4613:
3655:
4608:{\displaystyle {\begin{aligned}&H(s)={\frac {15}{s^{3}+6s^{2}+15s+15}}{\text{ (from the example below)}}\\&B_{arith}^{2}=10^{20/10}=0.01{\text{ (the arithmetic gain squared)}}\\&\\&{\text{Find }}H(s)'{\text{ such that }}|H(s)'|=-20{\text{ dB at }}\omega =1{\text{.}}\\&H(s)H(-s)={\frac {225}{-s^{6}+6s^{4}-45^{2}s+225}}\\&P(s)=225-B_{arith}^{2}(-s^{6}+6s^{4}-45^{2}s+225)=0.01s^{6}-0.06s^{4}+0.45s^{2}+222.75{\text{ (polynomial to be factored)}}\\&R=j5.0771344{\text{ (the positive imaginary root for the above polynomial)}}\\&{\text{For even order filters, use the positive real root.}}\\&\\&\omega _{-20{\text{ dB atten}}}=\omega _{c}=5.0771344{\text{ rad/sec (20 dB attenuation frequency)}}\\&H(s)'=H(s)_{A=20{\text{ dB at }}\omega =1}={\frac {15}{(5.0771344^{3})s^{3}+(6\times 5.0771344^{2})s^{2}+(15\times 5.0771344)s+15}}\\&={\frac {15}{130.87478s^{3}+154.66376s^{2}+76.157016s+15}}\\&\\&{\text{Check:}}\\&|H(j)'|={\bigg |}{\frac {15}{130.87478j^{3}+154.66376j^{2}+76.157016j+15}}{\bigg |}=0.1=-20{\text{ dB Gain}}\end{aligned}}}
4631:
618:
4623:
281:
1606:
5699:(IIR) filters with comparable frequency response, IIR filters obtained by the bilinear transformation do not have constant group delay. Since the important characteristic of a Bessel filter is its maximally-flat group delay, the bilinear transform is inappropriate for converting an analog Bessel filter into a digital form.
1255:
There is no standard set attenuation value for Bessel filters. However, −3.0103 dB is a common choice. Some applications may use a higher or lower attenuation such as −1 dB or −20 dB. Setting the cut-off attenuation frequency involves first finding the frequency that achieves the
284:
A plot of the gain and group delay for a fourth-order low-pass Bessel filter. Note that the transition from the passband to the stopband is much slower than for other filters, but the group delay is practically constant in the passband. The Bessel filter maximizes the flatness of the group delay
5376:
1364:
260:, the Bessel filter has a small amount of overshoot, but still much less than other common frequency-domain filters, such as Butterworth filters. It has been noted that the impulse response of Bessel–Thomson filters tends towards a Gaussian as the order of the filter is increased.
5782:
5213:
4988:
2814:
5555:. This is the greatest number of terms that can be set to zero, since there are a total of four coefficients in the third-order Bessel polynomial, requiring four equations in order to be defined. One equation specifies that the gain be unity at
2984:
will then be virtually the exact desired value at 1 rad/sec. If performed properly, only a handful of iterations are needed to set the attenuation through a wide range of desired attenuation values for both small and very large order filters.
5470:
1245:
3626:
392:
1601:{\displaystyle {\begin{aligned}H(s)&={\frac {15}{s^{3}+6s^{2}+15s+15}}\\H(s)'&=H(s)_{{\text{desired dB at }}\omega =1}={\frac {15}{(\omega _{c}s)^{3}+6(\omega _{c}s)^{2}+15\omega _{c}s+15}}\\\end{aligned}}}
5228:
607:
1020:
263:
Compared to finite-order approximations of the
Gaussian filter, the Bessel filter has a slightly better shaping factor (i.e., how well a particular filter approximates the ideal lowpass response), flatter
1109:
916:
5052:
4761:
5702:
The digital equivalent is the Thiran filter, also an all-pole low-pass filter with maximally-flat group delay, which can also be transformed into an allpass filter, to implement fractional delays.
2561:
3660:
1369:
5089:
1776:
828:
3455:
2388:
3042:
does not contain any phase information, directly factoring the transfer function will not produce usable results. However, the transfer function may be modified by multiplying it with
4861:
756:
4872:
2459:
3040:
2180:
1702:
5605:
3150:
3111:
2599:
533:
431:
5078:
4673:
3380:
2313:
700:
497:
2237:
5527:
5500:
2924:
2897:
2870:
2843:
2588:
2486:
2207:
1806:
1634:
1337:
1281:
462:
5680:
5579:
5553:
4819:
3303:
3198:
1959:
1854:
3639:
Cutoff attenuations that are above the pass band ripple or below the stop band ripple will come back with multiple roots, so the correct root will have to be selected.
3336:
2269:
2122:
189:
3259:
3072:
2095:
2068:
2021:
1989:
1915:
3227:
2982:
2953:
1883:
1310:
5652:
5391:
4790:
5860:
5625:
2041:
1357:
644:
5756:
5607:, leaving two equations to specify two terms in the series expansion to be zero. This is a general property of the group delay for a Bessel filter of order
1120:
3463:
2182:, and this modification will allow the use of real numbers instead of complex numbers when evaluating the polynomial and its derivative. the real
303:
5371:{\displaystyle D(\omega )=-{\frac {d\phi }{d\omega }}={\frac {6\omega ^{4}+45\omega ^{2}+225}{\omega ^{6}+6\omega ^{4}+45\omega ^{2}+225}}.\,}
182:
3636:
Of the set of roots from above, select the positive imaginary root for odd order filters, and positive real root for even order filters.
538:
922:
175:
1031:
834:
5981:
5971:
4996:
4681:
2494:
5208:{\displaystyle \phi (\omega )=-\arg(H(j\omega ))=\arctan \left({\frac {15\omega -\omega ^{3}}{15-6\omega ^{2}}}\right).\,}
6143:
6181:
2872:
should be limited to prevent it from going negative early in the iterations for increased reliability. When complete,
1707:
5868:
6082:
5951:
535:
is indeterminate by the definition of reverse Bessel polynomials, but is a removable singularity, it is defined that
5760:
762:
5655:
terms in the series expansion of the group delay will be zero, thus maximizing the flatness of the group delay at
3385:
3152:
to be real at all frequencies, and then finding the frequency that result on the square of the desired attention.
2318:
6121:
237:(1784–1846), who developed the mathematical theory on which the filter is based. The filters are also called
4983:{\displaystyle G(\omega )=|H(j\omega )|={\frac {15}{\sqrt {\omega ^{6}+6\omega ^{4}+45\omega ^{2}+225}}}.\,}
464:
is a frequency chosen to give the desired cut-off frequency. The filter has a low-frequency group delay of
100:
4828:
1778:
and use the square of the desired cutoff gain, and is just as accurate, so the square terms will be used.
706:
6099:
5736:
4634:
Group delay plot of the third-order Bessel low-pass filter, illustrating flat unit delay in the passband.
2396:
272:
than a
Gaussian filter of the same order, although the Gaussian has lower time delay and zero overshoot.
219:
2809:{\displaystyle \omega _{a}=\omega _{a}-(|H_{2}(\omega _{a})H_{2}(-\omega _{a})|-B^{2})/(d/d\omega _{a})}
226:), which preserves the wave shape of filtered signals in the passband. Bessel filters are often used in
6186:
3650:
A 20-dB cut-off frequency attenuation example using the 3-pole Bessel example below is set as follows.
2996:
2127:
1658:
5998:
5696:
5584:
3116:
3077:
117:
70:
65:
33:
502:
400:
5057:
4645:
3341:
2594:
When steps 1) through 4) are complete, the expression involving Newton's method may be written as:
2274:
666:
467:
2212:
6176:
5505:
5478:
3631:
2902:
2875:
2848:
2821:
2566:
2464:
2185:
1784:
1641:
1612:
1315:
1259:
440:
111:
5659:
5558:
5532:
4795:
3264:
3159:
1920:
1815:
123:
83:
5889:
5465:{\displaystyle D(\omega )=1-{\frac {\omega ^{6}}{225}}+{\frac {\omega ^{8}}{1125}}+\cdots .}
3311:
2244:
2100:
3232:
3045:
2073:
2046:
1994:
1967:
1888:
3203:
2958:
2929:
1859:
1286:
8:
5631:
4769:
5941:
5819:
5716:
5692:
5610:
2026:
1652:
1637:
1342:
629:
622:
257:
128:
40:
4792:).The roots of the denominator polynomial, the filter's poles, include a real pole at
1312:
polynomials to the inverse of that frequency. To scale the polynomials, simply append
6078:
6053:
6018:
5977:
5947:
5890:"How to choose a filter? (Butterworth, Chebyshev, Inverse Chebyshev, Bessel–Thomson)"
5836:
4822:
657:
653:
434:
294:
207:
90:
22:
252:, and tends towards the same shape as filter order increases. While the time-domain
6045:
6036:
Thiran, J.-P. (1971). "Recursive digital filters with maximally flat group delay".
6010:
5797:
5721:
242:
234:
95:
45:
6072:
6014:
5731:
5711:
4639:
4626:
Gain plot of the third-order Bessel low-pass filter, versus normalized frequency.
290:
249:
227:
60:
50:
3457:. For example, 3.010 dB converts to 0.5, 1 dB converts to 0.79432823 and so on.
2390:. For example, 3.010 dB converts to 0.5, 1 dB converts to 0.79432823 and so on.
5695:
is used to convert continuous-time (analog) filters to discrete-time (digital)
223:
1359:
term in each coefficient, as shown in the 3 pole Bessel filter example below.
