AN-649: Using the Analog Devices Active Filter Design Tool

Introduction

The Analog Devices Active Filter Design Tool assists the engineer in designing all-pole active filters.

The filter design process consists of two steps. In Step 1, the response of the filter is determined, meaning the attenuation and/or phase response of the filter is defined. In Step 2, the topology of the filter—how it is built—is defined. This application note is intended to help in Step 1. Several different standard responses are discussed, and their attenuation, group delay, step response, and impulse response are presented. The filter tool is then employed to design the filter. An example is provided.

Standard Responses

Many transfer functions may be used to satisfy the attenuation and/or phase requirements of a particular filter. The one that is selected will depend on the particular system. The importance of frequency domain response versus time domain response must be determined. Also, both of these might be traded off against filter complexity, and therefore cost.

Butterworth Filter

The Butterworth filter is the best compromise between attenuation and phase response. It has no ripple in the pass band or the stop band; because of this, it is sometimes called a maximally flat filter. The Butterworth filter achieves its flatness at the expense of a relatively wide transition region from pass band to stop band, with average transient characteristics.

The values of the elements of the Butterworth filter are more practical and less critical than many other filter types. The frequency response, group delay, impulse response, and step response are shown in Figure 1. The pole locations and corresponding ωo and α terms are tabulated in Table II.

Chebyshev Filter

The Chebyshev (or Chevyshev, Tschebychev, Tschebyscheff, or Tchevysheff, depending on the translation from Russian) filter has a smaller transition region than the same-order Butterworth filter, at the expense of ripples in its pass band. This filter gets its name from the Chebyshev criterion, which minimizes the height of the maximum ripple.

Chebyshev filters have 0 dB relative attenuation at dc. Odd-order filters have an attenuation band that extends from 0 dB to the ripple value. Even-order filters have a gain equal to the pass-band ripple. The number of cycles of ripple in the pass band is equal to the order of the filter.

The Chebyshev filters are typically normalized so that the edge of the ripple band is at ωo = 1.

The 3 dB bandwidth is given by

equation 1

This is tabulated in Table I.

Figures 2 through 6 show the frequency response, group delay, impulse response, and step response for the various Chebyshev filters. The pole locations and corresponding ωo and α terms are tabulated in Tables III through VII.

Table I. Chebyshev Cutoff Frequency to –3 dB Frequency
Order 0.01dB 0.1dB 0.25dB 0.5dB 1dB
2 3.30362 1.93432 1.59814 1.38974 1.21763
3 1.87718 1.38899 1.25289 1.16749 1.09487
4 1.46690 1.21310 1.13977 1.09310 1.05300
5 1.29122 1.13472 1.08872 1.05926 1.03381
6 1.19941 1.09293 1.06134 1.04103 1.02344
7 1.14527 1.06800 1.04495 1.03009 1.01721
8 1.11061 1.05193 1.03435 1.02301 1.01316
9 1.08706 1.04095 1.02711 1.01817 1.01040
10 1.07033 1.03313 1.02194 1.01471 1.00842

Bessel Filter

Butterworth filters have fairly good amplitude and transient behavior. The Chebyshev filters improve on the amplitude response at the expense of transient behavior. The Bessel filter is optimized to obtain better transient response due to a linear phase (i.e., constant delay) in the pass band. This means that there will be relatively poor frequency response (less amplitude discrimination).

The frequency response, group delay, impulse response, and step response for the Bessel filter are shown in Figure 7. The pole locations and corresponding ωo and α terms are tabulated in Table VIII.

Linear Phase With Equiripple Error

The linear phase filter offers linear phase response in the pass band, over a wider range than the Bessel, and superior attenuation far from cutoff. This is accomplished by letting the phase response have ripples, similar to the amplitude ripples of the Chebyshev. As the ripple is increased, the region of constant delay extends further into the stop band. This will also cause the group delay to develop ripples, since it is the derivative of the phase response. The step response will show slightly more overshoot than the Bessel and the impulse response will show a bit more ringing.

The frequency response, group delay, impulse response, and step response for equiripple filters with error of 0.05° and 0.5° are shown in Figures 8 and 9, respectively. The pole locations and corresponding ωo and α terms are tabulated in Tables IX and X.

