My notes on Hilbert transform

    Hilbert transform in frequency domain
    -It is a peculiar sort of filter that changes the phase of the spectral components depending on the sign of their frequency.
    -It only effects the phase of the signal. It has no effect on the amplitude at all.For any signal g(t), its Hilbert Transform has the following property:
    \mathbf{\hat{G}}(f)=\begin{cases} -j & \text{for} f>0 \\ j & \text{for} f<0\end{cases}

    Hilbert transform in time domain

    In time domain the Hilbert transform \hat{g(t)} of a function g(t) is its convolution with \frac{1}{\pi t} . The choice of the function \frac{1}{\pi t} is because the Fourier transform of the function gives:
    \mathbf{F}(\frac{1}{\pi t})=-j sgn(f)=\begin{cases} -j & \text{for} f>0 \\ j & \text{for} f<0\end{cases}
    -The signal and its Hilbert Transform are orthogonal. This is because by rotating the signal 90° we have now made it orthogonal to the original signal, that being the definition of orthogonality.
    -The signal and its Hilbert Transform have identical energy because phase shift do not change the energy of the signal only amplitude changes can do that.

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