---
title: "Fourier series of full-wave rectified sine"
subject: "Data Communication"
chapter: 3
key: "semester_5/data_communication:fs-full-wave-rectified"
source: https://www.bctengineer.com/q/semester_5/data_communication/fs-full-wave-rectified
site: BCT Engineering
---

# Fourier series of full-wave rectified sine

Data Communication · Chapter 3: Signal Analysis

- Asked in: 2070 Asar, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/fs-full-wave-rectified
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-3

## Question

Find the Fourier series representation of the full-wave rectified sine wave given below. [Figure: periodic full-wave rectified sine wave of amplitude 1, humps repeating every π, with axis marks at -3π, -2π, -π, 0, π, 2π, 3π]

## Short answer

The full‑wave rectified sine wave has DC component 2/π and only even cosine harmonics: x(t)=2/π – (4/π) Σ_{n=1}^{∞} cos(2nt)/(4n²−1). Its amplitude spectrum falls as 4/[π(4n²−1)].

## Answer

x(t) = |sin t| (amplitude 1), a periodic wave with period T₀ = π, so ω₀ = 2π/T₀ = 2 rad/s.

On 0 < t < π, x(t) = sin t. The wave is even, so only cosine terms exist (bₖ = 0).

#### Coefficients
a₀ = (1/π) ∫₀^π sin t dt = 2/π

aₙ = (2/π) ∫₀^π sin t cos(2nt) dt
= (1/π) ∫₀^π [sin((1+2n)t) + sin((1-2n)t)] dt
= (1/π) [ 2/(1+2n) + 2/(1-2n) ]
= (1/π) · 4/(1 - 4n²)

**aₙ = -4 / (π(4n² - 1))**

#### Result
**x(t) = 2/π - (4/π) Σ_{n=1}^{∞} cos(2nt) / (4n² - 1)**

= 2/π - (4/3π) cos 2t - (4/15π) cos 4t - (4/35π) cos 6t - ...

The mean value is 2/π = 0.637.

#### Exponential form
cₙ = 2 / (π(1 - 4n²)): c₀ = 0.637, c₁ = -0.212, c₂ = -0.0424, c₃ = -0.0182 (verified numerically). Only even harmonics of the original sine (2, 4, 6, ...) are present.

## Check yourself

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

1. The period of |sin t| is
   A. 2π
   B. π/2
   C. π
   D. 1
   Answer: C. Humps repeat every π. (Key point: Period)

2. ω0 =
   A. 1 rad/s
   B. π
   C. 2π
   D. 2 rad/s
   Answer: D. 2π/π. (Key point: Fundamental)

3. Only cosine terms because
   A. The wave is even
   B. The wave is odd
   C. Period is π
   D. DC is nonzero
   Answer: A. bk = 0. (Key point: Symmetry)

4. Mean value is
   A. 1/π
   B. 2/π
   C. 1/2
   D. 0
   Answer: B. (1/π)∫sin. (Key point: Mean)

## Related in Data Communication

- [Fourier series of half-wave rectified sine](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afs-half-wave-rectified) — Data Communication, chapter 3
- [Fourier series coefficients of a square wave](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afs-coefficients-square-wave) — Data Communication, chapter 3

## Related in other subjects

- [Average and RMS of full-wave rectified wave](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Aaverage-rms-full-wave-rectified) — Basic Electrical Engineering, chapter 5
- [RMS and average of half-wave rectified sine](https://www.bctengineer.com/question.md?key=semester_1%2Ffundamental_of_electrical_and_electronics_engineering%3Arms-average-half-wave) — Fundamental of Electrical and Electronics Engineering, chapter 2
- [Exponential Fourier series of sawtooth wave](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Afourier-exponential-form-sawtooth) — Electrical Circuit Theory, chapter 6
- [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