1240:{\displaystyle a_{k}={\frac {(2n-k)!}{2^{n-k}k!(n-k)!}}\quad k=0,1,\ldots ,n.}
6170:
6057:
6049:
6022:
5382:
3621:{\displaystyle P(S)=H_{num}(S)H_{num}(-S)-B_{arith}^{2}H_{den}(S)H_{den}(-S)}
253:
215:
105:
1655:
requires a known magnitude value and derivative magnitude value for the for
5801:
387:{\displaystyle H(s)={\frac {\theta _{n}(0)}{\theta _{n}(s/\omega _{0})}}\,}
4766:
where the numerator has been chosen to give unity gain at zero frequency (
5726:
5219:
4630:
617:
269:
265:
203:
161:
6155:
6071:
Madisetti, Vijay (1997). "Section 11.3.2.2 Classical IIR Filter Types".
4622:
6161:
6149:
6123:
Discrete-time modeling of acoustic tubes using fractional delay filters
5946:. Norwell, Massachusetts, US: Kluwer Academic Publishers. p. 224.
280:
156:
151:
146:
5999:"Design of maximally flat IIR filters with flat group delay responses"
5943:
Design and
Analysis of Analog Filters: A Signal Processing Perspective
5790:
Proceedings of the IEE - Part III: Radio and
Communication Engineering
3645:
1647:
602:{\displaystyle \theta _{n}(0)=\lim _{x\rightarrow 0}\theta _{n}(x)}
1015:{\displaystyle n=5:\quad s^{5}+15s^{4}+105s^{3}+420s^{2}+945s+945}
626:
5783:"Delay networks having maximally flat frequency characteristics"
2955:
transfer function denominator. The attenuation of the modified
5820:"Transient Response and Transforms: 3.1 Bessel-Thomson filters"
2988:
5913:
241:
in recognition of W. E. Thomson, who worked out how to apply
4638:
The transfer function for a third-order (three-pole) Bessel
4172: (the positive imaginary root for the above polynomial)
1104:{\displaystyle \theta _{n}(s)=\sum _{k=0}^{n}a_{k}s^{k},}
233:
The filter's name is a reference to German mathematician
911:{\displaystyle n=4:\quad s^{4}+10s^{3}+45s^{2}+105s+105}
5080:. This is conventionally called the cut-off frequency.
5047:{\displaystyle |H(j\omega )|={\frac {1}{\sqrt {2}}},\,}
4756:{\displaystyle H(s)={\frac {15}{s^{3}+6s^{2}+15s+15}},}
621:
The roots of the third-order Bessel polynomial are the
6156:
Bessel
Filters Polynomials, Poles and Circuit Elements
5662:
5634:
5613:
5587:
5561:
5535:
5508:
5481:
5394:
5231:
5092:
5060:
4999:
4875:
4831:
4798:
4772:
4684:
4648:
3658:
3466:
3388:
3344:
3314:
3267:
3235:
3206:
3162:
3119:
3080:
3048:
2999:
2961:
2932:
2905:
2878:
2851:
2824:
2602:
2569:
2556:{\displaystyle H_{2}(\omega _{a})H_{2}(-\omega _{a})}
2497:
2467:
2399:
2321:
2277:
2247:
2215:
2188:
2130:
2103:
2076:
2049:
2029:
1997:
1970:
1923:
1891:
1862:
1818:
1787:
1710:
1661:
1615:
1367:
1345:
1318:
1289:
1262:
1123:
1034:
925:
837:
765:
709:
669:
632:
541:
505:
470:
443:
403:
306:
2845:
with no complex arithmetic needed. The movement of
2590:. DO NOT take the absolute value of the derivative.
2124:, and so on. The modified function will be called
4182:
For even order filters, use the positive real root.
5674:
5646:
5619:
5599:
5573:
5547:
5529:are zero, resulting in a very flat group delay at
5521:
5494:
5464:
5370:
5207:
5072:
5046:
4982:
4855:
4813:
4784:
4755:
4667:
4607:
3646:Simple cut-off frequency example with root finding
3620:
3449:
3374:
3330:
3297:
3253:
3221:
3192:
3144:
3105:
3066:
3034:
2976:
2947:
2918:
2891:
2864:
2837:
2808:
2582:
2555:
2480:
2453:
2382:
2307:
2263:
2231:
2201:
2174:
2116:
2089:
2062:
2035:
2015:
1983:
1953:
1909:
1877:
1848:
1800:
1770:
1696:
1648:Finding attenuation frequency with Newton's method
1628:
1600:
1351:
1331:
1304:
1275:
1256:desired attenuation, which will be referred to as
1239:
1103:
1014:
910:
822:
750:
694:
638:
601:
527:
491:
456:
425:
386:
5969:
4576:
4516:
1771:{\displaystyle |H(j\omega _{c})H(-j\omega _{c})|}
1250:
6168:
5581:and a second specifies that the gain be zero at
652:The transfer function of the Bessel filter is a
565:
6162:Java source code to compute Bessel filter poles
5970:Bianchi, Giovanni; Sorrentino, Roberto (2007).
5935:
5933:
6064:
5965:
5963:
5749:
1025:The reverse Bessel polynomials are given by:
823:{\displaystyle n=3:\quad s^{3}+6s^{2}+15s+15}
183:
6129:(Thesis). Helsinki University of Technology.
6113:
5976:. McGraw–Hill Professional. pp. 31–43.
5837:"comp.dsp | IIR Gaussian Transition filters"
5813:
5811:
3450:{\displaystyle B_{arith}^{2}=10^{A_{dB}/10}}
2989:Finding attenuation frequency from the roots
2383:{\displaystyle B_{arith}^{2}=10^{A_{dB}/10}}
5930:
4233: rad/sec (20 dB attenuation frequency)
16:Type of analog linear filter in electronics
6029:
5960:
5774:
190:
176:
6097:
6091:
6070:
5990:
5973:Electronic filter simulation & design
5906:
5808:
5367:
5204:
5043:
4979:
2461:in Newton's method using the real value,
383:
248:The Bessel filter is very similar to the
6119:
5939:
4629:
4621:
2209:can now be used in place of the complex
616:
437:from which the filter gets its name and
279:
275:
5817:
5780:
3308:Convert the desired attenuation in dB,
2926:that can be used to scale the original
2241:Convert the desired attenuation in dB,
1704:. However, it is easier to operate on
625:of the filter transfer function in the
6169:
6074:The Digital Signal Processing Handbook
6035:
5853:
5829:
3338:, to a squared arithmetic gain value,
2491:Calculate the derivative the modified
2271:, to a squared arithmetic gain value,
5996:
5882:
612:
4856:{\displaystyle s=-1.8389\pm j1.7544}
751:{\displaystyle n=2:\quad s^{2}+3s+3}
6098:Smith III, Julius O. (2015-05-22).
6038:IEEE Transactions on Circuit Theory
3795: (the arithmetic gain squared)
3200:is not already available, multiply
2454:{\displaystyle |H_{2}(s)H_{2}(-s)|}
1856:is not already available, multiply
13:
5594:
2488:. Always take the absolute value.
14:
6198:
6137:
4150: (polynomial to be factored)
3035:{\displaystyle |H(j\omega _{a})|}
2175:{\displaystyle H_{2}(s)H_{2}(-s)}
1697:{\displaystyle |H(j\omega _{c})|}
5781:Thomson, W. E. (November 1949).
5385:expansion of the group delay is
2563:with respect to the real value,
256:of the Gaussian filter has zero
6144:Bessel and Linear Phase Filters
5600:{\displaystyle \omega =\infty }
3630:Find the roots of P(S) using a
3145:{\displaystyle H(j\omega _{a})}
3106:{\displaystyle H(j\omega _{a})}
3074:to eliminate all odd powers of
1206:
938:
850:
778:
722:
682:
656:whose denominator is a reverse
6100:"Thiran Allpass Interpolators"
5404:
5398:
5241:
5235:
5135:
5132:
5123:
5117:
5102:
5096:
5021:
5017:
5008:
5001:
4912:
4908:
4899:
4892:
4885:
4879:
4694:
4688:
4507:
4499:
4492:
4485:
4385:
4373:
4357:
4338:
4322:
4309:
4271:
4264:
4251:
4244:
4091:
4037:
3998:
3992:
3918:
3909:
3903:
3897:
3857:
3849:
3842:
3835:
3822:
3815:
3731: (from the example below)
3673:
3667:
3615:
3606:
3587:
3581:
3532:
3523:
3504:
3498:
3476:
3470:
3292:
3283:
3277:
3271:
3248:
3239:
3216:
3210:
3187:
3178:
3172:
3166:
3139:
3123:
3100:
3084:
3061:
3052:
3028:
3024:
3008:
3001:
2971:
2965:
2942:
2936:
2803:
2782:
2779:
2763:
2750:
2737:
2724:
2718:
2710:
2693:
2689:
2673:
2660:
2647:
2633:
2629:
2550:
2534:
2521:
2508:
2447:
2443:
2434:
2421:
2415:
2401:
2169:
2160:
2147:
2141:
2010:
1998:
1948:
1939:
1933:
1927:
1904:
1895:
1872:
1866:
1843:
1834:
1828:
1822:
1764:
1760:
1741:
1735:
1719:
1712:
1690:
1686:
1670:
1663:
1557:
1540:
1525:
1508:
1479:
1472:
1455:
1448:
1381:
1375:
1299:
1293:
1251:Setting the cutoff attenuation
1197:
1185:
1155:
1140:
1051:
1045:
596:
590:
572:
558:
552:
528:{\displaystyle \theta _{n}(0)}
522:
516:
426:{\displaystyle \theta _{n}(s)}
420:
414:
377:
356:
341:
335:
316:
310:
1:
5742:
5073:{\displaystyle \omega =1.756}
4668:{\displaystyle \omega _{0}=1}
3375:{\displaystyle B_{arith}^{2}}
2308:{\displaystyle B_{arith}^{2}}
695:{\displaystyle n=1:\quad s+1}
492:{\displaystyle 1/\omega _{0}}
66:Optimum "L" (Legendre) filter
6077:. CRC Press. p. 11-32.
