16.9. Filter Coefficient Tables

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16.9. Filter Coefficient Tables

The following tables contain the coefficients for the three filter types; Bessel, Butterworth, and Tschebyscheff. The Tschebyscheff tables (Table 16.11) are split into categories for the following passband ripples: 0.5, 1, 2, and 3 dB.

The table headers consist of the following quantities:

n is the filter order,

i is the number of the partial filter,

a_i and b_i are the filter coefficients,

Table 16.6

Bessel Coefficients

n	i	a_i	b_i	k_i = f_Ci/f_C	Q_i
1	1	1.0000	0.0000	1.000	–
2	1	1.3617	0.6180	1.000	0.58
3	1	0.7560	0.0000	1.323	–
3	2	0.9996	0.4772	1.414	0.69
4	1	1.3397	0.4889	0.978	0.52
4	2	0.7743	0.3890	1.797	0.81
5	1	0.6656	0.0000	1.502	–
	2	1.1402	0.4128	1.184	0.56
	3	0.6216	0.3245	2.138	0.92
6	1	1.2217	0.3887	1.063	0.51
	2	0.9686	0.3505	1.431	0.61
	3	0.5131	0.2756	2.447	1.02
7	1	0.5937	0.0000	1.648	–
	2	1.0944	0.3395	1.207	0.53
	3	0.8304	0.3011	1.695	0.66
	4	0.4332	0.2381	2.731	1.13
8	1	1.1112	0.3162	1.164	0.51
	2	0.9754	0.2979	1.381	0.56
	3	0.7202	0.2621	1.963	0.71
	4	0.3728	0.2087	2.992	1.23
9	1	0.5386	0.0000	1.857	–
	2	1.0244	0.2834	1.277	0.52
	3	0.8710	0.2636	1.574	0.59
	4	0.6320	0.2311	2.226	0.76
	5	0.3257	0.1854	3.237	1.32
10	1	1.0215	0.2650	1.264	0.50
	2	0.9393	0.2549	1.412	0.54
	3	0.7815	0.2351	1.780	0.62
	4	0.5604	0.2059	2.479	0.81
	5	0.2883	0.1665	3.466	1.42

Table 16.7

Butterworth Coefficients

n	i	a_i	b_i	k_i = f_Ci/f_C	Q_i
1	1	1.0000	0.0000	1.000	–
2	1	1.4142	1.0000	1.000	0.71
3	1	1.0000	0.0000	1.000	–
3	2	1.0000	1.0000	1.272	1.00
4	1	1.8478	1.0000	0.719	0.54
4	2	0.7654	1.0000	1.390	1.31
5	1	1.0000	0.0000	1.000	–
	2	1.6180	1.0000	0.859	0.62
	3	0.6180	1.0000	1.448	1.62
6	1	1.9319	1.0000	0.676	0.52
	2	1.4142	1.0000	1.000	0.71
	3	0.5176	1.0000	1.479	1.93
7	1	1.0000	0.0000	1.000	–
	2	1.8019	1.0000	0.745	0.55
	3	1.2470	1.0000	1.117	0.80
	4	0.4450	1.0000	1.499	2.25
8	1	1.9616	1.0000	0.661	0.51
	2	1.6629	1.0000	0.829	0.60
	3	1.1111	1.0000	1.206	0.90
	4	0.3902	1.0000	1.512	2.56
9	1	1.0000	0.0000	1.000	–
	2	1.8794	1.0000	0.703	0.53
	3	1.5321	1.0000	0.917	0.65
	4	1.0000	1.0000	1.272	1.00
	5	0.3473	1.0000	1.521	2.88
10	1	1.9754	1.0000	0.655	0.51
	2	1.7820	1.0000	0.756	0.56
	3	1.4142	1.0000	1.000	0.71
	4	0.9080	1.0000	1.322	1.10
	5	0.3129	1.0000	1.527	3.20

Table 16.8

Tschebyscheff Coefficients for 0.5 dB

n	i	a_i	b_i	k_i = f_Ci/f_C	Q_i
1	1	1.0000	0.0000	1.000	–
2	1	1.3614	1.3827	1.000	0.86
3	1	1.8636	0.0000	0.537	–
3	2	0.0640	1.1931	1.335	1.71
4	1	2.6282	3.4341	0.538	0.71
4	2	0.3648	1.1509	1.419	2.94
5	1	2.9235	0.0000	0.342	–
	2	1.3025	2.3534	0.881	1.18
	3	0.2290	1.0833	1.480	4.54
Table Continued

