It can be shown (Bracewell [2000]) that 1 ⇔ δ(µ) and δ(t) ⇔ 1. Use the first of these prop...

It can be shown (Bracewell [2000]) that 1 ⇔ δ(µ) and δ(t) ⇔ 1. Use the first of these properties and the translation property from Table 4.3 to show that the Fourier transform of the continuous function f(t) = sin(2πnt), where n is a real number,is F(µ) = (j/2)[δ(µ + n) − δ(µ − n)].

TABLE 4.3

Summary of DFT pairs.The closedform expressions in 12 and 13 are valid only for continuous variables.They can be used with discrete variables by sampling the closed-form, continuous expressions.