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The process of down sampling can be visualized as a two-step progression. The process starts as an input series x(n) that is processed by a filter h(n) to obtain the output sequence y(n) with reduced bandwidth. The sample rate of the output sequence is then reduced Q-to-1 to a rate commensurate with
411:
Fig 1: These graphs depict the spectral distributions of an oversampled function and the same function sampled at 1/3 the original rate. The bandwidth, B, in this example is just small enough that the slower sampling does not cause overlap (aliasing). Sometimes, a sampled function is resampled at a
412:
lower rate by keeping only every M sample and discarding the others, commonly called "decimation". Potential aliasing is prevented by lowpass-filtering the samples before decimation. The maximum filter bandwidth is tabulated in the bandwidth units used by the common filter design applications.
380:
outputs are being summed. This viewpoint offers a different implementation that might be advantageous in a multi-processor architecture. In other words, the input stream is demultiplexed and sent through a bank of M filters whose outputs are summed. When implemented that way, it is called a
1852:
Sliusar I.I., Slyusar V.I., Voloshko S.V., Smolyar V.G. Next
Generation Optical Access based on N-OFDM with decimation.// Third International Scientific-Practical Conference "Problems of Infocommunications. Science and Technology (PIC S&T'2016)". â Kharkiv. - October 3 â6, 2016.
475:
1398:
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Realizable low-pass filters have a "skirt", where the response diminishes from near one to near zero. In practice the cutoff frequency is placed far enough below the theoretical cutoff that the filter's skirt is contained below the theoretical
726:
1846:
T. Schilcher. RF applications in digital signal processing//" Digital signal processing". Proceedings, CERN Accelerator School, Sigtuna, Sweden, May 31-June 9, 2007. - Geneva, Switzerland: CERN (2008). - P. 258. - DOI: 10.5170/CERN-2008-003.
158:(i.e. high-frequency signal components will copy into the lower frequency band and be mistaken for lower frequencies). Step 1, when necessary, suppresses aliasing to an acceptable level. In this application, the filter is called an
670:{\displaystyle \underbrace {\sum _{n=-\infty }^{\infty }\overbrace {x(nT)} ^{x}\ \mathrm {e} ^{-\mathrm {i} 2\pi fnT}} _{\text{DTFT}}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X{\Bigl (}f-{\frac {k}{T}}{\Bigr )}.}
1841:
Sampling rate conversion systems are used to change the sampling rate of a signal. The process of sampling rate decrease is called decimation, and the process of sampling rate increase is called interpolation.
984:
1061:) the data rate, and step 2 requires a lowpass filter before decimation. Therefore, both operations can be accomplished by a single filter with the lower of the two cutoff frequencies. For the
1461:
292:
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Generally, this approach is applicable when the ratio Fy/Fx is a rational, or an irrational number, and is suitable for the sampling rate increase and for the sampling rate decrease.
1858:
Saska
Lindfors, Aarno Pärssinen, Kari A. I. Halonen. A 3-V 230-MHz CMOS Decimation Subsampler.// IEEE transactions on circuits and systemsâ Vol. 52, No. 2, February 2005. â P. 110.
1096:
1185:
903:{\displaystyle \sum _{n=-\infty }^{\infty }x(n\cdot MT)\ \mathrm {e} ^{-\mathrm {i} 2\pi fn(MT)}={\frac {1}{MT}}\sum _{k=-\infty }^{\infty }X\left(f-{\tfrac {k}{MT}}\right).}
388:
For completeness, we now mention that a possible, but unlikely, implementation of each phase is to replace the coefficients of the other phases with zeros in a copy of the
1028:
1671:
the reduced signal bandwidth. In reality the processes of bandwidth reduction and sample rate reduction are merged in a single process called a multirate filter.
702:
925:) overlap. The purpose of the anti-aliasing filter is to ensure that the reduced periodicity does not create overlap. The condition that ensures the copies of
348:
such subsequences (phases) multiplexed together. The dot product is the sum of the dot products of each subsequence with the corresponding samples of the
1883:
1600:
76:
or a continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate (or
17:
100:
only every tenth sample. This factor multiplies the sampling interval or, equivalently, divides the sampling rate. For example, if
936:
1876:
917:. An example of both these distributions is depicted in the two traces of Fig 1. Aliasing occurs when adjacent copies of
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1731:
1921:
53:
2121:
1869:
1911:
1812:
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1393:{\displaystyle X_{s}(i\Omega )\triangleq \sum _{n=-\infty }^{\infty }x(nT)\ \mathrm {e} ^{-\mathrm {i} \Omega nT},}
1130:
199:
2007:
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108:
by a factor of 5/4, the resulting sample rate is 35,280. A system component that performs decimation is called a
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Decimators can be used to reduce the sampling frequency, whereas interpolators can be used to increase it.