Guassian-to-6 dB And Guassian-to-12 dB Filter

Gaussian-to-6 dB and Gaussian-to-12 dB filters are a compromise between a Chebyshev filter and a Gaussian filter, which is similar to a Bessel filter. A transitional filter has nearly linear phase shift and smooth, monotonic roll-off in the pass band. Above the pass band and especially at higher values of n, there is a break point beyond which the attenuation increases dramatically compared to that of the Bessel.

The Gaussian-to-6 dB filter has better transient response in the pass band than does the Butterworth filter. Beyond the breakpoint, which occurs at ωo = 1.5, the roll-off is similar to that of the Butterworth filter.

The Gaussian-to-12 dB filter’s transient response in the pass band is much better than that of the Butterworth filter. Beyond the 12 dB breakpoint, which occurs at ωo = 2, the attenuation is less than that of the Butterworth filter.

The frequency response, group delay, impulse response, and step response for Gaussian-to-6 dB and Gaussianto-12 dB filters are shown in Figures 10 and 11, respectively. The pole locations and corresponding ωo and α terms are tabulated in Tables XI and XII.

Using The Prototype Response Curves

The response curves and design tables for several of the low-pass prototypes of the all-pole responses discussed previously are now cataloged. All of the curves are normalized to a –3 dB cutoff frequency of 1 Hz. This allows direct comparison of the various responses. In all cases, the amplitude response for the 2- through 10-pole cases for the frequency range of 0.1 Hz to 10 Hz will be shown. Then, a detail of the 0.1 Hz to 2 Hz pass band will be shown. The group delay from 0.1 Hz to 10 Hz, the impulse response, and the step response from 0 seconds to 5 seconds will also be shown.

Curves must be denormalized if they are to be used to determine the response of real life filters. In the case of the amplitude responses, this is accomplished by simply multiplying the frequency axis by the desired cutoff frequency, FC. To denormalize the group delay curves, divide the delay axis by 2π FC and multiply the frequency axis by FC. Denormalize the step response by dividing the time axis by 2π FC. Denormalize the impulse response by dividing the time axis by 2π FC and multiplying the amplitude axis by 2π FC.

For a high-pass filter, simply invert the frequency axis for the amplitude response. In transforming a low-pass filter into a high-pass filter, the transient behavior is not preserved. Zverev provides a computational method for calculating these responses.

In transforming a low-pass into a narrow-band bandpass, the 0 Hz axis is moved to the center frequency, F0. It stands to reason that the response of the band-pass case around the center frequency would then match the low-pass response around 0 Hz. The frequency response curve of a low-pass filter actually mirrors itself around 0 Hz, although we generally do not concern ourselves with negative frequency.

To denormalize the group delay curve for a band-pass filter, divide the delay axis by πBW, where BW is the 3 dB bandwidth in Hz. Then, multiply the frequency axis by BW/2. In general, the delay of the band-pass filter at F0 will be twice the delay of the low-pass prototype with the same bandwidth at 0 Hz. This is due to the fact that the low-pass to band-pass transformation results in a filter with order 2n, even though it is typically referred to as having the same order as the low-pass filter we derive it from. This approximation holds for narrow-band filters. As the bandwidth of the filter is increased, some distortion of the curve occurs. The delay becomes less symmetrical, peaking below F0.

The envelope of the response of a band-pass filter resembles the step response of the low-pass prototype. More exactly, it is almost identical to the step response of a low-pass filter with half the bandwidth. To determine the envelope response of the band-pass filter, divide the time axis of the low-pass prototype’s step response by πBW, where BW is the 3 dB bandwidth. The previous discussions of overshoot, ringing, and so on can now be applied to the carrier envelope.

The envelope of the response of a narrow-band band-pass filter to a short burst (where the burst width is much less than the rise time of the band-pass filter’s denormalized step response) of carrier can be determined by denormalizing the impulse response of the low-pass prototype. To do this, multiply the amplitude axis and divide the time axis by πBW, where BW is the 3 dB bandwidth. It is assumed that the carrier frequency is high enough so that many cycles occur during the burst interval.

