First, we assume that the function is a swept sinusoid such that we may relate the function in the time and frequency domains without the integrands.

Rearranging gives: 

.

Thus we can relate the amplitude spectrum of a swept sinusoid as being created by either the amplitude of the time response or the relationship of the time sampling and the inverse instantaneous frequency of the swept sinusoid. This is very remarkable result from such a general theorem.

Let us assume that we want a constant amplitude, then

  .

This defines the instantaneous frequency. In order to produce the argument of the cosine we must integrate the result over time:

This is the familiar linear sweep.

We define the argument of the cosine