Meta tags:

Headings (most frequently used words):

topology, butterworth, filter, implementation, cauer, contents, original, paper, overview, example, transfer, function, and, design, comparison, with, other, linear, filters, see, also, references, normalized, polynomials, maximal, flatness, high, frequency, roll, off, minimum, order, nonstandard, cutoff, attenuation, sallen, key, digital, proof, of, component, formulae, for,

Text of the page (most frequently used words):
the (242), displaystyle (149), #filter (118), and (73), butterworth (66), omega (60), frac (44), with (42), #filters (41), frequency (38), order (37), for (29), are (27), pass (24), that (24), edit (21), gain (21), right (21), function (21), left (18), response (17), topology (16), low (15), cutoff (15), design (14), transfer (14), log (14), this (13), poles (13), where (13), qquad (13), sqrt (13), pole (12), cos (12), sin (12), all (11), cauer (11), polynomials (11), may (10), linear (10), off (10), given (10), each (10), can (10), text (9), network (9), type (9), band (9), also (9), other (9), passband (9), component (9), will (9), attenuation (9), have (9), capacitor (8), chebyshev (8), but (8), third (8), odd (8), sallen (8), key (8), series (8), form (8), inductor (8), wikipedia (7), was (7), from (7), paper (7), impedance (7), image (7), these (7), such (7), values (7), shown (7), ldots (7), even (7), used (7), number (7), alpha (7), normalized (7), using (6), page (6), active (6), theory (6), analog (6), elliptic (6), delay (6), which (6), stopband (6), higher (6), first (6), proof (6), circle (6), high (6), prod (6), complex (6), plot (6), per (6), terms (5), isbn (5), different (5), synthesis (5), equal (5), networks (5), circuit (5), formulae (5), phase (5), roll (5), implementation (5), only (5), second (5), real (5), must (5), not (5), then (5), minimum (5), gamma (5), general (5), half (5), frequencies (5), obtained (5), his (5), toggle (4), table (4), contents (4), search (4), use (4), electronic (4), weinberg (4), 1951 (4), takahasi (4), ladder (4), original (4), signal (4), 1930 (4), amplifiers (4), see (4), ripple (4), both (4), digital (4), often (4), method (4), two (4), case (4), transform (4), one (4), passive (4), usually (4), capacitors (4), implement (4), voltage (4), element (4), desired (4), inductors (4), following (4), radius (4), through (4), stop (4), possible (4), flat (4), 2361 (4), coefficients (4), its (4), written (4), varphi (4), value (4), factors (4), negative (4), plane (4), magnitude (4), example (4), were (4), tube (4), constant (4), hide (4), move (4), sidebar (4), view (3), about (3), non (3), retrieved (3), single (3), huelsman (3), doi (3), realisation (3), lowpass (3), louis (3), processing (3), references (3), bessel (3), rolls (3), more (3), around (3), than (3), time (3), group (3), increasing (3), monotonic (3), orders (3), cascaded (3), polynomial (3), assuming (3), there (3), most (3), above (3), account (3), driven (3), start (3), current (3), unity (3), ideal (3), prototype (3), defined (3), equation (3), derived (3), elements (3), therefore (3), decade (3), octave (3), derivative (3), zero (3), maximal (3), flatness (3), ripples (3), maximally (3), has (3), product (3), phi (3), 414214s (3), 414214 (3), three (3), graph (3), simple (3), when (3), showed (3), could (3), coil (3), vacuum (3), tools (3), main (3), languages (2), contact (2), privacy (2), policy (2), available (2), apply (2), inc (2), categories (2), errors (2), date (2), short (2), description (2), wikidata (2), jurišić (2), george (2), sensitivity (2), amplifier (2), 1974 (2), valued (2), fourth (2), characteristic (2), bibcode (2), results (2), ire (2), orchard (2), give (2), published (2), analysis (2), help (2), cite (2), 279 (2), a001622 (2), oeis (2), golden (2), ratio (2), matthaei (2), mcgraw (2), hill (2), coupling (2), wireless (2), engineer (2), pdf (2), without (2), here (2), showing (2), common (2), types (2), considerable (2), comparison (2), matched (2), calculated (2), described (2), further (2), chosen (2), wish (2), denominator (2), quadratic (2), out (2), uses (2), components (2), resistors (2), implemented (2), stages (2), including (2), english (2), shunt (2), forms (2), terminated (2), source (2), load (2), unit (2), aligned (2), infty (2), factor (2), applied (2), lie (2), stated (2), top (2), nonstandard (2), lceil (2), cdot (2), rceil (2), bigr (2), required (2), follows (2), because (2), power (2), lim (2), slope (2), same (2), monotonically (2), respect (2), determine (2), 3925 (2), 4317 (2), 8021 (2), 8824 (2), 7588 (2), 5817 (2), 1634 (2), 9864 (2), 1258 (2), 1371 (2), 8462 (2), 4940 (2), 0978 (2), 5918 (2), 8637 (2), 4641 (2), 6131 (2), 4142 (2), four (2), decimal (2), places (2), formula (2), sum (2), 618033988749 (2), 618034s (2), 618034 (2), axis (2), laplace (2), overline (2), sigma (2), infinity (2), becomes (2), approximately (2), like (2), modified (2), placing (2), parallel (2), resonant (2), circuits (2), new (2), selected (2), resonate (2), old (2), space (2), plotted (2), angular (2), arg (2), figure (2), equations (2), towards (2), bode (2), overview (2), inches (2), separated (2), discovered (2), designed (2), should (2), rlc (2), composite (2), bridged (2), equaliser (2), lattice (2), zobel (2), linkwitz (2), riley (2), optimum (2), legendre (2), gaussian (2), appearance (2), upload (2), file (2), changes (2), links (2), history (2), read (2), article (2), subsection (2), create (2), donate (2), menu (2), add, topic, mobile, cookie, statement, statistics, developers, code, conduct, legal, safety, contacts, disclaimers, under, additional, site, you, agree, registered, trademark, profit, organization, wikimedia, foundation, creative, commons, attribution, sharealike, license, rendered, parsoid, last, edited, august, 2026, utc, hidden, cs1, articles, https, org, index, php, title, butterworth_filter, oldid, 1369387533, dražen, moschytz, mijat, neven, 2008, allpole, tables, 173, 159, automatika, december, realization, 1709, 1109, proc, 9689, proceedings, ieee, 1971, 3rd, 272, 1049, 19710185, 1971ell, 271h, 271, electronics, letters, slepian, paul, tchebycheff, 101, transactions, tchebysheff, inst, elec, commun, engrs, japan, 1957, explict, tschebyscheff, 1155, 1160, appl, phys, 1954, trans, william, bennett, transmission, march, 1932, 1849656, paarmann, larry, 2001, norwell, massachusetts, kluwer, academic, publishers, 117, 118, 7923, 7373, perspective, incompatibility, book, 1962, malabar, florida, robert, krieger, publishing, company, 1975, 496, 2022, 88275, 321, 2027, mdp, 39015000986086, hdl, 494, bosse, siebketten, ohne, dämpfungsschwankungen, durchlaßbereich, potenzketten, 284, 124123311, s2cid, 1515, freq, 1951freq, 279b, frequenz, weisstein, eric, 2020, mathworld, wolfram, com, young, leo, jones, 1964, 107, 105, 974, 64007937, lccn, 104, microwave, matching, structures, bianchi, giovanni, sorrentino, roberto, 2007, professional, 978, 149467, simulation, 541, 