While the group delay, step, and impulse curves cannot be used directly to predict the distortion to the waveform caused by the filter, they are a useful figure of merit when used to compare filters.

Figure 1. Butterworth Response

Figure 2. 0.01 dB Chebyshev Response

Figure 3. 0.1 dB Chebyshev Response

Figure 4. 0.25 dB Chebyshev Response

Figure 5. 0.5 dB Chebyshev Response

Figure 6. 1 dB Chebyshev Response

Figure 7. Bessel Response

Figure 8. Linear Phase with Equiripple Error of 0.05° Response

Figure 9. Linear Phase with Equiripple Error of 0.5° Response

Figure 10. Gaussian-to-12 dB Response

Figure 11. Gaussian-to-6 dB Response

Table II. Butterworth Design
Order Section Real Part Imaginary Part F0 α Q −3 dB
Frequency
Peaking Frequency Peaking Level
2 1 0.7071 0.7071 1.0000 1.4142 0.7071 1.0000    
3 1 0.5000 0.8660 1.0000 1.0000 1.0000   0.7071 1.2493
  2 1.0000   1.0000     1.0000    
4 1 0.9239 0.3827 1.0000 1.8478 0.5412 0.7195    
  2 0.3827 0.9239 1.0000 0.7654 1.3065   0.8409 3.0102
5 1 0.8090 0.5878 1.0000 1.6180 0.6180 0.8588    
  2 0.3090 0.9511 1.0000 0.6180 1.6182   0.8995 4.6163
  3 1.0000   1.0000
  1.0000    
6 1 0.9659 0.2588 1.0000 1.9319 0.5176 0.6758    
  2 0.7071 0.7071 1.0000 1.4142 0.7071 1.0000    
  3 0.2588 0.9659 1.0000 0.5176 1.9319   0.9306 6.0210
7 1 0.9010 0.4339 1.0000 1.8019 0.5550 0.7449    
  2 0.6235 0.7818 1.0000 1.2470 0.8019   0.4717 0.2204
  3 0.2225 0.9749 1.0000 0.4450 2.2471   0.9492 7.2530
  4 1.0000   1.0000     1.0000    
8 1 0.9808 0.1951 1.0000 1.9616 0.5098 0.6615    
  2 0.8315 0.5556 1.0000 1.6629 0.6013 0.8295    
  3 0.5556 0.8315 1.0000 1.1112 0.9000   0.6186 0.6876
  4 0.1951 0.9808 1.0000 0.3902 2.5628   0.9612 8.3429
9 1 0.9397 0.3420 1.0000 1.8794 0.5321 0.7026    
  2 0.7660 0.6428 1.0000 1.5320 0.6527 0.9172    
  3 0.5000 0.8660 1.0000 1.0000 1.0000   0.7071 1.2493
  4 0.1737 0.9848 1.0000 0.3474 2.8785   0.9694 9.3165
  5 1.0000   1.0000     1.0000    
 10 1 0.9877 0.1564 1.0000 1.9754 0.5062 0.6549    
  2 0.8910 0.4540 1.0000 1.7820 0.5612 0.7564    
  3 0.7071 0.7071 1.0000 1.4142 0.7071 1.0000    
  4 0.4540 0.8910 1.0000 0.9080 1.1013   0.7667 1.8407
  5 0.1564 0.9877 1.0000 0.3128 3.1970   0.9752 10.2023
Table III. 0.01 dB Chebyshev Design
Order Section Real Part Imaginary Part F0 α Q −3 dB
Frequency
Peaking Frequency Peaking Level
