---
title: "Convolution property of CTFS"
subject: "Signals and Systems"
chapter: 2
key: "semester_4/signals_and_systems:ctfs-convolution-property"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ctfs-convolution-property
site: BCT Engineering
---

# Convolution property of CTFS

Signals and Systems · Chapter 2: Fourier Series

- Asked in: 2072 Magh, 2+4 marks
- Video: [Convolution Property of Continuous Time Fourier Series](https://www.youtube.com/watch?v=OEsOcvoo_kg) — FEMA 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/ctfs-convolution-property
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2

## Question

State and prove the convolution property of continuous-time Fourier series.

## Short answer

Periodic convolution z(t) = ∫_T x(τ) y(t-τ) dτ has Fourier coefficients c_k = T a_k b_k. Proof: swap integration order, use orthogonality of exponentials, and the coefficient definition of y.

## Answer

#### Statement
Let $x(t)$ and $y(t)$ be periodic with the same period $T$ ($\omega_0 = 2\pi/T$), with $x(t) \xleftrightarrow{FS} a_k$ and $y(t) \xleftrightarrow{FS} b_k$. The **periodic convolution** of the two signals has Fourier series coefficients $T a_k b_k$:
$$
z(t) = \int_T x(\tau)\,y(t-\tau)\,d\tau \;\xleftrightarrow{FS}\; c_k = T a_k b_k
$$
So convolution in time becomes multiplication of the coefficients.

#### Proof
$z(t)$ is periodic with period $T$, because $y(t - \tau)$ is periodic in $t$. Its coefficients are
$$
c_k = \frac{1}{T}\int_T z(t)e^{-jk\omega_0 t}dt = \frac{1}{T}\int_T\left[\int_T x(\tau)y(t-\tau)d\tau\right]e^{-jk\omega_0 t}dt
$$
Swap the order of integration, and write $e^{-jk\omega_0 t} = e^{-jk\omega_0 \tau}e^{-jk\omega_0 (t-\tau)}$:
$$
c_k = \frac{1}{T}\int_T x(\tau)e^{-jk\omega_0 \tau}\left[\int_T y(t-\tau)e^{-jk\omega_0 (t-\tau)}dt\right]d\tau
$$
Put $\sigma = t - \tau$. The integrand is periodic with period $T$, so integrating over any interval of length $T$ gives the same result:
$$
\int_T y(\sigma)e^{-jk\omega_0 \sigma}d\sigma = T b_k
$$
Therefore
$$
c_k = \frac{1}{T}\cdot T b_k\int_T x(\tau)e^{-jk\omega_0 \tau}d\tau = T b_k\cdot a_k
$$
$$
\boxed{c_k = T a_k b_k}
$$

**Use:** the output of an LTI system for a periodic input can be found by multiplying spectra instead of convolving. The dual result is the multiplication property: $x(t)y(t) \xleftrightarrow{FS} \sum_{l} a_l b_{k-l}$.

## Check yourself

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

1. For the periodic convolution z(t) of x and y (period T), the coefficients are:
   A. a_k b_k
   B. T a_k b_k
   C. a_k + b_k
   D. (1/T) a_k b_k
   Answer: B. c_k = T a_k b_k. (Key point: Statement)

2. In the proof, e^{-jk w0 t} is split into:
   A. e^{-jk w0 tau} e^{-jk w0 (t - tau)}
   B. e^{jk w0 tau} e^{-jk w0 t}
   C. e^{-jk w0 tau} e^{jk w0 (t - tau)}
   D. e^{-jk w0 tau} only
   Answer: A. This lets the integral separate into a_k and b_k pieces after substituting sigma = t - tau. (Key point: Proof manipulation)

3. Why can the integral over sigma = t - tau be taken over any interval of length T?
   A. The integrand is periodic with period T
   B. y is zero outside T
   C. tau is constant
   D. w0 is zero
   Answer: A. The integrand y(sigma) e^{-jk w0 sigma} is periodic with period T. (Key point: Why integration range can be shifted)

4. What is the practical use of this property?
   A. Finding the output of an LTI system for periodic input by multiplying spectra
   B. Finding the dc value
   C. Removing Gibbs ripples
   D. Finding the period
   Answer: A. Convolution in time becomes multiplication of coefficients, which is simpler. (Key point: Practical use)

## Related in Signals and Systems

- [Time-scaling property of CTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-time-scaling) — Signals and Systems, chapter 2, asked 4×
- [CTFT convolution property](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-convolution-property) — Signals and Systems, chapter 3, asked 12×
- [DTFT convolution property](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-convolution-property) — Signals and Systems, chapter 3, asked 12×
- [DTFT properties; convolution property](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-properties-convolution) — Signals and Systems, chapter 3, asked 2×
- [CTFS time shifting and time scaling](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-shift-scale-properties) — Signals and Systems, chapter 2, asked 2×
- [Convolution property of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-convolution-property) — Signals and Systems, chapter 2

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- [Properties of continuous-time Fourier series](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-series-properties) — Data Communication, chapter 3
- [State and prove circular convolution property](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Acircular-conv-property-prove) — Digital Signal Analysis and Processing, chapter 7
- [Fourier transform property: Convolution](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-convolution) — Communication Systems, chapter 1
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