CHAPTER 13
The Hilbert Transform
The Hilbert transform of a real-valued time-domain signal x(t) is another real-valued time-domain signal, denoted by 
such that 
is an analytic signal. The Fourier transform of x(t) is a complex-valued frequency domain signal X(f), which is clearly quite different from the Hilbert transform 
or the quantity z(t). From z(t), one can define a magnitude function A(t) and a phase function θ(t), where A(t) describes the envelope of the original function x(t) versus time, and θ(t) describes the instantaneous phase of x(t) versus time. Section 13.1 gives three equivalent mathematical definitions for the Hilbert transform, followed by examples and basic properties. The intrinsic nature of the Hilbert transform to causal functions and physically realizable systems is also shown. Section 13.2 derives special formulas for the Hilbert transform of correlation functions and their envelopes. Applications are outlined for both nondispersive and dispersive propagation problems. Section 13.3 discusses the computation of two envelope signals followed by correlation of the envelope signals. Further material on the Hilbert transform and its applications appears in Refs ...