2 1 0.6743 0.7075 0.9774 1.3798 0.7247   0.2142 0.0100
3 1 0.4233 0.8663 0.9642 0.8780 1.1389   0.7558 2.0595
  2 0.8467   0.8467     0.8467    
4 1 0.6762 0.3828 0.7770 1.7405 0.5746 0.6069    
  2 0.2801 0.9241 0.9656 0.5801 1.7237   0.8806 5.1110
5 1 0.5120 0.5879 0.7796 1.3135 0.7613   0.2889 0.0827
  2 0.1956 0.9512 0.9711 0.4028 2.4824   0.9309 8.0772
  3 0.6328   0.6328
  0.6328    
6 1 0.5335 0.2588 0.5930 1.7995 0.5557 0.4425    
  2 0.3906 0.7072 0.8079 0.9670 1.0342   0.7204 3.4077
  3 0.1430 0.9660 0.9765 0.2929 3.4144   0.9689 10.7605
7 1 0.4393 0.4339 0.6175 1.4229 0.7028 0.6136    
  2 0.3040 0.7819 0.8389 0.7247 1.3798   0.7204 3.4077
  3 0.1085 0.9750 0.9810 0.2212 4.5208   0.9689 13.1578
  4 0.4876   0.4876     0.4876    
8 1 0.4268 0.1951 0.4693 1.8190 0.5498 0.3451    
  2 0.3168 0.5556 0.6396 0.9907 1.0094   0.4564 1.3041
  3  0.2418 0.8315 0.8659 0.5585 1.7906   0.7956 5.4126
  4 0.0849 0.9808 0.9845 0.1725 5.7978   0.9771 15.2977
9 1 0.3686 0.3420 0.5028 1.4661 0.6821 0.4866    
  2 0.3005 0.6428 0.7096 0.8470 1.1807   0.5682 2.3008
  3 0.1961 0.8661 0.8880 0.4417 2.2642   0.8436 7.3155
  4 0.0681 0.9848 0.9872 0.1380 7.2478   0.9824 17.2249
  5 0.3923   0.3923  
0.3923    
 10 1 0.3522 0.1564 0.3854 1.8279 0.5471 0.2814    
  2 0.3178 0.454 0.5542 1.1469 0.8719   0.3242 0.5412
  3 0.2522 0.7071 0.7507 0.6719 1.4884   0.6606 3.9742
  4 0.1619 0.891 0.9056 0.3576 2.7968   0.8762 9.0742
  5 0.0558 0.9877 0.9893 0.1128 8.8645   0.9861 18.9669
Table IV. 0.1 dB Chebyshev Design
Order Section Real Part Imaginary Part F0 α Q −3 dB
Frequency
Peaking Frequency Peaking Level
2 1 0.6104 0.7106 0.9368 1.3032 0.7673   0.3638 0.0999
3 1 0.3490 0.8684 0.9359 0.7458 1.3403   0.7952 3.1978
  2 0.6970   0.6970     0.6970    
4 1 0.2177 0.9254 0.9507 0.4580 2.1834   0.8994 7.0167
  2 0.5257 0.3833 0.6506 1.6160 0.6188 0.5596    
5 1 0.3842 0.5884 0.7027 1.0935 0.9145   0.4457 0.7662
  2 0.1468 0.9521 0.9634 0.3048 3.2812   0.9407 10.4226
  3 0.4749   0.4749     0.4749    
6 1 0.3916 0.2590 0.4695 1.6682 0.5995 0.3879    
  2 0.2867 0.7077 0.7636 0.7509  1.3316   0.6470 3.1478
  3 0.1049 0.9667 0.9724 0.2158 4.6348   0.9610 13.3714
7 1 0.3178 0.4341 0.5380 1.1814 0.8464   0.2957 0.4157
  2 0.2200 0.7823 0.8126 0.5414 1.8469   0.7507 5.6595
  3 0.0785 0.9755 0.9787 0.1604 6.2335   0.9723 15.9226
