One-pole, one-zero high-pass filter

Sometimes an audio signal carries an unwanted constant offset, or in other words, a zero-frequency component. For example, the waveshaping spectra of Section 5.3 almost always contain a constant component. This is inaudible, but, since it specifes electrical power that is converted to heat in your speakers, its presence reduces the level of loudness you can reach without distortion. Another name for a constant signal component is ``DC", meaning ``direct current".

An easy and practical way to remove the zero-frequency component from an audio signal is to use a one-pole lowpass filter to extract it, and then subtract the result from the signal. The resulting transfer function is one minus the transfer function of the lowpass, giving:

$$H(Z) = 1 - {{{1-p} \over {1 - p{Z^{-1}}}}} = p{{{1-{Z^{-1}}} \over {1 - p{Z^{-1}}}}}$$

The factor of $1-p$ in the numerator of the low-pass transfer function is the normalization factor needed so that the gain is one at zero frequency.

By examining the right-hand side of the equation (comparing it to the general formula for compound filters), we see that there is still a pole at the real number $p$, and there is now also a zero at the point $1$. The pole-zero plot is as shown in Figure 8.13 part (a), and the frequency response in part (b). (From here on, we will only plot frequency responses to the Nyquist frequency $\pi$; in the previous example we plotted it all the way up to the sample rate, $2\pi$.)

Figure 8.13: One-pole, one-zero high-pass filter: (a) pole-zero diagram; (b) frequency response (from zero to Nyquist frequency).