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Hilbert Transform

\begin{displaymath}
\begin{array}{rcll}
y(t) &\isdef & (h_i \ast x)(t) & \hbox{(Hilbert transform of $x$)} \\ [20pt]
h_i(t) &\isdef & \dfrac{1}{\pi t} & \hbox{(Hilbert transform \lq\lq kernel'')} \\ [20pt]
H_i(\omega) &\isdef & \left\{\begin{array}{ll}
-j, & \omega>0 \\ [5pt]
\quad\! j, & \omega<0 \\ [5pt]
\quad\! 0, & \omega=0 \\
\end{array} \right. &
\hbox{(Hilbert frequency response)}\\ [30pt]
x_a(t) &\isdef & x(t) + j y(t) & \hbox{(Analytic signal from $x$)}\\ [20pt]
& = & \dfrac{1}{\pi}\displaystyle\int_0^{\infty} X(\omega)e^{j\omega t} d\omega
& \hbox{(Note: Lower limit usually $-\infty$)}\\ [20pt]
\emph{Proof:} & \\ [10pt]
X_a(\omega) &=& X(\omega) + j Y(\omega) & \hbox{(By linearity of Fourier transform)}\\ [20pt]
&=& X_++X_- & \hbox{(Apply frequency response)}\\
& & + j [-j X_+ + j X_-] \\ [20pt]
&=& 2X_+(\omega)
\end{array}\end{displaymath}


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``The Window Method for FIR Digital Filter Design}'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2020-06-27 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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