Bessel filter
A Bessel filter (also a Thomson or Bessel-Thomson filter) is a linear low-pass filter designed so that the delay it imposes on a signal is as uniform as possible across its passband. A signal made of many frequencies emerges with every component late by nearly the same time, so a pulse keeps its shape. The price is a slow roll-off, since the filter cuts off high frequencies less steeply than a Butterworth or Chebyshev filter of the same order. It is a common choice in patch clamp amplifiers and in nanopore recording chains, where it sits ahead of the converter.
History
The design comes from a paper of 1949 by W. E. Thomson, "Delay networks having maximally flat frequency characteristics".[1] The denominator of its transfer function turned out to be a Bessel polynomial. Those polynomials were studied as a family in the same year by Krall and Frink, and Storch gave the synthesis of ladder networks from them in 1954.[2][3] The polynomials take their name from their link to Bessel functions, which are named for the astronomer Friedrich Bessel.
Design
A signal passes through a filter with a delay for each frequency, called the group delay. A filter with a flat group delay applies one delay to everything in the band, which is the same as delaying the whole waveform without altering it. The Bessel design makes the delay flat at zero frequency to the highest order that the number of poles allows. Normalized to a delay of one second, the transfer function of the four-pole filter is
and its response falls by 3 dB at 2.114 rad/s. Set to that corner, the delay at low frequencies is 0.336 divided by the corner frequency in hertz.
| Property | Bessel | Butterworth |
|---|---|---|
| Overshoot after a step | 0.84% | 10.8% |
| Rise time, 10% to 90% | 0.350 / fc | 0.387 / fc |
| Group delay at low frequency | 0.336 / fc | 0.416 / fc |
| Group delay at fc, relative to low frequency | −1.8% | +41% |
| Attenuation at 2 fc | 13.4 dB | 24.1 dB |
| Attenuation at 4 fc | 34.4 dB | 48.2 dB |
The values were calculated from the analog prototypes with SciPy. The Butterworth filter cuts more sharply, and it pays with an overshoot of about a tenth of the step and a delay that grows by two-fifths as the frequency approaches the corner. A pulse through it rings. Through the Bessel filter it comes out smoothed and a little late.
Short events
A filter at fc also shrinks events that are brief compared with its time constant. A rectangular blockade lasting T reaches only part of its depth, and the fraction depends on T fc.
| T fc | Four-pole Bessel | One-pole RC |
|---|---|---|
| 0.1 | 28% | 47% |
| 0.2 | 53% | 72% |
| 0.3 | 73% | 85% |
| 0.5 | 95% | 96% |
| 1.0 | 101% | 100% |
At 10 kHz a blockade of 20 µs, which is T fc = 0.2, reaches 53% of its depth through four Bessel poles and 72% through one RC stage.[a] The higher order cuts noise faster beyond the corner and costs the shortest events more. Pedone and colleagues recovered the width and height of pulses as short as 0.3 / fc, as described in dwell time.[4]
Use with converters
A converter that samples at fs folds anything above fs/2 into the band below, so a filter ahead of it removes what would fold. At 100 kS/s a component at 80 kHz appears at 20 kHz. A four-pole Bessel filter with its corner at 20 kHz would attenuate that component by 34.4 dB, a factor of about 50 in amplitude. The Axopatch 200B passes the signal through a four-pole Bessel filter with settings of 1, 2, 5, 10 and 100 kHz.[5] A study of the noise of solid-state pores used an external eight-pole Bessel filter at 49.9 kHz and digitized at 250 or 500 kHz.[6] How a filter and a sampling rate are chosen together, and what a given bandwidth does to the noise, is treated in bandwidth (signal processing) and nanopore noise.
At Molecular Reality
The analog chain drawn for the MR1 Nanopore Reader has no anti-aliasing filter, and it samples at 100 kS/s.[7] The filter does appear in one of the company's essays. The Nanopore World Model, published in June 2026, describes a 200 kHz current trace as "mostly thermal noise, capacitive ringing, and the amplifier's Bessel filter talking to itself".[8]
See also
Notes
- ^ The RC figure is 1 − exp(−2π × 0.2). The Bessel values come from a numerical step response. The 101% at T fc = 1.0 is the overshoot.
References
- ^ Thomson, W. E. (1949). "Delay networks having maximally flat frequency characteristics". Proceedings of the IEE, Part III: Radio and Communication Engineering 96 (44): 487-490. doi:10.1049/pi-3.1949.0101.
- ^ Krall, H. L.; Frink, O. (1949). "A new class of orthogonal polynomials: the Bessel polynomials". Transactions of the American Mathematical Society 65 (1): 100-115. doi:10.1090/S0002-9947-1949-0028473-1.
- ^ Storch, L. (1954). "Synthesis of constant-time-delay ladder networks using Bessel polynomials". Proceedings of the IRE 42 (11): 1666-1675. doi:10.1109/JRPROC.1954.274662.
- ^ Pedone, D.; et al. (2009). "Data analysis of translocation events in nanopore experiments". Analytical Chemistry 81 (23): 9689-9694. doi:10.1021/ac901877z.
- ^ "Axon Axopatch 200B Microelectrode Amplifier (data sheet)". Molecular Devices. July 2012.
- ^ Smeets, R. M. M.; et al. (2008). "Noise in solid-state nanopores". Proceedings of the National Academy of Sciences 105 (2): 417-421. doi:10.1073/pnas.0705349105.
- ^ "MR1 Nanopore Reader". GitHub. October 16, 2026.
- ^ "Keltar's Critique v2: The Nanopore World Model (Hardened)". Molecular Reality Corporation. June 2026.
Further reading
- Colquhoun, D.; Sigworth, F. J. (1995). "Fitting and statistical analysis of single-channel records". In Colquhoun, D.; Neher, E. (ed.). Single-Channel Recording. 2nd ed. Plenum Press. pp. 483-587. doi:10.1007/978-1-4419-1229-9_19.