Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

Table 1

Basic properties of quaternion convolution.

Basic property	Quaternion convolution
Linearity	(𝜅1𝑓+𝜅2𝑔)⋆ℎ=𝜅1(𝑓⋆ℎ)+𝜅2(𝑓⋆ℎ),𝜅1,𝜅2∈ℍ,
	ℎ⋆(𝜅1𝑓+𝜅2𝑔)=𝜅1(ℎ⋆𝑓)+𝜅2(ℎ⋆𝑔),𝜅1,𝜅2∈ℝ
Shifting	(𝜏𝐚𝑓⋆𝑔)=𝜏𝐚(𝑓⋆𝑔),
	(𝑓⋆𝜏𝐚𝑔)=𝜏𝐚(𝑓⋆𝑔)
Conjugation	(𝑓⋆𝑔)=(𝑔⋆𝑓)
Associativity	(𝑓⋆𝑔)⋆ℎ=𝑓⋆(𝑔⋆ℎ)
Distributivity	𝑓⋆(𝑔+ℎ)=(𝑓⋆𝑔)+(𝑓⋆ℎ)
Vector differential	𝐚⋅∇(𝑓⋆𝑔)=(𝐚⋅∇𝑓)⋆𝑔=𝑓⋆(𝐚⋅∇𝑔)
Impulse convolution	𝑓⋆𝛿=𝑓