116:
1183:
308:
583:
725:
815:
1055:
97:
is to create a realizable filter by shortening and/or time-shifting a non-causal impulse response. If shortening is necessary, it is often accomplished as the product of the impulse-response with a
1330:
1017:
931:
1268:
1225:
1363:
452:
416:
164:
378:
182:
91:
460:
351:
331:
612:
119:
Each component of the causal filter output begins when its stimulus begins. The outputs of the non-causal filter begin before the stimulus begins.
70:) must be causal because such systems cannot act on a future input. In effect that means the output sample that best represents the input at time
736:
1178:{\displaystyle H(\omega )=\left(\delta (\omega )-{i \over \pi \omega }\right)*G(\omega )=G(\omega )-i\cdot {\widehat {G}}(\omega )\,}
1353:
1279:
115:
958:
1385:
858:
47:
1273:
Taking the
Hilbert transform of the above equation yields this relation between "H" and its Hilbert transform:
1234:
1191:
1390:
1342:
Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (September 2007),
67:
1027:
945:
1349:
1344:
1228:
421:
36:
28:
600:
391:
303:{\displaystyle f(x)=\int _{x-1}^{x+1}s(\tau )\,d\tau \ =\int _{-1}^{+1}s(x+\tau )\,d\tau \,}
139:
356:
578:{\displaystyle f(t-1)=\int _{-2}^{0}s(t+\tau )\,d\tau =\int _{0}^{+2}s(t-\tau )\,d\tau \,}
98:
73:
336:
316:
133:
109:
94:
1379:
720:{\displaystyle f(t)=(h*s)(t)=\int _{-\infty }^{\infty }h(\tau )s(t-\tau )\,d\tau .\,}
105:
39:
17:
604:
591:
Any linear filter (such as a moving average) can be characterized by a function
46:
indicates that the filter output depends only on past and present inputs. A
1231:
done in the frequency domain (rather than the time domain). The sign of
333:
could represent a spatial coordinate, as in image processing. But if
112:, anti-causal filter whose inverse is also stable and anti-causal.
810:{\displaystyle f(t)=\int _{0}^{\infty }h(\tau )s(t-\tau )\,d\tau }
114:
1365:
Determining a System's
Causality from its Frequency Response
848:) be a causal filter with corresponding Fourier transform
836:
Characterization of causal filters in the frequency domain
1348:(3rd ed.), Cambridge University Press, p. 767,
588:
which is a delayed version of the non-realizable output.
952:(ω) is real-valued. We now have the following relation
93:
comes out slightly later. A common design practice for
1270:
may depend on the definition of the
Fourier Transform.
820:
and general equality of these two expressions requires
1282:
1237:
1194:
1058:
961:
861:
739:
615:
463:
424:
394:
359:
339:
319:
185:
142:
76:
1325:{\displaystyle {\widehat {H}}(\omega )=iH(\omega )}
1324:
1262:
1219:
1177:
1011:
925:
809:
719:
577:
446:
410:
372:
345:
325:
302:
158:
85:
1012:{\displaystyle h(t)=2\,\Theta (t)\cdot g(t)\,}
50:whose output also depends on future inputs is
926:{\displaystyle g(t)={h(t)+h^{*}(-t) \over 2}}
8:
380:, then a moving average defined that way is
1033:This means that the Fourier transforms of
1284:
1283:
1281:
1263:{\displaystyle {\widehat {G}}(\omega )\,}
1259:
1239:
1238:
1236:
1220:{\displaystyle {\widehat {G}}(\omega )\,}
1216:
1196:
1195:
1193:
1174:
1154:
1153:
1094:
1057:
1008:
980:
960:
948:and, consequently, its Fourier transform
936:which is non-causal. On the other hand,
899:
877:
860:
800:
764:
759:
738:
716:
706:
670:
662:
614:
574:
567:
540:
535:
521:
497:
489:
462:
443:
423:
407:
393:
369:
358:
338:
318:
299:
292:
265:
257:
240:
216:
205:
184:
155:
141:
104:An example of an anti-causal filter is a
75:
54:, whereas a filter whose output depends
62:. Systems (including filters) that are
7:
730:In those terms, causality requires
981:
765:
671:
666:
418:depends on future inputs, such as
108:filter, which can be defined as a
25:
1319:
1313:
1301:
1295:
1256:
1250:
1213:
1207:
1171:
1165:
1141:
1135:
1126:
1120:
1088:
1082:
1068:
1062:
1005:
999:
990:
984:
971:
965:
914:
905:
889:
883:
871:
865:
797:
785:
779:
773:
749:
743:
703:
691:
685:
679:
652:
646:
643:
631:
625:
619:
564:
552:
518:
506:
479:
467:
440:
428:
404:
398:
366:
360:
289:
277:
237:
231:
195:
189:
152:
146:
128:The following definition is a
1:
1028:Heaviside unit step function
176:is omitted for simplicity:
1407:
852:(ω). Define the function
454:. A realizable output is
1049:) are related as follows
37:linear and time-invariant
447:{\displaystyle s(t+1)\,}
166:. A constant factor of
1362:Rowell (January 2009),
1326:
1264:
1221:
1179:
1013:
927:
811:
721:
579:
448:
412:
411:{\displaystyle f(t)\,}
374:
347:
327:
304:
160:
159:{\displaystyle s(x)\,}
120:
87:
66:(i.e. that operate in
1327:
1265:
1222:
1180:
1014:
928:
812:
722:
603:. Its output is the
580:
449:
413:
375:
373:{\displaystyle (t)\,}
348:
328:
305:
161:
118:
88:
1371:, MIT OpenCourseWare
1280:
1235:
1192:
1056:
959:
859:
737:
613:
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357:
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183:
140:
74:
58:on future inputs is
832: < 0.
