You can infer from Problem 4.3 that 1 ⇔ δ(μ, v) and μ(t, z) ⇔ 1. Use the first of these pr...

You can infer from Problem 4.3 that 1 ⇔ δ(μ, v) and μ(t, z) ⇔ 1. Use the first of these properties and the translation property in Table 4.3 to show that the Fourier transform of the continuous function f(t, z) = A sin(2πμ0t + 2πv0 z) is

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.

4.3 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.