n	i	a_i	b_i	k_i = f_Ci/f_C	Q_i
6	1	3.8645	6.9797	0.366	0.68
	2	0.7528	1.8573	1.078	1.81
	3	0.1589	1.0711	1.495	6.51
7	1	4.0211	0.0000	0.249	–
	2	1.8729	4.1795	0.645	1.09
	3	0.4861	1.5676	1.208	2.58
	4	0.1156	1.0443	1.517	8.84
8	1	5.1117	11.960	0.276	0.68
	1	5.1117	7	0.276	0.68
	2	1.0639	2.9365	0.844	1.61
	3	0.3439	1.4206	1.284	3.47
	4	0.0885	1.0407	1.521	11.53
9	1	5.1318	0.0000	0.195	–
	2	2.4283	6.6307	0.506	1.06
	3	0.6839	2.2908	0.989	2.21
	4	0.2559	1.3133	1.344	4.48
	5	0.0695	1.0272	1.532	14.58
10	1	6.3648	18.369	0.222	0.67
	1	6.3648	5	0.222	0.67
	2	1.3582	4.3453	0.689	1.53
	3	0.4822	1.9440	1.091	2.89
	4	0.1994	1.2520	1.381	5.61
	5	0.0563	1.0263	1.533	17.99

Table 16.9

Tschebyscheff Coefficients for 1 dB Pa

n	i	a_i	b_i	k_i = f_Ci/f_C	Q_i
1	1	1.0000	0.0000	1.000	–
2	1	1.3022	1.5515	1.000	0.96
3	1	2.2156	0.0000	0.451	–
3	2	0.5442	1.2057	1.353	2.02
4	1	2.5904	4.1301	0.540	0.78
4	2	0.3039	1.1697	1.417	3.56
5	1	3.5711	0.0000	0.280	–
	2	1.1280	2.4896	0.894	1.40
	3	0.1872	1.0814	1.486	5.56
6	1	3.8437	8.5529	0.366	0.76
	2	0.6292	1.9124	1.082	2.20
	3	0.1296	1.0766	1.493	8.00
7	1	4.9520	0.0000	0.202	–
	2	1.6338	4.4899	0.655	1.30
	3	0.3987	1.5834	1.213	3.16
	4	0.0937	1.0432	1.520	10.90
Table Continued

n	i	a_i	b_i	k_i = f_Ci/f_C	Q_i
8	1	5.1019	14.760	0.276	0.75
	1	5.1019	8	0.276	0.75
	2	0.8916	3.0426	0.849	1.96
	3	0.2806	1.4334	1.285	4.27
	4	0.0717	1.0432	1.520	14.24
9	1	6.3415	0.0000	0.158	–
	2	2.1252	7.1711	0.514	1.26
	3	0.5624	2.3278	0.994	2.71
	4	0.2076	1.3166	1.346	5.53
	5	0.0562	1.0258	1.533	18.03
10	1	6.3634	22.746	0.221	0.75
	1	6.3634	8	0.221	0.75
	2	1.1399	4.5167	0.694	1.86
	3	0.3939	1.9665	1.093	3.56
	4	0.1616	1.2569	1.381	6.94
	5	0.0455	1.0277	1.532	22.26

Table 16.10

Tschebyscheff Coefficients for 2 dB Pa

n	i	a_i	b_i	k_i = f_Ci/f_C	Q_i
1	1	1.0000	0.0000	1.000	–
2	1	1.1813	1.7775	1.000	1.13
3	1	2.7994	0.0000	0.357	–
3	2	0.4300	1.2036	1.378	2.55
4	1	2.4025	4.9862	0.550	0.93
4	2	0.2374	1.1896	1.413	4.59
5	1	4.6345	0.0000	0.216	–
	2	0.9090	2.6036	0.908	1.78
	3	0.1434	1.0750	1.493	7.23
6	1	3.5880	10.464	0.373	0.90
	1	3.5880	8	0.373	0.90
	2	0.4925	1.9622	1.085	2.84
	3	0.0995	1.0826	1.491	10.46
7	1	6.4760	0.0000	0.154	–
	2	1.3258	4.7649	0.665	1.65
	3	0.3067	1.5927	1.218	4.12
	4	0.0714	1.0384	1.523	14.28
8	1	4.7743	18.151	0.282	0.89
	1	4.7743	0	0.282	0.89
	2	0.6991	3.1353	0.853	2.53
	3	0.2153	1.4449	1.285	5.58
	4	0.0547	1.0461	1.518	18.39
Table Continued

n	i	a_i	b_i	k_i = f_Ci/f_C	Q_i
9	1	8.3198	0.0000	0.120	–
	2	1.7299	7.6580	0.522	1.60
	3	0.4337	2.3549	0.998	3.54
	4	0.1583	1.3174	1.349	7.25
	5	0.0427	1.0232	1.536	23.68
10	1	5.9618	28.037	0.226	0.89
	1	5.9618	6	0.226	0.89
	2	0.8947	4.6644	0.697	2.41
	3	0.3023	1.9858	1.094	4.66
	4	0.1233	1.2614	1.380	9.11
	5	0.0347	1.0294	1.531	29.27

