# Butterworth filter

A Butterworth filter is an electronic or digital signal filter whose magnitude response is maximally flat in the passband: it has no ripple, falls off steadily beyond the cutoff frequency, and attenuates exactly 3 dB at the cutoff. It is one of the most commonly used frequency-domain filter designs and is also called a maximally flat filter.<sup>[1](https://www.sciencedirect.com/topics/engineering/butterworth-filter)</sup>

| Key fact | Detail |
|---|---|
| Origin | Stephen Butterworth, "On the Theory of Filter Amplifiers," *Experimental Wireless & the Wireless Engineer*, vol. 7, pp. 536–541, October 1930, written at the Admiralty Research Laboratory<sup>[2](https://bibbase.org/network/publication/butterworth-onthetheoryoffilteramplifiers-1930)</sup> |
| Magnitude response | \( \lvert H(j\omega) \rvert^2 = 1 / (1 + (\omega/\omega_c)^{2N}) \), with cutoff \( \omega_c \) and order \( N \)<sup>[3](https://people.eecs.ku.edu/~demarest/212/Butterworth%20Filters.pdf)</sup> |
| Cutoff behavior | Gain drops to \( 1/\sqrt{2} \) of the passband value (the −3 dB point) at the cutoff frequency, consistently for every order<sup>[4](https://docs.scipy.org/doc/scipy-1.13.0/reference/generated/scipy.signal.butter.html)</sup> |
| Roll-off | Attenuation of −20·N dB/decade: −20 dB/decade for first order, −40 for second, and so on<sup>[5](https://cpjobling.github.io/eg-247-textbook/filter_design/analogue_filters.html)</sup> |
| Pole structure | An all-pole transfer function: all \( N \) poles lie equally spaced on a circle in the left half of the s-plane, and all zeros sit at infinity<sup>[1](https://www.sciencedirect.com/topics/engineering/butterworth-filter)</sup><sup> • </sup><sup>[6](https://uu.diva-portal.org/smash/get/diva2%3A617728/FULLTEXT01.pdf)</sup> |
| Trade-off | Flatness costs steepness: Chebyshev and elliptic designs of the same order roll off faster, and Butterworth needs a higher order to meet a given magnitude specification<sup>[7](https://www.mathworks.com/help/signal/ref/butter.html)</sup><sup> • </sup><sup>[6](https://uu.diva-portal.org/smash/get/diva2%3A617728/FULLTEXT01.pdf)</sup> |
| Typical uses | Audio processing, ADC anti-aliasing, and general-purpose filtering where in-band amplitude variation must be minimized<sup>[8](https://rftools.io/compare/butterworth-vs-chebyshev/)</sup> |

## Origin

Stephen Butterworth (M.Sc.), working at the Admiralty Research Laboratory, published the design in October 1930 under the title "On the Theory of Filter Amplifiers" in *Experimental Wireless & the Wireless Engineer*.<sup>[2](https://bibbase.org/network/publication/butterworth-onthetheoryoffilteramplifiers-1930)</sup> His approach attacked filtering "from a new angle": instead of the orthodox wave-filter theory of the day, he combined simple filter units separated by vacuum-tube amplifier stages (valves), so that one amplifier provided both filtering and amplification. He noted that low-pass filter units could be built little larger than the anode resistance of an ordinary resistance-capacity amplifier, and described units whose resistances and inductances were wound on a cylinder 3 inches long and 1¼ inches in diameter with the capacitors inside the core.<sup>[9](https://people.eecs.ku.edu/~demarest/212/Buttorworth%20Orig.%20Paper.pdf)</sup> The paper was later reprinted in the Benchmark Papers in Electrical Engineering and Computer Science series edited by M. E. Van Valkenburg (1974).<sup>[10](https://onlinelibrary.wiley.com/doi/10.1002/047134608X.W2215)</sup>

