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Unit 4 · Lesson 4-4
Filtering
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Filtering

Why ideal filters cannot be built, filter specifications, Butterworth and Chebyshev responses, order, and distortionless transmission

1930 — a filter as flat as possible
Stephen Butterworth, a British engineer and physicist, publishes a paper with a modest title, "On the Theory of Filter Amplifiers". At the time, designing a filter took a great deal of experience, because the theory in use did not say how to get a response you wanted. Butterworth sets out a goal and meets it exactly: a response that is as flat as possible in its passband, and then falls away steadily (Butterworth filter). His filter was not in common use for over thirty years after its publication. Today it is the default choice wherever a smooth, ripple-free passband matters, from audio crossovers to anti-aliasing filters in front of every analog-to-digital converter. This lesson is about filter design: what we ask of a filter, why we can never have all of it, and how Butterworth's and Chebyshev's answers trade one thing for another.
The ideal low-pass filter passes every frequency below the cut-off ωc\omega_c unchanged and blocks everything above: ∣H∣=1|H| = 1 for ∣ω∣<ωc|\omega| < \omega_c and 00 beyond, with zero phase. By duality (Lesson 4-1) its impulse response is a sinc,
h(t)=sin⁡(ωct)πt,h(t) = \frac{\sin(\omega_c t)}{\pi t},
which rings on forever in both directions — including before t=0t = 0. The ideal filter would have to respond before its input arrived. It is non-causal, and no real circuit or real-time program can be it. Every practical filter is an approximation, and design is the choice of which way to fall short.
💡Filter specifications
A low-pass specification marks out where the response may go:
  • passband up to ωp\omega_p: the gain must stay within a ripple ApA_p dB of 0 dB (for example, 1 dB);
  • stopband from ωs\omega_s: the gain must be at least AsA_s dB down (for example, 40 dB);
  • transition band between them: free.
The narrower the transition band and the stricter the ripple and attenuation, the higher the order — the number of poles, and so of capacitors and inductors, or of coefficients in software.
∣H(jω)∣2=11+(ω/ωc)2N|H(j\omega)|^2 = \frac{1}{1 + (\omega/\omega_c)^{2N}}

The Butterworth magnitude response of order N: −3 dB at ωc, maximally flat at ω = 0, and falling 20N dB per decade above ωc.

Derivation
The order a specification needs
Ask for at most ApA_p dB of loss at ωp\omega_p and at least AsA_s dB at ωs\omega_s. In decibels the Butterworth loss is 10log⁡10(1+(ω/ωc)2N)10\log_{10}\left(1 + (\omega/\omega_c)^{2N}\right), so
(ωp/ωc)2N≤10Ap/10−1,(ωs/ωc)2N≥10As/10−1.(\omega_p/\omega_c)^{2N} \le 10^{A_p/10} - 1, \qquad (\omega_s/\omega_c)^{2N} \ge 10^{A_s/10} - 1 .
Divide the second by the first to eliminate ωc\omega_c:
(ωsωp)2N≥10As/10−110Ap/10−1⟹N≥log⁡10 ⁣[10As/10−110Ap/10−1]2log⁡10(ωs/ωp).\left(\frac{\omega_s}{\omega_p}\right)^{2N} \ge \frac{10^{A_s/10} - 1}{10^{A_p/10} - 1} \quad\Longrightarrow\quad N \ge \frac{\log_{10}\!\left[\dfrac{10^{A_s/10} - 1}{10^{A_p/10} - 1}\right]}{2\log_{10}(\omega_s/\omega_p)} .
Round up. The order grows with the attenuation asked for and blows up as the transition ratio ωs/ωp\omega_s/\omega_p approaches 1.
📐One kilohertz in, two kilohertz out
Pass up to 1 kHz with at most 1 dB of loss, reject from 2 kHz by at least 40 dB: N≥log⁡10 ⁣[104−1100.1−1]2log⁡102=log⁡10(38 618)0.602=4.5870.602=7.6    ⟹    N=8.N \ge \frac{\log_{10}\!\left[\dfrac{10^{4} - 1}{10^{0.1} - 1}\right]}{2\log_{10}2} = \frac{\log_{10}(38\,618)}{0.602} = \frac{4.587}{0.602} = 7.6 \;\;\Longrightarrow\;\; N = 8 . An eighth-order Butterworth filter: eight poles, falling 160 dB per decade beyond its cut-off. A Chebyshev filter meets the same specification with fewer poles, as the next paragraph and the lab show.
Trading flatness for steepness. Butterworth spends all of its freedom on a flat passband. A Chebyshev type I filter allows the passband to ripple, equally, up to the permitted ApA_p, and in exchange falls much faster just past the cut-off; for the specification above it needs order 5. An elliptic (Cauer) filter ripples in both bands and is the steepest of all for a given order, order 4 here. The price of steepness is paid in the phase: the sharper the edge, the more unevenly different frequencies are delayed, and the longer the filter rings after a step.
💡Distortionless transmission and group delay
A filter passes a signal without distorting its shape only if, across the signal's band, its gain is constant and its phase is linear in frequency, ∠H=−ωt0\angle H = -\omega t_0: then every component is delayed by the same t0t_0 (Lesson 4-2). How far a filter departs from that is measured by its group delay, τg(ω)=−d∠H(jω)dω,\tau_g(\omega) = -\frac{d\angle H(j\omega)}{d\omega}, the delay suffered by a narrow group of frequencies around ω\omega. A flat group delay means no phase distortion. Butterworth's group delay rises to a peak near the cut-off, and Chebyshev's peak is higher still — which is why sharp analog filters smear pulses, and why digital FIR filters with exactly linear phase (Lesson 9-1) are so valued.
An ideal electrical filter should not only completely reject the unwanted frequencies but should also have uniform sensitivity for the wanted frequencies.
– Stephen Butterworth, "On the Theory of Filter Amplifiers" (1930), quoted in Butterworth filter
🕰Thirty years on the shelf
Butterworth's paper showed how to build his filter from amplifier stages with simple networks between them — a practical design method at a time when filter design was largely trial and experience. It nevertheless took over thirty years to come into common use (Butterworth filter); today its formulas are in every filter-design library, including SciPy's butter.
⚠−3 dB is half the power, not half the amplitude
A Butterworth filter's cut-off ωc\omega_c is where ∣H∣2=12|H|^2 = \tfrac12: half the power, ∣H∣=0.707|H| = 0.707. Half the amplitude is −6 dB. And "the cut-off" of a Chebyshev filter is usually quoted at its ripple edge, not at −3 dB, so compare specifications, not names.
💡Where you will use this
Analog filters guard every sampler against aliasing (Lesson 5-2). The same prototypes become digital IIR filters through the bilinear transform (Lesson 9-2), and linear-phase FIR filters (Lesson 9-1) solve the delay problem at the cost of length. Group delay matters wherever pulse shape matters: data links, radar, audio crossovers.
📚Further reading
  • Oppenheim and Willsky, Signals and Systems, sections 6.3–6.6 — ideal and non-ideal filters, group delay.
  • Arthur B. Williams and Fred J. Taylor, Electronic Filter Design Handbook (4th ed., McGraw-Hill, 2006) — the practical reference for Butterworth, Chebyshev and elliptic designs.
  • Butterworth filter on Wikipedia.
↓But there is a problem
We can now shape a spectrum on purpose. Everything so far has been continuous: signals defined at every instant, filters made of capacitors and inductors. But the music on your phone, the photographs on your screen and the sensor readings in every aircraft are lists of numbers — samples. When does a list of samples hold all of a continuous signal, so that nothing is lost? The answer fixed the number 44,100 into every compact disc.