Discrete Convolution and the Discrete Fourier Transform

Discrete Convolution First of all we need to introduce what we might call the “wraparound” convention. Because the complex numbers wj = e i 2πj N have the property wj±N = wj, which readily extends to wj+mN = wj for any integer m, and since in the discrete Fourier context we represent all N -dimensional vectors as linear combinations of the Fourier vectors Wk whose components are wkj , we make the convention that for any vector Z ∈ E with components zk, k = 1, 2, · · · , N − 1, if we write zk with k lying outside the range 0 through N − 1 what is meant is zk+mN where m is the unique integer such that k +mN does lie in this range. Thus, for example, if N = 8 and we write z−13, what we mean is z−13+2×8 = z3. This convention may initially seem rather strange and arbitrary but, in fact, it is quite essential for effective use of and computation with the discrete Fourier transform.