4749:
4735:
4704:
155:
2419:) or the frequency domain (most common). Matched filters perform a cross-correlation between the input signal and a known pulse shape. The FIR convolution is a cross-correlation between the input signal and a time-reversed copy of the impulse response. Therefore, the matched filter's impulse response is "designed" by sampling the known pulse-shape and using those samples in reverse order as the coefficients of the filter.
4718:
127:
2568:, because the corresponding sinc function is zero at every other sample point (except the center one). The product with the window function does not alter the zeros, so almost half of the coefficients of the final impulse response are zero. An appropriate implementation of the FIR calculations can exploit that property to double the filter's efficiency.
5531:
3634:
3947:
2557:) and the height of the ripples, and thereby derive the frequency-domain parameters of an appropriate window function. Continuing backward to an impulse response can be done by iterating a filter design program to find the minimum filter order. Another method is to restrict the solution set to the parametric family of
2508:
response is corrected according to the desired specs, and the inverse DFT is then computed. In the time-domain, only the first N coefficients are kept (the other coefficients are set to zero). The process is then repeated iteratively: the DFT is computed once again, correction applied in the frequency domain and so on.
1599:
1179:
2507:
Equiripple FIR filters can be designed using the DFT algorithms as well. The algorithm is iterative in nature. The DFT of an initial filter design is computed using the FFT algorithm (if an initial estimate is not available, h=delta can be used). In the
Fourier domain, or DFT domain, the frequency
2561:, which provides closed form relationships between the time-domain and frequency domain parameters. In general, that method will not achieve the minimum possible filter order, but it is particularly convenient for automated applications that require dynamic, on-the-fly, filter design.
5336:
3275:
5581:(DFT) of the impulse response. And because of symmetry, filter design or viewing software often displays only the region. The magnitude plot indicates that the moving-average filter passes low frequencies with a gain near 1 and attenuates high frequencies, and is thus a crude
862:
5585:. The phase plot is linear except for discontinuities at the two frequencies where the magnitude goes to zero. The size of the discontinuities is π, representing a sign reversal. They do not affect the property of linear phase, as illustrated in the final figure.
5186:
388:
2405:
3673:
1414:
2540:. The result is a finite impulse response filter whose frequency response is modified from that of the IIR filter. Multiplying the infinite impulse by the window function in the time domain results in the frequency response of the IIR being
983:, especially when low frequency (relative to the sample rate) cutoffs are needed. However, many digital signal processors provide specialized hardware features to make FIR filters approximately as efficient as IIR for many applications.
1032:
2731:
2269:
1302:
4180:
5029:
2551:. The result of the frequency domain convolution is that the edges of the rectangle are tapered, and ripples appear in the passband and stopband. Working backward, one can specify the slope (or width) of the tapered region (
4459:
4688:
5195:. Zero frequency (DC) corresponds to (1, 0), positive frequencies advancing counterclockwise around the circle to the Nyquist frequency at (−1, 0). Two poles are located at the origin, and two zeros are located at
1962:
5526:{\displaystyle {\begin{aligned}H\left(e^{j\omega }\right)&={\frac {1}{3}}+{\frac {1}{3}}e^{-j\omega }+{\frac {1}{3}}e^{-j2\omega }\\&={\frac {1}{3}}e^{-j\omega }\left(1+2\cos(\omega )\right).\end{aligned}}}
3266:
3629:{\displaystyle {\text{MSE}}=\int _{-1/2}^{1/2}\sum _{n=0}^{k}s\cos(2\pi nF)\sum _{\tau =0}^{k}s\cos(2\pi \tau F)\,dF-2\int _{-1/2}^{1/2}\sum _{n=0}^{k}s\cos(2\pi nF)H_{d}\,dF+\int _{-1/2}^{1/2}H_{d}(F)^{2}\,dF}
3143:
4522:
5315:
5254:
721:
5341:
2444:
is commonly used to find an optimal equiripple set of coefficients. Here the user specifies a desired frequency response, a weighting function for errors from this response, and a filter order
199:
4043:
5043:
909:
Require no feedback. This means that any rounding errors are not compounded by summed iterations. The same relative error occurs in each calculation. This also makes implementation simpler.
194:
3942:{\displaystyle {\frac {\partial {\text{MSE}}}{\partial s}}=2\sum _{\tau =0}^{k}s\int _{-1/2}^{1/2}\cos(2\pi nF)\cos(2\pi \tau F)\,dF-2\int _{-1/2}^{1/2}H_{d}(F)^{2}\cos(2\pi nF)\,dF=0}
2277:
2954:
1825:
5575:
1703:
4872:
2544:
with the
Fourier transform (or DTFT) of the window function. If the window's main lobe is narrow, the composite frequency response remains close to that of the ideal IIR filter.
1363:
2163:
979:
The main disadvantage of FIR filters is that considerably more computation power in a general purpose processor is required compared to an IIR filter with similar sharpness or
4821:
1594:{\displaystyle H_{2\pi }(\omega )\ \triangleq \sum _{n=-\infty }^{\infty }h\cdot \left({e^{i\omega }}\right)^{-n}=\sum _{n=0}^{N}b_{n}\cdot \left({e^{i\omega }}\right)^{-n},}
1329:
2122:
2077:
1778:
2474:
coefficients that minimize the maximum deviation from the ideal. Intuitively, this finds the filter that is as close as possible to the desired response given that only
2826:
1396:
617:
590:
954:
5321:
1648:
2759:
1748:
1645:
1174:{\displaystyle \underbrace {{\mathcal {F}}\{x*h\}} _{Y(\omega )}=\underbrace {{\mathcal {F}}\{x\}} _{X(\omega )}\cdot \underbrace {{\mathcal {F}}\{h\}} _{H(\omega )}}
2789:
2028:
1999:
1881:
1625:
4556:
4904:
891:
97:
5706:
5679:
644:
554:
4242:
4213:
3667:
3015:
2986:
2871:
1854:
1723:
1018:
2498:
2472:
525:
2587:
2175:
690:
499:
479:
457:
426:
1184:
4049:
2536:
In the window design method, one first designs an ideal IIR filter and then truncates the infinite impulse response by multiplying it with a finite length
4918:
5756:
Rabiner, Lawrence R., and Gold, Bernard, 1975: Theory and
Application of Digital Signal Processing (Englewood Cliffs, New Jersey: Prentice-Hall, Inc.)
4248:
2500:
coefficients can be used. This method is particularly easy in practice since at least one text includes a program that takes the desired filter and
963:
by making the coefficient sequence symmetric. This property is sometimes desired for phase-sensitive applications, for example data communications,
5772:
A. E. Cetin, O.N. Gerek, Y. Yardimci, "Equiripple FIR filter design by the FFT algorithm," IEEE Signal
Processing Magazine, pp. 60–64, March 1997.
4564:
2415:
FIR filters are designed by finding the coefficients and filter order that meet certain specifications, which can be in the time domain (e.g. a
5740:
Oppenheim, Alan V., Willsky, Alan S., and Young, Ian T.,1983: Signals and
Systems, p. 256 (Englewood Cliffs, New Jersey: Prentice-Hall, Inc.)
712:
The impulse response of the filter as defined is nonzero over a finite duration. Including zeros, the impulse response is the infinite sequence
3154:
5594:
1890:
3022:
5787:
4748:
5761:
5745:
4529:
In addition, we can treat the importance of passband and stopband differently according to our needs by adding a weighted function,
107:
4464:
705:
that in many implementations or block diagrams provides the delayed inputs to the multiplication operations. One may speak of a
2582:
To design FIR filter in the MSE sense, we minimize the mean square error between the filter we obtained and the desired filter.
857:{\displaystyle h=\sum _{i=0}^{N}b_{i}\cdot \delta ={\begin{cases}b_{n}&0\leq n\leq N\\0&{\text{otherwise}}.\end{cases}}}
5718:
5259:
5198:
2166:
1366:
4770:
4709:
Block diagram of a simple FIR filter (second-order/3-tap filter in this case, implementing a moving average smoothing filter)
5181:{\displaystyle H(z)={\frac {1}{3}}+{\frac {1}{3}}z^{-1}+{\frac {1}{3}}z^{-2}={\frac {1}{3}}{\frac {z^{2}+z+1}{z^{2}}}.}
3956:
5034:
The block diagram on the right shows the second-order moving-average filter discussed below. The transfer function is
2437:
383:{\displaystyle {\begin{aligned}y&=b_{0}x+b_{1}x+\cdots +b_{N}x\\&=\sum _{i=0}^{N}b_{i}\cdot x,\end{aligned}}}
5614:
5578:
47:
4734:
2400:{\displaystyle H_{2\pi }(\omega )=\left.{\widehat {H}}(z)\,\right|_{z=e^{j\omega }}={\widehat {H}}(e^{j\omega }).}
916:, since the output is a sum of a finite number of finite multiples of the input values, so can be no greater than
5629:
902:
59:
2876:
62:(IIR) filters, which may have internal feedback and may continue to respond indefinitely (usually decaying).
