825:
1664:
is difficult, and not well understood. The effect will be amplitude-dependent, among other things. But generally, waveshapers—particularly those with sharp corners (e.g., some derivatives are discontinuous) -- tend to introduce large numbers of high frequency harmonics. If these introduced harmonics exceed the
524:
1663:
The sound produced by digital waveshapers tends to be harsh and unattractive, because of problems with aliasing. Waveshaping is a non-linear operation, so it's hard to generalize about the effect of a waveshaping function on an input signal. The mathematics of non-linear operations on audio signals
1675:
With relatively simple, and relatively smooth waveshaping functions (sin(a*x), atan(a*x), polynomial functions, for example), this procedure may reduce aliased content in the harmonic signal to the point that it is musically acceptable. But waveshaping functions other than polynomial waveshaping
206:
Sin, arctan, polynomial functions, or piecewise functions (such as the hard clipping function) are commonly used as waveshaping transfer functions. It is also possible to use table-driven functions, consisting of discrete points with some degree of interpolation or linear segments.
408:
820:{\displaystyle \sum _{n=0}^{N}a_{n}{\Bigg (}\alpha {\frac {e^{j(\omega t+\phi )}+e^{-j(\omega t+\phi )}}{2}}{\Bigg )}^{n}=a_{0}+\sum _{n=1}^{N}{\frac {a_{n}\alpha ^{n}}{2^{n-1}}}{\frac {(e^{j(\omega t+\phi )}+e^{-j(\omega t+\phi )})^{n}}{2}}}
1268:
509:
1653:
1491:
198:. In practice, the input to the waveshaper, x, is considered on for digitally sampled signals, and f will be designed such that y is also on to prevent unwanted clipping in software.
224:
169:
1523:
1594:
1553:
840:
63:
of a guitar or bass. Some synthesizers or virtual software instruments have built-in waveshapers. The effect can make instruments sound noisy or
94:
that changes an audio signal by mapping an input signal to the output signal by applying a fixed or variable mathematical function, called the
1676:
functions will introduce an infinite number of harmonics into the signal, some which may audibly alias even at the supersampled frequency.
426:
1495:
From the above equation, several observations can be made about the effect of a polynomial shaping function on a single sinusoid:
412:
Polynomial functions are convenient as shaping functions because, when given a single sinusoid as input, a polynomial of degree
1599:
74:, waveshaping is used to introduce a static, or memoryless, nonlinearity to approximate the transfer characteristic of a
1713:
Yeh, David T. and
Pakarinen, Jyri (2009). "A Review of Digital Techniques for Modeling Vacuum-Tube Guitar Amplifiers",
1701:
1275:
1672:
can somewhat but not completely alleviate this problem, depending on how fast the introduced harmonics fall off.
1566:
The shape of the spectrum produced by each monomial term is fixed and determined by the binomial coefficients .
1668:, then they will be heard as harsh inharmonic content with a distinctly metallic sound in the output signal.
420:
th harmonic of the sinusoid. To prove this, consider a sinusoid used as input to the general polynomial.
403:{\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}=\sum _{n=0}^{N}a_{n}x^{n}}
1750:
1726:
1768:
118:
91:
1505:
55:
to achieve an extra-abrasive sound. This effect is most used to enhance the sound of a music
17:
1572:
1531:
32:
102:, to the input signal (the term shaping function is preferred to avoid confusion with the
8:
52:
515:
64:
59:
by altering the waveform or vowel. Rock musicians may also use a waveshaper for heavy
1697:
103:
831:
24:
1559:
down to the fundamental, and all even monomial terms generate even harmonics from
834:
to transform back to trigonometric form and find coefficients for each harmonic.
71:
1569:
The weight of that spectrum in the overall output is determined solely by its
1499:
All of the sinusoids generated are harmonically related to the original input.