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2001:
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sequence at the input rate (which means multiplying by zeros), and decimate the output by a factor of
1995:
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174:
31:
400:. The equivalence of this inefficient method and the implementation described above is known as the
1942:
1804:
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1011:
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can be explained as a two-step process, with an equivalent implementation that is more efficient:
1937:
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represents the input sequence being downsampled. In a general purpose processor, after computing
88:
1120:
337:, where almost half of the coefficients are zero and need not be included in the dot products.
47:
1695:
1530:
1796:
2061:
1990:
1973:
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used for the downsampling process has been accepted and used in many textbooks and fields.
8:
1797:
72:) and sample-rate reduction. When the process is performed on a sequence of samples of a
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The periodic summation has been reduced in amplitude and periodicity by a factor of
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design, it relies on feedback from output to input, prior to the second step. With
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The process of reducing a sampling rate by an integer factor is referred to as
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output. The calculation performed by a decimating FIR filter for the
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167:
155:
1756:
Digital Signal
Processing: Principles, Algorithms and Applications
1694:(2 ed.). Wellesley, MA: Wellesley-Cambridge Press. pp.
1529:(2nd ed.). Upper Saddle River, N.J.: Prentice Hall. p.
68:, or they can describe an entire process of bandwidth reduction (
716:
in the formulas above gives the DTFT of the decimated sequence,
404:. It is sometimes used in derivations of the polyphase method.
364:
dot products is never involved in the other dot products. Thus
1658:. Upper Saddle River, NJ: Prentice Hall PTR. pp. 20â21.
705:
1158:
General techniques for sample-rate conversion by factor R â
979:{\displaystyle B<{\tfrac {0.5}{T}}\cdot {\tfrac {1}{M}},}
132:
Reduce high-frequency signal components with a digital
1575:
of a data sequence. We also refer to downsampling as
1077:
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947:
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1656:
Multirate Signal
Processing for Communication Systems
1628:
Handbook of
Formulas and Tables for Signal Processing
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Decreasing the sampling rate is known as decimation.
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Impulse response coefficients taken at intervals of
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1605:. Englewood Cliffs, NJ: Prentice-Hall. p. 32.
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Step 1 requires a lowpass filter after increasing (
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978:
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352:sequence. Furthermore, because of downsampling by
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1827:Multirate Filtering for Digital Signal Processing
1724:Multirate Filtering for Digital Signal Processing
1521:; Schafer, Ronald W.; Buck, John R. (1999). "4".
659:
636:
112:. Decimation by an integer factor is also called
2113:
1594:
1035:Increase (resample) the sequence by a factor of
368:low-order FIR filters are each filtering one of
1456:{\displaystyle X_{c}(i2\pi f)\triangleq X(f).}
162:, and its design is discussed below. Also see
1891:
1877:
325:, and recompute the dot product. In the case
181:, it is an easy matter to compute only every
1682:
1069:case, the anti-aliasing filter cutoff,
287:{\displaystyle y=\sum _{k=0}^{K-1}x\cdot h,}
1626:Poularikas, Alexander D. (September 1998).
1884:
1870:
1625:
1630:(1 ed.). CRC Press. pp. 42â48.
1619:
1280:{\displaystyle \Omega \triangleq 2\pi f,}
1167:
1016:
46:are terms associated with the process of
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1511:
406:
317:is to advance the starting index in the
1775:Understanding Digital Signal Processing
1753:
1644:
997:
416:
14:
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1758:(3rd ed.). India: Prentice-Hall.
1650:
301:sequence is the impulse response, and
86:is a term that historically means the
1865:
1824:
1772:
1721:
1715:
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1006:denote the decimation factor, where:
1547:
173:When the anti-aliasing filter is an
124:Rate reduction by an integer factor
54:multi-rate digital signal processing
1710:No sensible engineer would do that.
