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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)$ .
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--(5.777,3.411)--(5.782,3.404)--(5.786,3.398)--(5.791,3.392)--(5.795,3.386)--(5.800,3.379)%
--(5.805,3.373)--(5.809,3.367)--(5.814,3.361)--(5.818,3.355)--(5.823,3.349)--(5.827,3.342)%
--(5.832,3.336)--(5.836,3.330)--(5.841,3.324)--(5.845,3.318)--(5.850,3.312)--(5.854,3.306)%
--(5.858,3.300)--(5.863,3.294)--(5.867,3.288)--(5.872,3.282)--(5.876,3.276)--(5.880,3.270)%
--(5.884,3.264)--(5.889,3.258)--(5.893,3.252)--(5.897,3.247)--(5.902,3.241)--(5.906,3.235)%
--(5.910,3.229)--(5.914,3.223)--(5.918,3.217)--(5.923,3.212)--(5.927,3.206)--(5.931,3.200)%
--(5.935,3.194)--(5.939,3.189)--(5.943,3.183)--(5.948,3.177)--(5.952,3.172)--(5.956,3.166)%
--(5.960,3.160)--(5.964,3.155)--(5.968,3.149)--(5.972,3.143)--(5.976,3.138)--(5.980,3.132)%
--(5.984,3.127)--(5.988,3.121)--(5.992,3.116)--(5.996,3.110)--(6.000,3.105)--(6.004,3.099)%
--(6.008,3.094)--(6.012,3.088)--(6.016,3.083)--(6.019,3.077)--(6.023,3.072)--(6.027,3.066)%
--(6.031,3.061)--(6.035,3.056)--(6.039,3.050)--(6.043,3.045)--(6.046,3.039)--(6.050,3.034)%
--(6.054,3.029)--(6.058,3.023)--(6.062,3.018)--(6.065,3.013)--(6.069,3.008)--(6.073,3.002)%
--(6.077,2.997)--(6.080,2.992)--(6.084,2.987)--(6.088,2.981)--(6.091,2.976)--(6.095,2.971)%
--(6.099,2.966)--(6.102,2.961)--(6.106,2.956)--(6.110,2.950)--(6.113,2.945)--(6.117,2.940)%
--(6.121,2.935)--(6.124,2.930)--(6.128,2.925)--(6.131,2.920)--(6.135,2.915)--(6.138,2.910)%
--(6.142,2.905)--(6.146,2.900)--(6.149,2.895)--(6.153,2.890)--(6.156,2.885)--(6.160,2.880)%
--(6.163,2.875)--(6.167,2.870)--(6.170,2.865)--(6.174,2.860)--(6.177,2.855)--(6.181,2.850)%
--(6.184,2.845)--(6.187,2.840)--(6.191,2.836)--(6.194,2.831)--(6.198,2.826)--(6.201,2.821)%
--(6.204,2.816)--(6.208,2.811)--(6.211,2.807)--(6.215,2.802)--(6.218,2.797)--(6.221,2.792)%
--(6.225,2.787)--(6.228,2.783)--(6.231,2.778)--(6.235,2.773)--(6.238,2.769)--(6.241,2.764)%
--(6.244,2.759)--(6.248,2.754)--(6.251,2.750)--(6.254,2.745)--(6.258,2.740)--(6.261,2.736)%
--(6.264,2.731)--(6.267,2.726)--(6.270,2.722)--(6.274,2.717)--(6.277,2.713)--(6.280,2.708)%
--(6.283,2.703)--(6.286,2.699)--(6.290,2.694)--(6.293,2.690)--(6.296,2.685)--(6.299,2.681)%
--(6.302,2.676)--(6.305,2.672)--(6.309,2.667)--(6.312,2.663)--(6.315,2.658)--(6.318,2.654)%
--(6.321,2.649)--(6.324,2.645)--(6.327,2.640)--(6.330,2.636)--(6.333,2.631)--(6.336,2.627)%
--(6.340,2.623)--(6.343,2.618)--(6.346,2.614)--(6.349,2.609)--(6.352,2.605)--(6.355,2.601)%
--(6.358,2.596)--(6.361,2.592)--(6.364,2.588)--(6.367,2.583)--(6.370,2.579)--(6.373,2.575)%
--(6.376,2.570)--(6.379,2.566)--(6.382,2.562)--(6.385,2.558)--(6.388,2.553)--(6.391,2.549)%
--(6.393,2.545)--(6.396,2.541)--(6.399,2.536)--(6.402,2.532)--(6.405,2.528)--(6.408,2.524)%
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--(6.428,2.494)--(6.431,2.490)--(6.434,2.486)--(6.437,2.482)--(6.440,2.478)--(6.442,2.474)%
