96:
2254:
2229:
583:
558:
216:
20:
3003:
2979:
1860:
1836:
1891:
3513:
1268:
2771:
1590:
1054:
197:
playing the same signal at different distances from the listener, create a comb filtering effect on the audio. In any enclosed space, listeners hear a mixture of direct sound and reflected sound. The reflected sound takes a longer, delayed path compared to the direct sound, and a comb filter is
1065:
2532:
1442:
2622:
2218:
910:
547:
2406:
895:
773:
1263:{\displaystyle {\begin{aligned}\left|H(e^{j\omega })\right|&={\sqrt {(1+\alpha \cos(\omega K))^{2}+(\alpha \sin(\omega K))^{2}}}\\&={\sqrt {(1+\alpha ^{2})+2\alpha \cos(\omega K)}}\end{aligned}}}
2417:
3392:
2079:
413:
2926:
2627:
1772:
1447:
1070:
3320:
3165:
2766:{\displaystyle {\begin{aligned}f&=0,{\frac {1}{K}},{\frac {2}{K}},{\frac {3}{K}}\cdots \\\omega &=0,{\frac {2\pi }{K}},{\frac {4\pi }{K}},{\frac {6\pi }{K}}\cdots \end{aligned}}}
1585:{\displaystyle {\begin{aligned}f&={\frac {1}{2K}},{\frac {3}{2K}},{\frac {5}{2K}}\cdots \\\omega &={\frac {\pi }{K}},{\frac {3\pi }{K}},{\frac {5\pi }{K}}\cdots \,\end{aligned}}}
3234:
1351:
3078:
1310:
650:
2312:
1631:
2090:
1971:
296:
2611:
2585:
1431:
1381:
2562:
1678:
1658:
694:
1049:{\displaystyle \left|H\left(e^{j\omega }\right)\right|={\sqrt {\Re \left\{H\left(e^{j\omega }\right)\right\}^{2}+\Im \left\{H\left(e^{j\omega }\right)\right\}^{2}}}}
1994:
670:
319:
428:
2326:
3471:
3490:
788:
2537:
Again, the response is periodic, as the graphs demonstrate. The feedback comb filter has some properties in common with the feedforward form:
107:
706:
95:
3411:
3088:
for the delay element. Continuous-time implementations share all the properties of the respective discrete-time implementations.
2527:{\displaystyle \left|H\left(e^{j\omega }\right)\right|={\frac {1}{\sqrt {\left(1+\alpha ^{2}\right)-2\alpha \cos(\omega K)}}}\,.}
198:
created where the two mix at the listener. Similarly, comb filtering may result from mono mixing of multiple mics, hence the 3:1
171:
44:
119:
3331:
2008:
342:
3452:
2867:
1713:
156:
160:
2847:
filter. If stable, the response simply consists of a repeating series of impulses decreasing in amplitude over time.
3533:
36:
1433:
the first minimum occurs at half the delay period and repeats at even multiples of the delay frequency thereafter:
3257:
3102:
2844:
3182:
3517:
1814:
1798:
1690:
1315:
144:
3538:
3475:
2971:; these are the poles of the transfer function. This leads to a pole–zero plot like the ones shown below.
3048:
202:
that neighboring mics should be separated at least three times the distance from its source to the mic.
3494:
2786:
The levels of the maxima and minima are no longer equidistant from 1. The maxima have an amplitude of
2213:{\displaystyle H(z)={\frac {Y(z)}{X(z)}}={\frac {1}{1-\alpha z^{-K}}}={\frac {z^{K}}{z^{K}-\alpha }}}
1276:
616:
148:
111:
2778:
However, there are also some important differences because the magnitude response has a term in the
2281:
1604:
1899:
224:
1908:
1825:
233:
48:
779:
76:; which refer to the direction in which signals are delayed before they are added to the input.
3240:
3085:
2613:
the first maximum occurs at 0 and repeats at even multiples of the delay frequency thereafter:
2593:
2567:
2253:
2228:
1413:
1363:
1354:
697:
582:
557:
51:
of a comb filter consists of a series of regularly spaced notches in between regularly spaced
40:
28:
2547:
1663:
1643:
679:
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filters. Its response is simply the initial impulse with a second impulse after the delay.
3081:
3034:
84:
542:{\displaystyle H(z)={\frac {Y(z)}{X(z)}}=1+\alpha z^{-K}={\frac {z^{K}+\alpha }{z^{K}}}}
215:
19:
3432:
3239:
The feedforward form consists of an infinite number of zeros spaced along the jω axis (
3042:
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1979:
673:
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304:
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1859:
1835:
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1794:
1396:
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199:
115:
103:
Comb filters are employed in a variety of signal processing applications, including:
80:
3397:
The feedback form consists of an infinite number of poles spaced along the jω axis.
1890:
194:
182:
163:. If the delay is set to a few milliseconds, a comb filter can model the effect of
2401:{\displaystyle H\left(e^{j\omega }\right)={\frac {1}{1-\alpha e^{-j\omega K}}}\,,}
3456:
2779:
2541:
The response periodically drops to a local minimum and rises to a local maximum.