536, experimental, comb, slowly, discrete, next, fifth, largely, dependent, slightly, worsens, step, ringing, overshoot, quick, improves, amplitude, properties, implementations, based, methods, discretize, equivalent, sensitive, quantization, they, plus, section, biquad, sections, impulse, invariance, bilinear, amp, leaves, undefined, mean, noninverting, buffers, stage, implements, conjugate, pair, overall, cascading, separately, amps, quote, reference, issue, result, strongly, suggested, calculations, degree, does, constitue, mathematical, provides, rigorous, limiting, chebchev, historical, found, language, useful, multiplexers, diplexers, doubly, scaled, singly, usefully, combined, making, divided, immittance, having, realised, several, analogue, topologies, etc, lies, relocated, positioned, set, begin, times, leq, end, whereupon, continue, longer, algebraic, manipulation, defining, denotes, ceiling, proportional, square, rule, decibels, rightarrow, again, large, words, derivatives, resulting, requirement, limited, allowed, flatter, inverse, decreasing, since, always, positive, expansion, transformation, bandforms, any, cut, 2334, 6884, 1416, rounded, recursion, greek, letter, represents, solution, irrational, exact, 312869s, 907981s, 782013s, 975377s, 975377, 782013, 907981, 312869, 347296s, 532089s, 879385s, 879385, 532089, 347296, 390181s, 111140s, 662939s, 961571s, 961571, 662939, 111140, 390181, 445042s, 246980s, 801938s, 801938, 246980, 445042, 517638s, 931852s, 931852, 517638, 765367s, 847759s, 847759, 765367, six, multiplying, pairs, conjugates, setting, operator, sequence, textstyle, system, hence, expression, occur, equally, spaced, points, symmetric, stability, contains, specified, property, transforms, select, seen, approaches, rectangle, below, passed, while, suppressed, smaller, less, sharp, typical, into, placed, others, versions, note, rejected, interest, replacing, arranged, symmetrical, complete, density, absolute, shift, measure, distortion, introduced, differences, curve, either, taking, yield, device, looks, compared, slower, thus, require, particular, specification, achieve, viewed, logarithmic, slopes, linearly, decreases, changing, unlike, basic, functionality, diameter, length, plug, terminals, associated, contained, inside, wound, formed, part, plate, resistor, grid, solved, how, latter, enabling, construction, despite, losses, loss, core, materials, had, been, air, cored, audio, rather, lossy, adjust, compensate, winding, resistance, molypermalloy, radians, reactive, 7071, dealt, though, created, addition, output, built, respectively, radian, point, cannot, achieved, successively, closer, approximations, numbers, generated, substantial, choice, highly, interactive, whose, electrical, completely, reject, unwanted, uniform, wanted, amount, designer, experience, due, limitations, over, years, after, publication, referred, british, physicist, entitled, stephen, free, encyclopedia, item, projects, printable, version, download, print, export, switch, legacy, parser, get, shortened, url, information, permanent, link, related, what, actions, talk, українська, türkçe, svenska, српски, srpski, русский, português, polski, nederlands, 한국어, 日本語, italiano, français, فارسی, euskara, eesti, español, esperanto, deutsch, čeština, català, العربية, personal, special, pages, recent, community, portal, learn, contribute, random, events, navigation, jump, content,

Text of the page (random words):