## Frequency response

The defining property is maximal flatness at zero frequency: the magnitude function is flat at the origin and decreases monotonically in both the passband and the stopband, with no ripple anywhere.<sup>[6](https://uu.diva-portal.org/smash/get/diva2%3A617728/FULLTEXT01.pdf)</sup> The magnitude-squared response of the normalized low-pass prototype is

\[ \lvert H(j\omega) \rvert^2 = \frac{1}{1 + (\omega/\omega_c)^{2N}}, \]

where \( N \) is the filter order and \( \omega_c \) the cutoff frequency.<sup>[3](https://people.eecs.ku.edu/~demarest/212/Butterworth%20Filters.pdf)</sup> At \( \omega = \omega_c \) the gain is \( 1/\sqrt{2} \), the −3 dB point, regardless of order.<sup>[4](https://docs.scipy.org/doc/scipy-1.13.0/reference/generated/scipy.signal.butter.html)</sup> Well into the stopband the response approaches \( (\omega_c/\omega)^{2N} \), giving an attenuation slope of −20·N dB per decade; a plot of attenuation curves shows −20 dB/decade for \( N = 1 \), −40 dB/decade for \( N = 2 \), and so on, with the transition sharpening as the order grows.<sup>[5](https://cpjobling.github.io/eg-247-textbook/filter_design/analogue_filters.html)</sup>

**Poles and the Butterworth polynomial.** The denominator of the transfer function is an all-pole (no finite zeros) polynomial whose \( N \) poles are the left-half-plane roots of \( 1 + (-1)^N s^{2N} = 0 \), spaced equally around a circle of radius \( \omega_c \).<sup>[1](https://www.sciencedirect.com/topics/engineering/butterworth-filter)</sup><sup> • </sup><sup>[3](https://people.eecs.ku.edu/~demarest/212/Butterworth%20Filters.pdf)</sup> The polynomials formed from these roots are called Butterworth polynomials, and all but one pole (when \( N \) is odd) occur as complex-conjugate pairs.<sup>[3](https://people.eecs.ku.edu/~demarest/212/Butterworth%20Filters.pdf)</sup>

## Comparison with other filter approximations

Practical filters cannot realize the ideal brick-wall response, so designers choose among standard approximations. The usual alternatives are Chebyshev, inverse Chebyshev (Chebyshev Type II), elliptic, and Bessel.<sup>[11](https://www.youtube.com/watch?v=JXcbbBMrLYE)</sup>

- **Chebyshev** accepts ripple in the passband (Type I) or stopband (Type II) in exchange for a steeper roll-off than Butterworth of the same order; its stopband attenuation increases with the allowed passband ripple.<sup>[12](https://ez.analog.com/precision-technology-signal-chains/precision-wide-bandwidth-signal-chains/a/fast-precision-faqs/FP746/how-can-i-decide-what-filter-approximation-to-use-chebyshev-butterworth-or-bessel-etc)</sup> Passband ripple makes plain Chebyshev unsuitable for audio systems.<sup>[13](https://www.analog.com/en/resources/technical-articles/analog-filter-design-demystified.html)</sup>
- **Elliptic** filters ripple in both bands and have the fastest roll-off of the common approximations.<sup>[11](https://www.youtube.com/watch?v=JXcbbBMrLYE)</sup>
- **Bessel** filters have a flat passband like Butterworth but the slowest roll-off; their constant group delay (time delay a filter applies across frequencies) gives minimal waveform distortion for pulses and square waves, with lower overshoot and shorter settling time than a Butterworth of equal order.<sup>[12](https://ez.analog.com/precision-technology-signal-chains/precision-wide-bandwidth-signal-chains/a/fast-precision-faqs/FP746/how-can-i-decide-what-filter-approximation-to-use-chebyshev-butterworth-or-bessel-etc)</sup><sup> • </sup><sup>[13](https://www.analog.com/en/resources/technical-articles/analog-filter-design-demystified.html)</sup>

In [Texas Instruments](https://www.edgepedia.org/texas-instruments)' summary table, the Butterworth sits between these extremes: maximally flat passband magnitude, a transition steeper than Bessel but not as good as Chebyshev, and a step response with some overshoot and ringing, less than the Chebyshev's.<sup>[14](https://www.ti.com/lit/an/slyt550/slyt550.pdf)</sup> Analog Devices describes it as a good "all rounder" that is simple to understand and suitable for applications such as audio processing.<sup>[13](https://www.analog.com/en/resources/technical-articles/analog-filter-design-demystified.html)</sup> The cost of flatness is order: meeting a given magnitude specification requires a larger filter order than the other approximations.<sup>[6](https://uu.diva-portal.org/smash/get/diva2%3A617728/FULLTEXT01.pdf)</sup>