  4 0.3528   0.3528     0.3528    
8 1 0.3058 0.1952 0.3628 1.6858 0.5932 0.2956    
  2 0.2529 0.5558 0.6106 0.8283 1.2073   0.4949 2.4532
  3 0.1732 0.8319 0.8497 0.4077 2.4531   0.8137 7.9784
  4 0.0608 0.9812 0.9831 0.1237 8.0819   0.9793 18.1669
9 1 0.2622 0.3421 0.4310 1.2166 0.8219   0.2197 0.3037
  2 0.2137 0.6430 0.6776 0.6308 1.5854   0.6064 4.4576
  3 0.1395 0.8663 0.8775 0.3180 3.1450    0.8550 10.0636
  4 0.0485 0.9852 0.9864 0.0982 10.1795   0.9840 20.1650
  5 0.2790   0.2790     0.2790    
 10 1 0.2493 0.1564 0.2943 1.6942 0.5902 0.2382    
  2 0.2249 0.4541 05067 0.8876 1.1266   0.3945 1.9880
  3 0.1785 0.7073 0.7295  0.4894 2.0434   0.6844 6.4750
  4 0.1146 0.8913 0.8986 0.2551 3.9208   0.8839 11.9386
  5 0.0395 0.9880 0.9888 0.0799 12.5163   0.9872  21.9565
Table V. 0.25 dB Chebyshev Design
Order Section Real Part Imaginary Part F0 α Q −3 dB
Frequency
Peaking Frequency Peaking Level
2 1 0.5621 0.7154 0.9098 1.2356 0.8093   0.4425 0.2502
3 1 0.3062 0.8712 0.9234 0.6632 1.5079   0.8156 4.0734
  2 0.6124   0.6125     0.6124    
4 1 0.4501 0.3840 0.5916 1.5215 0.6572 0.5470    
  2 0.1865 0.9272 0.9458 0.3944 2.5356   0.9082 8.2538
5 1 0.3247 0.5892 0.6727 0.9653 1.0359   0.4917 1.4585
  2 0.1240 0.9533 0.9613 0.2580 3.8763   0.9452 11.8413
  3 0.4013   0.4013     0.4013    
6 1 0.3284 0.2593 0.4184 1.5697 0.6371 0.3730    
  2 0.2404 0.7083 0.7480 0.6428 1.5557   0.6663 4.3121
  3 0.0880 0.9675 0.9715 0.1811 5.5205   0.9635 14.8753
7 1 0.2652 0.4344 0.5090 1.0421 0.9596   0.3441 1.0173
  2 0.1835 0.7828 0.8040 0.4565 2.1908   0.7610 7.0443
  3 0.0655 0.9761 0.9783 0.1339 7.4679   0.9739 17.4835
  4 0.2944   0.2944     0.2944    
8 1 0.2543 0.1953 0.3206 1.5862 0.6304 0.2822    
  2 0.2156 0.5561 0.5964 0.7230 1.3832   0.5126 3.4258
  3 0.1441 0.8323 0.8447 0.3412 2.9309   0.8197 9.4683
  4 0.0506 0.9817 0.9830 0.1029 9.7173   0.9804 19.7624
9 1  0.2176 0.3423 0.4056 1.0730 0.9320   0.2642 0.8624
  2 0.1774 0.6433 0.6673 0.5317 1.8808   0.6184 5.8052
  3 0.1158 0.8667 0.8744 0.2649 3.7755   0.8589 11.6163
  4 0.0402 0.9856 0.9864 0.0815 12.2659   0.9848 21.7812
  5 0.2315   0.2315     0.2315    
 10 1 0.2065 0.1565 0.2591 1.5940 0.6274 0.2267    
  2 0.1863 0.4543 0.4910 0.7588 1.3178   0.4143 3.0721
  3 0.1478 0.7075 0.7228 0.4090 2.4451   0.6919 7.9515
