What is Hilbert transform: Definition and 12 Discussions

In mathematics and in signal processing, the Hilbert transform is a specific linear operator that 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)}
(see § Definition). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° (π⁄2 radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see § Relationship with the Fourier transform). The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal u(t). 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.

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  1. A

    Hilbert Transform, Causality, PI Controller

    I was told that PI controller is a causal filter, and has frequency response represented by H(w) = Ki/(iw)+ Kp. I was also told that causal filter should satisfy this relationship H(w) = G(w) -i G_hat(w) where G_hat(w) is the Hilbert transform of G(w). Does this mean that we cannot freely...
  2. A

    Engineering I/Q of Signals and Hilbert Transform

    Hello, would anyone be willing to provide help to the following problem? I can find the Fourier Transform of the complex envelope of s(t) and the I/Q can be found by taking the Real and imaginary parts of that complex envelope, but how can I approach the actual question of finding the carrier...
  3. S

    Mathematica Solving 2-D partial integro-differential equation

    While reproducing a research paper, I came across the following equation, ∂f/∂t−(H(f)(∂f/∂x)=0 where [H(f)] is hilbert transform of 'f.' and f=f(x,t) and initial condition is f(x,0)=cos(x) and also has periodic boundary conditions given by F{H{f(x′,t)}}=i⋅sgn(k)F{f(x,t)}, where F(f(x,t) is...
  4. H

    MATLAB Trying to compute Hilbert transform numerically

    I know the result: \widehat{\mathscr{H}(f)}(k)=-i\sgn (k)\hat{f}(k) I want to use this to compute the Hilbert transform. I have written code for Fourier transform,inverse Fourier transform and that the Hilbert transform. My code is the following: function y=ft(x,f,k) n=length(k); %See now long...
  5. R

    What Is the Hilbert Transform of the Sinc Function?

    Homework Statement Show that the Hilbert transform of ##\frac{\sin(at)}{at}## is given by $$\frac{\sin^2(at/2)}{at/2}.$$ Homework Equations The analytic signal of a function is given by ##f_a(t) = 2 \int^\infty_0 F(\nu) \exp(j2 \pi \nu t) \ d\nu,## where ##F(\nu)## is the Fourier transform...
  6. R

    Hilbert Transform Homework: Show Envelope is |m(t)|

    Homework Statement For a real, band-limited function ##m(t)## and ##\nu_v > \nu_m,## show that the Hilbert transform of $$h(t) = m(t) cos(2\pi \nu_c t)$$ is $$\hat{h}(t) = m(t) sin(2 \pi \nu_c t),$$ and therefore the envelope of ##h(t)## is ##|m(t)|.## Homework Equations Analytic signal...
  7. N

    Remove spurious peaks of Hilbert transform for LP residual

    Homework Statement I want to eliminate spurious peaks of Hilbert transform for finding Glottal closure in LP residual. I have 4 step : Homework Equations 1-down-sample. 2-Hilbert Transform. 3-Identify Peaks in Hilbert Transform. 4-consider this hypothesis that time gap between two...
  8. H

    Computing the Hilbert transform via Fourier transform

    I know the result: \widehat{H(f)}=i\textrm{sgn}\hspace{1mm}(k)\hat{f} I thought I could use fft, and ifft to compute the transform easily, is there a MATLAB command for sgn? Mat
  9. E

    On the domain of the function that undergoes the Hilbert transform

    Hi all, This question is on the Hilbert transform, particularly on the domain of the input and output functions of the Hilbert transform. Before rising the question, consider the Fourier transform. The input is f(t) and the output is F(\omega). The function f and F are defined over...
  10. N

    Harmonic Functions, conjugates and the Hilbert Transform

    Hi, I am currently confused about something I've run across in the literature. Given that \nabla^2\phi = \phi_{xx}+\phi_{zz} = 0 for z\in (-\infty, 0] and \phi_z = \frac{\partial}{\partial x} |A|^2 at z=0. for A= a(x)e^{i \theta(x)} . The author claims that...
  11. R

    Hilbert transform filter phase

    We know that hilbert transform is a linear filter whose frequency response is given as -j*sgn(f), where f is the baseband frequency. Hence magnitude response of this filter is 1 and phase response is -pi/2 for f > 0 and pi/2 for f < 0. Hence phase response curve is like a staircase function (...
  12. enigma

    What is a Hilbert Transform and How Does it Work?

    Anybody know what a Hilbert transform does? The NB4 function is looking at how the frequency of noise from a gearbox changes as a damaged tooth passes the sensor. I understand the concept, but I don't understand what the math is actually computing...