---
title: "Dirichlet conditions for Fourier series"
subject: "Signals and Systems"
chapter: 2
key: "semester_4/signals_and_systems:dirichlet-conditions"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/dirichlet-conditions
site: BCT Engineering
---

# Dirichlet conditions for Fourier series

Signals and Systems · Chapter 2: Fourier Series

- Asked in: 2074 Bhadra, 3 marks
- Video: [Conditions for Existence of Fourier Series (Dirichlet Conditions)](https://www.youtube.com/watch?v=blS_OImUJ-c) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/dirichlet-conditions
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2

## Question

Explain the Dirichlet conditions for convergence of Fourier series.

## Short answer

Sufficient conditions for Fourier series convergence: absolute integrability over one period, finite number of maxima and minima, finite number of finite discontinuities. At a discontinuity the series converges to the average of left and right limits.

## Answer

The **Dirichlet conditions** are a set of sufficient conditions on a periodic signal $x(t)$ (period $T$) which guarantee that its Fourier series converges to $x(t)$ at every point of continuity.

1. **Absolute integrability over one period:** the signal must be absolutely integrable over a period,
$$
\int_{T} |x(t)|\, dt < \infty
$$
This makes every coefficient finite, since $|a_k| \le \frac{1}{T}\int_T |x(t)|\,dt$.
   - Violating example: $x(t) = 1/t$ for $0 < t \le 1$, repeated with period 1.
2. **Finite number of maxima and minima:** in any one period, $x(t)$ has only a finite number of maxima and minima (bounded variation).
   - Violating example: $x(t) = \sin(2\pi/t)$ for $0 < t \le 1$.
3. **Finite number of discontinuities:** in any finite interval there are only a finite number of discontinuities, and each one is finite (a finite jump).
   - Violating example: a staircase signal whose steps get halved again and again within one period, giving infinitely many jumps.

**Result:** if these conditions hold, the Fourier series equals $x(t)$ wherever $x(t)$ is continuous. At a jump it converges to the average of the left and right limits, $\frac{1}{2}[x(t^-) + x(t^+)]$. Near a jump, the partial sums of the series still overshoot by about 9% of the jump (the **Gibbs phenomenon**).

The conditions are sufficient but not necessary. Signals that break them are mostly artificial, so almost every practical signal has a convergent Fourier series.

## Check yourself

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

1. What does absolute integrability over a period guarantee?
   A. Every coefficient a_k is finite
   B. The signal is continuous
   C. The signal has no maxima
   D. The series has no ripples
   Answer: A. |a_k| is at most (1/T) times the integral of |x(t)|, so finite integral means finite coefficients. (Key point: Absolute integrability)

2. Which signal violates the finite maxima and minima condition?
   A. 1/t on (0,1]
   B. A square wave
   C. sin(2 pi/t) on (0,1]
   D. A triangular wave
   Answer: C. sin(2 pi/t) oscillates infinitely often near t = 0. (Key point: Violating examples)

3. At a discontinuity of a signal satisfying the Dirichlet conditions, the series converges to:
   A. The value to the right
   B. The value to the left
   C. Zero
   D. The average of the left and right limits
   Answer: D. The series gives (1/2)[x(t-) + x(t+)] at a jump. (Key point: Behaviour at a jump)

4. The Dirichlet conditions are:
   A. Necessary and sufficient
   B. Sufficient but not necessary
   C. Necessary but not sufficient
   D. Required only for DT signals
   Answer: B. They guarantee convergence but some signals that violate them may still have a series. (Key point: Nature of conditions)

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