  4 0.0949 0.8915 0.8965 0.2117 4.7236   0.8864 13.5344
  5 0.0327 0.9883 0.9888 0.0661 15.1199   0.9878 23.5957
Table VI. 0.5 dB Chebyshev Design
Order Section Real Part Imaginary Part F0 α Q −3 dB
Frequency
Peaking Frequency Peaking Level
2 1 0.5129 0.7225 1.2314 1.1577 0.8638   0.7072 0.5002
3 1 0.2683 0.8753 1.0688 0.5861 1.7061   0.9727 5.0301
  2 0.5366   0.6255
  0.6265    
4 1 0.3872 0.3850 0.5969 1.4182 0.7051 0.5951    
  2 0.1605 0.9297 1.0313 0.3402 2.9391   1.0010 9.4918
5 1 0.2767 0.5902 0.6905 0.8490 1.1779   0.5522 2.2849
  2 0.1057 0.9550 1.0178 0.2200 4.5451   1.0054 13.2037
  3 0.3420   0.3623     0.3623    
6 1 0.2784 0.2596 0.3963 1.4627 0.6836 0.3827    
  2 0.2037 0.7091 0.7680 0.5522 1.8109   0.7071 5.5025
  3 0.0746 0.9687 1.0114 0.1536 6.5119   1.0055 16.2998
7 1 0.2241 0.4349 0.5040 0.9161 1.0916   0.3839 1.7838
  2 0.1550 0.7836 0.8228 0.3881 2.5767   0.7912 8.3880
  3 0.0553 0.9771 1.0081 0.1130 8.8487   1.0049 18.9515
  4 0.2487   0.2562     0.2562    
8 1 0.2144 0.1955 0.2968 1.4779 0.6767 0.2835    
  2 0.1817 0.5565 0.5989 0.6208 1.6109   0.5381 4.5815
  3 0.1214 0.8328 0.8610 0.2885 3.4662   0.8429 10.8885
  4 0.0426 0.9824 1.0060 0.0867 11.5305   1.0041 21.2452
9 1  0.1831 0.3425 0.3954 0.9429 1.0605   0.2947 1.6023
  2 0.1493 0.6436 0.6727 0.4520 2.2126   0.6374 7.1258
  3 0.0974 0.8671 0.8884 0.2233 4.4779   0.8773 13.0759
  4 0.0338 0.9861 1.0046 0.0686 14.5829   1.0034 23.2820
  5 0.1949   0.1984     0.1984    
 10 1 0.1736 0.1566 0.2338 1.4851 0.6734 0.2221    
  2 0.1566 0.4545 0.4807 0.6515 1.5349   0.4267 4.2087
  3 0.1243 0.7078 0.7186 0.3459 2.8907   0.6968 9.3520
  4 0.0798 0.8919 0.8955 0.1782 5.6107   0.8883 15.0149
  5 0.0275 0.9887 0.9891 0.0556 17.9833   0.9883 25.1008
Table VII. 1 dB Chebyshev Design
Order Section Real Part Imaginary Part F0 α Q −3 dB
Frequency
Peaking Frequency Peaking Level
2 1 0.4508 0.7351 0.8623 1.0456 0.9564   0.5806 0.9995
3 1 0.2257 0.8822 0.9106 0.4957 2.0173   0.8528 6.3708
  2 0.4513   0.4513
  0.4513    
4 1 0.3199 0.3868 0.5019 1.2746 0.7845   0.2174  0.1557
  2 0.1325 0.9339 0.9433 0.2809 3.5594   0.9245 11.1142
5 1 0.2265 0.5918 0.6337 0.7149 1.3988   0.5467 3.5089
  2 0.0865 0.9575 0.9614 0.1800 5.5559   0.9536 14.9305
  3 0.2800   0.2800
  0.2800    
6 1 0.2268 0.2601 0.3451 1.3144 0.7608   0.1273 0.0813
  2 0.1550 0.7106 0.7273 0.4262 2.3462   0.6935 7.6090