5539:
4829:
5792:
5192:
4723:
1787:
1334:
5624:
2129:
1658:
5708:, a convenient choice for plotting software that displays the interval from 0 to the Nyquist frequency.
4780:
867:
If an FIR filter is non-causal, the range of nonzero values in its impulse response can start before
651:
1310:
800:
1021:
980:
2082:
2037:
1757:
99:
samples (from first nonzero element through last nonzero element) before it then settles to zero.
5604:
1399:
702:
2794:
1372:
595:
563:
5757:
5741:
5609:
4703:
2422:
When a particular frequency response is desired, several different design methods are common:
1781:
919:
31:
2736:
1630:
2764:
2565:
2003:
1974:
66:
51:
1859:
1607:
5599:
5582:
4766:
4532:
3640:
Step 3: Minimize the mean square error by doing partial derivative of MSE with respect to
2553:
2537:
2441:
972:
968:
115:
70:
4883:
901:
An FIR filter has a number of useful properties which sometimes make it preferable to an
870:
76:
5688:
5661:
2726:{\displaystyle {\text{MSE}}=f_{s}^{-1}\int _{-f_{s}/2}^{f_{s}/2}|H(f)-H_{d}(f)|^{2}\,df}
2264:{\displaystyle {\widehat {H}}(z)\ \triangleq \sum _{n=-\infty }^{\infty }h\cdot z^{-n}.}
1728:
4762:
4218:
4189:
3643:
2991:
2962:
2847:
2416:
1830:
1708:
1403:
994:
913:
622:
532:
1297:{\displaystyle y=x*h={\mathcal {F}}^{-1}{\big \{}X(\omega )\cdot H(\omega ){\big \}},}
5781:
5619:
4175:{\displaystyle s=\int _{-1/2}^{1/2}\cos(2\pi nF)H_{d}(F)\,dF,\ {\text{ for }}n\neq 0}
2558:
2548:
2477:
2451:
504:
185:, each value of the output sequence is a weighted sum of the most recent input values
178:
175:
103:
17:
5024:{\displaystyle h={\frac {1}{3}}\delta +{\frac {1}{3}}\delta +{\frac {1}{3}}\delta }
960:
660:
484:
464:
433:
402:
111:
4454:{\displaystyle h=s,h=s/2,h=s/2,\;for\;n=1,2,3,\ldots ,k,{\text{ where }}k=(N-1)/2}
154:
5577:
are plotted in the figure. But plots like these can also be generated by doing a
5638:
5634:
4774:
167:
147:
1369:(DTFT) and its inverse. Therefore, the complex-valued, multiplicative function
4754:
Amplitude and phase responses of the example second-order FIR smoothing filter
4740:
Magnitude and phase responses of the example second-order FIR smoothing filter
2517:
964:
2541:
58:
duration, because it settles to zero in finite time. This is in contrast to
4717:
4683:{\displaystyle {\text{MSE}}=\int _{-1/2}^{1/2}W(F)|R(F)-H_{d}(F)|^{2}\,dF}
1971:). Conversely, if one wants to design a filter for ordinary frequencies
2547:
The ideal response is often rectangular, and the corresponding IIR is a
1754:, which is favored by many filter design applications. The value
126:
3261:{\displaystyle {\text{MSE}}=\int _{-1/2}^{1/2}|R(F)-H_{d}(F)|^{2}\,dF}
2564:
The window design method is also advantageous for creating efficient
2521:
2513:
1957:{\displaystyle f=f'\cdot f_{s}={\tfrac {\omega }{2\pi }}\cdot f_{s}}
3138:{\displaystyle R(F)=e^{j2\pi Fk}H(F)=\sum _{n=0}^{k}s\cos(2\pi nF)}
619:-order FIR filter. If the filter is a direct form FIR filter then
2525:
153:
125:
4877:
To provide a more specific example, we select the filter order:
2440:(also known as the equiripple, optimal, or minimax method). The
4765:
filter is a very simple FIR filter. It is sometimes called a
2988:
even symmetric. Then, the discrete time
Fourier transform of
4517:{\displaystyle h=0{\text{ for }}n<0{\text{ and }}n\geq N}
1887:), ordinary frequency is related to normalized frequency by
1341:
1316:
1236:
1136:
1092:
1042:
73:
input) of an N-order discrete-time FIR filter lasts exactly
2308:
1968:
850:
5310:{\textstyle z_{2}=-{\frac {1}{2}}-j{\frac {\sqrt {3}}{2}}}
5249:{\textstyle z_{1}=-{\frac {1}{2}}+j{\frac {\sqrt {3}}{2}}}
2528:
provide convenient ways to apply these different methods.
5658:
An exception is MATLAB, which prefers a periodicity of
893:, with the defining formula appropriately generalized.
5542:
5262:
5201:
4783:
2480:
2454:
1925:
1803:
922:
663:
625:
598:
566:
535:
507:
487:
467:
436:
405:
5691:
5664:
5339:
5046:
4921:
4886:
4832:
4567:
4535:
4467:
4251:
4221:
4192:
4052:
3959:
3676:
3646:
3278:
3157:
3025:
2994:
2965:
2879:
2850:
2797:
2767:
2739:
2590:
2280:
2178:
2132:
2085:
2040:
2006:
1977:
1893:
1862:
1833:
1790:
1760:
1731:
1711:
1661:
1633:
1610:
1417:
1375:
1337:
1313:
1187:
1035:
997:
873:
724:
197:
79:
5700:
5673:
5569:
5525:
5309:
5248:
5180:
5023:
4898:
4866:
4815:
4682:
4550:
4516:
4453:
4236:
4207:
4174:
4037:
3941:
3661:
3628:
3260:
3137:
3009:
2980:
2948:
2865:
2820:
2783:
2753:
2725:
2492:
2466:
2399:
2263:
2157:
2116:
2071:
2022:
1993:
1956:
1875:
1848:
1819:
1772:
1742:
1717:
1697:
1639:
1619:
1593:
1390:
1357:
1323:
1296:
1173:
1012:
948:
885:
856:
684:
638:
611:
584:
548:
519:
493:
473:
451:
420:
382:
91:
158:A lattice-form discrete-time FIR filter of order
4726:of the example second-order FIR smoothing filter
4038:{\displaystyle s=\int _{-1/2}^{1/2}H_{d}(F)\,dF}
2791:is the spectrum of the filter we obtained, and
130:A direct form discrete-time FIR filter of order
5639:Linear constant-coefficient difference equation
4909:The impulse response of the resulting filter is
956:times the largest value appearing in the input.
