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DC Blocker Frequency Response

Figure B.11 shows the frequency response of the dc blocker for several values of $ R$ . The same plots are given over a log-frequency scale in Fig.B.12. The corresponding pole-zero diagrams are shown in Fig.B.13. As $ R$ approaches $ 1$ , the notch at dc gets narrower and narrower. While this may seem ideal, there is a drawback, as shown in Fig.B.14 for the case of $ R=0.9$ : The impulse response duration increases as $ R\to 1$ . While the ``tail'' of the impulse response lengthens as $ R$ approaches 1, its initial magnitude decreases. At the limit, $ R=1$ , the pole and zero cancel at all frequencies, the impulse response becomes an impulse, and the notch disappears.

Figure B.11: Frequency response overlays for the dc blocker defined by $ H(z) = (1-z^{-1})/(1-Rz^{-1})$ for various values of pole radius $ R$ . (a) Amplitude response. (b) Phase response.
\includegraphics[width=\twidth ]{eps/dcblockerfr}

Figure B.12: Log-frequency response overlays for the dc blocker defined by $ H(z) = (1-z^{-1})/(1-Rz^{-1})$ for various values of pole radius $ R$ . (a) Amplitude response. (b) Phase response.
\includegraphics[width=\twidth ]{eps/dcblockerfrlf}

Figure B.13: Pole-zero diagram overlays for the dc blocker defined by $ H(z) = (1-z^{-1})/(1-Rz^{-1})$ for various values of pole radius $ R$ .
\includegraphics[width=\twidth]{eps/dcblockerpz}

Figure B.14: Impulse response of the dc blocker defined by $ H(z) = (1-z^{-1})/(1-0.9z^{-1})$ .
\includegraphics[width=\twidth ]{eps/dcblockerir}

Note that the amplitude response in Fig.B.11a and Fig.B.12a exceeds 1 at half the sampling rate. This maximum gain is given by $ H(-1)=2/(1+R)$ . In applications for which the gain must be bounded by 1 at all frequencies, the dc blocker may be scaled by the inverse of this maximum gain to yield

\begin{eqnarray*}
H(z) &=& g\frac{1-z^{-1}}{1-Rz^{-1}}\\
y(n) &=& g[x(n) - x(n-1)] + R\, y(n-1), \quad\hbox{where}\\
g &\isdef & \frac{1+R}{2}.
\end{eqnarray*}


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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