  3 0.0608 0.9707 0.9726 0.1249 8.0036   0.9688 18.0827
7 1 0.1819 0.4354 0.4719 0.7710 1.2971   0.3956 2.9579
  2 0.1259 0.7846 0.7946 0.3169 3.1558   0.7744 10.0927
  3 0.0449 0.9785 0.9795 0.0918 10.8982   0.9775 20.7563
  4 0.2019   0.2019     0.2019    
8 1 0.1737 0.1956 0.2616 1.3280 0.7530   0.0899 0.0611
  2 0.1473 0.5571 0.5762 0.5112 1.9560   0.5373 6.1210
  3 0.0984 0.8337 0.8395 0.2344 4.2657   0.8279 12.6599
  4 0.0346 0.9836 0.9842 0.0702  14.2391   0.9830 23.0750
9 1 0.1482 0.3427 0.3734 0.7938 1.2597   0.3090 2.7498
  2 0.1208 0.6442 0.6554 0.3686 2.7129   0.6328 8.8187
  3 0.0788 0.8679 0.8715 0.1809 5.5268   0.8643 14.8852
  4 0.0274 0.9869 0.9873 0.0555 18.0226   0.9865 25.1197
  5 0.1577   0.1577     0.1577    
 10 1 0.1403 0.1567 0.2103 1.3341 0.7496   0.0698 0.0530
  2 0.1266 0.4548 0.4721 0.5363 1.8645   0.4368 5.7354
  3 0.1005 0.7084 0.7155 0.2809 3.5597   0.7012 11.1147
  4 0.0645 0.8926 0.8949 0.1441 6.9374   0.8903 16.8466
  5 0.0222 0.9895 0.9897 0.0449 22.2916   0.9893 26.9650
Table VIII. Bessel Design
Order Section Real Part Imaginary Part F0 α Q −3 dB
Frequency
Peaking Frequency Peaking Level
2 1   1.1050 0.6368 1.2754 1.7328 0.5771 1.0020    
3 1   1.0509  1.0025 1.4524 1.4471 0.6910 1.4185    
  2 1.3270   1.3270     1.3270  
4 1 1.3596 0.4071 1.4192 1.9160 0.5219 0.9705    
  2 0.9877 1.2476 1.5912 1.2414 0.8055   0.7622 0.2349
5 1 1.3851 0.7201 1.5611 1.7745 0.5635 1.1876    
  2 0.9606 1.4756 1.7607 1.0911 0.9165   1.1201 0.7768
  3 1.5069   1.5069     1.5069     
6 1 1.5735 0.3213 1.6060 1.9596 0.5103 1.0638    
  2 1.3836 0.9727 1.6913 1.6361 0.6112 1.4323    
  3  0.9318 1.6640 1.9071 0.9772 1.0234   1.3786 1.3851
7 1 1.6130 0.5896 1.7174 1.8784 0.5324 1.2074    
  2 1.3797 1.1923 1.8235 1.5132 0.6608 1.6964    
  3 0.9104 1.8375 2.0507 0.8879 1.1262   1.5961 1.9860
  4 1.6853   1.6853     1.6853    
8 1  1.7627 0.2737 1.7838 1.9763 0.5060 1.1675    
  2 0.8955 2.0044 2.1953 0.8158 1.2258   1.7932 2.5585
  3 1.3780 1.3926 1.9591 1.4067 0.7109   0.2011 0.0005
  4 1.6419 0.8256 1.8378 1.7868 0.5597 1.3849    
9 1 1.8081 0.5126 1.8794 1.9242 0.5197 1.2774    
  2 1.6532 1.0319 1.9488 1.6966 0.5894 1.5747    
  3 1.3683 1.5685 2.0815 1.3148 0.7606   0.7668 0.0807
  4 0.8788 2.1509 2.3235 0.7564 1.3220   1.9632 3.0949
  5 1.8575   1.8575     1.8575
 