1898:
Similarly, the general structure of a feedback comb filter is described by the
1633:
the minima have zero amplitude. In this case, the minima are sometimes known as
419:
328:
123:
3406:
3176:
is the delay (measured in seconds). This has the following transfer function:
178:
129:
2D and 3D comb filters implemented in hardware (and occasionally software) in
190:
164:
138:
3512:
890:{\displaystyle H\left(e^{j\omega }\right)={\bigl }-j\alpha \sin(\omega K)}
152:
72:
223:
The general structure of a feedforward comb filter is described by the
1360:
The graphs show the periodic magnitude response for various values of
193:, comb filtering can arise as an unwanted artifact. For instance, two
3001:
2977:
2834:
increases, the amplitude of the maxima rises increasingly rapidly.
2252:
2227:
1889:
1858:
1834:
1598:
The levels of the maxima and minima are always equidistant from 1.
768:{\displaystyle H\left(e^{j\omega }\right)=1+\alpha e^{-j\omega K}}
607:
The frequency response of a discrete-time system expressed in the
581:
556:
214:
94:
18:
134:
60:
1357:. Hence the magnitude response of the comb filter is periodic.
130:
2824:
is strictly less than 1. As can be seen from the graphs, as
1707:-domain transfer function of the feedforward comb filter:
99:
Advanced PAL Comb Filter-II (APCF-II, Motorola MC141627FT)
3096:
The feedforward form may be described by the equation:
2861:-domain transfer function of the feedback comb filter:
327:
is a scaling factor applied to the delayed signal. The
3387:{\displaystyle H(s)={\frac {1}{1-\alpha e^{-s\tau }}}}
3334:
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3185:
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3051:
2870:
2625:
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2420:
2329:
2284:
2093:
2074:{\displaystyle \left(1-\alpha z^{-K}\right)Y(z)=X(z)}
2011:
1982:
1976:
This equation can be rearranged so that all terms in
1911:
1801:
of the transfer function. The denominator is zero at
1716:
1666:
1646:
1607:
1445:
1416:
1366:
1318:
1279:
1068:
1059:
In the case of the feedforward comb filter, this is:
913:
791:
709:
682:
658:
619:
431:
408:{\displaystyle Y(z)=\left(1+\alpha z^{-K}\right)X(z)}
345:
307:
236:
137:
analog television decoders, reduce artifacts such as
2921:{\displaystyle H(z)={\frac {z^{K}}{z^{K}-\alpha }}}
1767:{\displaystyle H(z)={\frac {z^{K}+\alpha }{z^{K}}}}
1689:The feedforward comb filter is one of the simplest
904:response, which ignores phase. This is defined as:
3386:
3314:
3228:
3159:
3072:
2920:
2765:
2605:
2579:
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2526:
2400:
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1988:
1965:
1766:
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1425:
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1345:
1304:
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1048:
889:
767:
688:
664:
644:
541:
407:
313:
290:
219:Feedforward comb filter structure in discrete time
2967:solutions, equally spaced around a circle in the
1793:solutions, equally spaced around a circle in the
2564:coincide with the minima for negative values of
1660:coincide with the minima for negative values of
43:to itself, causing constructive and destructive
3455:. Acoustic Sciences Corporation. Archived from
1996:are on the left-hand side, and then taking the
1894:Feedback comb filter structure in discrete time
1863:Pole–zero plot of feedforward comb filter with
1839:Pole–zero plot of feedforward comb filter with
321:is the delay length (measured in samples), and
181:promises to increase the precision of existing
700:. Therefore, for the feedforward comb filter:
2843:The feedback comb filter is a simple type of
855:
821:
39:implemented by adding a delayed version of a
8:
3006:Pole–zero plot of feedback comb filter with
2982:Pole–zero plot of feedback comb filter with
2951:. The denominator is equal to zero whenever
3315:{\displaystyle y(t)=x(t)+\alpha y(t-\tau )}
3160:{\displaystyle y(t)=x(t)+\alpha x(t-\tau )}
586:Feedforward magnitude response for various
561:Feedforward magnitude response for various
782:, the frequency response is also given by
3369:
3350:
3333:
3259:
3229:{\displaystyle H(s)=1+\alpha e^{-s\tau }}
3214:
3184:
3104:
3050:
2903:
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2705:
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1981:
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1606:
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531:
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379:
344:
306:
235:
3033:Comb filters may also be implemented in
2257:Feedback magnitude response for various
2232:Feedback magnitude response for various
1777:the numerator is equal to zero whenever
3423:
3241:which corresponds to the Fourier domain
1346:{\displaystyle 2\alpha \cos(\omega K)}
336:of both sides of the equation yields:
3433:"Hearing, Columns and Comb Filtering"
3325:and the following transfer function:
1387:The response periodically drops to a
7:
3251:The feedback form has the equation:
2931:This time, the numerator is zero at
2084:The transfer function is therefore:
613:-domain is obtained by substitution
79:Comb filters may be implemented in
3073:{\displaystyle s=\sigma +j\omega }
2544:The maxima for positive values of
1640:The maxima for positive values of
1000:
953:
14:
110:(CIC) filters, commonly used for
66:Comb filters exist in two forms,
23:Feedforward comb filter structure
3511:
2411:the magnitude response becomes:
2314:into the feedback comb filter's
1395:), and periodically rises to a
1305:{\displaystyle (1+\alpha ^{2})}
645:{\displaystyle z=e^{j\omega },}
3344:
3338:
3309:
3297:
3285:
3279:
3270:
3264:
3195:
3189:
3154:
3142:
3130:
3124:
3115:
3109:
3037:which can be expressed in the
3029:Continuous time implementation
2880:
2874:
2514:
2505:
2307:{\displaystyle z=e^{j\omega }}
2135:
2129:
2121:
2115:
2103:
2097:
2068:
2062:
2053:
2047:
1960:
1948:
1936:
1930:
1921:
1915:
1726:
1720:
1626:{\displaystyle \alpha =\pm 1,}
1340:
1331:
1312:term is constant, whereas the
1299:
1280:
1251:
1242:
1224:
1205:
1182:
1178:
1169:
1157:
1145:
1141:
1132:
1114:
1097:
1081:
884:
875:
850:
841:
473:
467:
459:
453:
441:
435:
402:
396:
355:
349:
285:
273:
261:
255:
246:
240:
87:forms which are very similar.
