1812:
75:
1430:
2809:
922:
2238:
1839:. The second graph depicts the application of a lowpass filter at a higher data-rate, implemented by inserting zero-valued samples between the original ones. And the third graph is the DTFT of the filter output. The bottom table expresses the maximum filter bandwidth in various frequency units used by filter design tools.
3431:
The process of up sampling can be visualized as a two-step progression. The process starts by increasing the sample-rate of an input series x(n) by resampling . The zero-packed time series is processed by a filter h(n). In reality the processes of sample-rate increase and bandwidth reduction are
78:
Fig 1: Depiction of one dot product, resulting in one output sample (in green), for the case L=4, n=9, j=3. Three conceptual "inserted zeros" are depicted between each pair of input samples. Omitting them from the calculation is what distinguishes a multirate filter from a monorate
2532:
1425:{\displaystyle {\begin{aligned}y&=\sum _{k=-\infty }^{\infty }x_{L}\left\cdot h{\Bigl }\\&=\sum _{k=-\infty }^{\infty }x\left\cdot h\left\quad {\stackrel {m\ \triangleq \ j+nL}{\longrightarrow }}\quad y=\sum _{k=0}^{K}x\cdot h,\ \ j=0,1,\ldots ,L-1\end{aligned}}}
2042:
3013:
Upsampling requires a lowpass filter after increasing the data rate, and downsampling 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
2467:
3126:
where the response diminishes from near unity to near zero. So in practice the cutoff frequency is placed far enough below the theoretical cutoff that the filter's transition band is contained below the theoretical
888:
3238:
467:
2804:{\displaystyle \sum _{n=0,\pm L,\pm 2L,...,\pm \infty }{}x(nT/L)\ e^{-i2\pi fnT/L}\quad {\stackrel {m\ \triangleq \ n/L}{\longrightarrow }}\sum _{m=0,\pm 1,\pm 2,...,\pm \infty }{}x(mT)\ e^{-i2\pi fmT},}
724:
310:
calculations. It is an easy matter to omit them from both the data stream and the calculations. The calculation performed by a multirate interpolating FIR filter for each output sample is a dot product
2871:, which increases the number of periodic spectral images within the new bandwidth. Some authors describe that as new frequency components. The second graph also depicts a lowpass filter and
927:
2233:{\displaystyle \underbrace {\sum _{n=-\infty }^{\infty }\overbrace {x(nT)} ^{x}\ e^{-i2\pi fnT}} _{\text{DTFT}}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X{\Bigl (}f-{\frac {k}{T}}{\Bigr )}.}
1799:
zeros between the useful outputs of a phase and adding them to a sum is effectively decimation. It's the same result as not computing them at all. That equivalence is known as the
282:
2963:
3186:
3049:
1605:
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
3276:
2383:
156:
2031:
1906:
1659:
769:
2898:
2524:
1870:
1468:
2869:
2349:
1797:
1771:
1705:
795:
214:
2841:
2372:
1929:
911:
818:
490:
188:
2931:
2321:
2297:
2277:
1991:
1958:
1745:
1725:
1679:
1623:
1596:
1576:
1556:
1532:
1512:
1488:
607:
569:
549:
529:
106:
3584:
63:
or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a higher rate (or
1494:, where almost half of the coefficients are zero and need not be included in the dot products. Impulse response coefficients taken at intervals of
3361:
1598:
sequential output values. In some multi-processor architectures, these dot products are performed simultaneously, in which case it is called a
826:
3191:
2843:
That equivalence is depicted in the second graph of Fig.2. The only difference is that the available digital bandwidth is expanded to
323:
3577:
633:
3521:
3622:
3068:
40:
3822:
3570:
3612:
3340:
1811:
2492:
of interpolation. An example of both these distributions is depicted in the first and third graphs of Fig 2.
3786:
3767:
3063:
3003:
3708:
1961:
3464:
3424:
3396:
3371:
3653:
3083:
3484:
3749:
1873:
223:
3098:
3729:
3088:
2966:
3555:
2936:
3827:
3702:
3679:
3153:
3025:
3696:
3593:
3103:
71:
audio at 44,100 samples/second is upsampled by a factor of 5/4, the resulting sample-rate is 55,125.
20:
3643:
3638:
303:
3471:
the Noble
Identities apply to each polyphase component ... they don't apply to the whole filter.