1762:
1665:
1263:{\displaystyle a_{0}+\sum _{n=1}^{N}{\Bigg }}=a_{0}+\sum _{n=1}^{N}{\Bigg }}}
1669:
75:
56:
216:
60:
504:{\displaystyle \sum _{n=0}^{N}a_{n}(\alpha \cos(\omega t+\phi ))^{n}}
1751:
http://www.music.mcgill.ca/~gary/courses/2012/307/week12/node4.html
1727:
http://www.music.mcgill.ca/~gary/courses/2012/307/week12/node2.html
40:
36:
190:, which in general may vary as a function of time. This parameter
106:
from systems theory). The function can be any function at all.
79:
39:
are produced from simple tones by altering the shape of the
1737:
1694:
Computer Music: Synthesis, Composition, and
Performance
1658:
70:
In digital modeling of analog audio equipment such as
1602:
1575:
1534:
1508:
1278:
843:
527:
429:
227:
121:
1648:{\displaystyle {\frac {a_{n}\alpha ^{n}}{2^{n-1}}}}
194:is often used as a constant gain factor called the
1647:
1588:
1547:
1517:
1485:
1262:
819:
503:
402:
201:
163:
1477:
1418:
1405:
1318:
1254:
1192:
1179:
1106:
1061:
966:
953:
880:
643:
561:
1760:
109:Mathematically, the operation is defined by the
1486:{\displaystyle =a_{0}+\sum _{n=1}^{N}{\Bigg }}}
1596:coefficient and the amplitude of the input by
1396:
1382:
1692:Charles Dodge and Thomas A. Jersey (1997).
16:For the Swedish electronic musician, see
1739:Journal of the Audio Engineering Society
1761:
1659:Problems associated with waveshapers
13:
1409:
1183:
957:
14:
1780:
202:Commonly used shaping functions
85:
51:Waveshapers are used mainly by
1744:
1731:
1720:
1707:
1686:
1470:
1467:
1452:
1449:
1434:
1431:
1241:
1226:
1223:
1208:
1047:
1032:
1013:
998:
995:
983:
802:
796:
781:
762:
747:
736:
629:
614:
595:
580:
492:
488:
473:
461:
416:will only introduce up to the
237:
231:
210:
158:
155:
149:
143:
137:
131:
1:
518:to obtain complex sinusoids.
164:{\displaystyle y=f(a(t)x(t))}
1555:generate odd harmonics from
1502:The frequency never exceeds
7:
182:is the input function, and
10:
1785:
1679:
219:is a function of the form
15:
1563:down to DC (0 frequency).
178:is the shaping function,
1518:{\displaystyle N\omega }
1528:All odd monomial terms
46:
1715:Computer Music Journal
1649:
1590:
1549:
1519:
1487:
1400:
1315:
1264:
1174:
1103:
948:
877:
821:
690:
548:
514:Next, use the inverse
505:
450:
404:
379:
165:
1696:, "Glossary", p.438.
1650:
1591:
1589:{\displaystyle a_{n}}
1550:
1548:{\displaystyle x^{n}}
1520:
1488:
1366:
1295:
1265:
1154:
1083:
928:
857:
822:
670:
528:
506:
430:
405:
359:
166:
18:Waveshaper (musician)
1600:
1573:
1532:
1506:
1276:
841:
525:
427:
225:
119:
53:electronic musicians
33:distortion synthesis
111:waveshaper equation
90:A waveshaper is an
1645:
1586:
1545:
1515:
1483:
1260:
817:
501:
400:
161:
1717:, 33:2, pp. 89-90
1643:
1416:
1364:
1249:
1190:
1152:
1056:
964:
926:
830:Finally, use the
815:
731:
638:
104:transfer function
100:transfer function
35:in which complex
1776:
1769:Electronic music
1753:
1748:
1742:
1735:
1729:
1724:
1718:
1711:
1705:
1690:
1654:
1652:
1651:
1646:
1644:
1642:
1641:
1626:
1625:
1624:
1615:
1614:
1604:
1595:
1593:
1592:
1587:
1585:
1584:
1554:
1552:
1551:
1546:
1544:
1543:
1524:
1522:
1521:
1516:
1492:
1490:
1489:
1484:
1482:
1481:
1480:
1474:
1473:
1423:
1422:
1421:
1408:
1399:
1392:
1380:
1365:
1363:
1362:
1347:
1346:
1345:
1336:
1335:
1325:
1322:
1321:
1314:
1309:
1291:
1290:
1269:
1267:
1266:
1261:
1259:
1258:
1257:
1251:
1250:
1245:
1244:
1199:
1197:
1196:
1195:
1182:
1173:
1168:
1153:
1151:
1150:
1135:
1134:
1133:
1124:
1123:
1113:
1110:
1109:
1102:
1097:
1079:
1078:
1066:
1065:
1064:
1058:
1057:
1052:
1051:
1050:
1017:
1016:
973:
971:
970:
969:
956:
947:
942:
927:
925:
924:
909:
908:
907:
898:
897:
887:
884:
883:
876:
871:
853:
852:
832:binomial formula
826:
824:
823:
818:
816:
811:
810:
809:
800:
799:
766:
765:
734:
732:
730:
729:
714:
713:
712:
703:
702:
692:
689:
684:
666:
665:
653:
652:
647:
646:
639:
634:
633:
632:
599:
598:
570:
565:
564:
558:
557:
547:
542:
510:
508:
507:
502:
500:
499:
460:
459:
449:
444:
409:
407:
406:
401:
399:
398:
389:
388:
378:
373:
355:
354:
339:
338:
326:
325:
316:
315:
297:
296:
281:
280:
262:
261:
252:
251:
196:distortion index
170:
168:
167:
162:
96:shaping function
25:electronic music
1784:
1783:
1779:
1778:
1777:
1775:
1774:
1773:
1759:
1758:
1757:
1756:
1749:
1745:
1736:
1732:
1725:
1721:
1712:
1708:
1691:
1687:
1682:
1661:
1631:
1627:
1620:
1616:
1610:
1606:
1605:
1603:
1601:
1598:
1597:
1580:
1576:
1574:
1571:
1570:
1539:
1535:
1533:
1530:
1529:
1507:
1504:
1503:
1476:
1475:
1430:
1417:
1404:
1403:
1402:
1401:
1388:
1381:
1370:
1352:
1348:
1341:
1337:
1331:
1327:
1326:
1324:
1323:
1317:
1316:
1310:
1299:
1286:
1282:
1277:
1274:
1273:
1253:
1252:
1204:
1200:
1198:
1191:
1178:
1177:
1176:
1175:
1169:
1158:
1140:
1136:
1129:
1125:
1119:
1115:
1114:
1112:
1111:
1105:
1104:
1098:
1087:
1074:
1070:
1060:
1059:
1022:
1018:
979:
975:
974:
972:
965:
952:
951:
950:
949:
943:
932:
914:
910:
903:
899:
893:
889:
888:
886:
885:
879:
878:
872:
861:
848:
844:
842:
839:
838:
805:
801:
774:
770:
743:
739:
735:
733:
719:
715:
708:
704:
698:
694:
693:
691:
685:
674:
661:
657:
648:
642:
641:
640:
607:
603:
576:
572:
571:
569:
560:
559:
553:
549:
543:
532:
526:
523:
522:
516:Euler's formula
495:
491:
455:
451:
445:
434:
428:
425:
424:
394:
390:
384:
380:
374:
363:
350:
346:
334:
330:
321:
317:
311:
307:
286:
282:
270:
266:
257:
253:
247:
243:
226:
223:
222:
213:
204:
120:
117:
116:
88:
72:tube amplifiers
49:
21:
12:
11:
5:
1782:
1772:
1771:
1755:
1754:
1743:
1741:, 27:4, p. 250
1730:
1719:
1706:
1684:
1683:
1681:
1678:
1660:
1657:
1656:
1655:
1640:
1637:
1634:
1630:
1623:
1619:
1613:
1609:
1583:
1579:
1567:
1564:
1542:
1538:
1526:
1514:
1511:
1500:
1479:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1429:
1426:
1420:
1415:
1412:
1407:
1398:
1395:
1391:
1387:
1384:
1379:
1376:
1373:
1369:
1361:
1358:
1355:
1351:
1344:
1340:
1334:
1330:
1320:
1313:
1308:
1305:
1302:
1298:
1294:
1289:
1285:
1281:
1271:
1270:
1256:
1248:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1203:
1194:
1189:
1186:
1181:
1172:
1167:
1164:
1161:
1157:
1149:
1146:
1143:
1139:
1132:
1128:
1122:
1118:
1108:
1101:
1096:
1093:
1090:
1086:
1082:
1077:
1073:
1069:
1063:
1055:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1021:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
978:
968:
963:
960:
955:
946:
941:
938:
935:
931:
923:
920:
917:
913:
906:
902:
896:
892:
882:
875:
870:
867:
864:
860:
856:
851:
847:
828:
827:
814:
808:
804:
798:
795:
792:
789:
786:
783:
780:
777:
773:
769:
764:
761:
758:
755:
752:
749:
746:
742:
738:
728:
725:
722:
718:
711:
707:
701:
697:
688:
683:
680:
677:
673:
669:
664:
660:
656:
651:
645:
637:
631:
628:
625:
622:
619:
616:
613:
610:
606:
602:
597:
594:
591:
588:
585:
582:
579:
575:
568:
563:
556:
552:
546:
541:
538:
535:
531:
512:
511:
498:
494:
490:
487:
484:
481:
478:
475:
472:
469:
466:
463:
458:
454:
448:
443:
440:
437:
433:
397:
393:
387:
383:
377:
372:
369:
366:
362:
358:
353:
349:
345:
342:
337:
333:
329:
324:
320:
314:
310:
306:
303:
300:
295:
292:
289:
285:
279:
276:
273:
269:
265:
260:
256:
250:
246:
242:
239:
236:
233:
230:
212:
209:
203:
200:
188:index function
172:
171:
160:
157:
154:
151:
148:
145:
142:
139:
136:
133:
130:
127:
124:
87:
84:
48:
45:
9:
6:
4:
3:
2:
1781:
1770:
1767:
1766:
1764:
1752:
1747:
1740:
1734:
1728:
1723:
1716:
1710:
1703:
1702:0-02-864682-7
1699:
1695:
1689:
1685:
1677:
1673:
1671:
1667:
1666:Nyquist limit
1638:
1635:
1632:
1628:
1621:
1617:
1611:
1607:
1581:
1577:
1568:
1565:
1562:
1558:
1540:
1536:
1527:
1512:
1509:
1501:
1498:
1497:
1496:
1493:
1464:
1461:
1458:
1455:
1446:
1443:
1440:
1437:
1427:
1424:
1413:
1410:
1393:
1389:
1385:
1377:
1374:
1371:
1367:
1359:
1356:
1353:
1349:
1342:
1338:
1332:
1328:
1311:
1306:
1303:
1300:
1296:
1292:
1287:
1283:
1279:
1246:
1238:
1235:
1232:
1229:
1220:
1217:
1214:
1211:
1205:
1201:
1187:
1184:
1170:
1165:
1162:
1159:
1155:
1147:
1144:
1141:
1137:
1130:
1126:
1120:
1116:
1099:
1094:
1091:
1088:
1084:
1080:
1075:
1071:
1067:
1053:
1044:
1041:
1038:
1035:
1029:
1026:
1023:
1019:
1010:
1007:
1004:
1001:
992:
989:
986:
980:
976:
961:
958:
944:
939:
936:
933:
929:
921:
918:
915:
911:
904:
900:
894:
890:
873:
868:
865:
862:
858:
854:
849:
845:
837:
836:
835:
833:
812:
806:
793:
790:
787:
784:
778:
775:
771:
767:
759:
756:
753:
750:
744:
740:
726:
723:
720:
716:
709:
705:
699:
695:
686:
681:
678:
675:
671:
667:
662:
658:
654:
649:
635:
626:
623:
620:
617:
611:
608:
604:
600:
592:
589:
586:
583:
577:
573:
566:
554:
550:
544:
539:
536:
533:
529:
521:
520:
519:
517:
496:
485:
482:
479:
476:
470:
467:
464:
456:
452:
446:
441:
438:
435:
431:
423:
422:
421:
419:
415:
410:
395:
391:
385:
381:
375:
370:
367:
364:
360:
356:
351:
347:
343:
340:
335:
331:
327:
322:
318:
312:
308:
304:
301:
298:
293:
290:
287:
283:
277:
274:
271:
267:
263:
258:
254:
248:
244:
240:
234:
228:
220:
218:
208:
199:
197:
193:
189:
185:
181:
177:
152:
146:
140:
134:
128:
125:
122:
115:
114:
113:
112:
107:
105:
101:
97:
93:
83:
81:
77:
73:
68:
66:
62:
58:
54:
44:
42:
38:
34:
31:is a type of
30:
26:
19:
1746:
1738:
1733:
1722:
1714:
1709:
1693:
1688:
1674:
1670:Oversampling
1662:
1560:
1556:
1494:
1272:
829:
513:
417:
413:
411:
221:
214:
205:
195:
191:
187:
183:
179:
175:
173:
110:
108:
99:
95:
92:audio effect
89:
86:How it works
69:
50:
28:
22:
211:Polynomials
76:vacuum tube
57:synthesizer
29:waveshaping
217:polynomial
65:overdriven
61:distortion
1636:−
1618:α
1513:ω
1465:ϕ
1456:ω
1441:−
1428:
1397:⌋
1383:⌊
1368:∑
1357:−
1339:α
1297:∑
1239:ϕ
1230:ω
1215:−
1156:∑
1145:−
1127:α
1085:∑
1045:ϕ
1036:ω
1024:−
1011:ϕ
1002:ω
990:−
930:∑
919:−
901:α
859:∑
794:ϕ
785:ω
776:−
760:ϕ
751:ω
724:−
706:α
672:∑
627:ϕ
618:ω
609:−
593:ϕ
584:ω
567:α
530:∑
486:ϕ
477:ω
471:
465:α
432:∑
361:∑
302:⋯
291:−
275:−
82:limiter.
41:waveforms
1763:Category
1680:Sources
186:is the
37:spectra
1700:
174:where
80:diode
1698:ISBN
184:a(t)
180:x(t)
47:Uses
1425:cos
468:cos
98:or
78:or
23:In
1765::
215:A
67:.
43:.
27:,
1704:.
1639:1
1633:n
1629:2
1622:n
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1608:a
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1561:n
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1537:x
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1510:N
1478:]
1471:)
1468:)
1462:+
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1453:(
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1438:n
1435:(
1432:(
1419:)
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1406:(
1394:2
1390:/
1386:n
1378:0
1375:=
1372:k
1360:1
1354:n
1350:2
1343:n
1333:n
1329:a
1319:[
1312:N
1307:1
1304:=
1301:n
1293:+
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1280:=
1255:]
1247:2
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1236:+
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1131:n
1121:n
1117:a
1107:[
1100:N
1095:1
1092:=
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959:n
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945:n
940:0
937:=
934:k
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916:n
912:2
905:n
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891:a
881:[
874:N
869:1
866:=
863:n
855:+
850:0
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676:n
668:+
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636:2
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483:+
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436:n
418:N
414:N
396:n
392:x
386:n
382:a
376:N
371:0
368:=
365:n
357:=
352:0
348:a
344:+
341:x
336:1
332:a
328:+
323:2
319:x
313:2
309:a
305:+
299:+
294:1
288:n
284:x
278:1
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259:n
255:x
249:n
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241:=
238:)
235:x
232:(
229:f
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156:)
153:t
150:(
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141:t
138:(
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129:f
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123:y
20:.
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