1:
59:) giving the appearance of a
1966:{\displaystyle y=x+\alpha y}
291:{\displaystyle y=x+\alpha x}
206:Discrete time implementation
157:physical modelling synthesis
3080:analogous to the z domain.
3045:frequency domain parameter
1383:Some important properties:
170:in a cylindrical cavity or
161:digital waveguide synthesis
122:operations that change the
3555:
3472:"Feedforward Comb Filters"
3412:Fabry–Pérot interferometer
126:of a discrete-time system.
2845:infinite impulse response
900:Often of interest is the
2851:Pole–zero interpretation
2606:{\displaystyle \alpha ,}
2580:{\displaystyle \alpha ,}
1697:Pole–zero interpretation
1426:{\displaystyle \alpha ,}
1376:{\displaystyle \alpha .}
185:by nearly a hundredfold.
108:Cascaded integrator–comb
16:Signal processing filter
3491:"Feedback Comb Filters"
2590:For positive values of
2557:{\displaystyle \alpha }
1691:finite impulse response
1673:{\displaystyle \alpha }
1653:{\displaystyle \alpha }
1410:For positive values of
689:{\displaystyle \omega }
145:Audio signal processing
3388:
3316:
3230:
3161:
3074:
3021:
2997:
2922:
2767:
2607:
2581:
2558:
2528:
2402:
2308:
2275:
2250:
2214:
2075:
1990:
1967:
1895:
1878:
1854:
1768:
1674:
1654:
1627:
1586:
1427:
1399:(sometimes known as a
1391:(sometimes known as a
1377:
1347:
1306:
1264:
1050:
891:
769:
690:
666:
646:
604:
579:
543:
409:
315:
292:
220:
100:
24:
3389:
3317:
3231:
3162:
3075:
3041:as a function of the
3005:
2981:
2923:
2855:Looking again at the
2768:
2608:
2582:
2559:
2529:
2403:
2309:
2256:
2231:
2215:
2076:
1991:
1968:
1893:
1862:
1838:
1828:like the ones shown.
1769:
1701:Looking again at the
1675:
1655:
1628:
1587:
1428:
1378:
1348:
1307:
1265:
1051:
892:
770:
691:
667:
647:
585:
560:
544:
410:
316:
293:
218:
172:in a vibrating string
98:
22:
3520:at Wikimedia Commons
3332:
3258:
3183:
3103:
3049:
2868:
2623:
2594:
2568:
2548:
2418:
2327:
2320:-domain expression:
2282:
2091:
2009:
1980:
1909:
1714:
1664:
1644:
1605:
1443:
1414:
1364:
1316:
1277:
1066:
911:
789:
707:
680:
656:
617:
429:
343:
305:
234:
2810:The filter is only
1900:difference equation
225:difference equation
3384:
3312:
3226:
3157:
3070:
3022:
2998:
2918:
2763:
2761:
2603:
2577:
2554:
2524:
2398:
2304:
2276:
2251:
2224:Frequency response
2210:
2071:
1986:
1963:
1896:
1879:
1855:
1824:. This leads to a
1764:
1670:
1650:
1623:
1582:
1580:
1423:
1373:
1343:
1302:
1260:
1258:
1046:
887:
765:
686:
662:
642:
605:
580:
553:Frequency response
539:
405:
311:
288:
221:
101:
55:(sometimes called
49:frequency response
25:
3534:Signal processing
3516:Media related to
3453:"Acoustic Basics"
3382:
3086:analog delay line
3084:use some form of
3026:
3025:
2916:
2754:
2736:
2718:
2680:
2667:
2654:
2518:
2517:
2392:
2208:
2173:
2139:
1989:{\displaystyle y}
1883:
1882:
1762:
1680:, and vice versa.