769:
675:
548:
502:
273:
227:
1322:
1260:
1217:
1175:
1009:
923:
807:
755:
717:
658:
575:
531:
485:
444:
408:
370:
343:
323:
300:
253:
201:
156:
121:
86:{\displaystyle t,}
83:
18:Anti-causal filter
1386:Signal processing
1345:Numerical Recipes
1292:
1247:
1229:Hilbert transform
1204:
1162:
1107:
921:
346:{\displaystyle x}
326:{\displaystyle x}
249:
29:signal processing
16:(Redirected from
1398:
1372:
1370:
1358:
1331:
1329:
1328:
1323:
1294:
1293:
1285:
1269:
1267:
1266:
1261:
1249:
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1224:
1223:
1218:
1206:
1205:
1197:
1184:
1182:
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1164:
1163:
1155:
1113:
1109:
1108:
1106:
1095:
1018:
1016:
1015:
1010:
932:
930:
929:
924:
922:
917:
904:
903:
878:
816:
814:
813:
808:
768:
763:
726:
724:
723:
718:
674:
669:
601:impulse response
584:
582:
581:
576:
547:
539:
501:
496:
453:
451:
450:
445:
417:
415:
414:
409:
379:
377:
376:
371:
353:represents time
352:
350:
349:
344:
332:
330:
329:
324:
309:
307:
306:
301:
272:
264:
247:
226:
215:
175:
174:
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165:
163:
162:
157:
92:
90:
89:
84:
21:
1406:
1405:
1401:
1400:
1399:
1397:
1396:
1395:
1376:
1375:
1368:
1361:
1356:
1341:
1338:
1278:
1277:
1233:
1232:
1190:
1189:
1099:
1078:
1074:
1054:
1053:
957:
956:
895:
879:
857:
856:
838:
735:
734:
611:
610:
459:
458:
420:
419:
390:
389:
355:
354:
335:
334:
315:
314:
181:
180:
172:
168:
167:
138:
137:
126:
99:window function
95:digital filters
72:
71:
23:
22:
15:
12:
11:
5:
1404:
1402:
1394:
1393:
1388:
1378:
1377:
1374:
1373:
1359:
1354:
1337:
1334:
1333:
1332:
1321:
1318:
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1306:
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1300:
1297:
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1215:
1212:
1209:
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1200:
1186:
1185:
1173:
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1161:
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1152:
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1140:
1137:
1134:
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1128:
1125:
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1119:
1116:
1112:
1105:
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1098:
1093:
1090:
1087:
1084:
1081:
1077:
1073:
1070:
1067:
1064:
1061:
1020:
1019:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
979:
976:
973:
970:
967:
964:
934:
933:
920:
916:
913:
910:
907:
902:
898:
894:
891:
888:
885:
882:
876:
873:
870:
867:
864:
837:
834:
828:) = 0 for all
818:
817:
806:
803:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
767:
762:
758:
754:
751:
748:
745:
742:
728:
727:
715:
712:
709:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
673:
668:
665:
661:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
586:
585:
573:
570:
566:
563:
560:
557:
554:
551:
546:
543:
538:
534:
530:
527:
524:
520:
517:
514:
511:
508:
505:
500:
495:
492:
488:
484:
481:
478:
475:
472:
469:
466:
442:
439:
436:
433:
430:
427:
406:
403:
400:
397:
386:non-realizable
368:
365:
362:
342:
322:
311:
310:
298:
295:
291:
288:
285:
282:
279:
276:
271:
268:
263:
260:
256:
252:
246:
243:
239:
236:
233:
230:
225:
222:
219:
214:
211:
208:
204:
200:
197:
194:
191:
188:
154:
151:
148:
145:
136:of input data
134:moving average
125:
122:
82:
79:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1403:
1392:
1391:Filter theory
1389:
1387:
1384:
1383:
1381:
1367:
1366:
1360:
1357:
1355:9780521880688