54:(or response to any finite length input) is of
1286:
1252:
8:
1147:
1141:
1103:
1097:
1059:
1047:
1020:is described in the frequency domain by the
556:is the value of the impulse response at the
692:in these terms are commonly referred to as
5681:because the Nyquist frequency in units of
4380:
4370:
5690:
5663:
5554:
5541:
5470:
5456:
5431:
5417:
5402:
5388:
5375:
5355:
5340:
5338:
5295:
5279:
5267:
5261:
5234:
5218:
5206:
5200:
5167:
5144:
5137:
5127:
5115:
5101:
5089:
5075:
5062:
5045:
4993:
4962:
4937:
4920:
4885:
4846:
4837:
4831:
4807:
4788:
4782:
4673:
4667:
4662:
4646:
4622:
4600:
4596:
4587:
4580:
4568:
4566:
4534:
4500:
4486:
4466:
4443:
4417:
4359:
4315:
4250:
4220:
4191:
4158:
4145:
4130:
4092:
4088:
4079:
4072:
4051:
4028:
4013:
3999:
3995:
3986:
3979:
3958:
3926:
3896:
3880:
3866:
3862:
3853:
3846:
3829:
3771:
3767:
3758:
3751:
3729:
3718:
3683:
3677:
3675:
3645:
3619:
3613:
3597:
3583:
3579:
3570:
3563:
3549:
3543:
3497:
3486:
3472:
3468:
3459:
3452:
3435:
3393:
3382:
3336:
3325:
3311:
3307:
3298:
3291:
3279:
3277:
3251:
3245:
3240:
3224:
3200:
3190:
3186:
3177:
3170:
3158:
3156:
3093:
3082:
3045:
3024:
2993:
2964:
2922:
2878:
2849:
2817:
2802:
2796:
2780:
2766:
2750:
2744:
2738:
2716:
2710:
2705:
2689:
2665:
2655:
2649:
2644:
2635:
2629:
2621:
2608:
2603:
2591:
2589:
2479:
2453:
2426:
2382:
2364:
2363:
2349:
2338:
2332:
2312:
2311:
2285:
2279:
2249:
2224:
2210:
2180:
2179:
2177:
2137:
2131:
2105:
2096:
2090:
2084:
2060:
2051:
2045:
2039:
2011:
2005:
1982:
1976:
1948:
1924:
1915:
1892:
1867:
1861:
1832:
1802:
1789:
1759:
1730:
1710:
1666:
1660:
1632:
1609:
1579:
1565:
1560:
1546:
1536:
1525:
1509:
1495:
1490:
1464:
1450:
1422:
1416:
1374:
1346:
1340:
1339:
1336:
1315:
1314:
1312:
1285:
1284:
1251:
1250:
1241:
1235:
1234:
1186:
1156:
1135:
1134:
1131:
1112:
1091:
1090:
1087:
1068:
1041:
1040:
1037:
1034:
996:
941:
935:
926:
921:
872:
839:
807:
795:
765:
755:
744:
723:
662:
630:
624:
603:
597:
565:
540:
534:
506:
486:
466:
435:
404:
346:
336:
325:
287:
250:
225:
198:
196:
78:
5191:The next figure shows the corresponding
4823:, are found via the following equation:
2949:{\displaystyle r=h,k={\frac {(N-1)}{2}}}
2030:etc., using an application that expects
5733:
5651:
5570:{\textstyle H\left(e^{j\omega }\right)}
2504:, and returns the optimum coefficients.
2448:. The algorithm then finds the set of
5630:Infinite impulse response (IIR) filter
5536:The magnitude and phase components of
4867:{\displaystyle b_{i}={\frac {1}{N+1}}}
2828:is the spectrum of the desired filter.
2165:can also be expressed in terms of the
69:(that is, the output in response to a
3149:Step 2: Calculate mean square error.
7:
5320:The frequency response, in terms of
4769:filter, especially when followed by
2572:Least mean square error (MSE) method
2434:Least MSE (mean square error) method
991:The filter's effect on the sequence
646:is also a coefficient of the filter.
1820:{\displaystyle f'={\tfrac {1}{2}}.}
1358:{\displaystyle {\mathcal {F}}^{-1}}
27:Type of filter in signal processing
3690:
3680:
2225:
2220:
2158:{\displaystyle H_{2\pi }(\omega )}
1698:{\displaystyle H_{2\pi }(2\pi f')}
1604:where the added subscript denotes
1465:
1460:
650:This computation is also known as
25:
4816:{\textstyle b_{0},\ldots ,b_{N}}
4747:
4733:
4716:
4702:
2169:of the filter impulse response:
1181: and
142:+ 1 taps. Each unit delay is a
5595:Cascaded integrator–comb filter
1367:discrete-time Fourier transform
5508:
5502:
5056:
5050:
5018:
5006:
4987:
4975:
4956:
4950:
4931:
4925:
4663:
4658:
4652:
4636:
4630:
4623:
4619:
4613:
4545:
4539:
4477:
4471:
4440:
4428:
4356:
4350:
4341:
4329:
4312:
4306:
4297:
4285:
4276:
4270:
4261:
4255:
4231:
4225:
4202:
4196:
4142:
4136:
4123:
4108:
4062:
4056:
4025:
4019:
3969:
3963:
3923:
3908:
3893:
3886:
3826:
3811:
3802:
3787:
3744:
3738:
3702:
3696:
3656:
3650:
3610:
3603:
3536:
3521:
3512:
3506:
3432:
3417:
3408:
3402:
3375:
3360:
3351:
3345:
3241:
3236:
3230:
3214:
3208:
3201:
3132:
3117:
3108:
3102:
3072:
3066:
3035:
3029:
3004:
2998:
2975:
2969:
2937:
2925:
2910:
2898:
2889:
2883:
2860:
2854:
2814:
2808:
2777:
2771:
2706:
2701:
2695:
2679:
2673:
2666:
2391:
2375:
2329:
2323:
2300:
2294:
2239:
2233:
2197:
2191:
2152:
2146:
1843:
1837:
1692:
1675:
1479:
1473:
1437:
1431:
1385:
1379:
1324:{\displaystyle {\mathcal {F}}}
1281:
1275:
1266:
1260:
1227:
1221:
1212:
1206:
1197:
1191:
1166:
1160:
1122:
1116:
1078:
1072:
1007:
1001:
942:
927:
789:
777:
734:
728:
701:, based on the structure of a
679:
667:
446:
440:
415:
409:
370:
358:
308:
296:
271:
259:
240:
234:
211:
205:
1:
959:Can easily be designed to be
4558:Then, the MSE error becomes
4215:back to the presentation of
2117:{\displaystyle f_{2}/f_{s},}
2072:{\displaystyle f_{1}/f_{s},}
1773:{\displaystyle \omega =\pi }
527:terms on the right-hand side
4777:. The filter coefficients,
3951:After organization, we have
1827: When the sequence
905:(IIR) filter. FIR filters:
5809:
5615:Filter (signal processing)
5579:discrete Fourier transform
2821:{\displaystyle H_{d}(f)\,}
2512:Software packages such as
1856:has a known sampling-rate
1391:{\displaystyle H(\omega )}
612:{\textstyle N^{\text{th}}}
585:{\textstyle 0\leq i\leq N}
5788:Digital signal processing
2431:Frequency sampling method
949:{\textstyle \sum |b_{i}|}
903:infinite impulse response
60:infinite impulse response
5719:§ Sampling the DTFT
2442:Remez exchange algorithm
1647:represents frequency in
1365:respectively denote the
481:is the filter order; an
162:. Each unit delay is a
2761:is sampling frequency,
2754:{\displaystyle f_{s}\,}
1640:{\displaystyle \omega }
138:-stage delay line with
36:finite impulse response
5702:
5675:
5571:
5527:
5311:
5250:
5182:
5025:
4900:
4868:
4817:
4694:Moving average example
4684:
4552:
4518:
4455:
4238:
4209:
4176:
4039:
3943:
3734:
3663:
3630:
3502:
3398:
3341:
3262:
3139:
3098:
3011:
2982:
2950:
2867:
2822:
2785:
2784:{\displaystyle H(f)\,}
2755:
2727:
2494:
2468:
2438:Parks–McClellan method
2401:
2265:
2229:
2159:
2118:
2073:
2024:
2023:{\displaystyle f_{2},}
1995:
1994:{\displaystyle f_{1},}
1958:
1877:
1850:
1821:
1774:
1744:
1719:
1699:
1641:
1621:
1595:
1541:
1469:
1402:. It is defined by a
1392:
1359:
1325:
1298:
1175:
1014:
950:
887:
858:
760:
707:5th order/6-tap filter
686:
640:
613:
586:
550:
521:
495:
475:
453:
422:
384:
341:
171:
151:
134:. The top part is an
93:
5703:
5676:
5625:FIR transfer function
5572:
5528:
5312:
5251:
5183:
5026:
4901:
4869:
4818:
4685:
4553:
4519:
4456:
4239:
4210:
4177:
4040:
3944:
3714:
3664:
3631:
3482:
3378:
3321:
3263:
3140:
3078:
3012:
2983:
2951:
2868:
2823:
2786:
2756:
2728:
2495:
2469:
2402:
2266:
2206:
2160:
2119:
2074:
2025:
1996:
1959:
1878:
1876:{\displaystyle f_{s}}
1851:
1822:
1775:
1745:
1720:
1705:has a periodicity of
1700:
1642:
1622:
1620:{\displaystyle 2\pi }
1596:
1521:
1446:
1393:
1360:
1326:
1299:
1176:
1015:
951:
888:
859:
740:
687:
641:
614:
587:
551:
522:
496:
476:
459:is the output signal,
454:
423:
385:
321:
157:
129:
94:
5689:
5662:
5540:
5337:
5322:normalized frequency
5260:
5199:
5044:
4919:
4884:
4830:
4781:
4565:
4551:{\displaystyle W(f)}
4533:
4465:
4249:
4219:
4190:
4050:
3957:
3674:
3644:
3276:
3155:
3023:
2992:
2963:
2877:
2848:
2795:
2765:
2737:
2588:
2532:Window design method
2478:
2452:
2427:Window design method
2278:
2176:
2130:
2083:
2038:
2004:
1975:
1891:
1860:
1831:
1788:
1758:
1729:
1709:
1659:
1631:
1608:
1415:
1373:
1335:
1311:
1185:
1033:
995:
920:
871:
722:
661:
652:discrete convolution
623:
596:
564:
533:
505:
485:
465:
434:
428:is the input signal,
403:
195:
181:FIR filter of order
77:
18:Window design method
4899:{\displaystyle N=2}
4609:
4101:
4008:
3875:
3780:
3592:
3481:
3320:
3199:
2664:
2616:
1627:-periodicity. Here
1022:convolution theorem
886:{\displaystyle n=0}
102:FIR filters can be
92:{\displaystyle N+1}
5701:{\displaystyle 1,}
5698:
5683:half-cycles/sample
5674:{\displaystyle 2,}
5671:
5605:Digital delay line
5567:
5523:
5521:
5307:
5246:
5178:
5021:
4896:
4864:
4813:
4680:
4576:
4548:
4514:
4451:
4234:
4205:
4172:
4068:
4035:
3975:
3939:
3842:
3747:
3659:
3626:
3559:
3448:
3287:
3258:
3166:
3135:
3007:
2978:
2946:
2863:
2844:-point FIR filter
2818:
2781:
2751:
2723:
2617:
2599:
2490:
2464:
2397:
2261:
2155:
2114:
2069:
2034:, one would enter
2020:
1991:
1954:
1939:
1885:samples per second
1873:
1846:
1817:
1812:
1770:
1743:{\displaystyle f'}
1740:
1715:
1695:
1653:radians per sample
1637:
1617:
1591:
1400:frequency response
1388:
1355:
1321:
1294:
1171:
1170:
1154:
1126:
1110:
1082:
1066:
1010:
987:Frequency response
946:
883:
854:
849:
682:
639:{\textstyle b_{i}}
636:
609:
582:
549:{\textstyle b_{i}}
546:
517:
501:-order filter has
491:
471:
449:
418:
380:
378:
172:
152:
89:
5610:Electronic filter
5464:
5425:
5396:
5383:
5305:
5301:
5287:
5244:
5240:
5226:
5193:pole–zero diagram
5173:
5135:
5109:
5083:
5070:
5001:
4970:
4945:
4862:
4775:sinc-in-frequency
4724:Pole–zero diagram
4571:
4503:
4489:
4420:
4419: where
4237:{\displaystyle h}
4208:{\displaystyle s}
4161:
4157:
3706:
3686:
3662:{\displaystyle s}
3282:
3161:
3010:{\displaystyle r}
2981:{\displaystyle h}
2944:
2866:{\displaystyle h}
2594:
2566:half-band filters
2372:
2320:
2202:
2188:
2032:cycles per sample
1965:cycles per second
1938:
1849:{\displaystyle x}
1811:
1784:, corresponds to
1782:Nyquist frequency
1752:cycles per sample
1718:{\displaystyle 1}
1442:
1132:
1130:
1088:
1086:
1038:
1036:
1013:{\displaystyle x}
969:crossover filters
842:
703:tapped delay line
606:
32:signal processing
16:(Redirected from
5800:
5773:
5770:
5764:
5754:
5748:
5738:
5722:
5715:
5709:
5707:
5705:
5704:
5699:
5680:
5678:
5677:
5672:
5656:
5576:
5574:
5573:
5568:
5566:
5562:
5561:
5532:
5530:
5529:
5524:
5522:
5515:
5511:
5481:
5480:
5465:
5457:
5449:
5445:
5444:
5426:
5418:
5413:
5412:
5397:
5389:
5384:
5376:
5367:
5363:
5362:
5316:
5314:
5313:
5308:
5306:
5297:
5296:
5288:
5280:
5272:
5271:
5255:
5253:
5252:
5247:
5245:
5236:
5235:
5227:
5219:
5211:
5210:
5187:
5185:
5184:
5179:
5174:
5172:
5171:
5162:
5149:
5148:
5138:
5136:
5128:
5123:
5122:
5110:
5102:
5097:
5096:
5084:
5076:
5071:
5063:
5030:
5028:
5027:
5022:
5002:
4994:
4971:
4963:
4946:
4938:
4905:
4903:
4902:
4897:
4873:
4871:
4870:
4865:
4863:
4861:
4847:
4842:
4841:
4822:
4820:
4819:
4814:
4812:
4811:
4793:
4792:
4751:
4737:
4720:
4706:
4689:
4687:
4686:
4681:
4672:
4671:
4666:
4651:
4650:
4626:
4608:
4604:
4595:
4591:
4572:
4569:
4557:
4555:
4554:
4549:
4523:
4521:
4520:
4515:
4504:
4501:
4490:
4487:
4460:
4458:
4457:
4452:
4447:
4421:
4418:
4363:
4319:
4243:
4241:
4240:
4235:
4214:
4212:
4211:
4206:
4181:
4179:
4178:
4173:
4162:
4159:
4155:
4135:
4134:
4100:
4096:
4087:
4083:
4044:
4042:
4041:
4036:
4018:
4017:
4007:
4003:
3994:
3990:
3948:
3946:
3945:
3940:
3901:
3900:
3885:
3884:
3874:
3870:
3861:
3857:
3779:
3775:
3766:
3762:
3733:
3728:
3707:
3705:
3688:
3687:
3684:
3678:
3668:
3666:
3665:
3660:
3635:
3633:
3632:
3627:
3618:
3617:
3602:
3601:
3591:
3587:
3578:
3574:
3548:
3547:
3501:
3496:
3480:
3476:
3467:
3463:
3397:
3392:
3340:
3335:
3319:
3315:
3306:
3302:
3283:
3280:
3267:
3265:
3264:
3259:
3250:
3249:
3244:
3229:
3228:
3204:
3198:
3194:
3185:
3181:
3162:
3159:
3144:
3142:
3141:
3136:
3097:
3092:
3062:
3061:
3016:
3014:
3013:
3008:
2987:
2985:
2984:
2979:
2959:Step 1: Suppose
2955:
2953:
2952:
2947:
2945:
2940:
2923:
2872:
2870:
2869:
2864:
2827:
2825:
2824:
2819:
2807:
2806:
2790:
2788:
2787:
2782:
2760:
2758:
2757:
2752:
2749:
2748:
2732:
2730:
2729:
2724:
2715:
2714:
2709:
2694:
2693:
2669:
2663:
2659:
2654:
2653:
2643:
2639:
2634:
2633:
2615:
2607:
2595:
2592:
2499:
2497:
2496:
2493:{\textstyle N+1}
2491:
2473:
2471:
2470:
2467:{\textstyle N+1}
2465:
2406:
2404:
2403:
2398:
2390:
2389:
2374:
2373:
2365:
2359:
2358:
2357:
2356:
2337:
2333:
2322:
2321:
2313:
2293:
2292:
2270:
2268:
2267:
2262:
2257:
2256:
2228:
2223:
2200:
2190:
2189:
2181:
2164:
2162:
2161:
2156:
2145:
2144:
2123:
2121:
2120:
2115:
2110:
2109:
2100:
2095:
2094:
2078:
2076:
2075:
2070:
2065:
2064:
2055:
2050:
2049:
2029:
2027:
2026:
2021:
2016:
2015:
2000:
1998:
1997:
1992:
1987:
1986:
1963:
1961:
1960:
1955:
1953:
1952:
1940:
1937:
1926:
1920:
1919:
1907:
1882:
1880:
1879:
1874:
1872:
1871:
1855:
1853:
1852:
1847:
1826:
1824:
1823:
1818:
1813:
1804:
1798:
1779:
1777:
1776:
1771:
1749:
1747:
1746:
1741:
1739:
1724:
1722:
1721:
1716:
1704:
1702:
1701:
1696:
1691:
1674:
1673:
1655:). The function
1649:normalized units
1646:
1644:
1643:
1638:
1626:
1624:
1623:
1618:
1600:
1598:
1597:
1592:
1587:
1586:
1578:
1574:
1573:
1572:
1551:
1550:
1540:
1535:
1517:
1516:
1508:
1504:
1503:
1502:
1468:
1463:
1440:
1430:
1429:
1398:is the filter's
1397:
1395:
1394:
1389:
1364:
1362:
1361:
1356:
1354:
1353:
1345:
1344:
1330:
1328:
1327:
1322:
1320:
1319:
1307:where operators
1303:
1301:
1300:
1295:
1290:
1289:
1256:
1255:
1249:
1248:
1240:
1239:
1180:
1178:
1177:
1172:
1169:
1155:
1150:
1140:
1139:
1125:
1111:
1106:
1096:
1095:
1081:
1067:
1062:
1046:
1045:
1019:
1017:
1016:
1011:
955:
953:
952:
947:
945:
940:
939:
930:
892:
890:
889:
884:
863:
861:
860:
855:
853:
852:
843:
840:
812:
811:
770:
769:
759:
754:
709:, for instance.
