates over the whole time axis. (Key point: Signal type)

3. Why does the DTFT of a^n for all n (|a|<1) not exist?
   A. It is not causal
   B. It grows without bound as n → -∞, so it is not absolutely summable
   C. It is complex
   D. a is less than 1
   Answer: B. For negative n, a^n = (1/a)^|n| grows, so the sum diverges. (Key point: Literal a^n)

4. What is the DTFT of a^|n|?
   A. 1/(1 - a e^(-jω))
   B. (1 + a^2)/(1 - 2a cosω + a^2)
   C. (1 - a^2)/(1 - 2a cosω + a^2)
   D. 2a/(1 - 2a cosω + a^2)
   Answer: C. Adding the causal and anticausal parts gives a real, even function. (Key point: Two-sided a^|n|)

## Related in Signals and Systems

- [Derive the discrete-time Fourier series](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-derivation) — Signals and Systems, chapter 2, asked 15×
- [Fourier transform of a CT signal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-unspecified-ct-signal) — Signals and Systems, chapter 3
- [Dirichlet conditions for Fourier series](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adirichlet-conditions) — Signals and Systems, chapter 2
- [Periodic signal whose Fourier series fails](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afourier-series-existence-examples) — Signals and Systems, chapter 2
- [Continuous-time vs discrete-time signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-vs-dt-signals) — Signals and Systems, chapter 1, asked 3×
- [Derive DTFT; DTFT periodic with 2π](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-derivation-periodicity) — Signals and Systems, chapter 3, asked 3×

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