1602:Multirate Digital Signal Processing
1553:
441:), whose samples at some interval,
360:samples involved in any one of the
80:, as in the case of a photograph).
24:
1747:
1726:. New York: Hershey. p. 192.
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344:form a subsequence, and there are
104:audio at 44,100 samples/second is
25:
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1829:. New York: Hershey. p. 35.
1197:
1091:{\displaystyle {\tfrac {0.5}{M}}}
166:for information about decimating
154:Step 2 alone creates undesirable
120:Downsampling by an integer factor
1922:NyquistâShannon sampling theorem
1180:{\displaystyle \mathbb {R} ^{+}}
933:) do not overlap each other is:
2008:Discrete-time Fourier transform
1686:; Nguyen, Truong (1996-10-01).
1525:Discrete-Time Signal Processing
1489:
1477:
1465:
1239:
1152:
1142:
451:discrete-time Fourier transform
1777:. Prentice Hall. p. 304.
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1099:cycles per intermediate sample
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18:Decimation (signal processing)
13:
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1953:Statistical signal processing
1556:"Upsampling and downsampling"
1504:
1471:
1248:. "4.2". p 143. eq 4.6, where
1233:
1209:
376:of the input stream, and the
313:, the easiest way to compute
1131:VisvalingamâWhyatt algorithm
1023:{\displaystyle \mathbb {Z} }
392:array, process the original
94:decimation by a factor of 10
92:. But in signal processing,
7:
1104:
146:; that is, keep only every
10:
2143:
2002:Discrete Fourier transform
1979:Matched Z-transform method
1795:Antoniou, Andreas (2006).
1495:
1101:, is the lower frequency.
89:removal of every tenth one
2122:Digital signal processing
2047:
1996:Discrete cosine transform
1961:
1930:
1899:
1893:Digital signal processing
1799:Digital Signal Processing
1754:Proakis, John G. (2000).
1690:Wavelets and Filter Banks
1486:. "4.6". p 171. fig 4.22.
1191:and the Farrow structure.
32:digital signal processing
2029:Post's inversion formula
1943:Digital image processing
1825:Milic, Ljiljana (2009).
1722:MiliÄ, Ljiljana (2009).
1474:. "2.2". p 22. fig 2.10.
1189:polynomial interpolation
1136:
1050:Decimate by a factor of
1938:Audio signal processing
1803:. McGraw-Hill. p.
1773:Lyons, Richard (2001).
986:so that is the maximum
170:functions and signals.
142:the filtered signal by
64:can be synonymous with
1554:Tan, Li (2008-04-21).
1457:
1394:
1342:
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1121:Sample-rate conversion
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994:anti-aliasing filter.
980:
904:
860:
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698:
684:has units of seconds,
671:
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244:
1654:(2004-05-24). "2.2".
1484:Oppenheim and Schafer
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1395:
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1246:Oppenheim and Schafer
1222:Crochiere and Rabiner
1182:
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333:can be designed as a
305:is its length.
289:
218:
2062:Anti-aliasing filter
1991:Constant-Q transform
1974:Advanced z-transform
1498:. "1.2.1". fig 12.2.
1404:
1291:
1256:
1224:"2". p 32. eq 2.55a.
1162:
1073:
1012:
998:By a rational factor
937:
727:
688:
476:
457:representation of a
417:Anti-aliasing filter
402:first Noble identity
200:
160:anti-aliasing filter
1652:Harris, Frederic J.
449:sequence. Then the
189:output sample is a
2019:Integral transform
2014:Impulse invariance
1986:Bilinear transform
1519:Oppenheim, Alan V.
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459:periodic summation
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284:
2127:Signal processing
2109:
2108:
2034:Starred transform
2024:Laplace transform
1948:Speech processing
1917:Estimation theory
1836:978-1-60566-178-0
1733:978-1-60566-178-0
1595:Crochiere, R.E.;
1400: and
1360:
1085:
1039:. This is called
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835:
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697:{\displaystyle f}
655:
605:
591:
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433:of any function,
431:Fourier transform
27:Resampling method
16:(Redirected from
2134:
1907:Detection theory
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1236:. "2.2.1". p 25.
