For me, the most intuitive explanation of the Fourier transform is geometric and is as follows: If: . you have a vector space equipped with a dot product

For me, the most intuitive explanation of the Fourier transform is geometric and is as follows: If: . you have a vector space equipped with a dot product - say R³ . you have a specific orthonormal basis of said vector space - say (i, j, k) in R³ . you have a vector V - say V ∈ R³ . you want the coordinates of V in that basis . then all you have to do is take the dot product of V with each member of the basis (i.e. you *project* V onto each basis member). . in other words: V = Sum_over_basis_members[ individual_basis_member * Dot_product[V, individual_basis_member] ] And magically, that's all the various Fourier transforms are: . the discrete Fourier transform (both signal and basis are discrete and finite, dot product is the usual suspect) . the Fourier series (basis is discrete and infinite, signal is a continuous periodic function, dot product is the continuous version of the standard dot product, i.e. an integral) . the Fourier transform (basis is infinite and continuous, signal is any "well-behaved" function, and the reconstruction of the "vector" involves a double integral, one for the dot product and one for the summation over basis terms) For example, for the Fourier series: . The "Vector" is any well-behaved periodic function (don't recall exactly what well-behaved entails, but not the most important part of the story anyways) . The vector space is the set of well-behaved periodic functions . The basis is the family of functions Exp[2*π*i] where the parameter i = {0, 1, 2, 3, ..., infty} . The dot product is the cross-correlation between the function and a basis member: i.e. Integral[x=0, x=2*π, function*basis_member] . You rebuild the function as: function = Sum_over_basis_members[ basis_member Dot_product[function, basis_member] ] And that's it.