By G.A. Watson
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Additional resources for Conference on the Numerical Solution of Differential Equations
18 summation is equal to xoy0 but the second term is equal to x-1y1. Although in the original formulation of the finite Fourier transform, the x sequence was only specified for indices from 0 through N - 1, the periodicity property in (48) implies that x-1 be equal to xN-1. This leads to the name circular convolution since the undefined portions of the original sequence are replaced by a circular replication of the original data. The effect of circular convolution is shown in Fig. 9(a). Here we have shown an exponential sequence convolved with an impulse.
If , however, we use the same equation to evaluate F(±u, ± v), we discover that the periodicity properties 2 To be consistent with the notation in the one-dimensional case, we should express the space and frequency domain arrays as f m , n and Fu,v. However, we feel that for the two-dimensional case, the math looks a bit neater with the style chosen here, especially when one starts dealing with negative indices and other extensions. 3 where they represent continuously varying spatial frequencies.
If x and y represent spatial coordinates, (127) can be used to give a physical interpretation to the Fourier transform F(ut u) and to the coordinates u and v. Let us first examine the function The real and imaginary parts of this function are cos 2 (ux + vy) and sin 2 (ux + vy), respectively. In Fig. 18(a), we have shown cos 2 (ux + vy). y-axis goes through v cycles per unit distance. This is the reason why u and v are called the spatial frequencies along the x- and yaxes, respectively. Also, from the figure it can be seen that the spatial period of the pattern is (u2 + u 2 ) -1/2 .
Conference on the Numerical Solution of Differential Equations by G.A. Watson