Fourier Transforms

9.3 THE HILBERT TRANSFORM

9.3.1 The Fourier Transform of One-sided Functions

One-sided functions, which are nonzero only for t ≥ 0, represent several real phenomena. A signal that turns on at t = 0, for example, f(t) = cos 2πt U(t), is one sided. The impulse response of a causal linear system (one which does not produce an output before the input is applied) is one sided, for example, h(t) = e^{− t}U(t). Causality is an important constraint on the design of systems that operate in real time on streams of data.

The theme of this section is that the real and imaginary parts of the Fourier transforms of one-sided functions are not independent, but can be calculated one from the other. To see how this works, break the one-sided function f into its even and odd parts, and . Because f(t) = 0 for t < 0,

Thus, the even and odd parts are connected,

Now take the Fourier transform of both sides of this expression. The Fourier transform of a real and even function is real, and the transform of a real and odd function is imaginary. So, with , and , and is related to by

or,

(9.83a)

It ...