--(6.445,2.470)--(6.448,2.466)--(6.451,2.462)--(6.454,2.457)--(6.457,2.453)--(6.459,2.449)%
--(6.462,2.445)--(6.465,2.441)--(6.468,2.437)--(6.470,2.433)--(6.473,2.429)--(6.476,2.425)%
--(6.479,2.421)--(6.482,2.417)--(6.484,2.413)--(6.487,2.409)--(6.490,2.405)--(6.492,2.401)%
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--(6.511,2.374)--(6.514,2.370)--(6.517,2.366)--(6.519,2.362)--(6.522,2.358)--(6.525,2.354)%
--(6.527,2.350)--(6.530,2.346)--(6.533,2.342)--(6.535,2.339)--(6.538,2.335)--(6.541,2.331)%
--(6.543,2.327)--(6.546,2.323)--(6.549,2.319)--(6.551,2.316)--(6.554,2.312)--(6.556,2.308)%
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--(6.590,2.259)--(6.592,2.256)--(6.595,2.252)--(6.597,2.248)--(6.600,2.244)--(6.602,2.241)%
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--(7.338,1.149)--(7.339,1.147)--(7.340,1.145)--(7.342,1.143)--(7.343,1.141)--(7.345,1.138)%
--(7.346,1.136)--(7.348,1.134)--(7.349,1.132)--(7.350,1.130)--(7.352,1.128)--(7.353,1.126)%
--(7.355,1.123)--(7.356,1.121)--(7.358,1.119)--(7.359,1.117)--(7.360,1.115)--(7.362,1.113)%
--(7.363,1.111)--(7.365,1.109)--(7.366,1.106)--(7.368,1.104)--(7.369,1.102)--(7.370,1.100)%
--(7.372,1.098)--(7.373,1.096)--(7.375,1.094)--(7.376,1.092)--(7.377,1.090)--(7.379,1.087)%
--(7.380,1.085)--(7.382,1.083)--(7.383,1.081)--(7.384,1.079)--(7.386,1.077)--(7.387,1.075)%
--(7.389,1.073)--(7.390,1.071)--(7.391,1.069)--(7.393,1.067)--(7.394,1.064)--(7.395,1.062)%
--(7.397,1.060)--(7.398,1.058)--(7.400,1.056)--(7.401,1.054)--(7.402,1.052)--(7.404,1.050)%
--(7.405,1.048)--(7.406,1.046)--(7.408,1.044)--(7.409,1.042)--(7.411,1.040)--(7.412,1.038)%
--(7.413,1.036)--(7.415,1.034)--(7.416,1.031)--(7.417,1.029)--(7.419,1.027)--(7.420,1.025)%
--(7.421,1.023)--(7.423,1.021)--(7.424,1.019)--(7.426,1.017)--(7.427,1.015)--(7.428,1.013)%
--(7.430,1.011)--(7.431,1.009)--(7.432,1.007)--(7.434,1.005)--(7.435,1.003)--(7.436,1.001)%
--(7.438,0.999)--(7.439,0.997)--(7.440,0.995)--(7.442,0.993)--(7.443,0.991)--(7.444,0.989)%
--(7.446,0.987)--(7.447,0.985);
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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$

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

the Bode plot can be expressed as

$\displaystyle \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

$\displaystyle \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

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

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

$\displaystyle \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

$\displaystyle \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]$ .


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``Spectral Tilt Filters'', by Julius O. Smith IIIand Harrison F. Smith, adapted from arXiv.org publication arXiv:1606.06154 [cs.CE].
Copyright © 2021-06-18 by Julius O. Smith IIIand Harrison F. Smith
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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