1351:
1347:
1346:
1340:
1339:
1335:
1316:
1310:
1307:
1304:
1298:
1289:
1286:
1276:
1275:
1274:
1271:
1253:
1244:
1241:
1230:
1210:
1201:
1198:
1168:
1159:
1156:
1150:
1147:
1144:
1138:
1132:
1129:
1123:
1117:
1114:
1110:
1103:
1100:
1096:
1091:
1085:
1079:
1075:
1071:
1065:
1059:
1052:
1051:
1050:
1048:
1044:
1040:
1036:
1031:
1029:
1025:
1002:
996:
993:
987:
977:
974:
968:
962:
955:
954:
953:
951:
947:
943:
939:
918:
911:
908:
900:
896:
892:
886:
880:
874:
868:
862:
855:
854:
853:
851:
847:
843:
835:
833:
831:
827:
823:
804:
801:
794:
791:
788:
782:
776:
770:
760:
756:
752:
746:
740:
733:
732:
731:
713:
710:
707:
700:
697:
694:
688:
682:
676:
663:
659:
655:
649:
640:
637:
634:
628:
622:
616:
609:
608:
607:
606:
602:
599:) called its
598:
594:
589:
571:
568:
561:
558:
555:
549:
544:
541:
536:
532:
528:
525:
522:
515:
512:
509:
503:
498:
493:
490:
486:
482:
476:
473:
470:
464:
457:
456:
455:
437:
434:
431:
425:
401:
395:
387:
384:(also called
383:
363:
340:
320:
296:
293:
286:
283:
280:
274:
269:
266:
261:
258:
254:
250:
244:
241:
234:
228:
223:
220:
217:
212:
209:
206:
202:
198:
192:
186:
179:
178:
177:
149:
143:
135:
131:
123:
117:
113:
111:
107:
106:maximum phase
102:
100:
96:
80:
77:
69:
65:
61:
57:
53:
49:
45:
41:
40:causal system
38:
34:
33:causal filter
30:
19:
1364:
1343:
1272:
1187:
1046:
1042:
1038:
1034:
1032:
1023:
1021:
949:
941:
937:
935:
849:
845:
841:
839:
829:
825:
821:
819:
729:
596:
592:
590:
587:
385:
381:
312:
129:
127:
103:
63:
59:
55:
51:
43:
32:
26:
605:convolution
388:), because
60:anti-causal
42:. The word
1380:Categories
1336:References
382:non-causal
64:realizable
52:non-causal
1317:ω
1299:ω
1290:^
1254:ω
1245:^
1211:ω
1202:^
1169:ω
1160:^
1151:⋅
1145:−
1139:ω
1124:ω
1115:∗
1104:ω
1101:π
1092:−
1086:ω
1080:δ
1066:ω
1026:) is the
994:⋅
982:Θ
946:Hermitian
909:−
901:∗
805:τ
795:τ
792:−
777:τ
766:∞
757:∫
711:τ
701:τ
698:−
683:τ
672:∞
667:∞
664:−
660:∫
638:∗
572:τ
562:τ
559:−
533:∫
526:τ
516:τ
491:−
487:∫
474:−
297:τ
287:τ
259:−
255:∫
245:τ
235:τ
210:−
203:∫
68:real time
1022:where Θ(
171:⁄
130:sliding
124:Example
1352:
1188:where
1041:) and
313:where
248:
110:stable
48:filter
44:causal
1369:(PDF)
1227:is a
944:) is
35:is a
1350:ISBN
840:Let
56:only
31:, a
132:or
27:In
1382::
1030:.
101:.
1320:)
1314:(
1311:H
1308:i
1305:=
1302:)
1296:(
1287:H
1257:)
1251:(
1242:G
1214:)
1208:(
1199:G
1172:)
1166:(
1157:G
1148:i
1142:)
1136:(
1133:G
1130:=
1127:)
1121:(
1118:G
1111:)
1097:i
1089:)
1083:(
1076:(
1072:=
1069:)
1063:(
1060:H
1047:t
1045:(
1043:g
1039:t
1037:(
1035:h
1024:t
1006:)
1003:t
1000:(
997:g
991:)
988:t
985:(
978:2
975:=
972:)
969:t
966:(
963:h
950:G
942:t
940:(
938:g
919:2
915:)
912:t
906:(
897:h
893:+
890:)
887:t
884:(
881:h
875:=
872:)
869:t
866:(
863:g
850:H
846:t
844:(
842:h
830:t
826:t
824:(
822:h
802:d
798:)
789:t
786:(
783:s
780:)
774:(
771:h
761:0
753:=
750:)
747:t
744:(
741:f
714:.
708:d
704:)
695:t
692:(
689:s
686:)
680:(
677:h
656:=
653:)
650:t
647:(
644:)
641:s
635:h
632:(
629:=
626:)
623:t
620:(
617:f
597:t
595:(
593:h
569:d
565:)
556:t
553:(
550:s
545:2
542:+
537:0
529:=
523:d
519:)
513:+
510:t
507:(
504:s
499:0
494:2
483:=
480:)
477:1
471:t
468:(
465:f
441:)
438:1
435:+
432:t
429:(
426:s
405:)
402:t
399:(
396:f
367:)
364:t
361:(
341:x
321:x
294:d
290:)
284:+
281:x
278:(
275:s
270:1
267:+
262:1
251:=
242:d
238:)
232:(
229:s
224:1
221:+
218:x
213:1
207:x
199:=
196:)
193:x
190:(
187:f
173:2
169:1
153:)
150:x
147:(
144:s
81:,
78:t
20:)
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