698:
697:
691:
689:
688:
683:
645:
643:
642:
637:
635:
634:
618:
616:
615:
610:
608:
607:
604:
591:
589:
588:
583:
555:
553:
552:
547:
545:
544:
526:
524:
523:
520:{\textstyle N+1}
518:
500:
498:
497:
492:
480:
478:
477:
472:
458:
456:
455:
450:
427:
425:
424:
419:
389:
387:
386:
381:
379:
351:
350:
340:
335:
314:
292:
291:
255:
254:
230:
229:
98:
96:
95:
90:
67:impulse response
52:impulse response
21:
5808:
5807:
5803:
5802:
5801:
5799:
5798:
5797:
5778:
5777:
5776:
5771:
5767:
5755:
5751:
5739:
5735:
5731:
5726:
5725:
5716:
5712:
5687:
5686:
5660:
5659:
5657:
5653:
5648:
5600:Compact support
5591:
5583:low-pass filter
5550:
5546:
5538:
5537:
5520:
5519:
5486:
5482:
5466:
5447:
5446:
5427:
5398:
5368:
5351:
5347:
5335:
5334:
5263:
5258:
5257:
5202:
5197:
5196:
5163:
5140:
5139:
5111:
5085:
5042:
5041:
4917:
4916:
4882:
4881:
4851:
4833:
4828:
4827:
4803:
4784:
4779:
4778:
4759:
4758:
4757:
4756:
4755:
4752:
4743:
4742:
4741:
4738:
4729:
4728:
4727:
4721:
4712:
4711:
4710:
4707:
4696:
4661:
4642:
4563:
4562:
4531:
4530:
4502: and
4488: for
4463:
4462:
4247:
4246:
4217:
4216:
4188:
4187:
4186:Step 4: Change
4160: for
4126:
4048:
4047:
4009:
3955:
3954:
3892:
3876:
3689:
3679:
3672:
3671:
3642:
3641:
3609:
3593:
3539:
3274:
3273:
3239:
3220:
3153:
3152:
3041:
3021:
3020:
2990:
2989:
2961:
2960:
2924:
2875:
2874:
2846:
2845:
2798:
2793:
2792:
2763:
2762:
2740:
2735:
2734:
2704:
2685:
2645:
2625:
2586:
2585:
2574:
2554:transition band
2538:window function
2534:
2476:
2475:
2450:
2449:
2413:
2378:
2345:
2310:
2307:
2306:
2281:
2276:
2275:
2245:
2174:
2173:
2133:
2128:
2127:
2101:
2086:
2081:
2080:
2056:
2041:
2036:
2035:
2007:
2002:
2001:
1978:
1973:
1972:
1944:
1930:
1911:
1900:
1889:
1888:
1863:
1858:
1857:
1829:
1828:
1791:
1786:
1785:
1756:
1755:
1732:
1727:
1726:
1707:
1706:
1684:
1662:
1657:
1656:
1629:
1628:
1606:
1605:
1561:
1556:
1555:
1542:
1491:
1486:
1485:
1418:
1413:
1412:
1371:
1370:
1338:
1333:
1332:
1309:
1308:
1233:
1183:
1182:
1133:
1089:
1039:
1031:
1030:
993:
992:
989:
931:
918:
917:
912:Are inherently
899:
869:
868:
848:
847:
837:
831:
830:
813:
803:
796:
761:
720:
719:
695:
694:
659:
658:
626:
621:
620:
599:
594:
593:
562:
561:
560:th instant for
536:
531:
530:
503:
502:
483:
482:
463:
462:
432:
431:
401:
400:
377:
376:
342:
312:
311:
283:
246:
221:
214:
193:
192:
124:
108:continuous-time
75:
74:
71:Kronecker delta
28:
23:
22:
15:
12:
11:
5:
5806:
5804:
5796:
5795:
5790:
5780:
5779:
5775:
5774:
5765:
5749:
5732:
5730:
5727:
5724:
5723:
5710:
5697:
5694:
5670:
5667:
5650:
5649:
5647:
5644:
5643:
5642:
5637:(specifically
5632:
5627:
5622:
5617:
5612:
5607:
5602:
5597:
5590:
5587:
5565:
5560:
5557:
5553:
5549:
5545:
5534:
5533:
5518:
5514:
5510:
5507:
5504:
5501:
5498:
5495:
5492:
5489:
5485:
5479:
5476:
5473:
5469:
5463:
5460:
5455:
5452:
5450:
5448:
5443:
5440:
5437:
5434:
5430:
5424:
5421:
5416:
5411:
5408:
5405:
5401:
5395:
5392:
5387:
5382:
5379:
5374:
5371:
5369:
5366:
5361:
5358:
5354:
5350:
5346:
5343:
5342:
5304:
5300:
5294:
5291:
5286:
5283:
5278:
5275:
5270:
5266:
5243:
5239:
5233:
5230:
5225:
5222:
5217:
5214:
5209:
5205:
5189:
5188:
5177:
5170:
5166:
5161:
5158:
5155:
5152:
5147:
5143:
5134:
5131:
5126:
5121:
5118:
5114:
5108:
5105:
5100:
5095:
5092:
5088:
5082:
5079:
5074:
5069:
5066:
5061:
5058:
5055:
5052:
5049:
5032:
5031:
5020:
5017:
5014:
5011:
5008:
5005:
5000:
4997:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4969:
4966:
4961:
4958:
4955:
4952:
4949:
4944:
4941:
4936:
4933:
4930:
4927:
4924:
4907:
4906:
4895:
4892:
4889:
4875:
4874:
4860:
4857:
4854:
4850:
4845:
4840:
4836:
4810:
4806:
4802:
4799:
4796:
4791:
4787:
4763:moving average
4753:
4746:
4745:
4744:
4739:
4732:
4731:
4730:
4722:
4715:
4714:
4713:
4708:
4701:
4700:
4699:
4698:
4697:
4695:
4692:
4691:
4690:
4679:
4676:
4670:
4665:
4660:
4657:
4654:
4649:
4645:
4641:
4638:
4635:
4632:
4629:
4625:
4621:
4618:
4615:
4612:
4607:
4603:
4599:
4594:
4590:
4586:
4583:
4579:
4575:
4547:
4544:
4541:
4538:
4527:
4526:
4525:
4524:
4513:
4510:
4507:
4499:
4496:
4493:
4485:
4482:
4479:
4476:
4473:
4470:
4450:
4446:
4442:
4439:
4436:
4433:
4430:
4427:
4424:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4379:
4376:
4373:
4369:
4366:
4362:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4318:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4233:
4230:
4227:
4224:
4204:
4201:
4198:
4195:
4184:
4183:
4182:
4171:
4168:
4165:
4154:
4151:
4148:
4144:
4141:
4138:
4133:
4129:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4099:
4095:
4091:
4086:
4082:
4078:
4075:
4071:
4067:
4064:
4061:
4058:
4055:
4045:
4034:
4031:
4027:
4024:
4021:
4016:
4012:
4006:
4002:
3998:
3993:
3989:
3985:
3982:
3978:
3974:
3971:
3968:
3965:
3962:
3952:
3949:
3938:
3935:
3932:
3929:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3899:
3895:
3891:
3888:
3883:
3879:
3873:
3869:
3865:
3860:
3856:
3852:
3849:
3845:
3841:
3838:
3835:
3832:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3778:
3774:
3770:
3765:
3761:
3757:
3754:
3750:
3746:
3743:
3740:
3737:
3732:
3727:
3724:
3721:
3717:
3713:
3710:
3704:
3701:
3698:
3695:
3692:
3682:
3658:
3655:
3652:
3649:
3638:
3637:
3636:
3625:
3622:
3616:
3612:
3608:
3605:
3600:
3596:
3590:
3586:
3582:
3577:
3573:
3569:
3566:
3562:
3558:
3555:
3552:
3546:
3542:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3500:
3495:
3492:
3489:
3485:
3479:
3475:
3471:
3466:
3462:
3458:
3455:
3451:
3447:
3444:
3441:
3438:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3396:
3391:
3388:
3385:
3381:
3377:
3374:
3371:
3368:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3339:
3334:
3331:
3328:
3324:
3318:
3314:
3310:
3305:
3301:
3297:
3294:
3290:
3286:
3271:
3268:
3257:
3254:
3248:
3243:
3238:
3235:
3232:
3227:
3223:
3219:
3216:
3213:
3210:
3207:
3203:
3197:
3193:
3189:
3184:
3180:
3176:
3173:
3169:
3165:
3147:
3146:
3145:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3096:
3091:
3088:
3085:
3081:
3077:
3074:
3071:
3068:
3065:
3060:
3057:
3054:
3051:
3048:
3044:
3040:
3037:
3034:
3031:
3028:
3017:is defined as
3006:
3003:
3000:
2997:
2977:
2974:
2971:
2968:
2957:
2943:
2939:
2936:
2933:
2930:
2927:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2862:
2859:
2856:
2853:
2832:
2831:
2830:
2829:
2816:
2813:
2810:
2805:
2801:
2779:
2776:
2773:
2770:
2747:
2743:
2722:
2719:
2713:
2708:
2703:
2700:
2697:
2692:
2688:
2684:
2681:
2678:
2675:
2672:
2668:
2662:
2658:
2652:
2648:
2642:
2638:
2632:
2628:
2624:
2620:
2614:
2611:
2606:
2602:
2598:
2573:
2570:
2559:Kaiser windows
2533:
2530:
2510:
2509:
2505:
2489:
2486:
2483:
2463:
2460:
2457:
2435:
2432:
2429:
2417:matched filter
2412:
2409:
2408:
2407:
2396:
2393:
2388:
2385:
2381:
2377:
2371:
2368:
2362:
2355:
2352:
2348:
2344:
2341:
2336:
2331:
2328:
2325:
2319:
2316:
2309:
2305:
2302:
2299:
2296:
2291:
2288:
2284:
2272:
2271:
2260:
2255:
2252:
2248:
2244:
2241:
2238:
2235:
2232:
2227:
2222:
2219:
2216:
2213:
2209:
2205:
2199:
2196:
2193:
2187:
2184:
2154:
2151:
2148:
2143:
2140:
2136:
2113:
2108:
2104:
2099:
2093:
2089:
2068:
2063:
2059:
2054:
2048:
2044:
2019:
2014:
2010:
1990:
1985:
1981:
1951:
1947:
1943:
1936:
1933:
1929:
1923:
1918:
1914:
1910:
1906:
1903:
1899:
1896:
1870:
1866:
1845:
1842:
1839:
1836:
1816:
1810:
1807:
1801:
1797:
1794:
1769:
1766:
1763:
1738:
1735:
1714:
1694:
1690:
1687:
1683:
1680:
1677:
1672:
1669:
1665:
1636:
1616:
1613:
1602:
1601:
1590:
1585:
1582:
1577:
1571:
1568:
1564:
1559:
1554:
1549:
1545:
1539:
1534:
1531:
1528:
1524:
1520:
1515:
1512:
1507:
1501:
1498:
1494:
1489:
1484:
1481:
1478:
1475:
1472:
1467:
1462:
1459:
1456:
1453:
1449:
1445:
1439:
1436:
1433:
1428:
1425:
1421:
1404:Fourier series
1387:
1384:
1381:
1378:
1352:
1349:
1343:
1318:
1305:
1304:
1293:
1288:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1254:
1247:
1244:
1238:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1168:
1165:
1162:
1159:
1153:
1149:
1146:
1143:
1138:
1129:
1124:
1121:
1118:
1115:
1109:
1105:
1102:
1099:
1094:
1085:
1080:
1077:
1074:
1071:
1065:
1061:
1058:
1055:
1052:
1049:
1044:
1009:
1006:
1003:
1000:
988:
985:
977:
976:
957:
944:
938:
934:
929:
925:
910:
898:
895:
882:
879:
876:
865:
864:
851:
846:
838:
836:
833:
832:
829:
826:
823:
820:
817:
814:
810:
806:
802:
801:
799:
794:
791:
788:
785:
782:
779:
776:
773:
768:
764:
758:
753:
750:
747:
743:
739:
736:
733:
730:
727:
685:{\textstyle x}
681:
678:
675:
672:
669:
666:
648:
647:
633:
629:
602:
581:
578:
575:
572:
569:
543:
539:
528:
516:
513:
510:
494:{\textstyle N}
490:
474:{\textstyle N}
470:
460:
452:{\textstyle y}
448:
445:
442:
439:
429:
421:{\textstyle x}
417:
414:
411:
408:
391:
390:
375:
372:
369:
366:
363:
360:
357:
354:
349:
345:
339:
334:
331:
328:
324:
320:
317:
315:
313:
310:
307:
304:
301:
298:
295:
290:
286:
282:
279:
276:
273:
270:
267:
264:
261:
258:
253:
249:
245:
242:
239:
236:
233:
228:
224:
220:
217:
215:
213:
210:
207:
204:
201:
200:
123:
120:
88:
85:
82:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5805:
5794:
5793:Filter theory
5791:
5789:
5786:
5785:
5783:
5769:
5766:
5763:
5762:0-13-914101-4
5759:
5753:
5750:
5747:
5746:0-13-809731-3
5743:
5737:
5734:
5728:
5720:
5714:
5711:
5695:
5692:
5684:
5668:
5665:
5655:
5652:
5645:
5640:
5636:
5633:
5631:
5628:
5626:
5623:
5621:
5620:Filter design
5618:
5616:
5613:
5611:
5608:
5606:
5603:
5601:
5598:
5596:
5593:
5592:
5588:
5586:
5584:
5580:
5563:
5558:
5555:
5551:
5547:
5543:
5516:
5512:
5505:
5499:
5496:
5493:
5490:
5487:
5483:
5477:
5474:
5471:
5467:
5461:
5458:
5453:
5451:
5441:
5438:
5435:
5432:
5428:
5422:
5419:
5414:
5409:
5406:
5403:
5399:
5393:
5390:
5385:
5380:
5377:
5372:
5370:
5364:
5359:
5356:
5352:
5348:
5344:
5333:
5332:
5331:
5330:
5326:
5323:
5318:
5302:
5298:
5292:
5289:
5284:
5281:
5276:
5273:
5268:
5264:
5241:
5237:
5231:
5228:
5223:
5220:
5215:
5212:
5207:
5203:
5194:
5175:
5168:
5164:
5159:
5156:
5153:
5150:
5145:
5141:
5132:
5129:
5124:
5119:
5116:
5112:
5106:
5103:
5098:
5093:
5090:
5086:
5080:
5077:
5072:
5067:
5064:
5059:
5053:
5047:
5040:
5039:
5038:
5037:
5015:
5012:
5009:
5003:
4998:
4995:
4990:
4984:
4981:
4978:
4972:
4967:
4964:
4959:
4953:
4947:
4942:
4939:
4934:
4928:
4922:
4915:
4914:
4913:
4912:
4893:
4890:
4887:
4880:
4879:
4878:
4858:
4855:
4852:
4848:
4843:
4838:
4834:
4826:
4825:
4824:
4808:
4804:
4800:
4797:
4794:
4789:
4785:
4776:
4772:
4768:
4764:
4750:
4736:
4725:
4719:
4705:
4693:
4677:
4674:
4668:
4655:
4647:
4643:
4639:
4633:
4627:
4616:
4610:
4605:
4601:
4597:
4592:
4588:
4584:
4581:
4577:
4573:
4561:
4560:
4559:
4542:
4536:
4511:
4508:
4505:
4497:
4494:
4491:
4483:
4480:
4474:
4468:
4448:
4444:
4437:
4434:
4431:
4425:
4422:
4414:
4411:
4408:
4405:
4402:
4399:
4396:
4393:
4390:
4387:
4384:
4381:
4377:
4374:
4371:
4367:
4364:
4360:
4353:
4347:
4344:
4338:
4335:
4332:
4326:
4323:
4320:
4316:
4309:
4303:
4300:
4294:
4291:
4288:
4282:
4279:
4273:
4267:
4264:
4258:
4252:
4245:
4244:
4228:
4222:
4199:
4193:
4185:
4169:
4166:
4163:
4152:
4149:
4146:
4139:
4131:
4127:
4120:
4117:
4114:
4111:
4105:
4102:
4097:
4093:
4089:
4084:
4080:
4076:
4073:
4069:
4065:
4059:
4053:
4046:
4032:
4029:
4022:
4014:
4010:
4004:
4000:
3996:
3991:
3987:
3983:
3980:
3976:
3972:
3966:
3960:
3953:
3950:
3936:
3933:
3930:
3927:
3920:
3917:
3914:
3911:
3905:
3902:
3897:
3889:
3881:
3877:
3871:
3867:
3863:
3858:
3854:
3850:
3847:
3843:
3839:
3836:
3833:
3830:
3823:
3820:
3817:
3814:
3808:
3805:
3799:
3796:
3793:
3790:
3784:
3781:
3776:
3772:
3768:
3763:
3759:
3755:
3752:
3748:
3741:
3735:
3730:
3725:
3722:
3719:
3715:
3711:
3708:
3699:
3693:
3670:
3669:
3653:
3647:
3639:
3623:
3620:
3614:
3606:
3598:
3594:
3588:
3584:
3580:
3575:
3571:
3567:
3564:
3560:
3556:
3553:
3550:
3544:
3540:
3533:
3530:
3527:
3524:
3518:
3515:
3509:
3503:
3498:
3493:
3490:
3487:
3483:
3477:
3473:
3469:
3464:
3460:
3456:
3453:
3449:
3445:
3442:
3439:
3436:
3429:
3426:
3423:
3420:
3414:
3411:
3405:
3399:
3394:
3389:
3386:
3383:
3379:
3372:
3369:
3366:
3363:
3357:
3354:
3348:
3342:
3337:
3332:
3329:
3326:
3322:
3316:
3312:
3308:
3303:
3299:
3295:
3292:
3288:
3284:
3272:
3269:
3255:
3252:
3246:
3233:
3225:
3221:
3217:
3211:
3205:
3195:
3191:
3187:
3182:
3178:
3174:
3171:
3167:
3163:
3151:
3150:
3148:
3129:
3126:
3123:
3120:
3114:
3111:
3105:
3099:
3094:
3089:
3086:
3083:
3079:
3075:
3069:
3063:
3058:
3055:
3052:
3049:
3046:
3042:
3038:
3032:
3026:
3019:
3018:
3001:
2995:
2972:
2966:
2958:
2941:
2934:
2931:
2928:
2919:
2916:
2913:
2907:
2904:
2901:
2895:
2892:
2886:
2880:
2857:
2851:
2843:
2839:
2838:
2837:
2836:
2811:
2803:
2799:
2774:
2768:
2745:
2741:
2720:
2717:
2711:
2698:
2690:
2686:
2682:
2676:
2670:
2660:
2656:
2650:
2646:
2640:
2636:
2630:
2626:
2622:
2618:
2612:
2609:
2604:
2600:
2596:
2584:
2583:
2581:
2580:
2579:
2578:
2571:
2569:
2567:
2562:
2560:
2556:
2555:
2550:
2549:sinc function
2545:
2543:
2539:
2531:
2529:
2527:
2523:
2519:
2515:
2506:
2503:
2487:
2484:
2481:
2461:
2458:
2455:
2447:
2443:
2439:
2436:
2433:
2430:
2428:
2425:
2424:
2423:
2420:
2418:
2411:Filter design
2410:
2394:
2386:
2383:
2379:
2369:
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2360:
2353:
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2326:
2317:
2314:
2303:
2297:
2289:
2286:
2282:
2274:
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2258:
2253:
2250:
2246:
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2236:
2230:
2217:
2214:
2211:
2207:
2203:
2194:
2185:
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2125:
2111:
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2097:
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2087:
2066:
2061:
2057:
2052:
2046:
2042:
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2017:
2012:
2008:
1988:
1983:
1979:
1970:
1966:
1949:
1945:
1941:
1934:
1931:
1927:
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1916:
1912:
1908:
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1569:
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1434:
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1419:
1411:
1410:
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1408:
1405:
1401:
1382:
1376:
1368:
1350:
1347:
1291:
1278:
1272:
1269:
1263:
1257:
1245:
1242:
1230:
1224:
1218:
1215:
1209:
1203:
1200:
1194:
1188:
1163:
1157:
1151:
1144:
1127:
1119:
1113:
1107:
1100:
1083:
1075:
1069:
1063:
1056:
1053:
1050:
1029:
1028:
1027:
1026:
1023:
1004:
998:
986:
984:
982:
974:
970:
966:
962:
958:
936:
932:
923:
915:
911:
908:
907:
906:
904:
896:
894:
880:
877:
874:
844:
834:
827:
824:
821:
818:
815:
808:
804:
797:
792:
786:
783:
780:
774:
771:
766:
762:
756:
751:
748:
745:
741:
737:
731:
725:
718:
717:
716:
715:
710:
708:
704:
700:
676:
673:
670:
664:
655:
653:
631:
627:
600:
579:
576:
573:
570:
567:
559:
541:
537:
529:
514:
511:
508:
488:
468:
461:
443:
437:
430:
412:
406:
399:
398:
397:
396:
373:
367:
364:
361:
355:
352:
347:
343:
337:
332:
329:
326:
322:
318:
316:
305:
302:
299:
293:
288:
284:
280:
277:
274:
268:
265:
262:
256:
251:
247:
243:
237:
231:
226:
222:
218:
216:
208:
202:
191:
190:
189:
188:
184:
180:
179:discrete-time
177:
169:
165:
161:
156:
149:
145:
141:
137:
133:
128:
121:
119:
117:
113:
109:
105:
104:discrete-time
100:
86:
83:
80:
72:
68:
63:
61:
57:
53:
49:
45:
41:
37:
33:
19:
5768:
5752:
5736:
5713:
5682:
5654:
5535:
5328:
5324:
5319:
5190:
5035:
5033:
4910:
4908:
4876:
4760:
4528:
2841:
2834:
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2576:
2575:
2563:
2552:
2546:
2535:
2511:
2501:
2445:
2421:
2414:
2126:
2124: etc.
2031:
1964:
1884:
1751:
1750:in units of
1652:
1603:
1406:
1306:
1024:
990:
978:
961:linear phase
900:
866:
713:
711:
706:
693:
656:
649:
557:
394:
392:
186:
182:
173:
166:operator in
163:
159:
146:operator in
143:
139:
135:
131:
101:
64:
55:
43:
39:
35:
29:
5635:Z-transform
2167:Z-transform
981:selectivity
168:Z-transform
148:Z-transform
5782:Categories
5729:References
4771:decimation
3270:Therefore,
2518:GNU Octave
965:seismology
897:Properties
122:Definition
5559:ω
5506:ω
5500:
5478:ω
5472:−
5442:ω
5433:−
5410:ω
5404:−
5360:ω
5290:−
5277:−
5216:−
5117:−
5091:−
5013:−
5004:δ
4982:−
4973:δ
4948:δ
4798:…
4640:−
4582:−
4578:∫
4509:≥
4435:−
4406:…
4336:−
4167:≠
4115:π
4106:
4074:−
4070:∫
3981:−
3977:∫
3915:π
3906:
3848:−
3844:∫
3837:−
3821:τ
3818:π
3809:
3794:π
3785:
3753:−
3749:∫
3742:τ
3720:τ
3716:∑
3691:∂
3681:∂
3565:−
3561:∫
3528:π
3519:
3484:∑
3454:−
3450:∫
3443:−
3427:τ
3424:π
3415:
3406:τ
3384:τ
3380:∑
3367:π
3358:
3323:∑
3293:−
3289:∫
3218:−
3172:−
3168:∫
3124:π
3115:
3080:∑
3053:π
2932:−
2840:Given an
2683:−
2623:−
2619:∫
2610:−
2542:convolved
2387:ω
2370:^
2354:ω
2318:^
2298:ω
2290:π
2251:−
2243:⋅
2226:∞
2221:∞
2218:−
2208:∑
2204:≜
2186:^
2150:ω
2142:π
1942:⋅
1935:π
1928:ω
1909:⋅
1780:, called
1768:π
1762:ω
1682:π
1671:π
1635:ω
1615:π
1581:−
1570:ω
1553:⋅
1523:∑
1511:−
1500:ω
1483:⋅
1466:∞
1461:∞
1458:−
1448:∑
1444:≜
1435:ω
1427:π
1383:ω
1348:−
1279:ω
1270:⋅
1264:ω
1243:−
1216:∗
1164:ω
1152:⏟
1128:⋅
1120:ω
1108:⏟
1076:ω
1064:⏟
1054:∗
973:mastering
924:∑
841:otherwise
825:≤
819:≤
784:−
775:δ
772:⋅
742:∑
674:−
577:≤
571:≤
365:−
353:⋅
323:∑
303:−
278:⋯
266:−
170:notation.
150:notation.
