1653:
617:
96:
3490:
2553:
2780:
2340:
2567:
65:. The utility of a prototype filter comes from the property that all these other filters can be derived from it by applying a scaling factor to the components of the prototype. The filter design need thus only be carried out once in full, with other filters being obtained by simply applying a scaling factor.
2190:
1799:
2326:
1938:
2557:
The number of resonators in the expression corresponds to the number of passbands required. Lowpass and highpass filters can be viewed as special cases of the resonator expression with one or the other of the terms becoming zero as appropriate. Bandstop filters can be regarded as a combination of a
2558:
lowpass and a highpass filter. Multiple bandstop filters can always be expressed in terms of a multiple bandpass filter. In this way it, can be seen that this transformation represents the general case for any bandform, and all the other transformations are to be viewed as special cases of it.
2054:
1326:
2548:{\displaystyle {\frac {\omega _{\text{c}}'}{i\omega }}\to {\dfrac {1}{Q_{1}\left({\dfrac {i\omega }{\omega _{01}}}+{\dfrac {\omega _{01}}{i\omega }}\right)}}+{\dfrac {1}{Q_{2}\left({\dfrac {i\omega }{\omega _{02}}}+{\dfrac {\omega _{02}}{i\omega }}\right)}}+\cdots }
3458:
2801:. The Zobel prototypes do not, therefore, correspond to any particular bandform, but they can be transformed into any of them. Not giving special significance to any one bandform makes the method more mathematically pleasing; however, it is not in common use.
2775:{\displaystyle {\frac {i\omega }{\omega _{c}'}}\to {\dfrac {1}{Q_{1}\left({\dfrac {i\omega }{\omega _{01}}}+{\dfrac {\omega _{01}}{i\omega }}\right)}}+{\dfrac {1}{Q_{2}\left({\dfrac {i\omega }{\omega _{02}}}+{\dfrac {\omega _{02}}{i\omega }}\right)}}+\cdots }
1049:
712:
Impedance scaling by itself has no effect on the transfer function of the filter (providing that the terminating impedances have the same scaling applied to them). However, it is usual to combine the frequency and impedance scaling into a single step:
182:
In principle, any non-zero frequency point on the filter response could be used as a reference for the prototype design. For example, for filters with ripple in the passband, the corner frequency is usually defined as the highest frequency at maximum
305:
947:
855:
784:
872:. This in turn leads to the transformation of the impedance components of the filter into some other component(s). The frequency scaling above is a trivial case of bandform transformation corresponding to a lowpass to lowpass transformation.
1523:
is the resonant frequency of the resonators in the filter. Note that frequency scaling the prototype prior to lowpass to bandpass transformation does not affect the resonant frequency, but instead affects the final bandwidth of the filter.
3565:
and the order of the function is the order of the highest order polynomial. Any filter constructed from a finite number of discrete elements will be described by a rational function and in general, the order will be equal to the number of
1209:
1129:
3343:
3265:
398:
223:
To convert them to 50 Ohm multiply the given values by 50. To get the part value convert at the desired cut-off frequency (corner frequency). Example: The resistance shall be 75 Ohm and the corner frequency shall be 2 MHz.
510:
456:
519:
Impedance scaling is invariably a scaling to a fixed resistance. This is because the terminations of the filter, at least nominally, are taken to be a fixed resistance. To carry out this scaling to a nominal impedance
2561:
The same response can equivalently be obtained, sometimes with a more convenient component topology, by transforming to multiple stopbands instead of multiple passbands. The required transformation in those cases is:
1648:
2063:
1952:
1226:
1669:
2199:
1808:
1508:
1454:
1379:
3020:
68:
Especially useful is the ability to transform from one bandform to another. In this case, the transform is more than a simple scale factor. Bandform here is meant to indicate the category of
3126:
3184:
3350:
92:
filter is considered to be a type of multiple passband filter having two passbands. Most commonly, the prototype filter is expressed as a lowpass filter, but other techniques are possible.
708:
666:
612:
566:
3672:
Zobel, O J, "Electrical wave filters", US patent 1 850 146, filed 25 Nov 1930, issued 22 Mar 1932. Gives many useful formulae and a non-frequency domain basis for defining prototypes.
2934:
970:
2864:
243:
885:
789:
718:
3272:
3194:
335:
1138:
1058:
206:
prototype can be converted into any other fifth-order Bessel filter, but it cannot be transformed into a third-order Bessel filter or a fifth-order
461:
407:
2185:{\displaystyle L'\to L={\frac {\omega _{\text{c}}'}{\omega _{0}Q}}L'\,\lVert \,C={\frac {Q}{\omega _{0}\omega _{\text{c}}'}}{\frac {1}{L'}}}
1532:
2049:{\displaystyle {\frac {\omega _{\text{c}}'}{i\omega }}\to Q\left({\frac {i\omega }{\omega _{0}}}+{\dfrac {\omega _{0}}{i\omega }}\right)}
1321:{\displaystyle {\frac {i\omega }{\omega _{\text{c}}'}}\to Q\left({\frac {i\omega }{\omega _{0}}}+{\frac {\omega _{0}}{i\omega }}\right)}
1794:{\displaystyle L'\to L={\frac {\omega _{\text{c}}'Q}{\omega _{0}}}L'\,,\,C={\frac {1}{\omega _{0}\omega _{\text{c}}'Q}}{\frac {1}{L'}}}
2321:{\displaystyle C'\to C={\frac {\omega _{\text{c}}'}{\omega _{0}Q}}C'\,,\,L={\frac {Q}{\omega _{0}\omega _{\text{c}}'}}{\frac {1}{C'}}}
1933:{\displaystyle C'\to C={\frac {\omega _{c}'Q}{\omega _{0}}}C'\,\lVert \,L={\frac {1}{\omega _{0}\omega _{\text{c}}'Q}}{\frac {1}{C'}}}
230:
Filter types with adjustable ripple can not be easily tabulated as such as they depend on more than just the impedance and frequency.