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2040:
2039:Zak transform
2037:
2035:
2032:
2030:
2027:
2025:
2022:
2020:
2017:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1992:
1989:
1987:
1984:
1980:
1977:
1975:
1972:
1971:
1970:
1967:
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1960:
1954:
1951:
1949:
1946:
1944:
1941:
1939:
1936:
1935:
1933:
1929:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1904:
1902:
1898:
1894:
1887:
1882:
1880:
1875:
1873:
1868:
1867:
1864:
1857:
1855:
1851:
1849:
1845:
1842:
1838:
1832:
1828:
1823:
1820:
1816:
1814:0-07-145424-1
1810:
1806:
1801:
1800:
1793:
1790:
1786:
1784:0-201-63467-8
1780:
1776:
1771:
1767:
1761:
1757:
1752:
1751:
1739:
1735:
1729:
1725:
1718:
1711:
1707:
1701:
1697:
1692:
1691:
1685:
1679:
1672:
1667:
1661:
1657:
1653:
1647:
1639:
1633:
1629:
1622:
1614:
1608:
1604:
1603:
1599:(1983). "2".
1598:
1597:Rabiner, L.R.
1591:
1584:
1582:
1578:
1574:
1561:
1557:
1550:
1542:
1540:0-13-754920-2
1536:
1532:
1527:
1526:
1520:
1514:
1510:
1497:
1492:
1485:
1480:
1473:
1468:
1450:
1444:
1438:
1435:
1429:
1426:
1423:
1420:
1412:
1408:
1387:
1382:
1379:
1368:
1352:
1349:
1343:
1330:
1327:
1324:
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1316:
1307:
1299:
1295:
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1268:
1265:
1262:
1251:
1247:
1242:
1235:
1230:
1223:
1218:
1211:
1206:
1202:
1190:
1172:
1155:
1145:
1141:
1132:
1129:
1127:
1124:
1122:
1119:
1117:
1116:Posterization
1114:
1112:
1109:
1108:
1102:
1100:
1082:
1079:
1068:
1064:
1060:
1053:
1049:
1046:
1045:interpolation
1042:
1038:
1034:
1033:
1032:
1005:
995:
993:
989:
973:
967:
964:
958:
952:
949:
943:
940:
932:
928:
924:
920:
916:
897:
893:
885:
882:
878:
872:
869:
865:
861:
848:
845:
842:
838:
831:
828:
824:
819:
811:
808:
802:
799:
796:
793:
785:
769:
766:
763:
760:
754:
741:
738:
735:
731:
723:
722:
721:
719:
715:
711:
707:
704:has units of
691:
683:
664:
652:
649:
644:
641:
631:
618:
615:
612:
608:
602:
599:
594:
583:
577:
574:
571:
568:
565:
557:
539:
533:
527:
520:
517:
511:
494:
491:
488:
484:
472:
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468:
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460:
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409:
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371:
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328:
324:
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308:
304:
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281:
275:
269:
266:
260:
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229:
226:
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209:
203:
196:
195:
194:
192:
188:
184:
180:
179:FIR filtering
176:
171:
169:
165:
164:undersampling
161:
157:
149:
145:
141:
138:
135:
131:
130:
129:
127:
117:
115:
111:
107:
103:
99:
95:
91:
90:
85:
81:
79:
75:
71:
67:
63:
59:
56:system. Both
55:
51:
50:
45:
41:
37:
33:
19:
2086:Quantization
2081:Oversampling
2072:Nyquist rate
2067:Downsampling
2066:
1840:
1826:
1818:
1798:
1788:
1774:
1755:
1737:
1723:
1717:
1709:
1689:
1678:
1669:
1655:
1646:
1627:
1621:
1601:
1590:
1580:
1576:
1573:downsampling
1572:
1570:
1564:. Retrieved
1559:
1549:
1524:
1513:
1491:
1479:
1467:
1249:
1241:
1229:
1217:
1205:
1154:
1144:
1098:
1066:
1062:
1058:
1056:
1051:
1044:
1036:
1003:
1001:
991:
930:
926:
922:
918:
914:
912:
717:
713:
709:
708:. Replacing
681:
679:
466:
462:
453:(DTFT) is a
446:
445:, equal the
442:
438:
434:
426:
422:
420:
401:
397:
393:
389:
387:
382:
377:
373:
372:multiplexed
369:
365:
361:
357:
353:
349:
345:
341:
339:
330:
326:
322:
318:
314:
310:
306:
302:
298:
296:
186:
182:
172:
153:
147:
143:
139:
125:
123:
113:
109:
105:
102:compact disc
97:
93:
87:
83:
82:
73:
65:
61:
58:downsampling
57:
48:
43:
39:
36:downsampling
35:
29:
1969:Z-transform
1579:. The term
1560:eetimes.com
1472:Harris 2004
1234:Harris 2004
1210:Harris 2004
1030:; M > L.