1572:
1554:
1536:
1509:
1491:
1473:
1254:
1191:
1044:
698:angular frequency
665:{\displaystyle j}
537:
477:
420:transfer function
314:{\displaystyle K}
177:In astronomy the
29:signal processing
3546:
3515:
3499:
3498:
3493:. Archived from
3486:
3480:
3479:
3474:. Archived from
3467:
3461:
3460:
3449:
3443:
3442:
3440:
3439:
3428:
3393:
3391:
3390:
3385:
3383:
3381:
3380:
3379:
3351:
3321:
3319:
3318:
3313:
3235:
3233:
3232:
3227:
3225:
3224:
3175:
3166:
3164:
3163:
3158:
3092:Feedforward form
3079:
3077:
3076:
3071:
3020:in discrete time
3019:
3012:
2996:in discrete time
2995:
2988:
2974:
2973:
2966:
2960:
2950:
2943:
2937:
2927:
2925:
2924:
2919:
2917:
2915:
2908:
2907:
2897:
2896:
2887:
2860:
2839:Impulse response
2833:
2831:
2823:
2821:
2806:
2805:
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2802:
2796:
2793:
2772:
2770:
2769:
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2655:
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2586:
2584:
2583:
2578:
2563:
2561:
2560:
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2525:
2519:
2489:
2485:
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2483:
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2459:
2454:
2450:
2449:
2445:
2444:
2407:
2405:
2404:
2399:
2393:
2391:
2390:
2389:
2358:
2353:
2349:
2348:
2319:
2313:
2311:
2310:
2305:
2303:
2302:
2278:By substituting
2274:in discrete time
2273:
2266:
2249:in discrete time
2248:
2241:
2219:
2217:
2216:
2211:
2209:
2207:
2200:
2199:
2189:
2188:
2179:
2174:
2172:
2171:
2170:
2145:
2140:
2138:
2124:
2110:
2080:
2078:
2077:
2072:
2043:
2039:
2038:
2037:
2001:
1995:
1993:
1992:
1987:
1972:
1970:
1969:
1964:
1877:in discrete time
1876:
1869:
1853:in discrete time
1852:
1845:
1831:
1830:
1823:
1813:
1807:
1797:; these are the
1792:
1786:
1773:
1771:
1770:
1765:
1763:
1761:
1760:
1751:
1744:
1743:
1733:
1706:
1685:Impulse response
1679:
1677:
1676:
1671:
1659:
1657:
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1632:
1630:
1629:
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1508:
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1424:
1382:
1380:
1379:
1374:
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1350:
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1303:
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1269:
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1259:
1255:
1223:
1222:
1204:
1196:
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1190:
1189:
1153:
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1113:
1104:
1100:
1096:
1095:
1055:
1053:
1052:
1047:
1045:
1043:
1042:
1037:
1033:
1032:
1028:
1027:
996:
995:
990:
986:
985:
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952:
947:
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942:
938:
937:
896:
894:
893:
888:
859:
858:
825:
824:
815:
811:
810:
774:
772:
771:
766:
764:
763:
733:
729:
728:
695:
693:
692:
687:
671:
669:
668:
663:
651:
649:
648:
643:
638:
637:
612:
603:in discrete time
602:
595:
578:in discrete time
577:
570:
548:
546:
545:
540:
538:
536:
535:
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519:
518:
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502:
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392:
388:
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333:
326:
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318:
317:
312:
297:
295:
294:
289:
211:Feedforward form
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3553:
3549:
3548:
3547:
3545:
3544:
3543:
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3508:
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3502:
3488:
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3464:
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3450:
3446:
3437:
3435:
3431:Roger Russell.
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3255:
3249:
3210:
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3094:
3082:Analog circuits
3047:
3046:
3035:continuous time
3031:
3014:
3007:
2990:
2983:
2962:
2952:
2945:
2939:
2932:
2899:
2898:
2888:
2866:
2865:
2856:
2853:
2841:
2827:
2825:
2817:
2815:
2797:
2794:
2791:
2790:
2788:
2787:
2760:
2759:
2743:
2725:
2707:
2692:
2686:
2685:
2633:
2621:
2620:
2592:
2591:
2587:and vice versa.
2566:
2565:
2546:
2545:
2475:
2468:
2464:
2433:
2429:
2425:
2421:
2416:
2415:
2372:
2362:
2337:
2333:
2325:
2324:
2315:
2291:
2280:
2279:
2268:
2262:
2243:
2237:
2226:
2191:
2190:
2180:
2159:
2149:
2125:
2111:
2089:
2088:
2026:
2016:
2012:
2007:
2006:
1997:
1978:
1977:
1907:
1906:
1888:
1871:
1864:
1847:
1840:
1818:
1809:
1802:
1788:
1778:
1752:
1735:
1734:
1712:
1711:
1702:
1699:
1687:
1662:
1661:
1642:
1641:
1603:
1602:
1579:
1578:
1561:
1543:
1521:
1515:
1514:
1501:
1483:
1465:
1453:
1441:
1440:
1412:
1411:
1362:
1361:
1314:
1313:
1289:
1275:
1274:
1257:
1256:
1214:
1194:
1193:
1181:
1144:
1105:
1084:
1077:
1073:
1064:
1063:
1016:
1012:
1008:
1004:
1003:
969:
965:
961:
957:
956:
926:
922:
918:
914:
909:
908:
799:
795:
787:
786:
780:Euler's formula
746:
717:
713:
705:
704:
678:
677:
654:
653:
626:
615:
614:
608:
597:
591:
572:
566:
555:
527:
510:
509:
491:
463:
449:
427:
426:
422:is defined as:
375:
365:
361:
341:
340:
329:
322:
303:
302:
232:
231:
213:
208:
93:
85:continuous time
17:
12:
11:
5:
3552:
3550:
3542:
3541:
3536:
3526:
3525:
3522:
3521:
3507:
3506:External links
3504:
3501:
3500:
3497:on 2011-06-06.