88:, but others are possible. In particular, it is possible for a filter to have multiple passbands. In fact, in some treatments, the
1459:
26:
designs that are used as a template to produce a modified filter design for a particular application. They are an example of a
3065:
regardless of the bandform of the filter being designed. To obtain the required bandform, the following transforms are used:
1409:
2946:), constant k being those filters for which Z/Y is a constant. For this reason, filters of all classes are given in terms of
1339:
3536:. Image parameter filters are not rational and hence do not have a polynomial class. Such filters are classified by type (
2808:
rather than a two-terminal inductor or capacitor. The transfer function is expressed in terms of the product of the series
202:
The prototype filter can only be used to produce other filters of the same class and order. For instance, a fifth-order
3453:{\displaystyle U_{k}(\omega )\to Q^{2}\left({\frac {i\omega }{\omega _{0}}}+{\frac {\omega _{0}}{i\omega }}\right)^{2}}
2955:
2804:
The Zobel prototype considers filter sections, rather than components. That is, the transformation is carried out on a
3077:
3135:
3706:
184:
402:
It can readily be seen that to achieve this, the non-resistive components of the filter must be transformed by:
2883:
1656:
The prototype filter above, transformed to a 50 Ω, 6 MHz bandpass filter with 100 kHz bandwidth
671:
629:
575:
529:
62:
624:
It can readily be seen that to achieve this, the non-resistive components of the filter must be scaled as:
1044:{\displaystyle A(i\omega )\to A\left({\frac {\omega _{\text{c}}\,\omega _{\text{c}}'}{i\omega }}\right)}
3721:
3468:
192:
3716:
1397:
are the lower and upper frequency points (respectively) of the bandpass response corresponding to
300:{\displaystyle i\omega \to \left({\frac {\omega _{\text{c}}'}{\omega _{\text{c}}}}\right)i\omega }
3701:
2829:
942:{\displaystyle {\frac {i\omega }{\omega _{\text{c}}'}}\to {\frac {\omega _{\text{c}}}{i\omega }}}
35:
3711:
3558:
3483:
2943:
2790:
188:
199:
rather than the 3 dB point since cut-off is a well-defined point in this type of filter.
2821:
27:
850:{\displaystyle C\to \,{\frac {\omega _{\text{c}}'}{\omega _{\text{c}}}}\,{\frac {R'}{R}}\,C}
779:{\displaystyle L\to \,{\frac {\omega _{\text{c}}'}{\omega _{\text{c}}}}\,{\frac {R}{R'}}\,L}
238:
The prototype filter is scaled to the frequency required with the following transformation:
213:
A passive lumped low-pass prototype filter of fifth order and the T-topology might have the
3567:
2809:
214:
113:
58:
8:
3338:{\displaystyle U_{k}(\omega )\to \left({\frac {\omega _{\text{c}}}{i\omega }}\right)^{2}}
3260:{\displaystyle U_{k}(\omega )\to \left({\frac {i\omega }{\omega _{\text{c}}}}\right)^{2}}
316:′ is the value of the frequency parameter (e.g. cut-off frequency) for the prototype and
2335:
Filters with multiple passbands may be obtained by applying the general transformation:
121:
3548:
serves as the class name for image filters and is based on the filter circuit topology.
117:
3533:
3529:
3473:
3058:
2939:
2797:
provided an alternative basis for constructing a prototype which is not based in the
196:
151:
100:
47:
39:
23:
46:. However, in principle, the method can be applied to any kind of linear filter or
2938:
With image filters, it is possible to obtain filters of different classes from the
2805:
2798:
620:
The prototype filter above, transformed to a 600 Ω, 16 kHz lowpass filter
393:{\displaystyle A(i\omega )\to A\left(i{\frac {\omega }{\omega _{\text{c}}}}\right)}
207:
140:
125:
172:′ = 1. Likewise, the nominal or characteristic impedance of the filter is set to
2817:
3661:
Zobel, O J, "Theory and Design of
Uniform and Composite Electric Wave Filters",
1204:{\displaystyle C'\to L={\frac {1}{\omega _{\text{c}}\,\omega _{\text{c}}'\,C'}}}
1124:{\displaystyle L'\to C={\frac {1}{\omega _{\text{c}}\,\omega _{\text{c}}'\,L'}}}
1652:
112:
Parts of this article or section rely on the reader's knowledge of the complex
3695:
3541:
3537:
3478:
2824:, adding to the prototype's generality. Generally, ZY is a complex quantity,
203:
43:
570:
It may be more convenient on some elements to scale the admittance instead:
330:′ = 1 then the transfer function of the filter is transformed as:
31:
3561:
of the filter's rational function. A rational function is a ratio of two
3528:
The class of a filter is the mathematical class of the polynomials in the
616:
3024:
In the case of dissipationless networks, i.e. no resistors, the quantity
1334:
is the Q-factor and is equal to the inverse of the fractional bandwidth:
505:{\displaystyle C\to {\frac {\omega _{\text{c}}'}{\omega _{\text{c}}}}\,C}
451:{\displaystyle L\to {\frac {\omega _{\text{c}}'}{\omega _{\text{c}}}}\,L}
3677:
Microwave
Filters, Impedance-Matching Networks, and Coupling Structures
3562:
2813:
2794:
95:
1643:{\displaystyle A(i\omega )\to A\left(\omega _{\text{c}}'Q\left\right)}
3509:
3505:
3501:
1661:
54:
3489:
3497:
3062:
3054:
89:
85:
81:
77:
69:
1947:
The required frequency transformation for lowpass to bandstop is:
1213:
the primed quantities being the component value in the prototype.
