Bandwidth (signal processing)
Bandwidth in signal processing is the width of a range of frequencies. It is used of three things: the band that a filter or amplifier passes, the band that a signal occupies, and, loosely, the rate at which a channel carries data. In a measurement the first meaning governs everything else. A wider band lets faster events through, and it lets in more noise, so every instrument that records a current or a voltage works at a compromise between the two.
Definitions
The bandwidth of a low-pass system is the frequency fc at which its response has fallen by 3 dB, to 71% of the amplitude at low frequency and half of the power. For a single resistor and capacitor in series fc is 1 / (2πRC). Higher-order filters, such as a four-pole Bessel filter, reach the same −3 dB point and then fall faster.
The term was already in use in the telegraph era. In 1928 Nyquist analyzed how fast pulses can be sent through a channel without running into one another, and Hartley argued in the same year that the amount of information a system can send is proportional to the product of its bandwidth and the time of transmission.[1][2]
Speed
A system of bandwidth B cannot change faster than a time of order 1/B. For a single pole the time to rise from 10% to 90% of a step is 2.197 RC, which is 0.35 / fc. An event reaches full height only if it lasts long enough compared with that. Through a four-pole Bessel filter, a rectangular pulse of width T keeps 95% of its depth when T fc is 0.5, so the bandwidth for events of a given duration is about 0.5 / T.
| Event duration | Bandwidth |
|---|---|
| 1 ms | 500 Hz |
| 100 µs | 5 kHz |
| 10 µs | 50 kHz |
| 1 µs | 500 kHz |
| 100 ns | 5 MHz |
Noise
A resistor produces a current noise whose spectral density does not depend on frequency. Over a band of width B the root-mean-square current is the density times √B, and for noise-equivalent bandwidth the filter matters: it is π/2 times fc for a single pole and about 1.05 times fc for a four-pole Bessel filter. At 298 K a 1 MΩ resistor has a density of 128 fA/√Hz and a 1 GΩ resistor one of 4.1 fA/√Hz.
| Bandwidth | 1 MΩ | 1 GΩ |
|---|---|---|
| 1 kHz | 4.1 | 0.13 |
| 10 kHz | 12.8 | 0.41 |
| 100 kHz | 40.6 | 1.3 |
| 1 MHz | 128 | 4.1 |
Other noise sources scale differently. Noise from the voltage noise of an amplifier across an input capacitance rises as B3/2, so doubling the bandwidth multiplies it by 2.83.[3] The details are in nanopore noise.
Sampling
Digitizing a signal adds a second limit. Nyquist's analysis and Shannon's treatment of 1949 give the sampling theorem: a signal with no components above B is determined by samples taken at 2B per second, and components above half the sampling rate fold into the band below.[4] A filter ahead of the converter removes what would fold.
Recordings usually sample well above twice the filter corner. One study of pores sampled at 250 or 500 kHz behind a filter at 49.9 kHz, five to ten times the corner.[5] See Sampling (signal processing).
In nanopore work
DNA crosses a bare pore in an ultrathin solid-state membrane at 3 to 50 bases per microsecond.[6] Reading one base at that speed calls for megahertz and brings the noise that goes with them. A CMOS amplifier placed beside the pore reached a signal-to-noise ratio above 5 at 1 MHz in 2012, and an amplifier built for the purpose recorded at 10 MHz in 2016.[7][8] A motor enzyme that slows the strand to hundreds of bases a second brings the requirement down to a few kilohertz.
At Molecular Reality
The company has a name for the squeeze. Its whisper in a thunderstorm problem is that the current step between two DNA bases is smaller than the noise of a measurement fast enough to see it, and the speeding bullet problem is that DNA crosses a pore too fast.[9] The company's live stream samples its EDR4 instrument at 200 kHz, one value every 5 µs,[10] and the MR1 Nanopore Reader is designed to sample at 100 kS/s.[11] In The Nanopore World Model the same arithmetic is applied to a camera: a frame every 33,000 µs against an amplifier sample every 5 µs, a ratio of 6,600.[12]
See also
References
- ^ Nyquist, H. (1928). "Certain topics in telegraph transmission theory". Transactions of the American Institute of Electrical Engineers 47 (2): 617-644. doi:10.1109/T-AIEE.1928.5055024.
- ^ Hartley, R. V. L. (1928). "Transmission of information". Bell System Technical Journal 7 (3): 535-563. doi:10.1002/j.1538-7305.1928.tb01236.x.
- ^ Uram, J. D.; Ke, K.; Mayer, M. (2008). "Noise and bandwidth of current recordings from submicrometer pores and nanopores". ACS Nano 2 (5): 857-872. doi:10.1021/nn700322m.
- ^ Shannon, C. E. (1949). "Communication in the presence of noise". Proceedings of the IRE 37 (1): 10-21. doi:10.1109/JRPROC.1949.232969.
- ^ 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.
- ^ Feng, J.; et al. (2015). "Identification of single nucleotides in MoS2 nanopores". Nature Nanotechnology 10 (12): 1070-1076. doi:10.1038/nnano.2015.219.
- ^ Rosenstein, J. K.; et al. (2012). "Integrated nanopore sensing platform with sub-microsecond temporal resolution". Nature Methods 9 (5): 487-492. doi:10.1038/nmeth.1932.
- ^ Shekar, S.; et al. (2016). "Measurement of DNA translocation dynamics in a solid-state nanopore at 100 ns temporal resolution". Nano Letters 16 (7): 4483-4489. doi:10.1021/acs.nanolett.6b01661.
- ^ "Frequently asked questions". EQBio-OS. Molecular Reality Corporation. 2026.
- ^ "Keltar's Field Guide to Molecular Reality". Molecular Reality Corporation. August 2026.
- ^ "MR1 Nanopore Reader". GitHub. October 16, 2026.
- ^ "Keltar's Critique v2: The Nanopore World Model (Hardened)". Molecular Reality Corporation. June 2026.