3481:
3478:on 2011-06-06.
3462:
3459:on 2010-05-07.
3444:
3422:
3421:
3419:
3416:
3415:
3414:
3409:
3402:
3399:
3395:
3394:
3378:
3375:
3372:
3368:
3364:
3361:
3358:
3354:
3349:
3346:
3343:
3340:
3337:
3323:
3322:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3248:
3245:
3237:
3236:
3223:
3220:
3217:
3213:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3188:
3168:
3167:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3093:
3090:
3069:
3066:
3063:
3060:
3057:
3054:
3039:Laplace domain
3030:
3027:
3024:
3023:
2999:
2929:
2928:
2914:
2911:
2906:
2902:
2895:
2891:
2885:
2882:
2879:
2876:
2873:
2852:
2849:
2840:
2837:
2836:
2835:
2808:
2776:
2775:
2774:
2773:
2758:
2753:
2749:
2746:
2740:
2735:
2731:
2728:
2722:
2717:
2713:
2710:
2704:
2701:
2698:
2695:
2693:
2691:
2688:
2687:
2684:
2679:
2676:
2671:
2666:
2663:
2658:
2653:
2650:
2645:
2642:
2639:
2636:
2634:
2632:
2629:
2628:
2615:
2614:
2602:
2599:
2588:
2576:
2573:
2553:
2542:
2535:
2534:
2523:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2488:
2482:
2478:
2474:
2471:
2467:
2462:
2457:
2453:
2448:
2443:
2440:
2436:
2432:
2428:
2424:
2409:
2408:
2397:
2388:
2385:
2382:
2379:
2375:
2371:
2368:
2365:
2361:
2356:
2352:
2347:
2344:
2340:
2336:
2332:
2301:
2298:
2294:
2290:
2287:
2225:
2222:
2221:
2220:
2206:
2203:
2198:
2194:
2187:
2183:
2177:
2169:
2166:
2162:
2158:
2155:
2152:
2148:
2143:
2137:
2134:
2131:
2128:
2123:
2120:
2117:
2114:
2108:
2105:
2102:
2099:
2096:
2082:
2081:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2042:
2036:
2033:
2029:
2025:
2022:
2019:
2015:
1985:
1974:
1973:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1887:
1884:
1881:
1880:
1856:
1826:pole–zero plot
1775:
1774:
1759:
1755:
1750:
1747:
1742:
1738:
1731:
1728:
1725:
1722:
1719:
1698:
1695:
1686:
1683:
1682:
1681:
1669:
1649:
1638:
1622:
1619:
1616:
1613:
1610:
1599:
1595:
1594:
1593:
1592:
1576:
1571:
1567:
1564:
1558:
1553:
1549:
1546:
1540:
1535:
1532:
1527:
1524:
1522:
1520:
1517:
1516:
1513:
1507:
1504:
1500:
1495:
1489:
1486:
1482:
1477:
1471:
1468:
1464:
1459:
1456:
1454:
1452:
1449:
1448:
1435:
1434:
1422:
1419:
1408:
1372:
1369:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1301:
1296:
1292:
1288:
1285:
1282:
1271:
1270:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1221:
1217:
1213:
1210:
1207:
1202:
1199:
1197:
1195:
1188:
1184:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1151:
1147:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1111:
1108:
1106:
1103:
1099:
1094:
1091:
1087:
1083:
1080:
1076:
1072:
1071:
1057:
1056:
1041:
1036:
1031:
1026:
1023:
1019:
1015:
1011:
1007:
1002:
999:
994:
989:
984:
979:
976:
972:
968:
964:
960:
955:
950:
946:
941:
936:
933:
929:
925:
921:
917:
898:
897:
886:
883:
880:
877:
874:
871:
868:
865:
862:
857:
852:
849:
846:
843:
840:
837:
834:
831:
828:
823:
818:
814:
809:
806:
802:
798:
794:
776:
775:
762:
759:
756:
753:
749:
745:
742:
739:
736:
732:
727:
724:
720:
716:
712:
685:
674:imaginary unit
661:
641:
636:
633:
629:
625:
622:
554:
551:
550:
549:
534:
530:
525:
522:
517:
513:
506:
501:
498:
494:
490:
487:
484:
481:
475:
472:
469:
466:
461:
458:
455:
452:
446:
443:
440:
437:
434:
416:
415:
404:
401:
398:
395:
391:
385:
382:
378:
374:
371:
368:
364:
360:
357:
354:
351:
348:
310:
299:
298:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
212:
209:
207:
204:
187:
186:
175:
168:standing waves
142:
127:
92:
89:
15:
13:
10:
9:
6:
4:
3:
2:
3551:
3540:
3539:Filter theory
3537:
3535:
3532:
3531:
3529:
3519:
3514:
3510:
3509:
3505:
3496:
3492:
3485:
3482:
3477:
3473:
3470:Smith, J. O.