1527:
The transfer function of the filter is transformed according to:
864:
In general, the bandform of a filter is transformed by replacing
73:
965:′ on the prototype. The transfer function then transforms as:
139:
The prototype is most often a low-pass filter with a 3 dB
2942:
prototype by means of a different kind of transformation (see
227:+75jΩ -48jΩ +150jΩ -48jΩ +75jΩ 6μH 1.66nF 12μH 1.66nF 6μH
162:
1503:{\displaystyle \omega _{0}={\sqrt {\omega _{1}\omega _{2}}}}
868:
where it occurs in the transfer function with a function of
524:, each impedance element of the filter is transformed by:
1803:
and capacitors are transformed into parallel resonators,
1449:{\displaystyle \Delta \omega =\omega _{2}-\omega _{1}\,}
1221:
In this case, the required frequency transformation is:
1053:
Inductors are transformed into capacitors according to,
2194:
and capacitors are transformed into series resonators,
1374:{\displaystyle Q={\frac {\omega _{0}}{\Delta \omega }}}
880:
The frequency transformation required in this case is:
3188:
The bandform transformations from this prototype are,
50:, including mechanical, acoustic and optical filters.
3353:
3275:
3197:
3138:
3080:
2958:
2886:
2832:
2820:
for a description of half-sections. This quantity is
2734:
2707:
2685:
2649:
2622:
2600:
2570:
2507:
2480:
2458:
2422:
2395:
2373:
2343:
2202:
2066:
2018:
1955:
1811:
1672:
1535:
1462:
1412:
1342:
1229:
1141:
1061:
973:
958:
is the point on the highpass filter corresponding to
888:
792:
721:
674:
632:
578:
532:
464:
410:
338:
246:
2058:
Inductors are transformed into parallel resonators,
3061:and then continues to increase negatively into the
72:that the filter possesses. The usual bandforms are
3452:
3337:
3259:
3178:
3130:the independent variable of the response plot is,
3120:
3014:
2928:
2858:
2774:
2547:
2320:
2184:
2048:
1932:
1793:
1642:
1502:
1448:
1373:
1320:
1203:
1123:
1043:
941:
849:
778:
702:
660:
606:
560:
504:
450:
392:
299:
53:Filters are required to operate at many different
3175:
3015:{\displaystyle ZY=U_{k}(\omega )+iV_{k}(\omega )}
2855:
3693:
3121:{\displaystyle R_{0}=1\,,\,\omega _{\text{c}}=1}
3179:{\displaystyle U_{k}(\omega )=-\omega ^{2}\,\!}
1133:and capacitors are transformed into inductors,
191:(an older design method than the more modern
2816:Y of a filter half-section. See the article
2125:
1870:
220:+1jΩ -0.64jΩ +2jΩ -0.64jΩ +1jΩ (exemplary)
30:design from which the desired filter can be
187:rather than 3 dB. Another case is in
3684:An Introduction to Linear Network Analysis
859:
3174:
3101:
3097:
2929:{\displaystyle ZY=U(\omega )+iV(\omega )}
2854:
2264:
2260:
2128:
2124:
1873:
1869:
1734:
1730:
1445:
1189:
1175:
1109:
1095:
1012:
843:
827:
799:
772:
756:
728:
696:
668: and,
654:
600:
554:
498:
444:
3488:
2784:
2330:
1651:
1456: and
615:
94:
38:. They are most often seen in regard to
2950:for a constant k, which is notated as,
703:{\displaystyle C\to {\frac {R'}{R}}\,C}
661:{\displaystyle L\to {\frac {R}{R'}}\,L}
607:{\displaystyle Y\to {\frac {R'}{R}}\,Y}
561:{\displaystyle Z\to {\frac {R}{R'}}\,Z}
3694:
1942:
1660:Inductors are transformed into series
1216:
875:
3053:) ranges from 0 at the centre of the
134:
16:Template for electronic filter design
514:
233:
3686:, English Universities Press, 1961.
2874:are both, in general, functions of
13:
1413:
1362:
14:
3733:
3655:
1516:is the absolute bandwidth, and
44:linear analogue passive filters
3643:
3630:
3621:
3608:
3595:
3582:
3551:
3522:
3373:
3370:
3364:
3295:
3292:
3286:
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3003:
2984:
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2211:
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1981:
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582:
536:
468:
414:
354:
351:
342:
253:
1:
3663:Bell System Technical Journal
3576:
3557:The order of a filter is the
786: and,
458: and,
323:is the desired value. So if
3516:
7:
3462:
2859:{\displaystyle ZY=U+iV\,\!}
154:. Occasionally, frequency
10:
3738:
3469:Electronic filter topology
3072:prototype that is scaled:
2878:we should properly write,
1404:′ of the prototype, then,
193:network synthesis filters
128:representation of signals
124:and on knowledge of the
3707:Image impedance filters
3675:Matthaei, Young, Jones
3570:elements that are used.
3068:For a lowpass constant
3040:) need be considered.
860:Bandform transformation
189:image parameter filters
3669:(1923), pp. 1–46.
3513:
3484:Composite image filter
3454:
3339:
3261:
3180:
3122:
3016:
2944:composite image filter
2930:
2860:
2776:
2549:
2322:
2186:
2050:
1934:
1795:
1657:
1644:
1504:
1450:
1375:
1322:
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1125:
1045:
943:
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662:
621:
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562:
506:
452:
394:
301:
107:
3492:
3455:
3340:
3262:
3181:
3123:
3017:
2931:
2861:
2785:Alternative prototype
2777:
2550:
2331:Lowpass to multi-band
2323:
2187:
2051:
1935:
1796:
1655:
1645:
1505:
1451:
1376:
1323:
1206:
1126:
1046:
944:
852:
781:
705:
663:
619:
609:
563:
507:
453:
395:
302:
143:of angular frequency
99:A low pass prototype
98:
3351:
3273:
3195:
3136:
3078:
2956:
2884:
2830:
2789:In his treatment of
2568:
2341:
2200:
2064:
1953:
1809:
1670:
1533:
1460:
1410:
1340:
1227:
1139:
1059:
971:
886:
790:
719:
672:
630:
576:
530:
462:
408:
336:
244:
3032:) is zero and only
2812:, Z, and the shunt
2593:
2358:
2299:
2234:
2163:
2098:
1970:
1943:Lowpass to bandstop
1908:
1844:
1769:
1705:
1574:
1252:
1217:Lowpass to bandpass
1188:
1108:
1025:
911:
876:Lowpass to highpass
814:
743:
485:
431:
274:
165:is used instead of
3649:Zobel, 1930, p. 3.