2462:{\displaystyle {\frac {L}{T}}\sum _{k=-\infty }^{\infty }X\left(f-k\cdot {\frac {L}{T}}\right),}
36:
3456:
3327:
108:
can be explained as a 2-step process, with an equivalent implementation that is more efficient
74:
3243:
2900:
resulting in the desired spectral distribution (third graph). The filter's bandwidth is the
1827:. The entirety of the first graph depicts the discrete-time Fourier transform of a sequence
122:
3762:
3691:
3674:
2004:
1879:
1628:
732:
2874:
2498:
1846:
1444:
8:
2846:
2326:
1776:
1750:
1684:
774:
193:
3546:
2823:
2354:
1911:
893:
800:
472:
161:
3719:
3714:
3686:
2907:
2306:
2282:
2262:
1998:
1967:
1934:
1730:
1710:
1664:
1608:
1581:
1561:
1541:
1517:
1497:
1473:
574:
554:
534:
514:
91:
3776:
3734:
3724:
3648:
3617:
3460:
3449:
3420:
3412:
3392:
3367:
3336:
2901:
2495:
When the additional samples are inserted zeros, they decrease the sample-interval to
3607:
3357:
3322:
3073:
1491:
3514:"Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images"
306:
type, its efficiency can be improved, because the zeros contribute nothing to its
3513:
3123:
3093:
3444:
1994:
291:
1815:
Fig 2: The first triangle of the first graph represents the
Fourier transform
1558:
phases of the impulse response is filtering the same sequential values of the
3816:
3796:
3791:
3739:
64:
3562:
2526:
Omitting the zero-valued terms of the
Fourier series, it can be written as:
3781:
3772:
3078:
68:
3669:
307:
302:, and its design is discussed below. When the interpolation filter is an
883:{\displaystyle m-r={\bigl \lfloor }{\tfrac {m}{L}}{\bigr \rfloor }L-kL}
3335:(2nd ed.). Upper Saddle River, N.J.: Prentice Hall. p. 172.
624:
The interpolation filter output sequence is defined by a convolution
3757:
3233:{\displaystyle n={\bigl \lfloor }{\tfrac {m}{L}}{\bigr \rfloor },}
531:
sequence is the impulse response of the interpolation filter, and
3455:(2 ed.). Wellesley, MA: Wellesley-Cambridge Press. p.
462:{\displaystyle y=\sum _{k=0}^{K}x\cdot h,\ \ j=0,1,\ldots ,L-1,}
1803:. It is sometimes used in derivations of the polyphase method.
59:). When upsampling is performed on a sequence of samples of a
3556:"Matlab example of using polyphase filters for interpolation"
3419:. Upper Saddle River, NJ: Prentice Hall PTR. pp. 20–21.
2300:
1773:
terms of the each sum are identically zero. Computing
719:{\displaystyle y=\sum _{r=-\infty }^{\infty }x_{L}\cdot h}
3366:. Englewood Cliffs, NJ: Prentice-Hall. pp. 35–36.
3209:
3030:
2965: but filter design applications usually require
2941:
1215:
1161:
1062:
996:
850:
3551:(discusses a technique for bandlimited interpolation)
3432:
merged in a single process called a multirate filter.
3417:
Multirate Signal
Processing for Communication Systems
3389:
Handbook of
Formulas and Tables for Signal Processing
3246:
3194:
3156:
3028:
2939:
2910:
2877:
2849:
2826:
2535:
2501:
2386:
2357:
2329:
2309:
2285:
2265:
2045:
2007:
1970:
1937:
1914:
1882:
1849:
1779:
1753:
1733:
1713:
1687:
1667:
1631:
1611:
1584:
1564:
1544:
1520:
1500:
1476:
1447:
925:
896:
829:
803:
777:
735:
636:
577:
557:
537:
517:
475:
326:
226:
196:
164:
125:
94:
3321:
2972:
67:, as in the case of a photograph). For example, if
3448:
3326:
3270:
3232:
3180:
3043:
2957:
2925:
2892:
2863:
2835:
2803:
2518:
2461:
2366:
2343:
2315:
2291:
2271:
2232:
2025:
1985:
1952:
1923:
1900:
1864:
1791:
1765:
1739:
1719:
1699:
1673:
1653:
1617:
1590:
1570:
1550:
1526:
1506:
1482:
1462:
1424:
905:
882:
812:
789:
763:
718:
601:
563:
543:
523:
484:
461:
276:
208:
182:
150:
100:
83:
2222:
2199:
1106:
1039:
3814:
3355:
1681:times faster than the original input rate. Then
1806:
913: and the convolution can be rewritten as
3592:
3578:
3222:
3203:
3022:case, the interpolation filter cutoff,
2933:sequence. In units of Hz that value is
1228:
1209:
1174:
1155:
1075:
1056:
1009:
990:
863:
844:
298:In this application, the filter is called an
216:zeros. A notation for this operation is
3443:
3325:; Schafer, Ronald W.; Buck, John R. (1999).
3387:Poularikas, Alexander D. (September 1998).
2351:) increases the periodicity by a factor of
1831:formed by sampling the continuous function
3585:
3571:
3386:
51:, or it can describe an entire process of
3391:(1 ed.). CRC Press. pp. 42–48.