3466:
3463:
3458:
3454:
3448:
3445:
3434:
3427:
3424:
3417:
3413:
3410:
3408:
3405:
3404:
3400:
3398:
3376:
3373:
3370:
3366:
3362:
3359:
3356:
3352:
3347:
3341:
3335:
3328:
3327:
3326:
3306:
3303:
3300:
3294:
3291:
3288:
3282:
3276:
3273:
3267:
3261:
3254:
3253:
3252:
3247:Feedback form
3246:
3244:
3242:
3221:
3218:
3215:
3211:
3207:
3204:
3201:
3198:
3192:
3186:
3179:
3178:
3177:
3174:
3151:
3148:
3145:
3139:
3136:
3133:
3127:
3121:
3118:
3112:
3106:
3099:
3098:
3097:
3091:
3089:
3087:
3083:
3067:
3064:
3061:
3058:
3055:
3052:
3044:
3040:
3036:
3028:
3017:
3010:
3004:
3000:
2993:
2986:
2980:
2976:
2975:
2972:
2970:
2969:complex plane
2965:
2959:
2955:
2948:
2942:
2935:
2912:
2909:
2904:
2900:
2893:
2889:
2883:
2877:
2871:
2864:
2863:
2862:
2859:
2850:
2848:
2846:
2838:
2830:
2820:
2813:
2809:
2801:
2785:
2784:
2783:
2781:
2756:
2751:
2747:
2744:
2738:
2733:
2729:
2726:
2720:
2715:
2711:
2708:
2702:
2699:
2696:
2694:
2689:
2682:
2677:
2674:
2669:
2664:
2661:
2656:
2651:
2648:
2643:
2640:
2637:
2635:
2630:
2619:
2618:
2617:
2616:
2600:
2597:
2589:
2574:
2571:
2551:
2543:
2540:
2539:
2538:
2521:
2511:
2508:
2502:
2499:
2496:
2493:
2490:
2486:
2480:
2476:
2472:
2469:
2465:
2460:
2455:
2451:
2446:
2441:
2438:
2434:
2430:
2426:
2422:
2414:
2413:
2412:
2395:
2386:
2383:
2380:
2377:
2373:
2369:
2366:
2363:
2359:
2354:
2350:
2345:
2342:
2338:
2334:
2330:
2323:
2322:
2321:
2318:
2299:
2296:
2292:
2288:
2285:
2271:
2265:
2260:
2255:
2246:
2240:
2235:
2230:
2223:
2204:
2201:
2196:
2192:
2185:
2181:
2175:
2167:
2164:
2160:
2156:
2153:
2150:
2146:
2141:
2132:
2126:
2118:
2112:
2106:
2100:
2094:
2087:
2086:
2085:
2065:
2059:
2056:
2050:
2044:
2040:
2034:
2031:
2027:
2023:
2020:
2017:
2013:
2005:
2004:
2003:
2000:
1983:
1957:
1954:
1951:
1945:
1942:
1939:
1933:
1927:
1924:
1918:
1912:
1905:
1904:
1903:
1901:
1892:
1886:Feedback form
1885:
1874:
1867:
1861:
1857:
1850:
1843:
1837:
1833:
1832:
1829:
1827:
1821:
1816:
1812:
1805:
1800:
1796:
1795:complex plane
1791:
1785:
1781:
1757:
1753:
1748:
1745:
1740:
1736:
1729:
1723:
1717:
1710:
1709:
1708:
1705:
1696:
1694:
1692:
1684:
1667:
1647:
1639:
1636:
1620:
1617:
1614:
1611:
1608:
1600:
1597:
1596:
1574:
1569:
1565:
1562:
1556:
1551:
1547:
1544:
1538:
1533:
1530:
1525:
1523:
1518:
1511:
1505:
1502:
1498:
1493:
1487:
1484:
1480:
1475:
1469:
1466:
1462:
1457:
1455:
1450:
1439:
1438:
1437:
1436:
1420:
1417:
1409:
1406:
1402:
1398:
1397:local maximum
1394:
1390:
1389:local minimum
1386:
1385:
1384:
1370:
1367:
1358:
1356:
1337:
1334:
1328:
1325:
1322:
1319:
1294:
1290:
1286:
1283:
1248:
1245:
1239:
1236:
1233:
1230:
1227:
1219:
1215:
1211:
1208:
1200:
1198:
1186:
1175:
1172:
1166:
1163:
1160:
1154:
1149:
1138:
1135:
1129:
1126:
1123:
1120:
1117:
1109:
1107:
1101:
1092:
1089:
1085:
1078:
1074:
1062:
1061:
1060:
1039:
1034:
1029:
1024:
1021:
1017:
1013:
1009:
1005:
997:
992:
987:
982:
977:
974:
970:
966:
962:
958:
948:
944:
939:
934:
931:
927:
923:
919:
915:
907:
906:
905:
903:
881:
878:
872:
869:
866:
863:
860:
847:
844:
838:
835:
832:
829:
826:
816:
812:
807:
804:
800:
796:
792:
785:
784:
783:
781:
760:
757:
754:
751:
747:
743:
740:
737:
734:
730:
725:
722:
718:
714:
710:
703:
702:
701:
699:
683:
675:
659:
639:
634:
631:
627:
623:
620:
611:
600:
594:
589:
584:
575:
569:
564:
559:
552:
532:
528:
523:
520:
515:
511:
504:
499:
496:
492:
488:
485:
482:
479:
470:
464:
456:
450:
444:
438:
432:
425:
424:
423:
421:
399:
393:
389:
383:
380:
376:
372:
369:
366:
362:
358:
352:
346:
339:
338:
337:
335:
332:
325:
308:
282:
279:
276:
270:
267:
264:
258:
252:
249:
243:
237:
230:
229:
228:
226:
217:
210:
205:
203:
201:
200:rule of thumb
196:
192:
184:
183:spectrographs
180:
176:
173:
169:
166:
162:
158:
154:
150:
146:
143:
140:
136:
132:
128:
125:
121:
117:
116:interpolation
113:
112:anti-aliasing
109:
106:
105:
104:
97:
90:
88:
86:
82:
81:discrete time
77:
75:
74:
69:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
21:
3518:Comb filters
3495:the original
3489:Smith, J.O.
