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 "title": "Notch filter",
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 "excerpt": "A notch filter is a band-stop filter with a very narrow stopband that rejects a narrow frequency band, such as 50 or 60 Hz mains hum.",
 "snippet": "A notch filter is a band-stop filter with a very narrow stopband that rejects a narrow frequency band, such as 50 or 60 Hz mains hum.",
 "node": "technology.computing.ai.numerical_string_geometry",
 "markdown": "# Notch filter\n\nA notch filter is a band-stop filter with a very narrow stopband and two passbands: it rejects a narrow frequency band, such as the 50 or 60 Hz mains component, and leaves the rest of the spectrum little changed.<sup>[1](https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.iirnotch.html)</sup><sup> • </sup><sup>[2](https://www.ias.ac.in/article/fulltext/sadh/015/03/0133-0155)</sup> The interfering component within the notch is strongly attenuated while the rest of the spectrum is little changed; the price is phase distortion near the notch<sup>[3](http://facta.junis.ni.ac.rs/eae/fu2k13/fu01.pdf)</sup> and possible ringing at the start of the filtered signal.<sup>[4](https://arxiv.org/html/2406.10706)</sup>\n\n| Key fact | Value |\n|---|---|\n| Definition | Band-stop filter with very narrow stopband and two passbands<sup>[2](https://www.ias.ac.in/article/fulltext/sadh/015/03/0133-0155)</sup> |\n| Quality factor | \\( Q = \\omega_0 / \\mathrm{BW} \\), center frequency divided by the −3 dB bandwidth<sup>[1](https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.iirnotch.html)</sup> |\n| Canonical digital form | Second-order IIR biquad: zeros at \\( e^{j\\omega_0} \\), poles at \\( r \\cdot e^{j\\omega_0} \\)<sup>[3](http://facta.junis.ni.ac.rs/eae/fu2k13/fu01.pdf)</sup> |\n| Two-parameter result | A second-order digital notch is fully characterized by notch frequency and 3-dB rejection bandwidth, realizable with two multipliers<sup>[5](https://doi.org/10.1109/tcs.1974.1083889)</sup> |\n| Practical analog depth | About 40–50 dB for active twin-T/Fliege notches; up to below −100 dB for a sixth-order Bainter topology<sup>[6](https://www.ti.com/lit/an/slyt235/slyt235.pdf)</sup><sup> • </sup><sup>[7](https://www.ti.com/lit/an/slyt613/slyt613.pdf)</sup> |\n| Main trade-off | Higher selectivity (higher \\( Q \\), poles nearer the unit circle) lengthens transients and raises round-off noise<sup>[8](https://www.sciencedirect.com/science/article/abs/pii/S0263224112001133)</sup><sup> • </sup><sup>[3](http://facta.junis.ni.ac.rs/eae/fu2k13/fu01.pdf)</sup> |\n| Typical uses | 50/60 Hz hum and harmonics in ECG and biomedical signals, grid PLLs, EMI reduction in motor inverters<sup>[9](https://link.springer.com/article/10.1186/s13634-015-0210-5)</sup><sup> • </sup><sup>[10](https://www.sciencedirect.com/science/article/abs/pii/S0378779617300664)</sup> |\n\n## How it works\n\nThe notch is produced by a pair of transmission zeros placed on the frequency of the interference. In the standard second-order digital prototype, a pair of complex conjugate zeros sits at \\( e^{j\\omega_0} \\) on the unit circle, while a pair of conjugate poles at \\( r \\cdot e^{j\\omega_0} \\), with \\( r < 1 \\) but close to it, keeps the passband response near unity and sets the notch width: increasing the pole radius \\( r \\) narrows the −3 dB bandwidth.<sup>[3](http://facta.junis.ni.ac.rs/eae/fu2k13/fu01.pdf)</sup> The quality factor \\( Q = \\omega_0 / \\mathrm{BW} \\) characterizes the −3 dB bandwidth relative to the center frequency.<sup>[1](https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.iirnotch.html)</sup>\n\nIn a biquad view, \\( H(z) = g \\cdot (1 + \\beta_1 z^{-1} + \\beta_2 z^{-2})/(1 + a_1 z^{-1} + a_2 z^{-2}) \\); the zero angle is the antiresonance frequency and the zero radius affects the depth and width of the notch, with the pole radius \\( R \\) determining \\( Q \\).