3532:that describe its
3514:
3450:
3347:and for bandpass,
3335:
3257:
3176:
3118:
3012:
2926:
2856:
2772:
2764:
2755:
2728:
2679:
2670:
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2501:
2452:
2443:
2416:
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2222:
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2151:
2086:
2046:
2039:
1958:
1930:
1896:
1832:
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1446:
1371:
1318:
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1201:
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802:
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731:
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658:
622:
604:
558:
502:
473:
448:
419:
390:
297:
262:
135:Low-pass prototype
116:representation of
108:
40:electronic filters
28:nondimensionalised
3722:Electronic design
3679:McGraw-Hill 1964.
3534:transfer function
3530:rational function
3474:Electronic filter
3437:
3412:
3323:
3312:
3245:
3242:
3109:
3059:cut-off frequency
2940:constant k filter
2763:
2754:
2727:
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2012:
1979:
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515:Impedance scaling
496:
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234:Frequency scaling
197:cut-off frequency
48:signal processing
24:electronic filter
20:Prototype filters
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3494:Filter bandforms
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2806:two-port network
2799:frequency domain
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2113:
2109:
2108:
2094:
2091:
2085:
2074:
2055:
2053:
2052:
2047:
2045:
2041:
2040:
2037:
2029:
2028:
2019:
2013:
2011:
2010:
2001:
1993:
1980:
1978:
1966:
1963:
1957:
1939:
1937:
1936:
1931:
1929:
1927:
1916:
1914:
1912:
1904:
1901:
1895:
1894:
1881:
1868:
1860:
1858:
1857:
1848:
1840:
1830:
1819:
1800:
1798:
1797:
1792:
1790:
1788:
1777:
1775:
1773:
1765:
1762:
1756:
1755:
1742:
1729:
1721:
1719:
1718:
1709:
1701:
1698:
1691:
1680:
1649:
1647:
1646:
1641:
1639:
1635:
1634:
1630:
1629:
1627:
1619:
1618:
1609:
1604:
1602:
1601:
1592:
1584:
1570:
1567:
1509:
1507:
1506:
1501:
1499:
1497:
1496:
1487:
1486:
1477:
1472:
1471:
1455:
1453:
1452:
1447:
1444:
1443:
1431:
1430:
1380:
1378:
1377:
1372:
1370:
1368:
1360:
1359:
1350:
1327:
1325:
1324:
1319:
1317:
1313:
1312:
1310:
1302:
1301:
1292:
1287:
1285:
1284:
1275:
1267:
1254:
1248:
1245:
1239:
1231:
1210:
1208:
1207:
1202:
1200:
1198:
1197:
1184:
1181:
1174:
1173:
1170:
1160:
1149:
1130:
1128:
1127:
1122:
1120:
1118:
1117:
1104:
1101:
1094:
1093:
1090:
1080:
1069:
1050:
1048:
1047:
1042:
1040:
1036:
1034:
1026:
1021:
1018:
1011:
1010:
1007:
1000:
948:
946:
945:
940:
938:
936:
928:
927:
924:
918:
913:
907:
904:
898:
890:
856:
854:
853:
848:
842:
837:
829:
826:
824:
823:
820:
810:
807:
801:
785:
783:
782:
777:
771:
769:
758:
755:
753:
752:
749:
739:
736:
730:
709:
707:
706:
701:
695:
690:
682:
667:
665:
664:
659:
653:
651:
640:
613:
611:
610:
605:
599:
594:
586:
567:
565:
564:
559:
553:
551:
540:
511:
509:
508:
503:
497:
495:
494:
491:
481:
478:
472:
457:
455:
454:
449:
443:
441:
440:
437:
427:
424:
418:
399:
397:
396:
391:
389:
385:
384:
382:
381:
378:
369:
306:
304:
303:
298:
290:
286:
284:
283:
280:
270:
267:
261:
208:Chebyshev filter
195:) which use the
178:
160:
141:corner frequency
126:frequency domain
3737:
3736:
3732:
3731:
3730:
3728:
3727:
3726:
3717:Analog circuits
3692:
3691:
3658:
3653:
3648:
3644:
3635:
3631:
3626:
3622:
3613:
3609:
3600:
3596:
3587:
3583:
3579:
3574:
3556:
3552:
3527:
3523:
3519:
3465:
3444:
3429:
3423:
3419:
3417:
3406:
3402:
3394:
3392:
3391:
3387:
3386:
3380:
3376:
3358:
3354:
3352:
3349:
3348:
3329:
3315:
3309:
3305:
3303:
3299:
3298:
3280:
3276:
3274:
3271:
3270:
3251:
3239:
3235:
3227:
3225:
3221:
3220:
3202:
3198:
3196:
3193:
3192:
3168:
3164:
3143:
3139:
3137:
3134:
3133:
3106:
3102:
3085:
3081:
3079:
3076:
3075:
3048:
2997:
2993:
2972:
2968:
2957:
2954:
2953:
2885:
2882:
2881:
2831:
2828:
2827:
2818:Image impedance
2787:
2746:
2740:
2736:
2733:
2721:
2717:
2709:
2706:
2705:
2701:
2695:
2691:
2690:
2684:
2661:
2655:
2651:
2648:
2636:
2632:
2624:
2621:
2620:
2616:
2610:
2606:
2605:
2599:
2585:
2573:
2571:
2569:
2566:
2565:
2519:
2513:
2509:
2506:
2494:
2490:
2482:
2479:
2478:
2474:
2468:
2464:
2463:
2457:
2434:
2428:
2424:
2421:
2409:
2405:
2397:
2394:
2393:
2389:
2383:
2379:
2378:
2372:
2359:
2350:
2344:
2342:
2339:
2338:
2333:
2308:
2303:
2291:
2281:
2277:
2276:
2271:
2252:
2240:
2236:
2235:
2226:
2220:
2203:
2201:
2198:
2197:
2172:
2167:
2155:
2145:
2141:
2140:
2135:
2116:
2104:
2100:
2099:
2090:
2084:
2067:
2065:
2062:
2061:
2030:
2024:
2020:
2017:
2006:
2002:
1994:
1992:
1991:
1987:
1971:
1962:
1956:
1954:
1951:
1950:
1945:
1920:
1915:
1900:
1890:
1886:
1885:
1880:
1861:
1853:
1849:
1836:
1831:
1829:
1812:
1810:
1807:
1806:
1781:
1776:
1761:
1751:
1747:
1746:
1741:
1722:
1714:
1710:
1697:
1692:
1690:
1673:
1671:
1668:
1667:
1620:
1614:
1610:
1608:
1597:
1593:
1585:
1583:
1582:
1578:
1566:
1561:
1557:
1534:
1531:
1530:
1522:
1492:
1488:
1482:
1478:
1476:
1467:
1463:
1461:
1458:
1457:
1439:
1435:
1426:
1422:
1411:
1408:
1407:
1403:
1396:
1389:
1361:
1355:
1351:
1349:
1341:
1338:
1337:
1303:
1297:
1293:
1291:
1280:
1276:
1268:
1266:
1265:
1261:
1244:
1232:
1230:
1228:
1225:
1224:
1219:
1190:
1180:
1169:
1165:
1164:
1159:
1142:
1140:
1137:
1136:
1110:
1100:
1089:
1085:
1084:
1079:
1062:
1060:
1057:
1056:
1027:
1017:
1006:
1002:
1001:
999:
995:
972:
969:
968:
964:
957:
929:
923:
919:
917:
903:
891:
889:
887:
884:
883:
878:
862:
830:
828:
819:
815:
806:
800:
791:
788:
787:
762:
757:
748:
744:
735:
729:
720:
717:
716:
683:
681:
673:
670:
669:
644:
639:
631:
628:
627:
587:
585:
577:
574:
573:
544:
539:
531:
528:
527:
517:
490:
486:
477:
471:
463:
460:
459:
436:
432:
423:
417:
409:
406:
405:
377:
373:
368:
364:
360:
337:
334:
333:
329:
322:
315:
279:
275:
266:
260:
256:
245:
242:
241:
236:
228:
221:
176:
171:
158:
149:
137:
42:and especially
17:
12:
11:
5:
3735:
3725:
3724:
3719:
3714:
3709:
3704:
3702:Linear filters
3688:
3687:
3680:
3673:
3670:
3657:
3654:
3652:
3651:
3642:
3640:, pp. 727–729.
3629:
3627:Farago, p. 69.
3620:
3618:, pp. 438–440.
3607:
3605:, pp. 412–413.
3594:
3580:
3578:
3575:
3573:
3572:
3550:
3520:
3518:
3515:
3487:
3486:
3481:
3476:
3471:
3464:
3461:
3447:
3442:
3435:
3432:
3426:
3422:
3416:
3409:
3405:
3400:
3397:
3390:
3383:
3379:
3375:
3372:
3369:
3366:
3361:
3357:
3332:
3327:
3321:
3318:
3308:
3302:
3297:
3294:
3291:
3288:
3283:
3279:
3269:for highpass,
3254:
3249:
3238:
3233:
3230:
3224:
3219:
3216:
3213:
3210:
3205:
3201:
3171:
3167:
3163:
3160:
3157:
3154:
3151:
3146:
3142:
3117:
3114:
3105:
3100:
3096:
3093:
3088:
3084:
3044:
3011:
3008:
3005:
3000:
2996:
2992:
2989:
2986:
2983:
2980:
2975:
2971:
2967:
2964:
2961:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2822:nondimensional
2786:
2783:
2771:
2768:
2760:
2752:
2749:
2743:
2739:
2732:
2724:
2720:
2715:
2712:
2704:
2698:
2694:
2689:
2683:
2675:
2667:
2664:
2658:
2654:
2647:
2639:
2635:
2630:
2627:
2619:
2613:
2609:
2604:
2598:
2592:
2588:
2584:
2579:
2576:
2544:
2541:
2533:
2525:
2522:
2516:
2512:
2505:
2497:
2493:
2488:
2485:
2477:
2471:
2467:
2462:
2456:
2448:
2440:
2437:
2431:
2427:
2420:
2412:
2408:
2403:
2400:
2392:
2386:
2382:
2377:
2371:
2365:
2362:
2357:
2349:
2332:
2329:
2314:
2311:
2307:
2298:
2290:
2284:
2280:
2275:
2270:
2267:
2263:
2258:
2255:
2248:
2243:
2239:
2233:
2225:
2219:
2216:
2213:
2209:
2206:
2178:
2175:
2171:
2162:
2154:
2148:
2144:
2139:
2134:
2131:
2127:
2122:
2119:
2112:
2107:
2103:
2097:
2089:
2083:
2080:
2077:
2073:
2070:
2044:
2036:
2033:
2027:
2023:
2016:
2009:
2005:
2000:
1997:
1990:
1986:
1983:
1977:
1974:
1969:
