Bode Plots

A Bode Plot of a filter frequency response $H(j\omega)$ separately graphs the log-magnitude and phase versus log-frequency.¹⁶We are only concerned here with log-magnitude plots, and will omit consideration of the Bode phase plot, which happen to behave as expected naturally. The usual choice of log-magnitude units is decibels (dB) $20\log_{10}\left[\vert H(j\omega)\vert/R\right]$ (relative to an arbitrary reference, such as $R=1$ ), and the log-frequency axis is typically either in octaves ( $\propto\log_2(\omega)$ ) or decades ( $\propto\log_{10}(\omega)$ ). Thus, a single pole is said to give a roll-off of $-6$ dB per octave or $-20$ dB per decade. Octaves are typical in audio signal processing while decades are typical in the field of automatic control.

Figure 1: Bode plot and its asymptotes (``stick diagram'') for a single pole at $s=-1$ , i.e., the transfer function $H(s)=1/(s+1)$ .

Figure 1 illustrates the Bode plot and its associated ``stick diagram'' (comprised of asymptotic gains) for a single pole at $s=-1$ . We see that the response is flat for low frequencies, drops to $-3$ dB at the break frequency $\omega=1$ , and approaches the $-20$ dB per decade asymptote, reaching the asymptote quite well by one decade above the break frequency at $\omega=10$ .

For a general filter transfer function having $N$ poles $p_n$ and $M$ zeros $z_m$

$$H(s) = g\frac{\prod_{m=1}^M (s-z_m)}{\prod_{n=1}^N (s-p_n)}$$ (4)

the Bode plot can be expressed as

$$\tilde{B}({\tilde{\omega}}) = g_1 \log_{b_1} H\left(jb_2^{\log_{b_2}(\omega)}\right) = \tilde{G}({\tilde{\omega}}) + j\Theta({\tilde{\omega}}),$$

where $g_1=20$ , $b_1=10$ , ${\tilde{\omega}}=\log_{b_2}(\omega)$ , and $b_2$ is typically $2$ or $10$ . For mathematical simplicity, however, we'll choose instead $g_1=1$ and $b_1=b_2=e$ , giving

$$\tilde{B}({\tilde{\omega}}) = \ln H\left(je^{\tilde{\omega}}\right) = \tilde{G}({\tilde{\omega}}) + j\Theta({\tilde{\omega}}),$$

where ${\tilde{\omega}}= \ln(\omega)$ . In this choice of units, $N$ integrators $H(s)=1/s^N$ give a magnitude roll-off of $-N$ ``nepers per neper'', while $M$ differentiators $H(s)=s^M$ give a slope of $+M$ in the Bode magnitude plot

$$\tilde{G}({\tilde{\omega}}) = \mbox{re\ensuremath{\left\{\tilde{B}({\tilde{\omega}})\right\}}} = \mbox{re\ensuremath{\left\{\ln H\left(je^{\tilde{\omega}}\right)\right\}}} .$$ (5)

Our problem is to find poles and zeros of $H(s)$ to minimize some norm of the error

$$\left\Vert W({\tilde{\omega}})\left[\tilde{G}'({\tilde{\omega}}) - \alpha \right]\right\Vert$$

where $\tilde{G}'({\tilde{\omega}})=d\,\tilde{G}({\tilde{\omega}})/d{\tilde{\omega}}$ denotes the derivative of $\tilde{G}({\tilde{\omega}})$ with respect to ${\tilde{\omega}}$ , $\alpha$ is the desired slope of the log-magnitude frequency-response $\tilde{G}'$ versus log frequency ${\tilde{\omega}}$ , and $W({\tilde{\omega}})$ denotes a real, nonnegative weighting function.

As a specific example, for the Chebyshev norm and a uniform weighting $W({\tilde{\omega}})\equiv 1$ between frequencies ${\tilde{\omega}}_1$ and ${\tilde{\omega}}_2$ , we have

$$\min_H\left\{\max_{{\tilde{\omega}}\in[{\tilde{\omega}}_1,{\tilde{\omega}}_2]} \left\vert\tilde{G}'({\tilde{\omega}}) - \alpha \right\vert\right\}.$$

That is, we wish to minimize the worst-case deviation between the desired slope $\alpha$ and the achieved slope $\tilde{G}'({\tilde{\omega}})$ over a specific (audio) band $[{\tilde{\omega}}_1,{\tilde{\omega}}_2]$ .