 10 1 1.9335 0.2451 1.9490 1.9841 0.5040 1.2685    
  2 1.8467 0.7335 1.9870 1.8587 0.5380 1.4177    
  3 1.6661 1.2246 2.0678 1.6115 0.6205 1.7848    
  4 1.3648 1.7395 2.2110 1.2346 0.8100   1.0785 0.2531
  5 0.8686 2.2294 2.4580 0.7067 1.4150    2.1291 3.5944
Table IX. Equiripple with 0.05° Error Design
Order Section Real Part Imaginary Part F0 α Q −3 dB
Frequency
Peaking Frequency Peaking Level
2 1  1.0087 0.6680 1.2098 1.6675 0.5997 0.9999    
3 1 0.8541 1.0725 1.3710 1.2459 0.8026   0.6487 0.2232
  2 1.0459   1.0459     1.0459    
4 1 0.9648 0.4748 1.0753 1.7945 0.5573 0.8056    
  2 0.7448 1.4008 1.5865 0.9389 1.0650
1.1864 1.6286
5 1  0.8915 0.8733 1.2480 1.4287 0.6999 1.2351    
  2 0.6731 1.7085 1.8363 0.7331 1.3641   1.5703 3.3234
  3 0.9430   0.9430     0.9430    
6 1 0.8904 0.4111 0.9807 1.8158 0.5507 0.7229    
  2 0.8233 1.2179 1.4701 1.1201 0.8928   0.8975 0.6495
  3 0.6152 1.9810 2.0743 0.5932 1.6859   1.8831 4.9365
7 1 0.8425 0.7791 1.1475 1.4684 0.6810 1.1036    
  2  0.7708 1.5351 1.7177 0.8975 1.1143   1.3276 1.9162
  3 0.5727 2.2456 2.3175 0.4942 2.0233   2.1713 6.3948
  4 0.8615   0.8615     0.8615    
8 1 0.8195 0.3711 0.8996 1.8219 0.5489 0.6600    
  2 0.7930 1.1054 1.3604 1.1658 0.8578   0.7701 0.4705
  3 0.7213 1.8134 1.9516 0.7392 1.3528   1.6638 3.2627
  4 0.5341 2.4761 2.5330 0.4217 2.3713   2.4178 7.6973
9 1 0.7853 0.7125 1.0604 1.4812 0.6751 1.0102    
  2 0.7555 1.4127 1.6020 0.9432 1.0602   1.1937 1.6005
  3 0.6849 2.0854 2.1950 0.6241 1.6024   1.9667 4.5404
  4 0.5060 2.7133 2.7601 0.3667 2.7274   2.6657 8.8633
  5 0.7983   0.7983     0.7983    
 10 1 0.7592  0.3413 0.8324 1.8241 0.5482 0.6096    
  2 0.7467 1.0195 1.2637 1.1818 0.8462   0.6941 0.4145
  3 0.7159 1.6836 1.8295 0.7826 1.2778   1.5238 2.8507
  4 0.6475 2.3198 2.4085 0.5377 1.8598   2.2276 5.7152
  5 0.4777 2.9128 2.9517 0.3237 3.0895   2.8734 9.9130
Table X. Equiripple with 0.5° Error Design
Order Section Real Part Imaginary Part F0 α Q −3 dB
Frequency
Peaking Frequency Peaking Level
2 1  0.8590 0.6981 1.1069 1.5521 0.6443 1.000    
3 1 0.6969 1.1318 1.3292 1.0486 0.9536   0.8918 0.9836
  2 0.8257   0.8257     0.8257    
4 1 0.7448 0.5133 0.9045 1.6468 0.6072 0.7597    
  2 0.6037 1.4983 1.6154 0.7475 1.3379   1.3713 3.1817
5 1 0.6675 0.9401 1.1588 1.1693 0.8552   0.6518 0.4579
  2 0.5412 1.8256 1.9041 0.5684 1.7592   1.7435 5.2720
  3 0.7056   0.7056     0.7056    
6 1 0.6519 0.4374 0.7850 1.6608 0.6021 0.6522    
  2 0.6167 1.2963 1.4355 0.8592 1.1639   1.1402 2.2042
  3 0.4893 2.0982 2.1545 0.4542 2.2016   2.0404 7.0846
7 1 0.6190 0.8338 1.0385 1.1922 0.8388   0.5586 0.3798
  2 0.5816 1.6455 1.7453 0.6665 1.5004   1.5393 4.0353
  3 0.4598 2.3994 2.4431 0.3764 2.6567   2.3549 8.6433
  4 0.6283     0.6283   0.6283    
8 1 0.5791 0.3857 0.6958 1.6646 0.6007 0.5764    
  2 0.5665 1.1505 1.2824 0.8835 1.1319   1.0014 2.0187
  3 0.5303 1.8914 1.9643 0.5399 1.8521   1.8155 5.6819
  4 0.4148 2.5780 2.6112 0.3177 3.1475   2.5444 10.0703
9 1  0.5688 0.7595 0.9489 1.1989 0.8341   0.5033 0.3581
  2 0.5545 1.5089 1.6076 0.6899 1.4496   1.4033 3.7748
  3 0.5179 2.2329 2.2922 0.4519 2.2130   2.1720 7.1270
  4 0.4080 2.9028 2.9313 0.2784 3.5923   2.8740 11.1925
  5 0.5728   0.5728
  0.5728    
 10 1 0.5249  0.3487 0.6302 1.6659 0.6003 0.5215