## Phase and time-domain behavior

Butterworth's magnitude flatness does not extend to delay. Its group delay is not constant across the passband, and the phase response is nonlinear.<sup>[15](https://www.riverpublishers.com/pdf/ebook/RP_E9788792982032.pdf)</sup> A study of third-order Butterworth group delay illustrates the variation that band-shaping filters of this type show; all-pass filters can be added to correct such group-delay variations.<sup>[16](https://uodiyala.edu.iq/uploads/PDF%20ELIBRARY%20UODIYALA/EL46/Analog_and_Digital_Filter_Design_2E.pdf)</sup> In step-response terms the filter is mildly underdamped: a three-pole digital Butterworth showed a maximum low-pass overshoot of 8 percent with a damping factor of 0.5.<sup>[17](https://pubmed.ncbi.nlm.nih.gov/11163409/)</sup>

One practical remedy for phase distortion is zero-phase filtering: running a Butterworth low- or high-pass filter forward and then backward through the data eliminates phase shifts, at the cost of doubling the roll-off slope. An EEG processing pipeline using a second-order non-causal IIR Butterworth this way obtains a −12 dB/octave roll-off per pass and keeps local maxima at their expected positions.<sup>[18](https://www.frontiersin.org/journals/neurology/articles/10.3389/fneur.2019.00325/full)</sup>

## Digital implementation

Digital Butterworth filters are usually derived from the analog prototype. The standard route designs the low-pass analog prototype with a normalized 1 rad/s cutoff, applies frequency transformations for high-pass, band-pass, or band-stop versions, and converts to discrete time with the bilinear transform \( s \rightarrow \frac{2}{T} \cdot \frac{z-1}{z+1} \), with frequency prewarping so the digital cutoff matches the specified one.<sup>[19](https://docs.scipy.org/doc/scipy/tutorial/signal.html)</sup><sup> • </sup><sup>[7](https://www.mathworks.com/help/signal/ref/butter.html)</sup>

**Numerical care.** Converting poles and zeros into transfer-function polynomial coefficients is numerically sensitive even for orders as low as \( N \geq 4 \), and high-order or narrowband designs in that form can become unstable or incorrect through floating-point error. Toolkits therefore recommend the second-order sections (SOS) representation, a cascade of biquad sections that realizes the higher-order filter with minimal numerical error; SciPy's `butter` function, for example, defaults to returning coefficients but recommends `output='sos'` for general-purpose filtering.<sup>[4](https://docs.scipy.org/doc/scipy-1.13.0/reference/generated/scipy.signal.butter.html)</sup> MATLAB's `butter` implements the same five-step pipeline from prototype poles through bilinear conversion.<sup>[7](https://www.mathworks.com/help/signal/ref/butter.html)</sup>

A published example for electroencephalography derives closed-form coefficients for three-pole (order 3) recursive low-pass and high-pass Butterworth filters of arbitrary cutoff, designed by bilinear transformation; the resulting digital filter attenuates about 60 dB/decade, loses 3 dB at cutoff, and filters 60,000 samples in roughly 50 ms on the era's desktop hardware, fast enough for on-line EEG use.<sup>[17](https://pubmed.ncbi.nlm.nih.gov/11163409/)</sup>

## Applications

Butterworth filters are used where in-band amplitude accuracy matters more than the steepest possible cutoff: audio signal chains, anti-aliasing filters ahead of analog-to-digital converters, and general signal conditioning.<sup>[8](https://rftools.io/compare/butterworth-vs-chebyshev/)</sup> In biomedical signal processing, Butterworth designs appear alongside Chebyshev and elliptic filters for speech, ECG, EEG, and EMG analysis, and bidirectional 40–250 Hz band-pass Butterworth filters have been compared against other designs for detecting ventricular late potentials in ECG.<sup>[20](https://arxiv.org/abs/2002.03130)</sup><sup> • </sup><sup>[21](https://pubmed.ncbi.nlm.nih.gov/12426627/)</sup>

