Abstract
An important problem in the theory of linear systems is the relationship between the frequency- and the time-domain characteristics. An interesting problem of this sort is to obtain the step response of a system when its amplitude function is given, which arises in the design of filters, the measurement of system characteristics, etc. Two conventional methods are (1) factorization method that uses the factorization of the amplitude function, and (2) ω- domain method that uses the Kramers-Kronig relation to get the phase function from the attenuation function. Both methods are poor as computer algorithms.
We introduced here a new algorithm that uses the Kramers-Kronig relation extended to complex frequency domain, and that can easily be carried out by a computer. As an application, we have analyzed the problem of negative group velocity, and showed that the group velocity of a wave packet is nothing but the phase velocity of the envelope.