with ω displaystyle omega unlike other filter types that have non monotonic ripple in the passband or the stopband compared with a chebyshev type i type ii filter or an elliptic filter the butterworth filter has a slower roll off and thus will require a higher order to implement a particular stopband specification but butterworth filters have a more linear phase response in the passband than chebyshev type i type ii and elliptic filters can achieve example edit a transfer function of a third order low pass butterworth filter design shown in the figure on the right looks like this v o s v i s r 4 s 3 l 1 c 2 l 3 s 2 l 1 c 2 r 4 s l 1 l 3 r 4 displaystyle frac v_ o s v_ i s frac r_ 4 s 3 l_ 1 c_ 2 l_ 3 s 2 l_ 1 c_ 2 r_ 4 s l_ 1 l_ 3 r_ 4 a third order low pass filter cauer topology the filter becomes a butterworth filter with cutoff frequency ω c displaystyle omega _ c 1 when for example c 2 displaystyle c_ 2 4 3 f r 4 displaystyle r_ 4 1 ω l 1 displaystyle l_ 1 3 2 h and l 3 displaystyle l_ 3 1 2 h a simple example of a butterworth filter is the third order low pass design shown in the figure on the right with c 2 displaystyle c_ 2 4 3 f r 4 displaystyle r_ 4 1 ω l 1 displaystyle l_ 1 3 2 h and l 3 displaystyle l_ 3 1 2 h 3 taking the impedance of the capacitors c displaystyle c to be 1 c s displaystyle 1 cs and the impedance of the inductors l displaystyle l to be l s displaystyle ls where s σ j ω displaystyle s sigma j omega is the complex frequency the circuit equations yield the transfer function for this device h s v o s v i s 1 1 2 s 2 s 2 s 3 displaystyle h s frac v_ o s v_ i s frac 1 1 2s 2s 2 s 3 the magnitude of the frequency response gain g ω displaystyle g omega is given by g ω h j ω 1 1 ω 6 displaystyle g omega h j omega frac 1 sqrt 1 omega 6 obtained from g 2 ω h j ω 2 h j ω h j ω 1 1 ω 6 displaystyle g 2 omega h j omega 2 h j omega cdot h j omega frac 1 1 omega 6 and the phase is given by φ ω arg h j ω displaystyle phi omega arg h j omega gain and group delay of the third order butterworth filter with ω c 1 displaystyle omega _ c 1 the group delay is defined as the negative derivative of the phase shift with respect to angular frequency and is a measure of the distortion in the signal introduced by phase differences for different frequencies the gain and the delay for this filter are plotted in the graph on the left there are no ripples in the gain curve in either the passband or the stopband the log of the absolute value of the transfer function h s displaystyle h s is plotted in complex frequency space in the second graph on the right the function is defined by the three poles in the left half of the complex frequency plane log density plot of the transfer function h s displaystyle h s in complex frequency space for the third order butterworth filter with ω c displaystyle omega _ c 1 the three poles lie on a circle of unit radius in the left half plane these are arranged on a circle of radius unity symmetrical about the real s displaystyle s axis the gain function will have three more poles on the right half plane to complete the circle by replacing each inductor with a capacitor and each capacitor with an inductor a high pass butterworth filter is obtained a band pass butterworth filter is obtained by placing a capacitor in series with each inductor and an inductor in parallel with each capacitor to form resonant circuits the value of each new component must be selected to resonate with the old component at the frequency of interest a band stop butterworth filter is obtained by placing a capacitor in parallel with each inductor and an inductor in series with each capacitor to form resonant circuits the value of each new component must be selected to resonate with the old component at the frequency that is to be rejected transfer function edit plot of the gain of butterworth low pass filters of orders 1 through 5 with cutoff frequency ω c 1 displaystyle omega _ c 1 note that the slope is 20 n displaystyle n db decade where n displaystyle n is the filter order like all filters the typical