Discrete Hilbert transform

The Hilbert transformH\{f(t)\}of a given waveformf(t)is defined with the convolutionH{\f(t)} = f(t) \ast (1/\pit). It is well known that the second type of Hilbert transformK_{0}{\f(x)\}=\phi(x) \ast (1/2\pi)\cot\frac{1}{2}xexists for the transformed functionf(tg\frac{1}{2}x)= \phi(x). If the functionf(t)is periodic, it can be proved that one period of theHtransform off(t)is given by the H1transform of one period off(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.