B.3 Numerical computation of Fourier transformation

Next, consider how to numerically compute the Fourier transformation of a function h(t). A simple way is to use the rectangle formula to approximate the integration in Eq. (102), i.e.,

(110)

where h_j = h(t_j) and t_j = jΔ with j = …,−2,−1,0,1,2,…. Note Eq. (110) is an approximation, which will become exact if Δ → 0. In practice, we can sample h(t) only with a nonzero Δ. Therefore Eq. (110) is usually an approximation. Do we have some rules to choose a suitable Δ so that Eq. (110) can become a good approximation or even an exact relation? This important question is answered by the sampling theorem, which sates that a suitable Δ to make Eq. (110) exact is given by Δ ≤ 1∕(2f_c), where f_c is the largest frequency contained in h(t) (i.e., H(f) = 0 for |f| > f_c).