DC Blocker Frequency Response

Figure 10.12 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.10.13. The corresponding pole-zero diagrams are shown in Fig.10.14. 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.10.15 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 10.12: 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.

Figure 10.13: 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.

Figure 10.14: Pole-zero diagram overlays for the dc blocker defined by $H(z) = (1-z^{-1})/(1-Rz^{-1})$ for various values of pole radius $R$.

Figure 10.15: Impulse response of the dc blocker defined by $H(z) = (1-z^{-1})/(1-0.9z^{-1})$.

Note that the amplitude response in Fig.10.12a and Fig.10.13a 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

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