Table 16.11

Tschebyscheff Coefficients for 3 dB Pa

n	i	a_i	b_i	k_i = f_Ci/f_C	Q_i
1	1	1.0000	0.0000	1.000	–
2	1	1.0650	1.9305	1.000	1.30
3	1	3.3496	0.0000	0.299	–
3	2	0.3559	1.1923	1.396	3.07
4	1	2.1853	5.5339	0.557	1.08
4	2	0.1964	1.2009	1.410	5.58
5	1	5.6334	0.0000	0.178	–
	2	0.7620	2.6530	0.917	2.14
	3	0.1172	1.0686	1.500	8.82
6	1	3.2721	11.677	0.379	1.04
	1	3.2721	3	0.379	1.04
	2	0.4077	1.9873	1.086	3.46
	3	0.0815	1.0861	1.489	12.78
7	1	7.9064	0.0000	0.126	–
	2	1.1159	4.8963	0.670	1.98
	3	0.2515	1.5944	1.222	5.02
	4	0.0582	1.0348	1.527	17.46
8	1	4.3583	20.294	0.286	1.03
	1	4.3583	8	0.286	1.03
	2	0.5791	3.1808	0.855	3.08
	3	0.1765	1.4507	1.285	6.83
	4	0.0448	1.0478	1.517	22.87
9	1	10.175	0.0000	0.098	–
	1	9	0.0000	0.098	–
	2	1.4585	7.8971	0.526	1.93
	3	0.3561	2.3651	1.001	4.32
	4	0.1294	1.3165	1.351	8.87
	5	0.0348	1.0210	1.537	29.00
10	1	5.4449	31.378	0.230	1.03
	1	5.4449	8	0.230	1.03
	2	0.7414	4.7363	0.699	2.94
	3	0.2479	1.9952	1.094	5.70
	4	0.1008	1.2638	1.380	11.15
	5	0.0283	1.0304	1.530	35.85

Table 16.12

All-Pass Coefficients

n	i	a_i	b_i	f_i/f_C	Q_i	T_gr0
1	1	0.6436	0.0000	1.554	–	0.204
1	1	0.6436	0.0000	1.554	–	9
2	1	1.6278	0.8832	1.064	0.58	0.518
2	1	1.6278	0.8832	1.064	0.58	1
3	1	1.1415	0.0000	0.876	–	0.843
	1	1.1415	0.0000	0.876	–	7
	2	1.5092	1.0877	0.959	0.69
4	1	2.3370	1.4878	0.820	0.52	1.173
	1	2.3370	1.4878	0.820	0.52	8
	2	1.3506	1.1837	0.919	0.81
5	1	1.2974	0.0000	0.771	–	1.506
	1	1.2974	0.0000	0.771	–	0
	2	2.2224	1.5685	0.798	0.56
	3	1.2116	1.2330	0.901	0.92
6	1	2.6117	1.7763	0.750	0.51	1.839
	1	2.6117	1.7763	0.750	0.51	5
	2	2.0706	1.6015	0.790	0.61
	3	1.0967	1.2596	0.891	1.02
7	1	1.3735	0.0000	0.728	–	2.173
	1	1.3735	0.0000	0.728	–	7
	2	2.5320	1.8169	0.742	0.53
	3	1.9211	1.6116	0.788	0.66
	4	1.0023	1.2743	0.886	1.13
8	1	2.7541	1.9420	0.718	0.51	2.508
	1	2.7541	1.9420	0.718	0.51	4
	2	2.4174	1.8300	0.739	0.56
	3	1.7850	1.6101	0.788	0.71
	4	0.9239	1.2822	0.883	1.23
9	1	1.4186	0.0000	0.705	–	2.843
	1	1.4186	0.0000	0.705	–	4
	2	2.6979	1.9659	0.713	0.52
	3	2.2940	1.8282	0.740	0.59
	4	1.6644	1.6027	0.790	0.76
	5	0.8579	1.2862	0.882	1.32
10	1	2.8406	2.0490	0.699	0.50	3.178
	1	2.8406	2.0490	0.699	0.50	6
	2	2.6120	1.9714	0.712	0.54
	3	2.1733	1.8184	0.742	0.62
	4	1.5583	1.5923	0.792	0.81
	5	0.8018	1.2877	0.881	1.42

k_i is the ratio of the corner frequency of a partial filter, f_Ci, to the corner frequency of the overall filter, f_C. This ratio is used to determine the unity-gain bandwidth of the op amp, as well as to simplify the test of a filter design by measuring f_Ci and comparing it to f_C,

Q_i is the quality factor of the partial filter,

f_i/f_C ratio is used for test purposes of the all-pass filters, where f_i is the frequency, at which the phase is 180 degrees for a second-order filter, respectively 90 degrees for a first-order all-pass.

T_gr0 is the normalized group delay of the overall all-pass filter.

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Table of Contents for 16.9. Filter Coefficient Tables

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16.9. Filter Coefficient Tables

Table of Contents for
16.9. Filter Coefficient Tables