Question

brief note on Hilbert transforms

Answer 1

In mathematics and in signal processing, the Hilbert transform is a specific linear operatorthat takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function {\displaystyle 1/(\pi t)}:

{\displaystyle H(u)(t)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {u(\tau )}{t-\tau }}\,d\tau ,}

the improper integral being understood in the principal value sense. The Hilbert transform has a particularly simple representation in the frequency domain: it imparts a phase shift of 90° to every Fourier component of a function. For example, the Hilbert transform of {\displaystyle \cos(\omega t)}, where ω > 0, is {\displaystyle \cos(\omega t-\pi /2)}.

The Hilbert transform is important in signal processing, where it derives the analytic representation of a real-valued signal u(t). Specifically, the Hilbert transform of u is its harmonic conjugate v, a function of the real variable t such that the complex-valued function u+iv admits an extension to the complex upper half-plane satisfying the Cauchy–Riemann equations. The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions.