Approximating Arbitrary Slopes

To approximate arbitrary slopes $\alpha$ , we may alternate poles and zeros so that the average slope of the stick diagram equals $\alpha$ .

For example, to achieve a slope of $\alpha =-1/2$ (``half an integrator''), we may start with a pole near $p_1=-2\pi f_1$ , where $f_1$ is our lowest frequency of interest (nominally $20$ -$40$ Hz for audio), which causes the slope to approach $-1$ . Then, half an octave to the right, e.g., we can locate a zero to cancel the pole's roll-off, pushing the slope from $-1$ back toward 0 . Continuing in this way, we may locate a pole at each octave point $f_12^k$ , $k=0,1,2,\ldots\,$ , with zeros interlacing at $f_12^{k+1/2}$ , in order to achieve an average slope of $\alpha =-1/2$ .

Figure 2: Bode plot and its asymptotes (``stick diagram'') for $N=5$ poles and zeros, arranged to approximate a $1/f$ power response.

Figure 2 shows the Bode plot and the corresponding stick diagram for $N=5$ poles located to give breakpoints distributed along octaves starting with $\omega=1$ . That is, the poles are at $s=p_n=-r^n$ , for $r=2$ and $n=0,2,\ldots,N-1$ . To approximate a $1/f$ power-response having slope $\alpha =-1/2$ nepers per neper, we place $N$ zeros at $s=z_m=p_mr^{-\alpha}$ , $m=0,2,\ldots,N-1$ , i.e., shifted half an octave toward higher frequency, interlacing the poles. We see that the response is flat for frequencies below the first break-frequency $\omega=1$ as before, but the gain drops by less the $2$ dB at $\omega=1$ due primarily to the influence of the upcoming first zero at $\omega=\sqrt{2}$ . Between the pole-zero frequencies $\omega=1$ and $\omega=r^{N-1-\alpha}$ , the Bode plot smoothly interpolates the stick diagram which alternates between slopes of 0 and $-20$ dB per decade, as the poles and zeros break in alternation, yielding an average roll-off of $-10$ dB per decade, as desired. After the last zero breaks, the response levels off to a final slope of 0 . Alternatively, the last zero could be omitted to have a final $-20$ dB per decade slope, etc. Additionally, we plot the pole symbol `X' and zero symbol `O' along the upper horizontal axis at their corresponding break frequencies. This plot suggest that we may be able to choose $N$ , $r$ , and $p_1$ to achieve any desired accuracy over any finite band.

More generally, for any desired slope $\alpha\in(-1,1)$ , we place the $k$ th zero on the negative-real axis of the complex $s$ plane at $s = z_k = p_kr^{-\alpha}$ , $k=0,1,2,\ldots,N-1$ , where $p_k=p_1r^k$ denotes the $k$ th pole, exponentially distributed along the negative-real axis with spacing ratio $r=p_{k+1}/p_k$ , starting at $p_1=-2\pi f_1$ .

Note that $\alpha=0$ cancels all of the poles with zeros, yielding a constant magnitude frequency response, while $\alpha=-1$ cancels all poles except the first $p_1$ , leaving a slope of $-1$ nepers per neper (an integrator) for $\omega\gg \omega_1$ . When $\alpha>0$ , the pole-zero sequence starts out from the origin of the $s$ plane with a zero, thereby giving a net positive slope to the Bode magnitude plot. In particular, at $\alpha=1$ , all of the poles are canceled by zeros, leaving behind a single zero at $s=p_1/r$ , yielding a slope of $+1$ (differentiator) for $\omega\gg \omega_1/r$ . Between these extremes, the poles and zeros interlace asymmetrically to approximate any desired slope $\alpha$ .

As examples of other slopes, Fig.3 shows a Bode plot analogous to Fig.2 for the case $\alpha=+1$ (``half a differentiator''), and Fig.4 shows $\alpha=-0.2$ , showing the resulting asymmetric pole-zero layout on a log-frequency scale.

Figure 3: Bode plot and its asymptotes for $N=5$ poles and zeros, arranged to approximate ``half a differentiator'' $\vert H_{1/2}(j\omega )\vert=\sqrt {\omega }$ .

Figure 4: Bode plot and its asymptotes for $N=5$ poles and zeros, arranged to yield a Bode plot with slope $\alpha =0.2$ over a chosen frequency range: $\vert H_{-0.2}(j\omega )\vert=\omega ^{-0.2}$ .

To reduce the maximum error, the interlacing pole-zero pattern can be made more dense, e.g., by placing poles every half octave, or third octave, etc. As an example, Fig.5 shows the improvement obtained over Fig.2 by increasing the order from $N=5$ to $12$ .

Figure 5: Bode plot and its asymptotes (``stick diagram'') for $N=12$ poles and zeros, arranged to approximate a $1/f$ filter response.

It is not necessary for the slope $\alpha$ of the spectral roll-off to be restricted between $-1$ and $1$ neper per neper. Any number of poles or zeros can be used in the low-frequency region to establish any integer part for the slope, or some number of the regularly spaced poles (zeros) can be skipped before the partial cancellation array of zeros (poles) begins. The subsequent interlacing poles and zeros then only need to set the fractional value of the slope above $f_1$ .

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