5589:See also
2733:, where
1905:′
1796:′
1737:′
1689:′
4773:, or a
2835:Method:
2079:
112:digital
5760:
5744:
4767:boxcar
4156:
2873:, and
2524:, and
2522:Scilab
2514:MATLAB
2201:
1441:
971:, and
914:stable
592:of an
176:causal
174:For a
116:analog
110:, and
56:finite
50:whose
48:filter
44:filter
5646:Notes
2577:Goal:
2526:SciPy
1725:with
393:where
46:is a
5758:ISBN
5742:ISBN
5717:See
5327:, is
4495:<
4461:and
1883:(in
1331:and
657:The
65:The
34:, a
5685:is
5497:cos
4570:MSE
4103:cos
3903:cos
3806:cos
3782:cos
3685:MSE
3516:cos
3412:cos
3355:cos
3281:MSE
3160:MSE
3112:cos
2593:MSE
696:tap
114:or
106:or
40:FIR
30:In
5784::
5317:.
5256:,
4761:A
2520:,
2516:,
1969:Hz
967:,
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605:th
558:i'
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5503:(
5494:2
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5484:(
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5349:(
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5325:ω
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4957:]
4954:n
4951:[
4943:3
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4935:=
4932:]
4929:n
4926:[
4923:h
4911::
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4891:=
4888:N
4859:1
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4853:N
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4839:i
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4809:N
4805:b
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4669:2
4664:|
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4631:(
4628:R
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4617:F
4614:(
4611:W
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4602:/
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4589:/
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4574:=
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4543:f
4540:(
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4506:n
4498:0
4492:n
4484:0
4481:=
4478:]
4475:n
4472:[
4469:h
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4445:/
4441:)
4438:1
4432:N
4429:(
4426:=
4423:k
4415:,
4412:k
4409:,
4403:,
4400:3
4397:,
4394:2
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4385:=
4382:n
4378:r
4375:o
4372:f
4368:,
4365:2
4361:/
4357:]
4354:n
4351:[
4348:s
4345:=
4342:]
4339:n
4333:k
4330:[
4327:h
4324:,
4321:2
4317:/
4313:]
4310:n
4307:[
4304:s
4301:=
4298:]
4295:n
4292:+
4289:k
4286:[
4283:h
4280:,
4277:]
4274:0
4271:[
4268:s
4265:=
4262:]
4259:k
4256:[
4253:h
4232:]
4229:n
4226:[
4223:h
4203:]
4200:n
4197:[
4194:s
4170:0
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4153:,
4150:F
4147:d
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4140:F
4137:(
4132:d
4128:H
4124:)
4121:F
4118:n
4112:2
4109:(
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4090:1
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4081:/
4077:1
4066:=
4063:]
4060:n
4057:[
4054:s
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4030:d
4026:)
4023:F
4020:(
4015:d
4011:H
4005:2
4001:/
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3992:2
3988:/
3984:1
3973:=
3970:]
3967:0
3964:[
3961:s
3937:0
3934:=
3931:F
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3921:F
3918:n
3912:2
3909:(
3898:2
3894:)
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3803:)
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3773:/
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3760:/
3756:1
3745:]
3739:[
3736:s
3731:k
3726:0
3723:=
3712:2
3709:=
3703:]
3700:n
3697:[
3694:s
3657:]
3654:n
3651:[
3648:s
3624:F
3621:d
3615:2
3611:)
3607:F
3604:(
3599:d
3595:H
3589:2
3585:/
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3534:F
3531:n
3525:2
3522:(
3513:]
3510:n
3507:[
3504:s
3499:k
3494:0
3491:=
3488:n
3478:2
3474:/
3470:1
3465:2
3461:/
3457:1
3446:2
3440:F
3437:d
3433:)
3430:F
3421:2
3418:(
3409:]
3403:[
3400:s
3395:k
3390:0
3387:=
3376:)
3373:F
3370:n
3364:2
3361:(
3352:]
3349:n
3346:[
3343:s
3338:k
3333:0
3330:=
3327:n
3317:2
3313:/
3309:1
3304:2
3300:/
3296:1
3285:=
3256:F
3253:d
3247:2
3242:|
3237:)
3234:F
3231:(
3226:d
3222:H
3215:)
3212:F
3209:(
3206:R
3202:|
3196:2
3192:/
3188:1
3183:2
3179:/
3175:1
3164:=
3133:)
3130:F
3127:n
3121:2
3118:(
3109:]
3106:n
3103:[
3100:s
3095:k
3090:0
3087:=
3084:n
3076:=
3073:)
3070:F
3067:(
3064:H
3059:k
3056:F
3050:2
3047:j
3043:e
3039:=
3036:)
3033:F
3030:(
3027:R
3005:]
3002:n
2999:[
2996:r
2976:]
2973:n
2970:[
2967:h
2956:.
2942:2
2938:)
2935:1
2929:N
2926:(
2920:=
2917:k
2914:,
2911:]
2908:k
2905:+
2902:n
2899:[
2896:h
2893:=
2890:]
2887:n
2884:[
2881:r
2861:]
2858:n
2855:[
2852:h
2842:N
2815:)
2812:f
2809:(
2804:d
2800:H
2778:)
2775:f
2772:(
2769:H
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2707:|
2702:)
2699:f
2696:(
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2657:/
2651:s
2647:f
2641:2
2637:/
2631:s
2627:f
2613:1
2605:s
2601:f
2597:=
2502:N
2488:1
2485:+
2482:N
2462:1
2459:+
2456:N
2446:N
2395:.
2392:)
2384:j
2380:e
2376:(
2367:H
2361:=
2351:j
2347:e
2343:=
2340:z
2335:|
2330:)
2327:z
2324:(
2315:H
2304:=
2301:)
2295:(
2287:2
2283:H
2259:.
2254:n
2247:z
2240:]
2237:n
2234:[
2231:h
2215:=
2212:n
2198:)
2195:z
2192:(
2183:H
2153:)
2147:(
2139:2
2135:H
2112:,
2107:s
2103:f
2098:/
2092:2
2088:f
2067:,
2062:s
2058:f
2053:/
2047:1
2043:f
2018:,
2013:2
2009:f
1989:,
1984:1
1980:f
1967:(
1950:s
1946:f
1932:2
1922:=
1917:s
1913:f
1902:f
1898:=
1895:f
1869:s
1865:f
1844:]
1841:n
1838:[
1835:x
1815:.
1809:2
1806:1
1800:=
1793:f
1765:=
1734:f
1713:1
1693:)
1686:f
1679:2
1676:(
1668:2
1664:H
1651:(
1612:2
1589:,
1584:n
1576:)
1567:i
1563:e
1558:(
1548:n
1544:b
1538:N
1533:0
1530:=
1527:n
1519:=
1514:n
1506:)
1497:i
1493:e
1488:(
1480:]
1477:n
1474:[
1471:h
1455:=
1452:n
1438:)
1432:(
1424:2
1420:H
1407::
1386:)
1380:(
1377:H
1351:1
1342:F
1317:F
1292:,
1287:}
1282:)
1276:(
1273:H
1267:)
1261:(
1258:X
1253:{
1246:1
1237:F
1231:=
1228:]
1225:n
1222:[
1219:h
1213:]
1210:n
1207:[
1204:x
1201:=
1198:]
1195:n
1192:[
1189:y
1167:)
1161:(
1158:H
1148:}
1145:h
1142:{
1137:F
1123:)
1117:(
1114:X
1104:}
1101:x
1098:{
1093:F
1084:=
1079:)
1073:(
1070:Y
1060:}
1057:h
1051:x
1048:{
1043:F
1025::
1008:]
1005:n
1002:[
999:x
975:.
943:|
937:i
933:b
928:|
881:0
878:=
875:n
845:.
835:0
828:N
822:n
816:0
809:n
805:b
798:{
793:=
790:]
787:i
781:n
778:[
767:i
763:b
757:N
752:0
749:=
746:i
738:=
735:]
732:n
729:[
726:h
714::
699:s
680:]
677:i
671:n
668:[
665:x
632:i
628:b
601:N
580:N
574:i
568:0
542:i
538:b
515:1
512:+
509:N
489:N
469:N
447:]
444:n
441:[
438:y
416:]
413:n
410:[
407:x
395::
374:,
371:]
368:i
362:n
359:[
356:x
348:i
344:b
338:N
333:0
330:=
327:i
319:=
309:]
306:N
300:n
297:[
294:x
289:N
285:b
281:+
275:+
272:]
269:1
263:n
260:[
257:x
252:1
248:b
244:+
241:]
238:n
235:[
232:x
227:0
223:b
219:=
212:]
209:n
206:[
203:y
187::
183:N
164:z
160:N
144:z
140:N
136:N
132:N
87:1
84:+
81:N
38:(
20:)
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