prototype is the low pass filter which can be modified into a high pass filter or placed in series with others to form band pass and band stop filters and higher order versions of these the gain g ω displaystyle g omega of an n displaystyle n th order butterworth low pass filter is given in terms of the transfer function h s displaystyle h s as g 2 ω h j ω 2 g 0 2 1 ω ω c 2 n displaystyle g 2 omega left h j omega right 2 frac g_ 0 2 1 left frac omega omega _ c right 2n where n displaystyle n is the order of filter ω c displaystyle omega _ c is the cutoff frequency approximately the 3 db frequency and g 0 displaystyle g_ 0 is the dc gain gain at zero frequency it can be seen that as n displaystyle n approaches infinity the gain becomes a rectangle function and frequencies below ω c displaystyle omega _ c will be passed with gain g 0 displaystyle g_ 0 while frequencies above ω c displaystyle omega _ c will be suppressed for smaller values of n displaystyle n the cutoff will be less sharp we wish to determine the transfer function h s displaystyle h s where s σ j ω displaystyle s sigma j omega from laplace transform because h s 2 h s h s displaystyle left h s right 2 h s overline h s and as a general property of laplace transforms at s j ω displaystyle s j omega h j ω h j ω displaystyle h j omega overline h j omega if we select h s displaystyle h s such that h s h s g 0 2 1 s 2 ω c 2 n displaystyle h s h s frac g_ 0 2 1 left frac s 2 omega _ c 2 right n then with s j ω displaystyle s j omega we have the frequency response of the butterworth filter the n displaystyle n poles of this expression occur on a circle of radius ω c displaystyle omega _ c at equally spaced points and symmetric around the negative real axis for stability the transfer function h s displaystyle h s is therefore chosen such that it contains only the poles in the negative real half plane of s displaystyle s the k displaystyle k th pole is specified by s k 2 ω c 2 1 1 n e j 2 k 1 π n k 1 2 3 n displaystyle frac s_ k 2 omega _ c 2 1 frac 1 n e frac j 2k 1 pi n qquad k 1 2 3 ldots n and hence s k ω c e j 2 k n 1 π 2 n k 1 2 3 n displaystyle s_ k omega _ c e frac j 2k n 1 pi 2n qquad k 1 2 3 ldots n the transfer or system function may be written in terms of these poles as h s g 0 k 1 n ω c s s k g 0 k 1 n ω c s ω c e j 2 k n 1 π 2 n displaystyle h s g_ 0 prod _ k 1 n frac omega _ c s s_ k g_ 0 prod _ k 1 n frac omega _ c s omega _ c e frac j 2k n 1 pi 2n where displaystyle textstyle prod is the product of a sequence operator the denominator is a butterworth polynomial in s displaystyle s normalized butterworth polynomials edit the butterworth polynomials may be written in complex form as above but are usually written with real coefficients by multiplying pole pairs that are complex conjugates such as s 1 displaystyle s_ 1 and s n displaystyle s_ n the polynomials are normalized by setting ω c 1 displaystyle omega _ c 1 the normalized butterworth polynomials then have the general product form b n s k 1 n 2 s 2 2 s cos 2 k n 1 2 n π 1 n even displaystyle b_ n s prod _ k 1 frac n 2 left s 2 2s cos left frac 2k n 1 2n pi right 1 right qquad n text even b n s s 1 k 1 n 1 2 s 2 2 s cos 2 k n 1 2 n π 1 n odd displaystyle b_ n s s 1 prod _ k 1 frac n 1 2 left s 2 2s cos left frac 2k n 1 2n pi right 1 right qquad n text odd factors of butterworth polynomials of order 1 through 10 are shown in the following table to six decimal places n factors of butterworth polynomials b n s displaystyle b_ n s 1 s 1 displaystyle s 1 2 s 2 1 414214 s 1 displaystyle s 2 1 414214s 1 3 s 1 s 2 s 1 displaystyle s 1 s 2 s 1 4 s 2 0 765367 