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# Introduction

This module relates circular convolution of periodic signals in the time domain to multiplication in the frequency domain.

# Signal Circular Convolution

Given a signal $f⁢t f t$ with Fourier coefficients $c n c n$ and a signal $g⁢t g t$ with Fourier coefficients $d n d n$, we can define a new signal, $v⁢t v t$, where $v⁢t=f⁢t⊛g⁢t v t ⊛ f t g t$ We find that the Fourier Series representation of $v⁢t v t$, $a n a n$, is such that $a n = c n ⁢ d n a n c n d n$. $f⁢t⊛g⁢t ⊛ f t g t$ is the circular convolution of two periodic signals and is equivalent to the convolution over one interval, i.e. $f⁢t⊛g⁢t=∫0T∫0Tf⁢τ⁢g⁢t−τd τ d t ⊛ f t g t t 0 T τ 0 T f τ g t τ$.

Note:
Circular convolution in the time domain is equivalent to multiplication of the Fourier coefficients.
This is proved as follows
$a n =1T⁢∫0Tv⁢t⁢e−(j⁢ ω 0 ⁢n⁢t)d t =1T2⁢∫0T∫0Tf⁢τ⁢g⁢t−τd τ ⁢e−(ω⁢ j 0 ⁢n⁢t)d t =1T⁢∫0Tf⁢τ⁢(1T⁢∫0Tg⁢t−τ⁢e−(j⁢ ω 0 ⁢n⁢t)d t )d τ =∀ ν ,ν=t−τ:1T⁢∫0Tf⁢τ⁢(1T⁢∫−τT−τg⁢ν⁢e−(j⁢ ω 0 ⁢(ν+τ))d ν )d τ =1T⁢∫0Tf⁢τ⁢(1T⁢∫−τT−τg⁢ν⁢e−(j⁢ ω 0 ⁢n⁢ν)d ν )⁢e−(j⁢ ω 0 ⁢n⁢τ)d τ =1T⁢∫0Tf⁢τ⁢dn⁢e−(j⁢ ω 0 ⁢n⁢τ)d τ = d n ⁢(1T⁢∫0Tf⁢τ⁢e−(j⁢ ω 0 ⁢n⁢τ)d τ )= c n ⁢ d n a n 1 T t 0 T v t j ω 0 n t 1 T 2 t 0 T τ 0 T f τ g t τ ω j 0 n t 1 T τ 0 T f τ 1 T t 0 T g t τ j ω 0 n t ν ν t τ 1 T τ 0 T f τ 1 T ν τ T τ g ν j ω 0 ν τ 1 T τ 0 T f τ 1 T ν τ T τ g ν j ω 0 n ν j ω 0 n τ 1 T τ 0 T f τ d n j ω 0 n τ d n 1 T τ 0 T f τ j ω 0 n τ c n d n$
1

# Exercise

Take a look at a square pulse with a period of T.

For this signal

Take a look at a triangle pulse train with a period of T.

This signal is created by circularly convolving the square pulse with itself. The Fourier coefficients for this signal are $a n = c n 2=14⁢sin2π2⁢nπ2⁢n2 a n c n 2 1 4 2 n 2 2 n 2$

Exercise 1

Find the Fourier coefficients of the signal that is created when the square pulse and the triangle pulse are convolved.

Solution

$a n = undefined n = 0 1 8 s i n 3 ( π 2 n ) ( π 2 n ) 3 otherwise a n = undefined n = 0 1 8 s i n 3 ( π 2 n ) ( π 2 n ) 3 otherwise$

# Conclusion

Circular convolution in the time domain is equivalent to multiplication of the Fourier coefficients in the frequency domain.