Delta Functions

            The basic definitions are

                   (1)

         (2)

Delta function in time

 

Insert (1) into (2)

                     (3)

Interchange the summations (and integrations) to utilize the fact that the f's can be separated.

                                    (4)

Thus

      (5)

Note that the integral in (5) is

                                          (6)

which becomes F=1/Δ_t in the limit that f_m-f_n  0

 

Delta function in frequency

Insert  (2) into (1)

                 (7)

Sort out the times

                               (8)

Thus

  (9)

Note that the integral in (9) is

                                           (10)

 which becomes T in the limit that t0.