---
title: "Why DTFS uses only one period"
subject: "Signals and Systems"
chapter: 2
key: "semester_4/signals_and_systems:ctfs-vs-dtfs-duration"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ctfs-vs-dtfs-duration
site: BCT Engineering
---

# Why DTFS uses only one period

Signals and Systems · Chapter 2: Fourier Series

- Asked in: 2077 Chaitra, 2 marks
- Video: [Introduction to the Discrete-Time Fourier Series (DTFS)](https://www.youtube.com/watch?v=pbZMUmHyNbs) — Adam Panagos
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/ctfs-vs-dtfs-duration
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2

## Question

A continuous time Fourier Series is evaluated over infinite duration while a discrete time Fourier series is evaluated over one time period only. Why?

## Short answer

In continuous time, all harmonics e^{j k ω0 t} are distinct, requiring an infinite sum to represent signals with sharp transitions. In discrete time, e^{j (k+N) (2π/N) n} = e^{j k (2π/N) n}, so only N distinct harmonics exist; the series is a finite sum over one period.

## Answer

The difference comes from how many distinct harmonics each signal can have:

- **Continuous time:** the exponentials $e^{jk\omega_0 t}$ are all different for every integer $k$; a higher $k$ is always a faster oscillation. A CT periodic signal (e.g. a square wave with sharp jumps) may need **infinitely many** harmonics, so the synthesis sum runs from $k = -\infty$ to $\infty$, and convergence must be examined (Dirichlet conditions).
- **Discrete time:** $e^{j(k+N)(2\pi/N)n} = e^{jk(2\pi/N)n}e^{j2\pi n} = e^{jk(2\pi/N)n}$, because $n$ is an integer. Only **$N$ distinct** exponentials exist, so both the synthesis and analysis sums need only one period ($N$ terms) and the coefficients repeat, $a_{k+N} = a_k$. The DTFS is a finite sum, so it always exists and has no convergence problem.

In both cases the analysis is done over one period of the signal; the "infinite" range in CT refers to the infinite number of harmonic terms needed.

## Check yourself

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

1. Why does the CT Fourier series sum run from k = -infinity to infinity?
   A. Each e^{jk w0 t} is a distinct signal, and a signal with sharp jumps may need infinitely many
   B. Because CT signals are always infinite in duration
   C. Because the period of a CT signal is infinite
   D. Because the coefficients are periodic in k
   Answer: A. In CT every harmonic is different, and sharp features can require infinitely many of them. (Key point: Why CT needs infinite harmonics)

2. Why does a DT Fourier series need only N terms?
   A. Because DT signals have no high frequencies
   B. Because e^{j(k+N)(2 pi/N)n} = e^{jk(2 pi/N)n}, so only N distinct exponentials exist
   C. Because the DTFS ignores the phase
   D. Because N is always small
   Answer: B. Since n is an integer, e^{j2 pi n} = 1, so the harmonics repeat every N. (Key point: Only N distinct DT harmonics)

3. Which statement about convergence is right?
   A. The DTFS needs the Dirichlet conditions
   B. The CT series always converges
   C. The DTFS is a finite sum so it always exists, while the CT series needs convergence checks
   D. Neither series ever has convergence issues
   Answer: C. The DTFS is a finite sum of N terms; the CT series is infinite and needs the Dirichlet conditions. (Key point: Convergence)

## 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×
- [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×
- [Time-scaling property of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-time-scaling) — Signals and Systems, chapter 2, asked 6×
- [Time-shifting property of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-time-shifting) — Signals and Systems, chapter 2, asked 5×
- [DTFS linearity and time shift](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-linearity-time-shift) — Signals and Systems, chapter 2
- [Derive exponential Fourier series (CT)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-exponential-derivation) — Signals and Systems, chapter 2, asked 15×

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