1961:
1944:
1941:
1926:
1923:
1919:
1911:
1907:
1899:
1893:
1889:
1884:
1879:
1876:
1872:
1867:
1864:
1856:
1852:
1847:
1843:
1839:
1835:
1828:
1825:
1822:
1818:
1815:
1787:
1784:
1780:
1772:
1768:
1760:
1754:
1750:
1745:
1740:
1737:
1733:
1728:
1725:
1717:
1713:
1708:
1704:
1696:
1689:
1686:
1683:
1679:
1676:
1638:
1633:
1626:
1623:
1617:
1613:
1607:
1600:
1596:
1591:
1588:
1581:
1577:
1573:
1565:
1560:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1520:
1495:
1491:
1485:
1481:
1475:
1470:
1466:
1442:
1438:
1434:
1429:
1425:
1421:
1418:
1415:
1401:
1394:
1387:
1367:
1364:
1358:
1354:
1348:
1345:
1316:
1309:
1306:
1300:
1296:
1290:
1283:
1279:
1274:
1271:
1264:
1260:
1257:
1251:
1243:
1238:
1235:
1218:
1215:
1196:
1193:
1187:
1179:
1168:
1163:
1158:
1155:
1152:
1148:
1145:
1116:
1113:
1107:
1099:
1088:
1083:
1078:
1075:
1072:
1068:
1065:
1039:
1033:
1030:
1024:
1016:
1005:
998:
994:
991:
988:
985:
982:
979:
976:
962:
955:
935:
932:
922:
916:
910:
902:
897:
894:
877:
874:
861:
858:
846:
840:
836:
833:
818:
813:
805:
798:
795:
775:
768:
765:
761:
747:
742:
734:
727:
724:
699:
693:
689:
686:
680:
677:
657:
650:
647:
643:
638:
635:
603:
597:
593:
590:
584:
581:
557:
550:
547:
543:
538:
535:
516:
513:
501:
489:
484:
476:
470:
467:
447:
435:
430:
422:
416:
413:
388:
376:
372:
367:
363:
359:
356:
353:
350:
347:
344:
341:
327:
320:
313:
296:
293:
289:
278:
273:
265:
259:
255:
252:
249:
235:
232:
226:
219:
169:
147:
136:
133:
132:
131:
15:
9:
6:
4:
3:
2:
3734:
3723:
3720:
3718:
3715:
3713:
3712:Filter theory
3710:
3708:
3705:
3703:
3700:
3699:
3697:
3690:
3685:
3682:Farago, P S,
3681:
3678:
3674:
3671:
3668:
3664:
3660:
3659:
3646:
3639:
3633:
3624:
3617:
3611:
3604:
3598:
3591:
3585:
3581:
3569:
3564:
3560:
3554:
3547:
3543:
3539:
3535:
3531:
3525:
3521:
3511:
3507:
3503:
3499:
3495:
3491:
3485:
3482:
3480:
3479:Linear filter
3477:
3475:
3472:
3470:
3467:
3466:
3460:
3445:
3440:
3433:
3430:
3424:
3420:
3414:
3407:
3403:
3398:
3395:
3388:
3381:
3377:
3367:
3359:
3355:
3345:
3330:
3325:
3319:
3316:
3306:
3300:
3289:
3281:
3277:
3267:
3252:
3247:
3236:
3231:
3228:
3222:
3211:
3203:
3199:
3191:for lowpass,
3189:
3186:
3169:
3165:
3161:
3158:
3152:
3144:
3140:
3131:
3128:
3115:
3112:
3103:
3098:
3094:
3091:
3086:
3082:
3073:
3071:
3066:
3064:
3060:
3057:to −1 at the
3056:
3052:
3047:
3043:
3039:
3035:
3031:
3027:
3022:
3006:
2998:
2994:
2990:
2987:
2981:
2973:
2969:
2965:
2962:
2959:
2951:
2949:
2945:
2941:
2936:
2920:
2914:
2911:
2908:
2902:
2896:
2893:
2890:
2887:
2879:
2877:
2873:
2869:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2825:
2823:
2819:
2815:
2811:
2807:
2802:
2800:
2796:
2792:
2791:image filters
2782:
2769:
2766:
2758:
2750:
2747:
2741:
2737:
2730:
2722:
2718:
2713:
2710:
2702:
2696:
2692:
2687:
2681:
2673:
2665:
2662:
2656:
2652:
2645:
2637:
2633:
2628:
2625:
2617:
2611:
2607:
2602:
2590:
2586:
2582:
2577:
2574:
2563:
2559:
2555:
2542:
2539:
2531:
2523:
2520:
2514:
2510:
2503:
2495:
2491:
2486:
2483:
2475:
2469:
2465:
2460:
2454:
2446:
2438:
2435:
2429:
2425:
2418:
2410:
2406:
2401:
2398:
2390:
2384:
2380:
2375:
2363:
2360:
2355:
2347:
2336:
2328:
2312:
2309:
2305:
2296:
2288:
2282:
2278:
2273:
2268:
2265:
2261:
2256:
2253:
2246:
2241:
2237:
2231:
2223:
2217:
2214:
2207:
2204:
2195:
2192:
2176:
2173:
2169:
2160:
2152:
2146:
2142:
2137:
2132:
2129:
2120:
2117:
2110:
2105:
2101:
2095:
2087:
2081:
2078:
2071:
2068:
2059:
2056:
2042:
2034:
2031:
2025:
2021:
2014:
2007:
2003:
1998:
1995:
1988:
1984:
1975:
1972:
1967:
1959:
1948:
1940:
1924:
1921:
1917:
1909:
1905:
1897:
1891:
1887:
1882:
1877:
1874:
1865:
1862:
1854:
1850:
1845:
1841:
1837:
1833:
1826:
1823:
1816:
1813:
1804:
1801:
1785:
1782:
1778:
1770:
1766:
1758:
1752:
1748:
1743:
1738:
1735:
1731:
1726:
1723:
1715:
1711:
1706:
1702:
1694:
1687:
1684:
1677:
1674:
1665:
1663:
1654:
1650:
1636:
1631:
1624:
1621:
1615:
1611:
1605:
1598:
1594:
1589:
1586:
1579:
1575:
1571:
1563:
1558:
1554:
1545:
1542:
1536:
1528:
1525:
1519:
1515:
1510:
1493:
1489:
1483:
1479:
1473:
1468:
1464:
1440:
1436:
1432:
1427:
1423:
1419:
1416:
1405:
1400:
1393:
1386:
1381:
1365:
1356:
1352:
1346:
1343:
1335:
1333:
1328:
1314:
1307:
1304:
1298:
1294:
1288:
1281:
1277:
1272:
1269:
1262:
1258:
1249:
1241:
1236:
1233:
1222:
1214:
1211:
1194:
1191:
1185:
1177:
1166:
1161:
1156:
1153:
1146:
1143:
1134:
1131:
1114:
1111:
1105:
1097:
1086:
1081:
1076:
1073:
1066:
1063:
1054:
1051:
1037:
1031:
1028:
1022:
1014:
1003:
996:
992:
983:
980:
974:
966:
961:
954:
949:
933:
930:
920:
908:
900:
895:
892:
881:
873:
871:
867:
857:
844:
838:
834:
831:
816:
811:
803:
793:
773:
766:
763:
759:
745:
740:
732:
722:
714:
710:
697:
691:
687:
684:
675:
655:
648:
645:
641:
633:
625:
618:
614:
601:
595:
591:
588:
579:
571:
568:
555:
548:
545:
541:
533:
525:
523:
512:
499:
487:
482:
474:
465:
445:
433:
428:
420:
411:
403:
400:
386:
374:
370:
365:
361:
357:
348:
345:
339:
331:
326:
319:
312:
307:
294:
291:
287:
276:
271:
263:
257:
250:
247:
239:
231:
225:
218:
216:
211:
209:
205:
204:Bessel filter
200:
198:
194:
190:
186:
180:
175:
168:
164:
157:
153:
146:
142:
129:
127:
123:
119:
115:
110:
109:
106:Π (pi) filter
105:
104:
97:
93:
91:
87:
83:
79:
75:
71:
66:
64:
60:
56:
51:
49:
45:
41:
37:
33:
29:
25:
21:
3689:
3683:
3676:
3666:
3662:
3656:Bibliography
3645:
3637:
3632:
3623:
3615:
3610:
3602:
3597:
3592:, pp. 96–97.
3589:
3584:
3553:
3545:
3524:
3493:
3346:
3268:
3190:
3187:
3132:
3129:
3074:
3069:
3067:
3050:
3045:
3041:
3037:
3033:
3029:
3025:
3023:
2952:
2947:
2937:
2880:
2875:
2871:
2867:
2826:
2803:
2788:
2564:
2560:
2556:
2337:
2334:
2196:
2193:
2060:
2057:
1949:
1946:
1805:
1802:
1666:
1659:
1529:
1526:
1517:
1513:
1511:
1406:
1398:
1391:
1384:
1382:
1336:
1331:
1329:
1223:
1220:
1212:
1135:
1132:
1055:
1052:
967:
959:
952:
950:
882:
879:
869:
865:
863:
715:
711:
626:
623:
572:
569:
526:
521:
518:
404:
401:
332:
324:
317:
310:
308:
240:
237:
229:
222:
212:
201:
181:
179:= 1 Ω.
173:
166:
155:
144:
138:
111:
102:
67:
52:
19:
18:
3563:polynomials
150:′ = 1
55:frequencies
36:transformed
3696:Categories
3577:References
2814:admittance
1662:resonators
118:capacitors
63:bandwidths
59:impedances
3636:Matthaei
3614:Matthaei
3601:Matthaei
3588:Matthaei
3517:Footnotes
3510:band-stop
3506:band-pass
3502:high-pass
3434:ω
3421:ω
3404:ω
3399:ω
3374:→
3368:ω
3320:ω
3307:ω
3296:→
3290:ω
3237:ω
3232:ω
3218:→
3212:ω
3166:ω
3162:−
3153:ω
3104:ω
3007:ω
2982:ω
2921:ω
2903:ω
2810:impedance
2770:⋯
2751:ω
2738:ω
2719:ω
2714:ω
2666:ω
2653:ω
2634:ω
2629:ω
2597:→
2583:ω
2578:ω
2543:⋯
2524:ω
2511:ω
2492:ω
2487:ω
2439:ω
2426:ω
2407:ω
2402:ω
2370:→
2364:ω
2348:ω
2289:ω
2279:ω
2238:ω
2224:ω
2212:→
2153:ω
2143:ω
2126:‖
2102:ω
2088:ω
2076:→
2035:ω
2022:ω
2004:ω
1999:ω
1982:→
1976:ω
1960:ω
1898:ω
1888:ω
1871:‖
1851:ω
1834:ω
1821:→
1759:ω
1749:ω
1712:ω
1695:ω
1682:→
1625:ω
1612:ω
1595:ω
1590:ω
1564:ω
1552:→
1546:ω
1490:ω
1480:ω
1465:ω
1437:ω
1433:−
1424:ω
1417:ω
1414:Δ
1366:ω
1363:Δ
1353:ω
1308:ω
1295:ω
1278:ω
1273:ω
1256:→
1242:ω
1237:ω
1178:ω
1167:ω
1151:→
1098:ω
1087:ω
1071:→
1032:ω
1015:ω
1004:ω
990:→
984:ω
934:ω
921:ω
915:→
901:ω
896:ω
817:ω
804:ω
797:→
746:ω
733:ω
726:→
679:→
637:→
583:→
537:→
488:ω
475:ω
469:→
434:ω
421:ω
415:→
375:ω
371:ω
355:→
349:ω
295:ω
277:ω
264:ω
254:→
251:ω
215:reactance
161:= 1
122:inductors
114:impedance
101:constant
3568:reactive
3498:low-pass
3463:See also
3063:stopband
3055:passband
2591:′
2356:′
2313:′
2297:′
2257:′
2232:′
2208:′
2177:′
2161:′
2121:′
2096:′
2072:′
1968:′
1925:′
1906:′
1866:′
1842:′
1817:′
1786:′
1767:′
1727:′
1703:′
1678:′
1572:′
1250:′
1195:′
1186:′
1147:′
1115:′
1106:′
1067:′
1023:′
909:′
835:′
812:′
767:′
741:′
688:′
649:′
592:′
549:′
483:′
429:′
272:′
90:bandstop
86:bandstop
82:bandpass
78:highpass
70:passband
3544:etc).