6015:10.1016/j.sigpro.2008.01.016
5914:"Free Analog Filter Program"
2232:{\displaystyle j\omega _{a}}
7:
5940:Paarmann, Larry D. (2001).
5737:Group delay and phase delay
5705:
5522:{\displaystyle \omega ^{4}}
5495:{\displaystyle \omega ^{2}}
5475:Note that the two terms in
2919:{\displaystyle \omega _{c}}
2892:{\displaystyle \omega _{a}}
2865:{\displaystyle \omega _{a}}
2838:{\displaystyle \omega _{a}}
2583:{\displaystyle \omega _{a}}
2481:{\displaystyle \omega _{a}}
2202:{\displaystyle \omega _{a}}
1801:{\displaystyle \omega _{c}}
1629:{\displaystyle \omega _{c}}
1332:{\displaystyle \omega _{c}}
1276:{\displaystyle \omega _{c}}
457:{\displaystyle \omega _{0}}
10:
6203:
5686:
4617:
1808:, follow the steps below.
648:, here plotted as crosses.
245:to filter design in 1949.
6182:Network synthesis filters
5818:Roberts, Stephen (2001).
5697:infinite impulse response
5675:{\displaystyle \omega =0}
5574:{\displaystyle \omega =0}
5548:{\displaystyle \omega =0}
4814:{\displaystyle s=-2.3222}
3298:{\displaystyle H(s)H(-s)}
3193:{\displaystyle H(s)H(-s)}
1954:{\displaystyle H(s)H(-s)}
1849:{\displaystyle H(s)H(-s)}
660:, such as the following:
118:Bridged T delay equaliser
34:Network synthesis filters
6050:10.1109/TCT.1971.1083363
5997:Zhang, Xi (2008-07-01).
293:is characterized by its
285:curve at zero frequency.
222:(i.e., maximally linear
6150:Bessel Filter Constants
6120:Välimäki, Vesa (1995).
4993:The −3-dB point, where
3632:root finding algorithm.
3113:, which in turn forces
2818:using a real value for
2393:Calculate the modified
1283:, and then scaling the
84:Image impedance filters
51:Elliptic (Cauer) filter
5802:10.1049/pi-3.1949.0101
5759:. 2013. Archived from
5676:
5648:
5621:
5601:
5575:
5549:
5523:
5496:
5466:
5372:
5209:
5074:
5048:
4984:
4857:
4823:complex-conjugate pair
4815:
4786:
4757:
4669:
4635:
4627:
4609:
3622:
3451:
3376:
3332:
3331:{\displaystyle A_{dB}}
3299:
3255:
3223:
3194:
3146:
3107:
3068:
3036:
2978:
2949:
2920:
2893:
2866:
2839:
2810:
2584:
2557:
2482:
2455:
2384:
2309:
2265:
2264:{\displaystyle A_{dB}}
2233:
2203:
2176:
2118:
2117:{\displaystyle s^{10}}
2091:
2064:
2037:
2017:
1985:
1955:
1911:
1879:
1850:
1802:
1772:
1698:
1630:
1602:
1353:
1333:
1306:
1277:
1241:
1105:
1077:
1016:
912:
824:
752:
696:
649:
640:
603:
529:
493:
458:
427:
388:
286:
239:Bessel–Thomson filters
218:with a maximally flat
124:Composite image filter
5677:
5649:
5622:
5602:
5576:
5550:
5524:
5497:
5467:
5373:
5210:
5075:
5049:
4985:
4858:
4816:
4787:
4758:
4670:
4633:
4625:
4610:
3831: such that
3623:
3452:
3377:
3333:
3300:
3256:
3254:{\displaystyle H(-s)}
3224:
3195:
3147:
3108:
3069:
3067:{\displaystyle H(-s)}
3037:
2979:
2950:
2921:
2894:
2867:
2840:
2811:
2585:
2558:
2483:
2456:
2385:
2310:
2266:
2234:
2204:
2177:
2119:
2092:
2090:{\displaystyle s^{6}}
2065:
2063:{\displaystyle s^{2}}
2038:
2018:
2016:{\displaystyle (n+2)}
1986:
1984:{\displaystyle s^{n}}
1956:
1912:
1910:{\displaystyle H(-s)}
1880:
1851:
1803:
1773:
1699:
1631:
1603:
1354:
1334:
1307:
1278:
1242:
1106:
1057:
1017:
913:
825:
753:
697:
641:
620:
604:
530:
494:
459:
428:
389:
283:
276:The transfer function
101:General image filters
71:Linkwitz–Riley filter
5660:
5632:
5611:
5585:
5559:
5533:
5506:
5479:
5392:
5229:
5090:
5058:
4997:
4873:
4829:
4796:
4770:
4682:
4646:
3656:
3464:
3386:
3342:
3312:
3265:
3233:
3222:{\displaystyle H(s)}
3204:
3160:
3117:
3078:
3046:
2997:
2977:{\displaystyle G(s)}
2959:
2948:{\displaystyle H(s)}
2930:
2903:
2876:
2849:
2822:
2600:
2567:
2495:
2465:
2397:
2319:
2275:
2245:
2213:
2186:
2128:
2101:
2074:
2047:
2027:
1995:
1968:
1964:negate all terms of
1921:
1889:
1878:{\displaystyle H(s)}
1860:
1816:
1785:
1708:
1659:
1613:
1365:
1343:
1316:
1305:{\displaystyle H(s)}
1287:
1260:
1121:
1032:
923:
835:
763:
707:
667:
630:
539:
503:
468:
441:
401:
304:
214:is a type of analog
5647:{\displaystyle n-1}
4785:{\displaystyle s=0}
4036:
3765:
3564:
3415:
3371:
2348:
2304:
1485:desired dB at
108:(constant R) filter
5918:www.kecktaylor.com
5861:"Gaussian Filters"
5841:www.dsprelated.com
5717:Butterworth filter
5693:bilinear transform
5672:
5644:
5617:
5597:
5571:
5545:
5519:
5492:
5462:
5368:
5205:
5070:
5044:
4980:
4853:
4811:
4782:
4753:
4665:
4636:
4628:
4605:
4603:
4010:
3739:
3618:
3538:
3447:
3389:
3372:
3345:
3328:
3295:
3251:
3219:
3190:
3142:
3103:
3064:
3032:
2974:
2945:
2916:
2889:
2862:
2835:
2806:
2580:
2553:
2478:
2451:
2380:
2322:
2305:
2278:
2261:
2229:
2199:
2172:
2114:
2087:
2060:
2033:
2013:
1981:
1951:
1907:
1875:
1846:
1798:
1768:
1694:
1636:may be found with
1626:
1598:
1596:
1349:
1329:
1302:
1273:
1237:
1101:
1012:
908:
820:
748:
692:
650:
636:
613:Bessel polynomials
599:
579:
525:
489:
454:
423:
384:
287:
41:Butterworth filter
25:electronic filters
6187:Electronic design
6003:Signal Processing
5983:978-0-07-149467-0
5894:www.etc.tuiasi.ro
5620:{\displaystyle n}
5451:
5431:
5362:
5268:
5195:
5038:
5037:
4974:
4973:
4866:The gain is then
4863:, plotted above.
4748:
4599:
4572:
4475:
4461:
4398:
4287:
4286: dB at
4234:
4208:
4183:
4173:
4151:
3982:
3887:
3873:
3872: dB at
3832:
3810:
3796:
3732:
3727:
2899:can used for the
2043:. That would be
2036:{\displaystyle 4}
1592:
1486:
1439:
1352:{\displaystyle s}
1204:
658:Bessel polynomial
654:rational function
639:{\displaystyle s}
564:
435:Bessel polynomial
381:
295:transfer function
208:signal processing
200:
199:
91:Constant k filter
6194:
6131:
6130:
6128:
6117:
6111:
6110:
6108:
6107:
6102:. W3K Publishing
6095:
6089:
6088:
6068:
6062:
6061:
6033:
6027:
6026:
6009:(7): 1792–1800.