3484:
3476:the original
3465:
3457:the original
3447:
3436:. Retrieved
3426:
3396:
3324:
3250:
3238:
3172:
3169:
3095:
3032:
3015:
3008:
2991:
2984:
2963:
2957:
2953:
2946:
2940:
2933:
2930:
2857:
2854:
2842:
2828:
2818:
2799:
2777:
2536:
2410:
2316:
2277:
2269:
2263:
2258:
2244:
2238:
2233:
2083:
1998:
1975:
1897:
1872:
1865:
1848:
1841:
1819:
1810:
1803:
1789:
1783:
1779:
1776:
1703:
1700:
1688:
1634:
1404:
1400:
1392:
1359:
1355:periodically
1353:term varies
1272:
1058:
901:
899:
777:
609:
606:
598:
592:
587:
573:
567:
562:
417:
330:
323:
300:
222:
195:loudspeakers
188:
147:, including
102:
91:Applications
78:
71:
67:
65:
56:
52:
45:interference
32:
26:
2961:. This has
2780:denominator
2002:transform:
1787:. This has
124:sample rate
68:feedforward
33:comb filter
3528:Categories
3438:2010-04-22
3418:References
3407:Dirac comb
2261:values of
2236:values of
590:values of
565:values of
179:astro-comb
120:decimation
3377:τ
3371:−
3363:α
3360:−
3307:τ
3304:−
3292:α
3222:τ
3216:−
3208:α
3152:τ
3149:−
3137:α
3068:ω
3059:σ
2944:zeros at
2938:, giving
2913:α
2910:−
2757:⋯
2748:π
2730:π
2712:π
2690:ω
2683:⋯
2598:α
2572:α
2552:α
2509:ω
2503:
2497:α
2491:−
2477:α
2442:ω
2384:ω
2378:−
2370:α
2367:−
2346:ω
2300:ω
2205:α
2202:−
2165:−
2157:α
2154:−
2032:−
2024:α
2021:−
1955:−
1943:α
1808:, giving
1749:α
1668:α
1648:α
1615:±
1609:α
1575:⋯
1566:π
1548:π
1531:π
1519:ω
1512:⋯
1418:α
1368:α
1335:ω
1329:
1323:α
1291:α
1246:ω
1240:
1234:α
1216:α
1173:ω
1167:
1161:α
1136:ω
1130:
1124:α
1093:ω
1025:ω
1001:ℑ
978:ω
954:ℜ
935:ω
902:magnitude
879:ω
873:
867:α
861:−
845:ω
839:
833:α
808:ω
758:ω
752:−
744:α
726:ω
684:ω
635:ω
524:α
497:−
489:α
381:−
373:α
334:transform
280:−
268:α
191:acoustics
139:dot crawl
3401:See also
2259:negative
2234:positive
588:negative
563:positive
165:acoustic
153:flanging
73:feedback
3043:complex
2804:
2789:
672:is the
114:during
3170:where
3018:= −0.5
2832:|
2826:|
2822:|
2816:|
2812:stable
1875:= −0.5
778:Using
652:where
301:where
47:. The
41:signal
37:filter
2994:= 0.5
1851:= 0.5
1815:poles
1799:zeros
1635:nulls
1601:When
1405:tooth
1403:or a
1393:notch
149:delay
57:teeth
53:peaks
35:is a
3013:and
2989:and
2798:1 −
2267:and
2242:and
1870:and
1846:and
1401:peak
1273:The
676:and
596:and
571:and
418:The
159:and
135:NTSC
133:and
118:and
70:and
61:comb
31:, a
3243:).