191:dot product
114:compression
66:compression
40:compression
2116:Categories
2101:Upsampling
1962:Techniques
1931:Sub-fields
1765:8120311299
1705:0961408871
1665:0131465112
1637:0849385792
1612:0136051626
1581:decimation
1577:decimation
1566:2017-04-10
1562:. EE Times
1505:References
1111:Upsampling
1041:Upsampling
297:where the
84:Decimation
62:decimation
49:resampling
44:decimation
2076:frequency
1436:≜
1427:π
1377:Ω
1369:−
1339:∞
1334:∞
1331:−
1321:∑
1317:≜
1311:Ω
1269:π
1263:≜
1260:Ω
1059:expanding
959:⋅
873:−
857:∞
852:∞
849:−
839:∑
797:π
786:−
764:⋅
750:∞
745:∞
742:−
732:∑
645:−
627:∞
622:∞
619:−
609:∑
584:⏟
569:π
558:−
528:⏞
503:∞
498:∞
495:−
485:∑
429:) be the
383:polyphase
321:array by
267:⋅
258:−
238:−
220:∑
110:decimator
106:decimated
70:filtering
2057:Aliasing
2049:Sampling
1496:Tan 2008
1187:include
1126:Aliasing
1105:See also
385:filter.
168:bandpass
156:aliasing
140:Decimate
1287:
1149:cutoff.
1008:M, L â
150:sample.
98:keeping
78:density
2010:(DTFT)
1900:Theory
1833:
1811:
1781:
1762:
1730:
1702:
1698:â101.
1662:
1634:
1609:
1537:
1359:
1252:
990:of an
776:
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374:phases
74:signal
42:, and
2004:(DFT)
1998:(DCT)
1137:Notes
1043:, or
992:ideal
712:with
706:hertz
680:When
52:in a
1831:ISBN
1809:ISBN
1779:ISBN
1760:ISBN
1728:ISBN
1700:ISBN
1660:ISBN
1632:ISBN
1607:ISBN
1535:ISBN
1002:Let
944:<
590:DTFT
421:Let
329:=2,
60:and
1805:830
1696:100
1531:168
1080:0.5
1004:M/L
950:0.5
469:):
461:of
175:IIR
30:In
2118::
2074:/
1839:.
1817:.
1807:.
1787:.
1736:.
1708:.
1668:.
1569:.
1558:.
1533:.
720::
714:MT
193::
116:.
38:,
34:,
1885:e
1878:t
1871:v
1768:.
1640:.
1615:.
1543:.
1451:.
1448:)
1445:f
1442:(
1439:X
1433:)
1430:f
1424:2
1421:i
1418:(
1413:c
1409:X
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1383:T
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1350:n
1347:(
1344:x
1328:=
1325:n
1314:)
1308:i
1305:(
1300:s
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1173:+
1168:R
1083:M
1067:L
1063:M
1052:M
1047:.
1037:L
1017:Z
974:,
968:M
965:1
953:T
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931:f
929:(
927:X
923:f
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919:X
915:M
898:.
894:)
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883:M
879:k
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832:T
829:M
825:1
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812:T
809:M
806:(
803:n
800:f
794:2
790:i
781:e
773:)
770:T
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758:(
755:x
739:=
736:n
718:x
710:T
692:f
682:T
665:.
660:)
653:T
650:k
642:f
637:(
632:X
616:=
613:k
603:T
600:1
595:=
578:T
575:n
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566:2
562:i
553:e
543:]
540:n
537:[
534:x
524:)
521:T
518:n
515:(
512:x
492:=
489:n
467:f
465:(
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447:x
443:T
439:t
437:(
435:x
427:f
425:(
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398:M
394:x
390:h
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342:M
331:h
327:M
323:M
319:x
315:y
311:y
307:x
303:K
299:h
282:,
279:]
276:k
273:[
270:h
264:]
261:k
255:M
252:n
249:[
246:x
241:1
235:K
230:0
227:=
224:k
216:=
213:]
210:n
207:[
204:y
187:n
183:M
148:M
144:M
136:.
126:M
20:)
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