5994:
5988:
5987:
5967:
5958:
5957:
5937:
5928:
5927:
5925:
5924:
5910:
5904:
5903:
5901:
5900:
5886:
5880:
5879:
5877:
5876:
5867:. Archived from
5857:
5851:
5850:
5848:
5847:
5833:
5827:
5826:
5824:
5815:
5806:
5805:
5787:
5778:
5772:
5771:
5769:
5768:
5753:
5722:Chebyshev filter
5682:
5681:
5679:
5678:
5673:
5654:
5653:
5651:
5650:
5645:
5626:
5624:
5623:
5618:
5606:
5604:
5603:
5598:
5580:
5578:
5577:
5572:
5554:
5552:
5551:
5546:
5528:
5526:
5525:
5520:
5518:
5517:
5501:
5499:
5498:
5493:
5491:
5490:
5471:
5469:
5468:
5463:
5452:
5447:
5446:
5437:
5432:
5427:
5426:
5417:
5377:
5375:
5374:
5369:
5363:
5361:
5354:
5353:
5338:
5337:
5322:
5321:
5311:
5304:
5303:
5288:
5287:
5274:
5269:
5267:
5259:
5251:
5214:
5212:
5211:
5206:
5200:
5196:
5194:
5193:
5192:
5173:
5172:
5171:
5152:
5079:
5077:
5076:
5071:
5053:
5051:
5050:
5045:
5039:
5033:
5029:
5024:
5004:
4989:
4987:
4986:
4981:
4975:
4966:
4965:
4950:
4949:
4934:
4933:
4924:
4920:
4915:
4895:
4862:
4860:
4859:
4854:
4820:
4818:
4817:
4812:
4791:
4789:
4788:
4783:
4762:
4760:
4759:
4754:
4749:
4747:
4731:
4730:
4715:
4714:
4701:
4674:
4672:
4671:
4666:
4658:
4657:
4614:
4612:
4611:
4606:
4604:
4600:
4597:
4580:
4579:
4573:
4571:
4555:
4554:
4539:
4538:
4522:
4520:
4519:
4510:
4505:
4488:
4480:
4476:
4473:
4470:
4467:
4466:
4462:
4460:
4444:
4443:
4428:
4427:
4411:
4403:
4399:
4397:
4369:
4368:
4356:
4355:
4334:
4333:
4321:
4320:
4304:
4299:
4298:
4288:
4285:
4257:
4239:
4235:
4232:
4224:
4223:
4211:
4210:
4209:
4206:
4192:
4189:
4188:
4184:
4181:
4178:
4174:
4171:
4156:
4152:
4149:
4141:
4140:
4125:
4124:
4109:
4108:
4081:
4080:
4068:
4067:
4052:
4051:
4035:
4030:
3987:
3983:
3981:
3971:
3970:
3958:
3957:
3942:
3941:
3925:
3892:
3888:
3885:
3874:
3871:
3860:
3855:
3838:
3833:
3830:
3828:
3811:
3808:
3805:
3802:
3801:
3797:
3794:
3786:
3785:
3781:
3764:
3759:
3737:
3733:
3730:
3728:
3726:
3710:
3709:
3694:
3693:
3680:
3662:
3627:
3625:
3624:
3619:
3605:
3604:
3580:
3579:
3563:
3558:
3522:
3521:
3497:
3496:
3456:
3454:
3453:
3448:
3446:
3445:
3441:
3436:
3435:
3414:
3409:
3381:
3379:
3378:
3373:
3370:
3365:
3337:
3335:
3334:
3329:
3327:
3326:
3304:
3302:
3301:
3296:
3260:
3258:
3257:
3252:
3228:
3226:
3225:
3220:
3199:
3197:
3196:
3191:
3151:
3149:
3148:
3143:
3138:
3137:
3112:
3110:
3109:
3104:
3099:
3098:
3073:
3071:
3070:
3065:
3041:
3039:
3038:
3033:
3031:
3023:
3022:
3004:
2983:
2981:
2980:
2975:
2954:
2952:
2951:
2946:
2925:
2923:
2922:
2917:
2915:
2914:
2898:
2896:
2895:
2890:
2888:
2887:
2871:
2869:
2868:
2863:
2861:
2860:
2844:
2842:
2841:
2836:
2834:
2833:
2815:
2813:
2812:
2807:
2802:
2801:
2789:
2778:
2777:
2762:
2761:
2749:
2748:
2736:
2735:
2717:
2709:
2708:
2696:
2688:
2687:
2672:
2671:
2659:
2658:
2646:
2645:
2636:
2625:
2624:
2612:
2611:
2589:
2587:
2586:
2581:
2579:
2578:
2562:
2560:
2559:
2554:
2549:
2548:
2533:
2532:
2520:
2519:
2507:
2506:
2487:
2485:
2484:
2479:
2477:
2476:
2460:
2458:
2457:
2452:
2450:
2433:
2432:
2414:
2413:
2404:
2389:
2387:
2386:
2381:
2379:
2378:
2374:
2369:
2368:
2347:
2342:
2314:
2312:
2311:
2306:
2303:
2298:
2270:
2268:
2267:
2262:
2260:
2259:
2238:
2236:
2235:
2230:
2228:
2227:
2208:
2206:
2205:
2200:
2198:
2197:
2181:
2179:
2178:
2173:
2159:
2158:
2140:
2139:
2123:
2121:
2120:
2115:
2113:
2112:
2096:
2094:
2093:
2088:
2086:
2085:
2069:
2067:
2066:
2061:
2059:
2058:
2042:
2040:
2039:
2034:
2023:is divisible by
2022:
2020:
2019:
2014:
1990:
1988:
1987:
1982:
1980:
1979:
1960:
1958:
1957:
1952:
1916:
1914:
1913:
1908:
1884:
1882:
1881:
1876:
1855:
1853:
1852:
1847:
1807:
1805:
1804:
1799:
1797:
1796:
1777:
1775:
1774:
1769:
1767:
1759:
1758:
1734:
1733:
1715:
1703:
1701:
1700:
1695:
1693:
1685:
1684:
1666:
1635:
1633:
1632:
1627:
1625:
1624:
1607:
1605:
1604:
1599:
1597:
1593:
1591:
1581:
1580:
1565:
1564:
1552:
1551:
1533:
1532:
1520:
1519:
1503:
1498:
1497:
1487:
1484:
1461:
1440:
1438:
1422:
1421:
1406:
1405:
1392:
1358:
1356:
1355:
1350:
1338:
1336:
1335:
1330:
1328:
1327:
1311:
1309:
1308:
1303:
1282:
1280:
1279:
1274:
1272:
1271:
1246:
1244:
1243:
1238:
1205:
1203:
1178:
1177:
1161:
1138:
1133:
1132:
1110:
1108:
1107:
1102:
1097:
1096:
1087:
1086:
1076:
1071:
1044:
1043:
1021:
1019:
1018:
1013:
996:
995:
980:
979:
964:
963:
948:
947:
917:
915:
914:
909:
892:
891:
876:
875:
860:
859:
829:
827:
826:
821:
804:
803:
788:
787:
757:
755:
754:
749:
732:
731:
701:
699:
698:
693:
645:
643:
642:
637:
608:
606:
605:
600:
589:
588:
578:
551:
550:
534:
532:
531:
526:
515:
514:
498:
496:
495:
490:
488:
487:
478:
463:
461:
460:
455:
453:
452:
432:
430:
429:
424:
413:
412:
393:
391:
390:
385:
382:
380:
376:
375:
366:
355:
354:
344:
334:
333:
323:
243:Bessel functions
235:Friedrich Bessel
192:
185:
178:
96:m-derived filter
46:Chebyshev filter
19:
18:
6202:
6201:
6197:
6196:
6195:
6193:
6192:
6191:
6167:
6166:
6140:
6135:
6134:
6126:
6118:
6114:
6105:
6103:
6096:
6092:
6085:
6069:
6065:
6034:
6030:
5995:
5991:
5984:
5968:
5961:
5954:
5938:
5931:
5922:
5920:
5912:
5911:
5907:
5898:
5896:
5888:
5887:
5883:
5874:
5872:
5865:www.nuhertz.com
5859:
5858:
5854:
5845:
5843:
5835:
5834:
5830:
5822:
5816:
5809:
5796:(44): 487–490.