<sup>[11](https://ccrma.stanford.edu/~jos/fp2/BiQuad_Section.html)</sup> An equivalent analog formulation writes a stable second-order notch as \\( H(s) = 1/[1 + A(s)] \\), where \\( A(s) \\) is an allpass filter whose phase decreases from 0 to −2π; bilinear transformation gives the digital form \\( G_1(z) = 1/[1 + A(z)] \\).<sup>[12](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2006/papers/1568980195.pdf)</sup>\n\n## How it is done\n\n**Digital IIR design.** The second-order prototype is specified by the normalized notch frequency \\( \\omega_0 \\) and the bandwidth; MATLAB's `iirnotch` takes \\( w_0 \\) and \\( bw \\) with \\( 0 < w_0 < 1 \\), where 1.0 corresponds to π radians per sample, and returns numerator and denominator coefficients, with an optional argument to set the bandwidth at a level other than −3 dB.<sup>[13](https://www.mathworks.com/help/dsp/ref/iirnotch.html)</sup> The direct-form I biquad computes \\( y(n) = v(n) + \\beta_1 v(n-1) + \\beta_2 v(n-2) - a_1 y(n-1) - a_2 y(n-2) \\).<sup>[11](https://ccrma.stanford.edu/~jos/fp2/BiQuad_Section.html)</sup> Hirano, Nishimura, and Mitra showed that such a filter is uniquely characterized by the notch frequency and the 3-dB rejection bandwidth, so it can be realized with only two multipliers, with design procedures for prescribed values of both parameters.<sup>[5](https://doi.org/10.1109/tcs.1974.1083889)</sup> Allpass-based designs using Butterworth polynomials give maximally flat magnitude at \\( \\omega = 0 \\) and \\( \\omega = \\pi \\), while Chebyshev and Elliptic polynomials give equiripple passbands.<sup>[12](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2006/papers/1568980195.pdf)</sup>\n\n**Digital FIR design.** Linear-phase FIR notch filters must have odd length \\( N = 2M+1 \\); a prefilter \\( n_p(z) = (1 - 2\\cos\\omega_0 z^{-1} + z^{-2})^r \\) places \\( r \\) zeros at the notch frequency.<sup>[2](https://www.ias.ac.in/article/fulltext/sadh/015/03/0133-0155)</sup>\n\n**Analog realizations.** The twin-T network implements a notch with a single op amp but is hard to tune, requiring six high-precision components; mismatch in the \\( R_0 \\) resistors quickly erodes notch depth to less than 10 dB.<sup>[6](https://www.ti.com/lit/an/slyt235/slyt235.pdf)</sup> The Fliege topology needs only four precision components, two identical resistors and two identical capacitors, tolerates slight mismatches without eroding depth, and allows independent adjustment of \\( Q \\) and center frequency.<sup>[6](https://www.ti.com/lit/an/slyt235/slyt235.pdf)</sup> Among higher-order bandstop topologies, a sixth-order Sallen-Key reached only about −15 dB in the notch and an MFB version about −36.6 dB, while a sixth-order Bainter filter reached less than −100 dB; the Bainter notch \\( Q \\) depends on amplifier gains rather than component matching, so depth is insensitive to temperature drift or aging.<sup>[7](https://www.ti.com/lit/an/slyt613/slyt613.pdf)</sup>\n\n## Origin\n\nEarly digital work includes R. Carney's 1963 paper on digital notch filter design with tracking requirements in *IEEE Transactions on Space Electronics and Telemetry*<sup>[14](https://doi.org/10.1109/tset.1963.4337624)</sup> and A.G. Constantinides's 1969 *Electronics Letters* note \"Digital notch filters\".<sup>[15](https://doi.org/10.1049/el:19690150)</sup> The two-multiplier second-order design was established in a paper in *IEEE Transactions on Circuits and Systems* (21(4):540–546).<sup>[5](https://doi.org/10.1109/tcs.1974.1083889)</sup> Yu, Mitra, and Babic note that IIR notch design was considered by Carney and later by Hirano and colleagues, while FIR notch design had received little attention before their 1990 prefilter-equalizer method in *Sadhana*.<sup>[2](https://www.ias.ac.in/article/fulltext/sadh/015/03/0133-0155)</sup><sup> • </sup><sup>[16](https://doi.org/10.1007/bf02812034)</sup> Later design lines include Joshi and Dutta Roy's 1998 different-passband-gain IIR design<sup>[17](https://doi.org/10.1049/ip-vis:19981687)</sup>, Tseng and Pei's 2001 stable design with optimal pole placement<sup>[18](https://doi.org/10.1109/78.960414)</sup>, and Pei and Tseng's 1997 allpass-based multiple-notch design.