## References

1. [Butterworth filter, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/engineering/butterworth-filter)
2. [S. Butterworth (1930), "On the Theory of Filter Amplifiers," bibliographic record](https://bibbase.org/network/publication/butterworth-onthetheoryoffilteramplifiers-1930)
3. [Butterworth Filters (course notes), University of Kansas EECS](https://people.eecs.ku.edu/~demarest/212/Butterworth%20Filters.pdf)
4. [scipy.signal.butter, SciPy v1.13.0 Manual](https://docs.scipy.org/doc/scipy-1.13.0/reference/generated/scipy.signal.butter.html)
5. [Unit 7.1: Designing Analogue Filters, EG-247 Digital Signal Processing](https://cpjobling.github.io/eg-247-textbook/filter_design/analogue_filters.html)
6. [Filter approximation thesis, Uppsala University (DiVA portal)](https://uu.diva-portal.org/smash/get/diva2%3A617728/FULLTEXT01.pdf)
7. [butter – Butterworth filter design, MATLAB Signal Processing Toolbox](https://www.mathworks.com/help/signal/ref/butter.html)
8. [Butterworth vs Chebyshev Filter: Which to Use, rftools.io](https://rftools.io/compare/butterworth-vs-chebyshev/)
9. [S. Butterworth (1930), original paper scan, University of Kansas EECS](https://people.eecs.ku.edu/~demarest/212/Buttorworth%20Orig.%20Paper.pdf)
10. [Butterworth Filters, Wiley Encyclopedia of Electrical and Electronics Engineering (Calôba)](https://onlinelibrary.wiley.com/doi/10.1002/047134608X.W2215)
11. [Filter Approximations: Butterworth, Chebyshev, Elliptic and Bessel (video description), All About Electronics](https://www.youtube.com/watch?v=JXcbbBMrLYE)
12. [How can I decide what filter approximation to use, Analog Devices EngineerZone](https://ez.analog.com/precision-technology-signal-chains/precision-wide-bandwidth-signal-chains/a/fast-precision-faqs/FP746/how-can-i-decide-what-filter-approximation-to-use-chebyshev-butterworth-or-bessel-etc)
13. [Analog Filter Design Demystified, Analog Devices](https://www.analog.com/en/resources/technical-articles/analog-filter-design-demystified.html)
14. [Getting started with WEBENCH Filter Designer, Texas Instruments Analog Applications Journal, 4Q 2013](https://www.ti.com/lit/an/slyt550/slyt550.pdf)
15. [An Introduction to Digital Signal Processing: A Focus on Implementation, River Publishers](https://www.riverpublishers.com/pdf/ebook/RP_E9788792982032.pdf)
16. [Analog and Digital Filter Design, Second Edition (chapter PDF), University of Diyala library](https://uodiyala.edu.iq/uploads/PDF%20ELIBRARY%20UODIYALA/EL46/Analog_and_Digital_Filter_Design_2E.pdf)
17. [G. Alarcon, C. N. Guy, C. D. Binnie (2000), "A simple algorithm for a digital three-pole Butterworth filter of arbitrary cut-off frequency," J. Neurosci. Methods 104(1):35–44](https://pubmed.ncbi.nlm.nih.gov/11163409/)
18. [EEG Source Imaging: A Practical Review of the Analysis Steps, Frontiers in Neurology (2019)](https://www.frontiersin.org/journals/neurology/articles/10.3389/fneur.2019.00325/full)
19. [Signal Processing (scipy.signal) tutorial, SciPy v1.18.0 Manual](https://docs.scipy.org/doc/scipy/tutorial/signal.html)
20. [Design and Implementation of Butterworth, Chebyshev-I and Elliptic Filter for Speech Signal Analysis, arXiv:2002.03130](https://arxiv.org/abs/2002.03130)
21. [Clinical assessment of the effect of digital filtering on the detection of ventricular late potentials, PubMed](https://pubmed.ncbi.nlm.nih.gov/12426627/)

---
*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Algorithms and computational methods › Numerical, string, and geometric algorithms › Fourier and signal transforms*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license