5785:
5779:
5775:
5766:
5764:
5757:"Bessel Filter"
5755:
5754:
5750:
5745:
5732:Elliptic filter
5712:Bessel function
5708:
5689:
5661:
5658:
5657:
5656:
5633:
5630:
5629:
5628:
5612:
5609:
5608:
5586:
5583:
5582:
5560:
5557:
5556:
5534:
5531:
5530:
5513:
5509:
5507:
5504:
5503:
5486:
5482:
5480:
5477:
5476:
5442:
5438:
5436:
5422:
5418:
5416:
5393:
5390:
5389:
5349:
5345:
5333:
5329:
5317:
5313:
5312:
5299:
5295:
5283:
5279:
5275:
5273:
5260:
5252:
5250:
5230:
5227:
5226:
5188:
5184:
5174:
5167:
5163:
5153:
5151:
5147:
5091:
5088:
5087:
5059:
5056:
5055:
5028:
5020:
5000:
4998:
4995:
4994:
4961:
4957:
4945:
4941:
4929:
4925:
4919:
4911:
4891:
4874:
4871:
4870:
4830:
4827:
4826:
4797:
4794:
4793:
4771:
4768:
4767:
4726:
4722:
4710:
4706:
4705:
4700:
4683:
4680:
4679:
4653:
4649:
4647:
4644:
4643:
4640:low-pass filter
4620:
4602:
4601:
4596:
4575:
4574:
4550:
4546:
4534:
4530:
4526:
4521:
4515:
4514:
4506:
4498:
4484:
4478:
4477:
4472:
4468:
4464:
4463:
4439:
4435:
4423:
4419:
4415:
4410:
4401:
4400:
4364:
4360:
4351:
4347:
4329:
4325:
4316:
4312:
4308:
4303:
4284:
4274:
4270:
4250:
4237:
4236:
4231:
4219:
4215:
4205:
4198:
4194:
4190:
4186:
4185:
4180:
4176:
4175:
4170:
4154:
4153:
4148:
4136:
4132:
4120:
4116:
4104:
4100:
4076:
4072:
4063:
4059:
4047:
4043:
4031:
4014:
3985:
3984:
3966:
3962:
3953:
3949:
3937:
3933:
3929:
3924:
3890:
3889:
3884:
3870:
3856:
3848:
3834:
3829:
3821:
3807:
3803:
3799:
3798:
3793:
3777:
3773:
3769:
3760:
3743:
3735:
3734:
3729:
3705:
3701:
3689:
3685:
3684:
3679:
3659:
3657:
3654:
3653:
3648:
3594:
3590:
3569:
3565:
3559:
3542:
3511:
3507:
3486:
3482:
3465:
3462:
3461:
3437:
3428:
3424:
3423:
3419:
3410:
3393:
3387:
3384:
3383:
3366:
3349:
3343:
3340:
3339:
3319:
3315:
3313:
3310:
3309:
3266:
3263:
3262:
3234:
3231:
3230:
3205:
3202:
3201:
3161:
3158:
3157:
3133:
3129:
3118:
3115:
3114:
3094:
3090:
3079:
3076:
3075:
3047:
3044:
3043:
3027:
3018:
3014:
3000:
2998:
2995:
2994:
2991:
2960:
2957:
2956:
2931:
2928:
2927:
2910:
2906:
2904:
2901:
2900:
2883:
2879:
2877:
2874:
2873:
2856:
2852:
2850:
2847:
2846:
2829:
2825:
2823:
2820:
2819:
2797:
2793:
2785:
2773:
2769:
2757:
2753:
2744:
2740:
2731:
2727:
2713:
2704:
2700:
2692:
2683:
2679:
2667:
2663:
2654:
2650:
2641:
2637:
2632:
2620:
2616:
2607:
2603:
2601:
2598:
2597:
2574:
2570:
2568:
2565:
2564:
2544:
2540:
2528:
2524:
2515:
2511:
2502:
2498:
2496:
2493:
2492:
2472:
2468:
2466:
2463:
2462:
2446:
2428:
2424:
2409:
2405:
2400:
2398:
2395:
2394:
2370:
2361:
2357:
2356:
2352:
2343:
2326:
2320:
2317:
2316:
2299:
2282:
2276:
2273:
2272:
2252:
2248:
2246:
2243:
2242:
2223:
2219:
2214:
2211:
2210:
2193:
2189:
2187:
2184:
2183:
2154:
2150:
2135:
2131:
2129:
2126:
2125:
2108:
2104:
2102:
2099:
2098:
2081:
2077:
2075:
2072:
2071:
2054:
2050:
2048:
2045:
2044:
2028:
2025:
2024:
1996:
1993:
1992:
1975:
1971:
1969:
1966:
1965:
1922:
1919:
1918:
1890:
1887:
1886:
1861:
1858:
1857:
1817:
1814:
1813:
1792:
1788:
1786:
1783:
1782:
1763:
1754:
1750:
1729:
1725:
1711:
1709:
1706:
1705:
1689:
1680:
1676:
1662:
1660:
1657:
1656:
1653:Newton's method
1650:
1638:Newton's method
1620:
1616:
1614:
1611:
1610:
1595:
1594:
1576:
1572:
1560:
1556:
1547:
1543:
1528:
1524:
1515:
1511:
1507:
1502:
1483:
1482:
1478:
1462:
1454:
1442:
1441:
1417:
1413:
1401:
1397:
1396:
1391:
1384:
1368:
1366:
1363:
1362:
1344:
1341:
1340:
1323:
1319:
1317:
1314:
1313:
1288:
1285:
1284:
1267:
1263:
1261:
1258:
1257:
1253:
1167:
1163:
1162:
1139:
1137:
1128:
1124:
1122:
1119:
1118:
1092:
1088:
1082:
1078:
1072:
1061:
1039:
1035:
1033:
1030:
1029:
991:
987:
975:
971:
959:
955:
943:
939:
924:
921:
920:
887:
883:
871:
867:
855:
851:
836:
833:
832:
799:
795:
783:
779:
764:
761:
760:
727:
723:
708:
705:
704:
668:
665:
664:
631:
628:
627:
615:
584:
580:
568:
546:
542:
540:
537:
536:
510:
506:
504:
501:
500:
483:
479:
474:
469:
466:
465:
448:
444:
442:
439:
438:
408:
404:
402:
399:
398:
371:
367:
362:
350:
346:
345:
329:
325:
324:
322:
305:
302:
301:
291:low-pass filter
278:
250:Gaussian filter
228:audio crossover
196:
167:
166:
142:
134:
133:
129:mm'-type filter
86:
76:
75:
61:Gaussian filter
36:
24:
17:
12:
11:
5:
6200:
6190:
6189:
6184:
6179:
6177:Linear filters
6165:
6164:
6159:
6153:
6147:
6139:
6138:External links
6136:
6133:
6132:
6112:
6090:
6083:
6063:
6044:(6): 659–664.
6028:
5989:
5982:
5959:
5952:
5929:
5905:
5881:
5852:
5828:
5807:
5773:
5747:
5746:
5744:
5741:
5740:
5739:
5734:
5729:
5724:
5719:
5714:
5707:
5704:
5688:
5685:
5671:
5668:
5665:
5643:
5640:
5637:
5616:
5596:
5593:
5590:
5570:
5567:
5564:
5544:
5541:
5538:
5516:
5512:
5489:
5485:
5473:
5472:
5461:
5458:
5455:
5450:
5445:
5441:
5435:
5430:
5425:
5421:
5415:
5412:
5409:
5406:
5403:
5400:
5397:
5379:
5378:
5366:
5360:
5357:
5352:
5348:
5344:
5341:
5336:
5332:
5328:
5325:
5320:
5316:
5310:
5307:
5302:
5298:
5294:
5291:
5286:
5282:
5278:
5272:
5266:
5263:
5258:
5255:
5249:
5246:
5243:
5240:
5237:
5234:
5216:
5215:
5203:
5199:
5191:
5187:
5183:
5180:
5177:
5170:
5166:
5162:
5159:
5156:
5150:
5146:
5143:
5140:
5137:
5134:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5095:
5069:
5066:
5063:
5042:
5036:
5032:
5027:
5023:
5019:
5016:
5013:
5010:
5007:
5003:
4991:
4990:
4978:
4972:
4969:
4964:
4960:
4956:
4953:
4948:
4944:
4940:
4937:
4932:
4928:
4923:
4918:
4914:
4910:
4907:
4904:
4901:
4898:
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4890:
4887:
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4881:
4878:
4852:
4849:
4846:
4843:
4840:
4837:
4834:
4810:
4807:
4804:
4801:
4781:
4778:
4775:
4764:
4763:
4752:
4746:
4743:
4740:
4737:
4734:
4729:
4725:
4721:
4718:
4713:
4709:
4704:
4699:
4696:
4693:
4690:
4687:
4664:
4661:
4656:
4652:
4619:
4616:
4595:
4592:
4589:
4586:
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4578:
4570:
4567:
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4558:
4553:
4549:
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4533:
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4525:
4518:
4513:
4509:
4504:
4501:
4497:
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4479:
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4465:
4459:
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4434:
4431:
4426:
4422:
4418:
4414:
4409:
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4402:
4396:
4393:
4390:
4387:
4384:
4381:
4378:
4375:
4372:
4367:
4363:
4359:
4354:
4350:
4346:
4343:
4340:
4337:
4332:
4328:
4324:
4319:
4315:
4311:
4307:
4302:
4297:
4294:
4291:
4283:
4280:
4277:
4273:
4269:
4266:
4263:
4260:
4256:
4253:
4249:
4246:
4243:
4240:
4238:
4230:
4227:
4222:
4218:
4214:
4207: dB atten
4204:
4201:
4197:
4193:
4191:
4187:
4179:
4177:
4169:
4166:
4163:
4160:
4157:
4155:
4147:
4144:
4139:
4135:
4131:
4128:
4123:
4119:
4115:
4112:
4107:
4103:
4099:
4096:
4093:
4090:
4087:
4084:
4079:
4075:
4071:
4066:
4062:
4058:
4055:
4050:
4046:
4042:
4039:
4034:
4029:
4026:
4023:
4020:
4017:
4013:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3986:
3980:
3977:
3974:
3969:
3965:
3961:
3956:
3952:
3948:
3945:
3940:
3936:
3932:
3928:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3891:
3883:
3880:
3877:
3869:
3866:
3863:
3859:
3854:
3851:
3847:
3844:
3841:
3837:
3827:
3824:
3820:
3817:
3814:
3806:
3804:
3800:
3792:
3789:
3784:
3780:
3776:
3772:
3768:
3763:
3758:
3755:
3752:
3749:
3746:
3742:
3738:
3736:
3725:
3722:
3719:
3716:
3713:
3708:
3704:
3700:
3697:
3692:
3688:
3683:
3678:
3675:
3672:
3669:
3666:
3663:
3661:
3647:
3644:
3643:
3642:
3641:
3640:
3634:
3628:
3617:
3614:
3611:
3608:
3603:
3600:
3597:
3593:
3589:
3586:
3583:
3578:
3575:
3572:
3568:
3562:
3557:
3554:
3551:
3548:
3545:
3541:
3537:
3534:
3531:
3528:
3525:
3520:
3517:
3514:
3510:
3506:
3503:
3500:
3495:
3492:
3489:
3485:
3481:
3478:
3475:
3472:
3469:
3458:
3444:
3440:
3434:
3431:
3427:
3422:
3418:
3413:
3408:
3405:
3402:
3399:
3396:
3392:
3369:
3364:
3361:
3358:
3355:
3352:
3348:
3325:
3322:
3318:
3306:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3250:
3247:
3244:
3241:
3238:
3218:
3215:
3212:
3209:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3141:
3136:
3132:
3128:
3125:
3122:
3102:
3097:
3093:
3089:
3086:
3083:
3063:
3060:
3057:
3054:
3051:
3030:
3026:
3021:
3017:
3013:
3010:
3007:
3003:
2990:
2987:
2973:
2970:
2967:
2964:
2944:
2941:
2938:
2935:
2913:
2909:
2886:
2882:
2859:
2855:
2832:
2828:
2805:
2800:
2796:
2792:
2788:
2784:
2781:
2776:
2772:
2768:
2765:
2760:
2756:
2752:
2747:
2743:
2739:
2734:
2730:
2726:
2723:
2720:
2716:
2712:
2707:
2703:
2699:
2695:
2691:
2686:
2682:
2678:
2675:
2670:
2666:
2662:
2657:
2653:
2649:
2644:
2640:
2635:
2631:
2628:
2623:
2619:
2615:
2610:
2606:
2592:
2591:
2577:
2573:
2552:
2547:
2543:
2539:
2536:
2531:
2527:
2523:
2518:
2514:
2510:
2505:
2501:
2489:
2475:
2471:
2449:
2445:
2442:
2439:
2436:
2431:
2427:
2423:
2420:
2417:
2412:
2408:
2403:
2391:
2377:
2373:
2367:
2364:
2360:
2355:
2351:
2346:
2341:
2338:
2335:
2332:
2329:
2325:
2302:
2297:
2294:
2291:
2288:
2285:
2281:
2258:
2255:
2251:
2239:
2226:
2222:
2218:
2196:
2192:
2171:
2168:
2165:
2162:
2157:
2153:
2149:
2146:
2143:
2138:
2134:
2111:
2107:
2084:
2080:
2057:
2053:
2032:
2012:
2009:
2006:
2003:
2000:
1978:
1974:
1962:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1906:
1903:
1900:
1897:
1894:
1874:
1871:
1868:
1865:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1795:
1791:
1766:
1762:
1757:
1753:
1749:
1746:
1743:
1740:
1737:
1732:
1728:
1724:
1721:
1718:
1714:
1692:
1688:
1683:
1679:
1675:
1672:
1669:
1665:
1649:
1646:
1623:
1619:
1590:
1587:
1584:
1579:
1575:
1571:
1568:
1563:
1559:
1555:
1550:
1546:
1542:
1539:
1536:
1531:
1527:
1523:
1518:
1514:
1510:
1506:
1501:
1496:
1493:
1490:
1481:
1477:
1474:
1471:
1468:
1465:
1463:
1460:
1457:
1453:
1450:
1447:
1444:
1443:
1437:
1434:
1431:
1428:
1425:
1420:
1416:
1412:
1409:
1404:
1400:
1395:
1390:
1387:
1385:
1383:
1380:
1377:
1374:
1371:
1370:
1348:
1326:
1322:
1301:
1298:
1295:
1292:
1270:
1266:
1252:
1249:
1248:
1247:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1176:
1173:
1170:
1166:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1136:
1131:
1127:
1112:
1111:
1100:
1095:
1091:
1085:
1081:
1075:
1070:
1067:
1064:
1060:
1056:
1053:
1050:
1047:
1042:
1038:
1023:
1022:
1011:
1008:
1005:
1002:
999:
994:
990:
986:
983:
978:
974:
970:
967:
962:
958:
954:
951:
946:
942:
937:
934:
931:
928:
918:
907:
904:
901:
898:
895:
890:
886:
882:
879:
874:
870:
866:
863:
858:
854:
849:
846:
843:
840:
830:
819:
816:
813:
810:
807:
802:
798:
794:
791:
786:
782:
777:
774:
771:
768:
758:
747:
744:
741:
738:
735:
730:
726:
721:
718:
715:
712:
702:
691:
688:
685:
681:
678:
675:
672:
635:
614:
611:
598:
595:
592:
587:
583:
577:
574:
571:
567:
563:
560:
557:
554:
549:
545:
524:
521:
518:
513:
509:
486:
482:
477:
473:
451:
447:
422:
419:
416:
411:
407:
395:
394:
379:
374:
370:
365:
361:
358:
353:
349:
343:
340:
337:
332:
328:
321:
318:
315:
312:
309:
277:
274:
268:, and flatter
224:phase response
198:
197:
195:
194:
187:
180:
172:
169:
168:
165:
164:
159:
154:
149:
143:
141:Simple filters
140:
139:
136:
135:
132:
131:
126:
121:
115:
112:Lattice filter
109:
103:
98:
93:
87:
82:
81:
78:
77:
74:
73:
68:
63:
58:
53:
48:
43:
37:
32:
31:
28:
27:
15:
9:
6:
4:
3:
2:
6199:
6188:
6185:
6183:
6180:
6178:
6175:
6174:
6172:
6163:
6160:
6157:
6154:
6151:
6148:
6145:
6142:
6141:
6125:
6124:
6116:
6101:
6094:
6086:
6084:9780849385728
6080:
6076:
6075:
6067:
6059:
6055:
6051:
6047:
6043:
6039:
6032:
6024:
6020:
6016:
6012:
6008:
6004:
6000:
5993:
5985:
5979:
5975:
5974:
5966:
5964:
5955:
5953:0-7923-7373-1
5949:
5945:
5944:
5936:
5934:
5919:
5915:
5909:
5895:
5891:
5885:
5871:on 2020-01-11
5870:
5866:
5862:
5856:
5842:
5838:
5832:
5821:
5814:
5812:
5803:
5799:
5795:
5791:
5784:
5777:
5763:on 2013-01-24
5762:
5758:
5752:
5748:
5738:
5735:
5733:
5730:
5728:
5725:
5723:
5720:
5718:
5715:
5713:
5710:
5709:
5703:
5700:
5698:
5694:
5691:Although the
5684:
5669:
5666:
5663:
5641:
5638:
5635:
5614:
5591:
5588:
5568:
5565:
5562:
5542:
5539:
5536:
5514:
5510:
5487:
5483:
5459:
5456:
5453:
5448:
5443:
5439:
5433:
5428:
5423:
5419:
5413:
5410:
5407:
5401:
5395:
5388:
5387:
5386:
5384:
5383:Taylor series
5364:
5358:
5355:
5350:
5346:
5342:
5339:
5334:
5330:
5326:
5323:
5318:
5314:
5308:
5305:
5300:
5296:
5292:
5289:
5284:
5280:
5276:
5270:
5264:
5261:
5256:
5253:
5247:
5244:
5238:
5232:
5225:
5224:
5223:
5221:
5201:
5197:
5189:
5185:
5181:
5178:
5175:
5168:
5164:
5160:
5157:
5154:
5148:
5144:
5141:
5138:
5129:
5126:
5120:
5114:
5111:
5108:
5105:
5099:
5093:
5086:
5085:
5084:
5083:The phase is
5081:
5067:
5064:
5061:
5040:
5034:
5030:
5025:
5014:
5011:
5005:
4976:
4970:
4967:
4962:
4958:
4954:
4951:
4946:
4942:
4938:
4935:
4930:
4926:
4921:
4916:
4905:
4902:
4896:
4888:
4882:
4876:
4869:
4868:
4867:
4864:
4850:
4847:
4844:
4841:
4838:
4835:
4832:
4824:
4808:
4805:
4802:
4799:
4779:
4776:
4773:
4750:
4744:
4741:
4738:
4735:
4732:
4727:
4723:
4719:
4716:
4711:
4707:
4702:
4697:
4691:
4685:
4678:
4677:
4676:
4662:
4659:
4654:
4650:
4641:
4632:
4624:
4615:
4598: dB Gain
4593:
4590:
4587:
4584:
4581:
4568:
4565:
4562:
4559:
4556:
4551:
4547:
4543:
4540:
4535:
4531:
4527:
4523:
4511:
4502:
4495:
4489:
4482:
4457:
4454:
4451:
4448:
4445:
4440:
4436:
4432:
4429:
4424:
4420:
4416:
4412:
4407:
4405:
4394:
4391:
4388:
4382:
4379:
4376:
4370:
4365:
4361:
4352:
4348:
4344:
4341:
4335:
4330:
4326:
4317:
4313:
4305:
4300:
4295:
4292:
4289:
4281:
4278:
4275:
4267:
4261:
4258:
4254:
4247:
4241:
4228:
4225:
4220:
4216:
4212:
4202:
4199:
4195:
4167:
4164:
4161:
4158:
4145:
4142:
4137:
4133:
4129:
4126:
4121:
4117:
4113:
4110:
4105:
4101:
4097:
4094:
4088:
4085:
4082:
4077:
4073:
4069:
4064:
4060:
4056:
4053:
4048:
4044:
4040:
4032:
4027:
4024:
4021:
4018:
4015:
4011:
4007:
4004:
4001:
3995:
3989:
3978:
3975:
3972:
3967:
3963:
3959:
3954:
3950:
3946:
3943:
3938:
3934:
3930:
3926:
3921:
3915:
3912:
3906:
3900:
3894:
3881:
3878:
3875:
3867:
3864:
3861:
3852:
3845:
3839:
3825:
3818:
3812:
3790:
3787:
3782:
3778:
3774:
3770:
3766:
3761:
3756:
3753:
3750:
3747:
3744:
3740:
3723:
3720:
3717:
3714:
3711:
3706:
3702:
3698:
3695:
3690:
3686:
3681:
3676:
3670:
3664:
3651:
3638:
3637:
3635:
3633:
3629:
3612:
3609:
3601:
3598:
3595:
3591:
3584:
3576:
3573:
3570:
3566:
3560:
3555:
3552:
3549:
3546:
3543:
3539:
3535:
3529:
3526:
3518:
3515:
3512:
3508:
3501:
3493:
3490:
3487:
3483:
3479:
3473:
3467:
3459:
3442:
3438:
3432:
3429:
3425:
3420:
3416:
3411:
3406:
3403:
3400:
3397:
3394:
3390:
3367:
3362:
3359:
3356:
3353:
3350:
3346:
3323:
3320:
3316:
3307:
3289:
3286:
3280:
3274:
3268:
3245:
3242:
3236:
3213:
3207:
3184:
3181:
3175:
3169:
3163:
3155:
3154:
3153:
3134:
3130:
3126:
3120:
3095:
3091:
3087:
3081:
3058:
3055:
3049:
3019:
3015:
3011:
3005:
2986:
2968:
2962:
2939:
2933:
2911:
2907:
2884:
2880:
2857:
2853:
2830:
2826:
2816:
2798:
2794:
2790:
2786:
2774:
2770:
2766:
2758:
2754:
2745:
2741:
2732:
2728:
2721:
2714:
2705:
2701:
2697:
2684:
2680:
2676:
2668:
2664:
2655:
2651:
2642:
2638:
2626:
2621:
2617:
2613:
2608:
2604:
2595:
2575:
2571:
2545:
2541:
2537:
2529:
2525:
2516:
2512:
2503:
2499:
2490:
2473:
2469:
2440:
2437:
2429:
2425:
2418:
2410:
2406:
2392:
2375:
2371:
2365:
2362:
2358:
2353:
2349:
2344:
2339:
2336:
2333:
2330:
2327:
2323:
2300:
2295:
2292:
2289:
2286:
2283:
2279:
2256:
2253:
2249:
2240:
2224:
2220:
2216:
2194:
2190:
2166:
2163:
2155:
2151:
2144:
2136:
2132:
2109:
2105:
2082:
2078:
2055:
2051:
2030:
2007:
2004:
2001:
1976:
1972:
1963:
1945:
1942:
1936:
1930:
1924:
1901:
1898:
1892:
1869:
1863:
1840:
1837:
1831:
1825:
1819:
1811:
1810:
1809:
1793:
1789:
1779:
1755:
1751:
1747:
1744:
1738:
1730:
1726:
1722:
1716:
1681:
1677:
1673:
1667:
1654:
1645:
1643:
1639:
1621:
1617:
1608:
1588:
1585:
1582:
1577:
1573:
1569:
1566:
1561:
1553:
1548:
1544:
1537:
1534:
1529:
1521:
1516:
1512:
1504:
1499:
1494:
1491:
1488:
1475:
1469:
1466:
1464:
1458:
1451:
1445:
1435:
1432:
1429:
1426:
1423:
1418:
1414:
1410:
1407:
1402:
1398:
1393:
1388:
1386:
1378:
1372:
1360:
1346:
1324:
1320:
1296:
1290:
1268:
1264:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1200:
1194:
1191:
1188:
1182:
1179:
1174:
1171:
1168:
1164:
1158:
1152:
1149:
1146:
1143:
1134:
1129:
1125:
1117:
1116:
1115:
1098:
1093:
1089:
1083:
1079:
1073:
1068:
1065:
1062:
1058:
1054:
1048:
1040:
1036:
1028:
1027:
1026:
1009:
1006:
1003:
1000:
997:
992:
988:
984:
981:
976:
972:
968:
965:
960:
956:
952:
949:
944:
940:
935:
932:
929:
926:
919:
905:
902:
899:
896:
893:
888:
884:
880:
877:
872:
868:
864:
861:
856:
852:
847:
844:
841:
838:
831:
817:
814:
811:
808:
805:
800:
796:
792:
789:
784:
780:
775:
772:
769:
766:
759:
745:
742:
739:
736:
733:
728:
724:
719:
716:
713:
710:
703:
689:
686:
683:
679:
676:
673:
670:
663:
662:
661:
659:
655:
647:
633:
624:
619:
610:
593:
585:
581:
575:
569:
561:
555:
547:
543:
519:
511:
507:
484:
480:
475:
471:
449:
445:
436:
433:is a reverse
417:
409:
405:
372:
368:
363:
359:
351:
347:
338:
330:
326:
319:
313:
307:
300:
299:
298:
296:
292:
282:
273:
271:
267:
261:
259:
255:
254:step response
251:
246:
244:
240:
236:
231:
229:
225:
221:
217:
216:linear filter
213:
212:Bessel filter
209:
205:
193:
188:
186:
181:
179:
174:
173:
171:
170:
163:
160:
158:
155:
153:
150:
148:
145:
144:
138:
137:
130:
127:
125:
122:
119:
116:
113:
110:
107:
106:Zobel network
104:
102:
99:
97:
94:
92:
89:
88:
85:
80:
79:
72:
69:
67:
64:
62:
59:
57:
56:Bessel filter
54:
52:
49:
47:
44:
42:
39:
38:
35:
30:
29:
26:
23:Linear analog
21:
20:
6122:
6115:
6104:. Retrieved
6093:
6073:
6066:
6041:
6037:
6031:
6006:
6002:
5992:
5972:
5942:
5921:. Retrieved
5917:
5908:
5897:. Retrieved
5893:
5884:
5873:. Retrieved
5869:the original
5864:
5855:
5844:. Retrieved
5840:
5831:
5793:
5789:
5776:
5765:. Retrieved
5761:the original
5751:
5701:
5690:
5627:: the first
5474:
5380:
5217:
5082:
4992:
4865:
4825:of poles at
4765:
4637:
3652:
3649:
2992:
2817:
2596:
2593:
1780:
1651:
1642:root finding
1609:
1361:
1254:
1113:
1024:
651:
396:
288:
262:
247:
238:
232:
211:
201:
55:
6158:— C.R. Bond
6152:— C.R. Bond
5727:Comb filter
5220:group delay
3382:, by using
2315:, by using
270:group delay
266:phase delay
220:group delay
204:electronics
6171:Categories
6106:2022-05-14
5923:2022-05-14
5899:2022-05-14
5875:2022-05-14
5846:2022-05-14
5767:2022-05-14
5743:References
5054:occurs at
3809:Find
3261:to obtain
1917:to obtain
1781:To obtain
1640:, or with
162:RLC filter
120:(all-pass)
114:(all-pass)
6146:— Nuhertz
6058:0018-9324
6023:0165-1684
5664:ω
5639:−
5595:∞
5589:ω
5563:ω
5537:ω
5511:ω
5484:ω
5457:⋯
5440:ω
5420:ω
5414:−
5402:ω
5347:ω
5331:ω
5315:ω
5297:ω
5281:ω
5265:ω
5257:ϕ
5248:−
5239:ω
5186:ω
5179:−
5165:ω
5161:−
5158:ω
5145:
5130:ω
5115:
5109:−
5100:ω
5094:ϕ
5062:ω
5015:ω
4959:ω
4943:ω
4927:ω
4906:ω
4883:ω
4845:±
4839:−
4806:−
4651:ω
4591:−
4560:76.157016
4544:154.66376
4528:130.87478
4449:76.157016
4433:154.66376
4417:130.87478
4383:5.0771344
4380:×
4349:5.0771344
4345:×
4314:5.0771344
4290:ω
4229:5.0771344
4217:ω
4200:−
4196:ω
4168:5.0771344
4111:−
4070:−
4041:−
4008:−
3960:−
3931:−
3913:−
3876:ω
3865:−
3610:−
3536:−
3527:−
3287:−
3243:−
3182:−
3131:ω
3092:ω
3056:−
3016:ω
2908:ω
2881:ω
2854:ω
2827:ω
2795:ω
2771:ω
2767:−
2742:ω
2698:−
2681:ω
2677:−
2652:ω
2627:−
2618:ω
2605:ω
2572:ω
2542:ω
2538:−
2513:ω
2470:ω
2438:−
2221:ω
2191:ω
2164:−
1943:−
1899:−
1838:−
1790:ω
1752:ω
1745:−
1727:ω
1678:ω
1618:ω
1574:ω
1545:ω
1513:ω
1489:ω
1321:ω
1265:ω
1226:…
1192:−
1172:−
1150:−
1059:∑
1037:θ
582:θ
573:→
544:θ
508:θ
481:ω
446:ω
406:θ
369:ω
348:θ
327:θ
289:A Bessel
258:overshoot
230:systems.
157:LC filter
152:RL filter
147:RC filter
5706:See also
4821:, and a
4503:′
4255:′
3853:′
3826:′
1459:′
499:. Since
5687:Digital
4618:Example
1339:to the
6081:
6056:
6021:
5980:
5950:
5142:arctan
4851:1.7544
4842:1.8389
4809:2.3222
4474:Check:
4146:222.75
2993:Since
1114:where
397:where
6127:(PDF)
5823:(PDF)
5786:(PDF)
5068:1.756
4642:with
3460:Find
1991:when
646:plane
623:poles
6079:ISBN
6054:ISSN
6019:ISSN
5978:ISBN
5948:ISBN
5502:and
5449:1125
5381:The
5218:The
4130:0.45
4114:0.06
4098:0.01
3791:0.01
210:, a
206:and
6046:doi
6011:doi
5798:doi
5429:225
5359:225
5309:225
5222:is
5112:arg
4971:225
4675:is
4585:0.1
4089:225
4005:225
3979:225
3927:225
3229:by
3156:If
1885:by
1812:If
1010:945
1001:945
985:420
969:105
906:105
897:105
566:lim
202:In
6173::
6052:.
6042:18
6040:.
6017:.
6007:88
6005:.
6001:.
5962:^
5932:^
5916:.
5892:.
5863:.
5839:.
5810:^
5794:96
5792:.
5788:.
5683:.
