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Highpass Filter from Lowpass

A highpass filter can be constructed by shifting a lowpass filter by an amount equal to half the sampling rate.

\epsfig{file=eps/lpf2hpf.eps,width=5in,height=3in}

This is equivalent to a modulation by $ e^{-j\pi n} $ in the time domain. Hence,

\begin{eqnarray*}
H_{\hbox{hp}}(\omega) &=& \hbox{\sc Shift}_{\pi} \{H_{\hbox{lp}}(\omega)\} \eqsp H_{\hbox{lp}}(\omega-\pi)\quad (T=1)\\
\leftrightarrow\quad h_{\hbox{hp}}(n) &=& h_{\hbox{lp}}(n) e^{-j\pi n} \\
&=& h_{\hbox{lp}}(n) (-1)^n
\end{eqnarray*}

Summarizing:

Alternatively, if the passband of the prototype lowpass filter $ \hat{H}(\omega)$ is very flat or equal-ripple about unity gain, a highpass filter can be designed by simply subtracting from $ 1$ :

\begin{eqnarray*}
H_{\mbox{hp}}(\omega) &\mathrel{\stackrel{\Delta}{=}}& 1 - H_{\mbox{lp}}(\omega) \\
&\leftrightarrow& \delta(n) - h_{lp}(n)
\end{eqnarray*}


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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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