The Clifford Fourier transform in real Clifford algebras

We use the recent comprehensive research [17, 19] on the manifolds of square roots of −1 in real Clifford’s geometric algebras Cl(p,q) in order to construct the Clifford Fourier transform. Basically in the kernel of the complex Fourier transform the imaginary unit j ∈ C is replaced by a square root of−1 in Cl(p,q). The Clifford Fourier transform (CFT) thus obtained generalizes previously known and applied CFTs [9, 13, 14], which replaced j ∈ C only by blades (usually pseudoscalars) squaring to −1. A major advantage of real Clifford algebra CFTs is their completely real geometric interpretation. We study (left and right) linearity of the CFT for constant multivector coefficients ∈Cl(p,q), translation (x-shift) and modulation (ω-shift) properties, and signal dilations. We show an inversion theorem. We establish the CFT of vector differentials, partial derivatives, vector derivatives and spatial moments of the signal. We also derive Plancherel and Parseval identities as well as a general convolution theorem.