s 1 s 2 1 847759 s 1 displaystyle s 2 0 765367s 1 s 2 1 847759s 1 5 s 1 s 2 0 618034 s 1 s 2 1 618034 s 1 displaystyle s 1 s 2 0 618034s 1 s 2 1 618034s 1 6 s 2 0 517638 s 1 s 2 1 414214 s 1 s 2 1 931852 s 1 displaystyle s 2 0 517638s 1 s 2 1 414214s 1 s 2 1 931852s 1 7 s 1 s 2 0 445042 s 1 s 2 1 246980 s 1 s 2 1 801938 s 1 displaystyle s 1 s 2 0 445042s 1 s 2 1 246980s 1 s 2 1 801938s 1 8 s 2 0 390181 s 1 s 2 1 111140 s 1 s 2 1 662939 s 1 s 2 1 961571 s 1 displaystyle s 2 0 390181s 1 s 2 1 111140s 1 s 2 1 662939s 1 s 2 1 961571s 1 9 s 1 s 2 0 347296 s 1 s 2 s 1 s 2 1 532089 s 1 s 2 1 879385 s 1 displaystyle s 1 s 2 0 347296s 1 s 2 s 1 s 2 1 532089s 1 s 2 1 879385s 1 10 s 2 0 312869 s 1 s 2 0 907981 s 1 s 2 1 414214 s 1 s 2 1 782013 s 1 s 2 1 975377 s 1 displaystyle s 2 0 312869s 1 s 2 0 907981s 1 s 2 1 414214s 1 s 2 1 782013s 1 s 2 1 975377s 1 factors of butterworth polynomials of order 1 through 6 are shown in the following table exact n factors of butterworth polynomials b n s displaystyle b_ n s 1 s 1 displaystyle s 1 2 s 2 2 s 1 displaystyle s 2 sqrt 2 s 1 3 s 1 s 2 s 1 displaystyle s 1 s 2 s 1 4 s 2 2 2 s 1 s 2 2 2 s 1 displaystyle s 2 sqrt 2 sqrt 2 s 1 s 2 sqrt 2 sqrt 2 s 1 5 s 1 s 2 φ 1 s 1 s 2 φ s 1 displaystyle s 1 s 2 varphi 1 s 1 s 2 varphi s 1 6 s 2 2 3 s 1 s 2 2 s 1 s 2 2 3 s 1 displaystyle s 2 sqrt 2 sqrt 3 s 1 s 2 sqrt 2 s 1 s 2 sqrt 2 sqrt 3 s 1 where the greek letter phi φ displaystyle varphi or ϕ displaystyle phi represents the golden ratio it is an irrational number that is a solution to the quadratic equation x 2 x 1 0 displaystyle x 2 x 1 0 with a value of 4 5 φ 1 5 2 1 618033988749 displaystyle varphi frac 1 sqrt 5 2 1 618033988749 oeis a001622 the n th butterworth polynomial can also be written as a sum b n s k 0 n a k s k displaystyle b_ n s sum _ k 0 n a_ k s k with its coefficients a k displaystyle a_ k given by the recursion formula 6 7 a k 1 a k cos k γ sin k 1 γ displaystyle frac a_ k 1 a_ k frac cos k gamma sin k 1 gamma and by the product formula a k μ 1 k cos μ 1 γ sin μ γ displaystyle a_ k prod _ mu 1 k frac cos mu 1 gamma sin mu gamma where a 0 1 and γ π 2 n displaystyle a_ 0 1 qquad text and qquad gamma frac pi 2n further a k a n k displaystyle a_ k a_ n k the rounded coefficients a k displaystyle a_ k for the first 10 butterworth polynomials b n s displaystyle b_ n s are butterworth coefficients a k displaystyle a_ k to four decimal places n a 0 displaystyle a_ 0 a 1 displaystyle a_ 1 a 2 displaystyle a_ 2 a 3 displaystyle a_ 3 a 4 displaystyle a_ 4 a 5 displaystyle a_ 5 a 6 displaystyle a_ 6 a 7 displaystyle a_ 7 a 8 displaystyle a_ 8 a 9 displaystyle a_ 9 a 10 displaystyle a_ 10 1 1 1 2 1 1 4142 1 3 1 2 2 1 4 1 2 6131 3 4142 2 6131 1 5 1 3 2361 5 2361 5 2361 3 2361 1 6 1 3 8637 7 4641 9 1416 7 4641 3 8637 1 7 1 4 4940 10 0978 14 5918 14 5918 10 0978 4 4940 1 8 1 5 1258 13 1371 21 8462 25 6884 21 8462 13 1371 5 1258 1 9 1 5 7588 16 5817 31 1634 41 9864 41 9864 31 1634 16 5817 5 7588 1 10 1 6 3925 20 4317 42 8021 64 8824 74 2334 64 8824 42 8021 20 4317 6 3925 1 the normalized butterworth polynomials can be used to determine the transfer function for any low pass filter cut off frequency ω c displaystyle omega _ c as follows h s g 0 b n a displaystyle h s frac g_ 0 b_ n a where a s ω c displaystyle a frac s omega _ c transformation to other bandforms are also possible see prototype filter maximal flatness edit assuming ω c 1 displaystyle omega _ c 1 and g 0 1 displaystyle g_ 0 1 the derivative of the gain with respect to frequency can be shown to be d g d ω n g 3 ω 2 n 1 displaystyle frac dg d omega ng 3 omega 2n 1 which is monotonically