<sup>[19](https://doi.org/10.1109/82.554450)</sup> Adaptive lines trace to Nehorai's 1985 minimal-parameter [adaptive notch filter](https://www.edgepedia.org/adaptive-notch-filter)<sup>[20](https://doi.org/10.1109/tassp.1985.1164643)</sup>, Ferdjallah and Barr's 1994 unit-circle CLMS design for powerline noise in biomedical signals<sup>[21](https://doi.org/10.1109/10.293240)</sup>, and Li, Takahashi, and Takebe's 1993 variable-\\( Q \\) adaptive algorithm.<sup>[22](https://doi.org/10.1002/ecjc.4430760802)</sup> Transient-suppression variants include Piskorowski's 2009 digital \\( Q \\)-varying notch<sup>[23](https://doi.org/10.1109/tim.2009.2026605)</sup> and Tan, Jiang, and Wang's 2012 pole-radius-varying notch<sup>[24](https://doi.org/10.1109/tim.2012.2184013)</sup>, the latter building on Fettweis's 1986 wave digital filter framework in the *Proceedings of the IEEE*.<sup>[25](https://doi.org/10.1109/proc.1986.13458)</sup>\n\n## Variants\n\n**Adaptive notch filters (ANFs)** track a moving interference frequency. The DEESHA biquad has transfer function \\( H_{N,I}(Z) = [1 - (1-r^2)p \\cdot Z^{-1} + z^{-2}]/[1 - pZ^{-1} + r^2 z^{-2}] \\), where \\( p = (1+r^2)\\cos(\\Omega_N) \\) is adapted to acquire the notch frequency and \\( r \\) controls the bandwidth; it is a constant-bandwidth design giving bias-free convergence in white noise.<sup>[26](https://www.ias.ac.in/article/fulltext/sadh/023/01/0073-0082)</sup> The Rao-Kung ANF and Nehorai's filter become identical when \\( a_2 = 1 \\), and the Cho-Choi-Lee lattice ANF is a reformulation of DEESHA.<sup>[26](https://www.ias.ac.in/article/fulltext/sadh/023/01/0073-0082)</sup> Li's 1993 algorithm varies the \\( Q \\)-factor inversely with the distance between the filter center frequency and the sinusoid being tracked.<sup>[22](https://doi.org/10.1002/ecjc.4430760802)</sup>\n\n**Multiple and cascaded notches.** Cascading second-order single-notch sections is restricted to few notch frequencies and very narrow bandwidths; Pei and Tseng's allpass method transforms multiple-notch specifications into an equivalent allpass filter requiring \\( \\theta_A(\\omega_{Ni}) = -(2i-1)\\pi \\).<sup>[9](https://link.springer.com/article/10.1186/s13634-015-0210-5)</sup> Time-varying designs use a time-varying pole radius \\( r(n) \\) to shorten transients without degrading long-term selectivity.<sup>[8](https://www.sciencedirect.com/science/article/abs/pii/S0263224112001133)</sup>\n\n**In control loops and PLLs.** Notch-based PLLs for grid connection use a hybrid of one adaptive notch (for unbalance) plus two non-adaptive notches (for harmonics) to obtain fast dynamics and high harmonic filtering with a simple PLL.<sup>[10](https://www.sciencedirect.com/science/article/abs/pii/S0378779617300664)</sup>\n\n## Applications\n\nMultiple-notch filters are the standard tool against powerline hum and its harmonics in low-voltage biomedical measurements.<sup>[8](https://www.sciencedirect.com/science/article/abs/pii/S0263224112001133)</sup> A documented ECG example sampled at 720 Hz uses notches at \\( [0.2778\\pi, 0.5556\\pi, 0.8333\\pi] \\) with \\( 0.01\\pi \\) bandwidths to remove the powerline fundamental and harmonics.<sup>[9](https://link.springer.com/article/10.1186/s13634-015-0210-5)</sup> In power electronics, adaptive notch digital active EMI filters reduce conducted emissions in motor inverters.<sup>[27](https://doi.org/10.1109/emceurope57790.2023.10274230)</sup>\n\n## Limitations and alternatives\n\n**Depth and tuning.** Real-world active notch depth is limited to about 40–50 dB even when simulations show far more; tuning over ±1% of center frequency gives 100:1 rejection, but over ±10% only 10:1.<sup>[6](https://www.ti.com/lit/an/slyt235/slyt235.pdf)</sup>\n\n**Drift and stability.** In PLL applications, grid frequency variations significantly reduce a fixed notch's rejection capability, which motivates adaptive notch filters; ANFs reject harmonics regardless of grid drift but at considerable implementation complexity and computational cost.