Discrete Hilbert transform

Abstract

The Hilbert transform H\{f(t)\} of a given waveform f(t) is defined with the convolution H{\f(t)} = f(t) \ast (1/\pit) . It is well known that the second type of Hilbert transform K_{0}{\f(x)\}=\phi(x) \ast (1/2\pi)\cot\frac{1}{2}x exists for the transformed function f(tg\frac{1}{2}x)= \phi(x) . If the function f(t) is periodic, it can be proved that one period of the H transform of f(t) is given by the H 1 transform of one period of f(t) without regard to the scale of tbe variable. On the base of the discrete Fourier transform (DFT), the discrete Hilbert transform (DHT) is introduced and the defining expression for it is given. It is proved that this expression of DHT is identical to the relation obtained by the use of the trapezoidal rule to the cotangent form of the Hilbert transform.