5343:45
5293:45
5176:15
5155:15
4955:45
4922:15
4745:15
4736:15
4703:15
4594:20
4569:15
4524:15
4458:15
4413:15
4395:15
4377:15
4306:15
4282:20
4203:20
4074:45
3964:45
3868:20
3783:10
3775:20
3771:10
3724:15
3715:15
3682:15
3443:10
3421:10
2376:10
2354:10
2110:10
2097:,
2070:,
1644:.
1589:15
1570:15
1505:15
1436:15
1427:15
1394:15
953:15
881:45
865:10
818:15
809:15
609:.
297::
6109:.
6087:.
6060:.
6048::
6025:.
6013::
5986:.
5956:.
5926:.
5902:.
5878:.
5849:.
5825:.
5804:.
5800::
5770:.
5670:0
5667:=
5642:1
5636:n
5615:n
5592:=
5569:0
5566:=
5543:0
5540:=
5515:4
5488:2
5460:.
5454:+
5444:8
5434:+
5424:6
5411:1
5408:=
5405:)
5399:(
5396:D
5365:.
5356:+
5351:2
5340:+
5335:4
5327:6
5324:+
5319:6
5306:+
5301:2
5290:+
5285:4
5277:6
5271:=
5262:d
5254:d
5245:=
5242:)
5236:(
5233:D
5202:.
5198:)
5190:2
5182:6
5169:3
5149:(
5139:=
5136:)
5133:)
5127:j
5124:(
5121:H
5118:(
5106:=
5103:)
5097:(
5065:=
5041:,
5035:2
5031:1
5026:=
5022:|
5018:)
5012:j
5009:(
5006:H
5002:|
4977:.
4968:+
4963:2
4952:+
4947:4
4939:6
4936:+
4931:6
4917:=
4913:|
4909:)
4903:j
4900:(
4897:H
4893:|
4889:=
4886:)
4880:(
4877:G
4848:j
4836:=
4833:s
4803:=
4800:s
4780:0
4777:=
4774:s
4751:,
4742:+
4739:s
4733:+
4728:2
4724:s
4720:6
4717:+
4712:3
4708:s
4698:=
4695:)
4692:s
4689:(
4686:H
4663:1
4660:=
4655:0
4588:=
4582:=
4577:|
4566:+
4563:j
4557:+
4552:2
4548:j
4541:+
4536:3
4532:j
4517:|
4512:=
4508:|
4500:)
4496:j
4493:(
4490:H
4486:|
4455:+
4452:s
4446:+
4441:2
4437:s
4430:+
4425:3
4421:s
4408:=
4392:+
4389:s
4386:)
4374:(
4371:+
4366:2
4362:s
4358:)
4353:2
4342:6
4339:(
4336:+
4331:3
4327:s
4323:)
4318:3
4310:(
4301:=
4296:1
4293:=
4279:=
4276:A
4272:)
4268:s
4265:(
4262:H
4259:=
4252:)
4248:s
4245:(
4242:H
4226:=
4221:c
4213:=
4165:j
4162:=
4159:R
4143:+
4138:2
4134:s
4127:+
4122:4
4118:s
4106:6
4102:s
4095:=
4092:)
4086:+
4083:s
4078:2
4065:4
4061:s
4057:6
4054:+
4049:6
4045:s
4038:(
4033:2
4028:h
4025:t
4022:i
4019:r
4016:a
4012:B
4002:=
3999:)
3996:s
3993:(
3990:P
3976:+
3973:s
3968:2
3955:4
3951:s
3947:6
3944:+
3939:6
3935:s
3922:=
3919:)
3916:s
3910:(
3907:H
3904:)
3901:s
3898:(
3895:H
3886:.
3882:1
3879:=
3862:=
3858:|
3850:)
3846:s
3843:(
3840:H
3836:|
3823:)
3819:s
3816:(
3813:H
3788:=
3779:/
3767:=
3762:2
3757:h
3754:t
3751:i
3748:r
3745:a
3741:B
3721:+
3718:s
3712:+
3707:2
3703:s
3699:6
3696:+
3691:3
3687:s
3677:=
3674:)
3671:s
3668:(
3665:H
3616:)
3613:S
3607:(
3602:n
3599:e
3596:d
3592:H
3588:)
3585:S
3582:(
3577:n
3574:e
3571:d
3567:H
3561:2
3556:h
3553:t
3550:i
3547:r
3544:a
3540:B
3533:)
3530:S
3524:(
3519:m
3516:u
3513:n
3509:H
3505:)
3502:S
3499:(
3494:m
3491:u
3488:n
3484:H
3480:=
3477:)
3474:S
3471:(
3468:P
3439:/
3433:B
3430:d
3426:A
3417:=
3412:2
3407:h
3404:t
3401:i
3398:r
3395:a
3391:B
3368:2
3363:h
3360:t
3357:i
3354:r
3351:a
3347:B
3324:B
3321:d
3317:A
3305:.
3293:)
3290:s
3284:(
3281:H
3278:)
3275:s
3272:(
3269:H
3249:)
3246:s
3240:(
3237:H
3217:)
3214:s
3211:(
3208:H
3188:)
3185:s
3179:(
3176:H
3173:)
3170:s
3167:(
3164:H
3140:)
3135:a
3127:j
3124:(
3121:H
3101:)
3096:a
3088:j
3085:(
3082:H
3062:)
3059:s
3053:(
3050:H
3029:|
3025:)
3020:a
3012:j
3009:(
3006:H
3002:|
2972:)
2969:s
2966:(
2963:G
2943:)
2940:s
2937:(
2934:H
2912:c
2885:a
2858:a
2831:a
2804:)
2799:a
2791:d
2787:/
2783:]
2780:)
2775:a
2764:(
2759:2
2755:H
2751:)
2746:a
2738:(
2733:2
2729:H
2725:[
2722:d
2719:(
2715:/
2711:)
2706:2
2702:B
2694:|
2690:)
2685:a
2674:(
2669:2
2665:H
2661:)
2656:a
2648:(
2643:2
2639:H
2634:|
2630:(
2622:a
2614:=
2609:a
2576:a
2551:)
2546:a
2535:(
2530:2
2526:H
2522:)
2517:a
2509:(
2504:2
2500:H
2474:a
2448:|
2444:)
2441:s
2435:(
2430:2
2426:H
2422:)
2419:s
2416:(
2411:2
2407:H
2402:|
2372:/
2366:B
2363:d
2359:A
2350:=
2345:2
2340:h
2337:t
2334:i
2331:r
2328:a
2324:B
2301:2
2296:h
2293:t
2290:i
2287:r
2284:a
2280:B
2257:B
2254:d
2250:A
2225:a
2217:j
2195:a
2170:)
2167:s
2161:(
2156:2
2152:H
2148:)
2145:s
2142:(
2137:2
2133:H
2106:s
2083:6
2079:s
2056:2
2052:s
2031:4
2011:)
2008:2
2005:+
2002:n
1999:(
1977:n
1973:s
1961:.
1949:)
1946:s
1940:(
1937:H
1934:)
1931:s
1928:(
1925:H
1905:)
1902:s
1896:(
1893:H
1873:)
1870:s
1867:(
1864:H
1844:)
1841:s
1835:(
1832:H
1829:)
1826:s
1823:(
1820:H
1794:c
1765:|
1761:)
1756:c
1748:j
1742:(
1739:H
1736:)
1731:c
1723:j
1720:(
1717:H
1713:|
1691:|
1687:)
1682:c
1674:j
1671:(
1668:H
1664:|
1622:c
1586:+
1583:s
1578:c
1567:+
1562:2
1558:)
1554:s
1549:c
1541:(
1538:6
1535:+
1530:3
1526:)
1522:s
1517:c
1509:(
1500:=
1495:1
1492:=
1480:)
1476:s
1473:(
1470:H
1467:=
1456:)
1452:s
1449:(
1446:H
1433:+
1430:s
1424:+
1419:2
1415:s
1411:6
1408:+
1403:3
1399:s
1389:=
1382:)
1379:s
1376:(
1373:H
1347:s
1325:c
1300:)
1297:s
1294:(
1291:H
1269:c
1235:.
1232:n
1229:,
1223:,
1220:1
1217:,
1214:0
1211:=
1208:k
1201:!
1198:)
1195:k
1189:n
1186:(
1183:!
1180:k
1175:k
1169:n
1165:2
1159:!
1156:)
1153:k
1147:n
1144:2
1141:(
1135:=
1130:k
1126:a
1099:,
1094:k
1090:s
1084:k
1080:a
1074:n
1069:0
1066:=
1063:k
1055:=
1052:)
1049:s
1046:(
1041:n
1007:+
1004:s
998:+
993:2
989:s
982:+
977:3
973:s
966:+
961:4
957:s
950:+
945:5
941:s
936::
933:5
930:=
927:n
903:+
900:s
894:+
889:2
885:s
878:+
873:3
869:s
862:+
857:4
853:s
848::
845:4
842:=
839:n
815:+
812:s
806:+
801:2
797:s
793:6
790:+
785:3
781:s
776::
773:3
770:=
767:n
746:3
743:+
740:s
737:3
734:+
729:2
725:s
720::
717:2
714:=
711:n
690:1
687:+
684:s
680::
677:1
674:=
671:n
634:s
597:)
594:x
591:(
586:n
576:0
570:x
562:=
559:)
556:0
553:(
548:n
523:)
520:0
517:(
512:n
485:0
476:/
472:1
450:0
421:)
418:s
415:(
410:n
378:)
373:0
364:/
360:s
357:(
352:n
342:)
339:0
336:(
331:n
320:=
317:)
314:s
311:(
308:H
191:e
184:t
177:v
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