Calculus defines the differential of a time function like this
Computationally, we think of a differential as a finite difference, namely, a function is delayed a bit and then subtracted from its original self. Expressed as a Z-transform, the finite difference operator is (1-Z) with an implicit .In the language of filters, the time derivative is the filter (+1,-1).The filter (1-Z) is often simply called a ``differentiator.'' It is displayed in Figure 6. Notice its amplitude spectrum increases with frequency.
Theoretically, the amplitude spectrum of a time derivative operator increases linearly with frequency. Here is why. Begin from Fourier representation of a time function (15).
(22) | ||
(23) |
In Figure 6 we notice the spectrum begins looking like a linear function of ,but at higher frequencies, it curves. This is because at high frequencies, a finite difference is different from a differential.