decreasing for all ω displaystyle omega since the gain g displaystyle g is always positive the gain function of the butterworth filter therefore has no ripple the series expansion of the gain is given by g ω 1 1 2 ω 2 n 3 8 ω 4 n displaystyle g omega 1 frac 1 2 omega 2n frac 3 8 omega 4n ldots in other words all derivatives of the gain up to but not including the 2 n displaystyle n th derivative are zero at ω 0 displaystyle omega 0 resulting in maximal flatness if the requirement to be monotonic is limited to the passband only and ripples are allowed in the stopband then it is possible to design a filter of the same order such as the inverse chebyshev filter that is flatter in the passband than the maximally flat butterworth high frequency roll off edit again assuming ω c 1 displaystyle omega _ c 1 the slope of the log of the gain for large ω displaystyle omega is lim ω d log g d log ω n displaystyle lim _ omega rightarrow infty frac d log g d log omega n in decibels the high frequency roll off is therefore 20 n displaystyle n db decade or 6 n displaystyle n db octave the factor of 20 is used because the power is proportional to the square of the voltage gain see 20 log rule minimum order edit to design a butterworth filter using the minimum required number of elements the minimum order of the butterworth filter may be calculated as follows 8 n log 10 α s 10 1 10 α p 10 1 2 log ω s ω p displaystyle n left lceil frac log bigr frac 10 alpha _ s 10 1 10 alpha _ p 10 1 bigr 2 log omega _ s omega _ p right rceil where ω p displaystyle omega _ p and α p displaystyle alpha _ p are the pass band frequency and attenuation at that frequency in db ω s displaystyle omega _ s and α s displaystyle alpha _ s are the stop band frequency and attenuation at that frequency in db n displaystyle n is the minimum number of poles the order of the filter displaystyle lceil cdot rceil denotes the ceiling function nonstandard cutoff attenuation edit the cutoff attenuation for butterworth filters is usually defined to be 3 01 db if it is desired to use a different attenuation at the cutoff frequency then the following factor may be applied to each pole whereupon the poles will continue to lie on a circle but the radius will no longer be unity 8 the cutoff attenuation equation may be derived through algebraic manipulation of the butterworth defining equation stated at the top of the page 9 p a p 1 10 α 10 1 1 2 n for 0 α displaystyle begin aligned p_ a p_ 1 times 10 alpha 10 1 1 2n qquad text for 0 leq alpha infty end aligned where p a displaystyle p_ a is the relocated pole positioned to set the desired cutoff attenuation p 1 displaystyle p_ 1 is a 3 01 db cutoff pole that lies on the unit circle α displaystyle alpha is the desired attenuation at the cutoff frequency in db 1 db 10 db etc n displaystyle n is the number of poles the order of the filter filter implementation and design edit there are several different filter topologies available to implement a linear analogue filter the most often used topology for a passive realisation is the cauer topology and the most often used topology for an active realisation is the sallen key topology cauer topology edit butterworth filter using cauer topology the cauer topology uses passive components shunt capacitors and series inductors to implement a linear analog filter the butterworth filter having a given transfer function can be realised using a cauer 1 form the k th element is given by 10 c k 2 sin 2 k 1 2 n π k odd displaystyle c_ k 2 sin left frac 2k 1 2n pi right qquad k text odd l k 2 sin 2 k 1 2 n π k even displaystyle l_ k 2 sin left frac 2k 1 2n pi...