**5. Concluding remarks and historical developments**

In this chapter, we first presented Hilbert transform as an analytic extension problem. Hilbert transform uniquely exists due to Cauchy-Riemann equations. We then reviewed several different ways to calculate Hilbert transform. A few important points stand out: First, Hilbert transform can be regarded as a 90-degree phase shifter. Secondly, the real part and imaginary part of a physically viable transfer function must satisfy Kramers-Krönig relations, which is the Hilbert transform applied in time-frequency duality. A good reference on these topics can be found in Oppenheim & Schafer (2010).

The construction of Hilbert transform pairs through Cauchy-Riemann equations in Sec. 2.1 was found in the appendices of an old text on microwave electronics (Scott, 1970). The original formulation was stated in terms of Kramers-Krönig relation, and in this chapter that formulation is adapted so the signal is defined on the real line instead of the frequency axis *jω*.

As a matter of fact, the definition of Hilbert transform was not given by David Hilbert himself. The name "Hilbert transform" was first given by the British mathematician G. H. Hardy in honor of Hilbert's pioneering work on integral equations (King, 2009a). Hardy's early work established mathematical rigor of the transform (Hardy, 1932), which we now apply in various areas such as physiology and telecommnication. A review of these contributions from brilliant mathematicians reminds us that the transform really is a heritage from the 20th century. Nevertheless, it is also amusing that nowadays we calculate Hilbert transform with super fast computers, which might never had been envisioned by 20th-century pioneers, including Hilbert, Hardy, Scott, or even Oppenheim.