3011:= 8
2987:= 8
2949:= 0
2936:= 0
2814:if
2500:cos
2272:= 2
2247:= 2
1868:= 8
1844:= 8
1822:= 0
1817:at
1806:= 0
1782:= −
1326:cos
1237:cos
1164:sin
1127:cos
870:sin
836:cos
696:is
601:= 1
576:= 1
227::
189:In
131:PAL
83:or
27:In
3530::
2956:=
2782::
1902::
1407:).
155:,
151:,
63:.
3441:.
3374:s
3367:e
3357:1
3353:1
3348:=
3345:)
3342:s
3339:(
3336:H
3310:)
3301:t
3298:(
3295:y
3289:+
3286:)
3283:t
3280:(
3277:x
3274:=
3271:)
3268:t
3265:(
3262:y
3219:s
3212:e
3205:+
3202:1
3199:=
3196:)
3193:s
3190:(
3187:H
3173:τ
3155:)
3146:t
3143:(
3140:x
3134:+
3131:)
3128:t
3125:(
3122:x
3119:=
3116:)
3113:t
3110:(
3107:y
3065:j
3062:+
3056:=
3053:s
3016:α
3009:K
2992:α
2985:K
2964:K
2958:α
2954:z
2947:z
2941:K
2934:z
2905:K
2901:z
2894:K
2890:z
2884:=
2881:)
2878:z
2875:(
2872:H
2858:z
2829:α
2819:α
2807:.
2800:α
2795:/
2792:1
2752:K
2745:6
2739:,
2734:K
2727:4
2721:,
2716:K
2709:2
2703:,
2700:0
2697:=
2678:K
2675:3
2670:,
2665:K
2662:2
2657:,
2652:K
2649:1
2644:,
2641:0
2638:=
2631:f
2601:,
2575:,
2522:.
2515:)
2512:K
2506:(
2494:2
2487:)
2481:2
2473:+
2470:1
2466:(
2461:1
2456:=
2452:|
2447:)
2439:j
2435:e
2431:(
2427:H
2423:|
2396:,
2387:K
2381:j
2374:e
2364:1
2360:1
2355:=
2351:)
2343:j
2339:e
2335:(
2331:H
2317:z
2297:j
2293:e
2289:=
2286:z
2270:K
2264:α
2245:K
2239:α
2197:K
2193:z
2186:K
2182:z
2176:=
2168:K
2161:z
2151:1
2147:1
2142:=
2136:)
2133:z
2130:(
2127:X
2122:)
2119:z
2116:(
2113:Y
2107:=
2104:)
2101:z
2098:(
2095:H
2069:)
2066:z
2063:(
2060:X
2057:=
2054:)
2051:z
2048:(
2045:Y
2041:)
2035:K
2028:z
2018:1
2014:(
1999:z
1984:y
1961:]
1958:K
1952:n
1949:[
1946:y
1940:+
1937:]
1934:n
1931:[
1928:x
1925:=
1922:]
1919:n
1916:[
1913:y
1873:α
1866:K
1849:α
1842:K
1820:z
1811:K
1804:z
1790:K
1784:α
1780:z
1758:K
1754:z
1746:+
1741:K
1737:z
1730:=
1727:)
1724:z
1721:(
1718:H
1704:z
1637:.
1621:,
1618:1
1612:=
1570:K
1563:5
1557:,
1552:K
1545:3
1539:,
1534:K
1526:=
1506:K
1503:2
1499:5
1494:,
1488:K
1485:2
1481:3
1476:,
1470:K
1467:2
1463:1
1458:=
1451:f
1421:,
1371:.
1341:)
1338:K
1332:(
1320:2
1300:)
1295:2
1287:+
1284:1
1281:(
1252:)
1249:K
1243:(
1231:2
1228:+
1225:)
1220:2
1212:+
1209:1
1206:(
1201:=
1187:2
1183:)
1179:)
1176:K
1170:(
1158:(
1155:+
1150:2
1146:)
1142:)
1139:K
1133:(
1121:+
1118:1
1115:(
1110:=
1102:|
1098:)
1090:j
1086:e
1082:(
1079:H
1075:|
1040:2
1035:}
1030:)
1022:j
1018:e
1014:(
1010:H
1006:{
998:+
993:2
988:}
983:)
975:j
971:e
967:(
963:H
959:{
949:=
945:|
940:)
932:j
928:e
924:(
920:H
916:|
885:)
882:K
876:(
864:j
856:]
851:)
848:K
842:(
830:+
827:1
822:[
817:=
813:)
805:j
801:e
797:(
793:H
761:K
755:j
748:e
741:+
738:1
735:=
731:)
723:j
719:e
715:(
711:H
660:j
640:,
632:j
628:e
624:=
621:z
610:z
599:K
593:α
574:K
568:α
533:K
529:z
521:+
516:K
512:z
505:=
500:K
493:z
486:+
483:1
480:=
474:)
471:z
468:(
465:X
460:)
457:z
454:(
451:Y
445:=
442:)
439:z
436:(
433:H
403:)
400:z
397:(
394:X
390:)
384:K
377:z
370:+
367:1
363:(
359:=
356:)
353:z
350:(
347:Y
331:z
324:α
309:K
286:]
283:K
277:n
274:[
271:x
265:+
262:]
259:n
256:[
253:x
250:=
247:]
244:n
241:[
238:y
174:.
141:.
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