3496:: see,
2866:and as
74:lowpass
3638:et al.
3616:et al.
3603:et al.
3590:et al.
3542:m-type
3538:k-type
1330:where
951:where
309:where
185:ripple
32:scaled
3667:vol.2
3559:order
2795:Zobel
152:rad/s
3546:Type
2948:U(ω)
2870:and
1390:and
120:and
84:and
61:and
22:are
1383:If
34:or
3698::
3665:,
3540:,
3508:,
3504:,
3500:,
2793:,
2742:02
2723:02
2657:01
2638:01
2515:02
2496:02
2430:01
2411:01
1664:,
870:iω
866:iω
217::
210:.
163:Hz
80:,
76:,
57:,
3512:.
3446:2
3441:)
3431:i
3425:0
3415:+
3408:0
3396:i
3389:(
3382:2
3378:Q
3371:)
3365:(
3360:k
3356:U
3331:2
3326:)
3317:i
3311:c
3301:(
3293:)
3287:(
3282:k
3278:U
3253:2
3248:)
3241:c
3229:i
3223:(
3215:)
3209:(
3204:k
3200:U
3170:2
3159:=
3156:)
3150:(
3145:k
3141:U
3116:1
3113:=
3108:c
3099:,
3095:1
3092:=
3087:0
3083:R
3070:k
3051:ω
3049:(
3046:k
3042:U
3038:ω
3036:(
3034:U
3030:ω
3028:(
3026:V
3010:)
3004:(
2999:k
2995:V
2991:i
2988:+
2985:)
2979:(
2974:k
2970:U
2966:=
2963:Y
2960:Z
2924:)
2918:(
2915:V
2912:i
2909:+
2906:)
2900:(
2897:U
2894:=
2891:Y
2888:Z
2876:ω
2872:V
2868:U
2852:V
2849:i
2846:+
2843:U
2840:=
2837:Y
2834:Z
2767:+
2759:)
2748:i
2731:+
2711:i
2703:(
2697:2
2693:Q
2688:1
2682:+
2674:)
2663:i
2646:+
2626:i
2618:(
2612:1
2608:Q
2603:1
2587:c
2575:i
2540:+
2532:)
2521:i
2504:+
2484:i
2476:(
2470:2
2466:Q
2461:1
2455:+
2447:)
2436:i
2419:+
2399:i
2391:(
2385:1
2381:Q
2376:1
2361:i
2352:c
2310:C
2306:1
2293:c
2283:0
2274:Q
2269:=
2266:L
2262:,
2254:C
2247:Q
2242:0
2228:c
2218:=
2215:C
2205:C
2174:L
2170:1
2157:c
2147:0
2138:Q
2133:=
2130:C
2118:L
2111:Q
2106:0
2092:c
2082:=
2079:L
2069:L
2043:)
2032:i
2026:0
2015:+
2008:0
1996:i
1989:(
1985:Q
1973:i
1964:c
1922:C
1918:1
1910:Q
1902:c
1892:0
1883:1
1878:=
1875:L
1863:C
1855:0
1846:Q
1838:c
1827:=
1824:C
1814:C
1783:L
1779:1
1771:Q
1763:c
1753:0
1744:1
1739:=
1736:C
1732:,
1724:L
1716:0
1707:Q
1699:c
1688:=
1685:L
1675:L
1637:)
1632:]
1622:i
1616:0
1606:+
1599:0
1587:i
1580:[
1576:Q
1568:c
1559:(
1555:A
1549:)
1543:i
1540:(
1537:A
1521:0
1518:ω
1514:ω
1512:Δ
1494:2
1484:1
1474:=
1469:0
1441:1
1428:2
1420:=
1402:c
1399:ω
1395:2
1392:ω
1388:1
1385:ω
1357:0
1347:=
1344:Q
1332:Q
1315:)
1305:i
1299:0
1289:+
1282:0
1270:i
1263:(
1259:Q
1246:c
1234:i
1192:C
1182:c
1171:c
1162:1
1157:=
1154:L
1144:C
1112:L
1102:c
1091:c
1082:1
1077:=
1074:C
1064:L
1038:)
1029:i
1019:c
1008:c
997:(
993:A
987:)
981:i
978:(
975:A
963:c
960:ω
956:c
953:ω
931:i
925:c
905:c
893:i
845:C
839:R
832:R
821:c
808:c
794:C
774:L
764:R
760:R
750:c
737:c
723:L
698:C
692:R
685:R
676:C
656:L
646:R
642:R
634:L
602:Y
596:R
589:R
580:Y
556:Z
546:R
542:R
534:Z
522:R
500:C
492:c
479:c
466:C
446:L
438:c
425:c
412:L
387:)
379:c
366:i
362:(
358:A
352:)
346:i
343:(
340:A
328:c
325:ω
321:c
318:ω
314:c
311:ω
292:i
288:)
281:c
268:c
258:(
248:i
177:′
174:R
170:c
167:ω
159:′
156:f
148:c
145:ω
130:.
103:k
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