<sup>[10](https://www.sciencedirect.com/science/article/abs/pii/S0378779617300664)</sup> For IIR designs, if \\( r \\) is too close to unity, round-off noise, proportional to \\( 1/\\varepsilon^2 \\) with \\( \\varepsilon = 1 - r \\), becomes very large, and finite-wordlength implementations introduce [limit cycles](https://www.edgepedia.org/limit-cycle); IIR filters are potentially unstable and lack linear phase, while FIR filters are unconditionally stable and can give exact linear phase.<sup>[3](http://facta.junis.ni.ac.rs/eae/fu2k13/fu01.pdf)</sup> In cascaded adaptive cells, a second sinusoidal component biases the detected frequency, and the closer the frequencies, the higher the bias.<sup>[28](https://ace.ucv.ro/sintes11/Volume2/5%20ELECTRONICS/E04%20-%20CIOCHINA%20Silviu.pdf)</sup>\n\n**Selectivity versus transients.** Sharper notches, from more FIR coefficients or IIR poles closer to the unit circle, increase the duration of the transient response, so selectivity and short transients are contradictory design goals.<sup>[8](https://www.sciencedirect.com/science/article/abs/pii/S0263224112001133)</sup> At fixed order, higher \\( Q \\) is achieved by pushing poles closer to the zeros, but stability limits this, so improving the brickwall approximation requires increasing the order.<sup>[29](https://www.mathworks.com/help/dsp/ug/design-of-peaking-and-notching-filters.html)</sup>\n\n**Alternatives.** At the same filter order, IIR notch filters have a narrower stopband and higher quality factor than FIR, while FIR offers linear phase and stability.<sup>[9](https://link.springer.com/article/10.1186/s13634-015-0210-5)</sup> A comb filter is a type of notch filter in which nulls occur periodically across the frequency band, used for rejection of powerline harmonics.<sup>[30](https://inspirajournals.com/uploads/Issues/1138111298.pdf)</sup>\n\n**Tooling.** MATLAB's `iirnotch`, introduced in R2011a, was deprecated in R2024a in favor of `designNotchPeakIIR`.<sup>[13](https://www.mathworks.com/help/dsp/ref/iirnotch.html)</sup>\n\n## References\n\n1. [iirnotch, SciPy v1.18.0 Manual](https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.iirnotch.html)\n2. [Design of linear phase FIR notch filters (Yu, Mitra, Babic), Sadhana, 1990](https://www.ias.ac.in/article/fulltext/sadh/015/03/0133-0155)\n3. [FIR notch filter design, a review (Dutta Roy, Kumar, Jain), Facta Universitatis, 2013](http://facta.junis.ni.ac.rs/eae/fu2k13/fu01.pdf)\n4. [Notch Filters without Transient Effects: A Constrained Optimization Design (arXiv, 2024)](https://arxiv.org/html/2406.10706)\n5. [Design of digital notch filters (Hirano, Nishimura, Mitra), IEEE Transactions on Circuits and Systems, 1974](https://doi.org/10.1109/tcs.1974.1083889)\n6. [High-speed notch filters (Texas Instruments application report slyt235)](https://www.ti.com/lit/an/slyt235/slyt235.pdf)\n7. [Bandstop filters and the Bainter topology (TI, Bonnie C. Baker)](https://www.ti.com/lit/an/slyt613/slyt613.pdf)\n8. [Suppressing harmonic powerline interference using multiple-notch filtering methods with improved transient behavior (Measurement, 2012)](https://www.sciencedirect.com/science/article/abs/pii/S0263224112001133)\n9. [A generalized design framework for IIR digital multiple notch filters, EURASIP Journal on Advances in Signal Processing, 2015](https://link.springer.com/article/10.1186/s13634-015-0210-5)\n10. [Analysis and design of notch filter-based PLLs for grid-connected applications (Electric Power Systems Research)](https://www.sciencedirect.com/science/article/abs/pii/S0378779617300664)\n11. [The BiQuad Section (Stanford CCRMA, Julius O. Smith)](https://ccrma.stanford.edu/~jos/fp2/BiQuad_Section.html)\n12. [Design of IIR Notch Filters with Maximally Flat or Equiripple Magnitude Characteristics (EUSIPCO 2006)](https://www.eurasip.org/Proceedings/Eusipco/Eusipco2006/papers/1568980195.pdf)\n13. [iirnotch - Second-order IIR notch filter - MATLAB](https://www.mathworks.com/help/dsp/ref/iirnotch.html)\n14. [R. Carney (1963). Design of a Digital Notch Filter with Tracking Requirements. IEEE Transactions on Space Electronics and Telemetry.](https://doi.org/10.1109/tset.1963.4337624)\n15. [A.G. Constantinides (1969). Digital notch filters. Electronics Letters.](https://doi.org/10.1049/el:19690150)\n16. [Tian-Hu Yu, Sanjit K Mitra, Hrvoje Babic (1990). Design of linear phase FIR notch filters. Sadhana.](https://doi.org/10.1007/bf02812034)\n17. [Y.V. Joshi, S.C. Dutta Roy (1998). Design of IIR notch filters with different passband gains. IEE Proceedings - Vision Image and Signal Processing.](https://doi.org/10.1049/ip-vis:19981687)\n18. [Chien-Cheng Tseng, Soo-Chang Pei (2001). Stable IIR notch filter design with optimal pole placement. IEEE Transactions on Signal Processing.](https://doi.org/10.1109/78.960414)\n19. [Soo-Chang Pei, Chien-Cheng Tseng (1997). IIR multiple notch filter design based on allpass filter. IEEE Transactions on Circuits and Systems II Analog and Digital Signal Processing.](https://doi.org/10.1109/82.554450)\n20. [A. Nehorai (1985). A minimal parameter adaptive notch filter with constrained poles and zeros. IEEE Transactions on Acoustics Speech and Signal Processing.](https://doi.org/10.1109/tassp.1985.1164643)\n21. [M. Ferdjallah, R.E. Barr (1994). Adaptive digital notch filter design on the unit circle for the removal of powerline noise from biomedical signals. IEEE Transactions on Biomedical Engineering.](https://doi.org/10.1109/10.293240)\n22. [Shang Li, Nobuaki Takahashi, Tsuyoshi Takebe (1993). Fast stabilized adaptive algorithm for IIR bandpass/notch filters for a single sinusoid detection. Electronics and Communications in Japan (Part III Fundamental Electronic Science).](https://doi.org/10.1002/ecjc.4430760802)\n23. [J. Piskorowski (2009). Digital $Q$-Varying Notch IIR Filter With Transient Suppression. IEEE Transactions on Instrumentation and Measurement.](https://doi.org/10.1109/tim.2009.2026605)\n24. [Li Tan, Jean Jiang, Liangmo Wang (2012). Pole-Radius-Varying IIR Notch Filter With Transient Suppression. IEEE Transactions on Instrumentation and Measurement.](https://doi.org/10.1109/tim.2012.2184013)\n25. [A. Fettweis (1986). Wave digital filters: Theory and practice. Proceedings of the IEEE.](https://doi.org/10.1109/proc.1986.13458)\n26. [Review of adaptive notch filters and line enhancers (Sādhanā, 1998)](https://www.ias.ac.in/article/fulltext/sadh/023/01/0073-0082)\n27. [Machine-Learning-Based Parameterization of Adaptive Notch Filters for CM Noise Reduction in Motor Inverters (2023, TU Dortmund)](https://doi.org/10.1109/emceurope57790.2023.10274230)\n28. [Adaptive Notch IIR Filters: An Implementation on Motorola SC140 (Ciochină, Ciochină, Roman)](https://ace.ucv.ro/sintes11/Volume2/5%20ELECTRONICS/E04%20-%20CIOCHINA%20Silviu.pdf)\n29. [Design Peak and Notch Filters - MATLAB](https://www.mathworks.com/help/dsp/ug/design-of-peaking-and-notching-filters.html)\n30. [Notch and Comb Filter Design: Analysis and Review (Inspira JMME)](https://inspirajournals.com/uploads/Issues/1138111298.pdf)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Algorithms and computational methods › Numerical, string, and geometric algorithms*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Oct 15, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n",
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 "credit_md": "\"[Notch filter](https://www.edgepedia.org/notch-filter)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/notch-filter](https://www.edgepedia.org/notch-filter). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).",
 "credit_html": "\"<a href=\"https://www.edgepedia.org/notch-filter\">Notch filter</a>\", Edgepedia (EdgeChat), <a href=\"https://www.edgepedia.org/notch-filter\">https://www.edgepedia.org/notch-filter</a>. <a href=\"https://www.edgepedia.org/edgepedia/license\">Edgepedia Community License 1.0</a>.",
 "speakable": "A notch filter is a band-stop filter with a very narrow stopband that rejects a narrow frequency band, such as 50 or 60 Hz mains hum."
}