3380:
1747:sequence is the sum of the phases, where
35:are terms associated with the process of
3349:
3315:
1810:
73:
3405:
290:Smooth out the discontinuities using a
3815:
3411:
3566:
3511:
3505:
3437:
277:{\displaystyle x_{L}=x_{\uparrow L}.}
3547:"Digital Audio Resampling Home Page"
3476:
3069:Multi-rate digital signal processing
2985:denote the upsampling factor, where
2376:
2035:
771:can be non-zero are those for which
316:
41:multi-rate digital signal processing
3482:
3363:Multirate Digital Signal Processing
3122:Realizable low-pass filters have a
88:Rate increase by an integer factor
13:
3539:
3150:"2.3". p 38. eq 2.80, where
2958:{\displaystyle {\tfrac {0.5}{T}},}
2745:
2589:
2416:
2411:
2189:
2184:
2069:
2064:
1514:form a subsequence, and there are
1140:
1135:
968:
963:
671:
666:
14:
3839:
3301:. "2.2". p 23. fig 2.12 (bottom).
3188: which also requires
3181:{\displaystyle m\triangleq j+nL,}
3134:
3044:{\displaystyle {\tfrac {0.5}{L}}}
2973:Upsampling by a fractional factor
1578:data stream and producing one of
158:comprising the original samples,
3623:Nyquist–Shannon sampling theorem
1908:whose samples at some interval,
1538:) multiplexed together. Each of
16:Digital signal resampling method
3709:Discrete-time Fourier transform
3524:from the original on 2023-09-30
3447:; Nguyen, Truong (1996-10-01).
3333:Discrete-Time Signal Processing
3116:
2816:
2658:
1962:discrete-time Fourier transform
1288:
1250:
84:Upsampling by an integer factor
3292:
3289:. "2.2". p 23. fig 2.12 (top).
3280:
3147:
3141:
3052:cycles per intermediate sample
2920:
2914:
2764:
2755:
2663:
2616:
2599:
2477:
2248:
2109:
2103:
2090:
2081:
2017:
2011:
1980:
1974:
1947:
1941:
1892:
1886:
1859:
1853:
1727:outputs are zero. The desired
1648:
1642:
1373:
1358:
1349:
1337:
1307:
1292:
1255:
939:
933:
758:
746:
713:
707:
698:
686:
646:
640:
596:
581:
500:
411:
396:
387:
375:
345:
330:
263:
259:
252:
243:
237:
174:
168:
142:
136:
1:
3654:Statistical signal processing
3485:"Upsampling and downsampling"
3308:
3298:
3286:
3084:Sampling (information theory)
890: for integer values of
3500:chapter 12.1.2, figure 12-5B
7:
3089:Signal (information theory)
3057:
2820:regardless of the value of
1823:) of a continuous function
1807:Interpolation filter design
294:, which replaces the zeros.
10:
3844:
3703:Discrete Fourier transform
3680:Matched Z-transform method
3512:Lyons, Rick (2015-03-23).
3054:, is the lower frequency.
2488:which is also the desired
2323:times faster (at interval
1534:such subsequences (called
797:is an integer multiple of
3823:Digital signal processing
3748:
3697:Discrete cosine transform
3662:
3631:
3600:
3594:Digital signal processing
3451:Wavelets and Filter Banks
3104:Poisson summation formula
729:The only terms for which
21:digital signal processing
3730:Post's inversion formula
3644:Digital image processing
3109:
2997:Upsample by a factor of
551:is the largest value of
3639:Audio signal processing
3271:{\displaystyle j=m-nL.}
2814:which is equivalent to
1625:array, and process the
47:can be synonymous with
3483:Tan, Li (2008-04-21).
3272:
3234:
3182:
3045:
2959:
2927:
2894:
2865:
2837:
2805:
2520:
2463:
2420:
2368:
2345:
2317:
2293:
2279:has units of seconds,
2273:
2234:
2193:
2073:
2027:
1987:
1954:
1925:
1902:
1866:
1840:
1793:
1767:
1741:
1721:
1701:
1675:
1655:
1619:
1592:
1572:
1552:
1528:
1508:
1484:
1464:
1426:
1333:
1144:
972:
907:
884:
814:
791:
765:
720:
675:
603:
565:
545:
525:
486:
463:
371:
278:
210:
184:
152:
151:{\displaystyle x_{L},}
102:
80:
3415:(2004-05-24). "2.2".
3273:
3235:
3183:
3148:Crochiere and Rabiner
3046:
2969:. (see Fig 2, table)
2960:
2928:
2895:
2866:
2838:
2806:
2521:
2464:
2397:
2369:
2346:
2318:
2294:
2274:
2235:
2170:
2050:
2028:
2026:{\displaystyle X(f):}
1988:
1955:
1926:
1903:
1901:{\displaystyle x(t),}
1867:
1814:
1801:second Noble identity
1794:
1768:
1742:
1722:
1702:
1676:
1656:
1654:{\displaystyle x_{L}}
1620:
1593:
1573:
1553:
1529:
1509:
1490:can be designed as a
1485:
1465:
1427:
1313:
1121:
949:
908:
885:
815:
792:
766:
764:{\displaystyle x_{L}}
721:
652:
604:
566:
546:
526:
487:
464:
351:
279:
211:
185:
153:
103:
77:
3763:Anti-aliasing filter
3692:Constant-Q transform
3675:Advanced z-transform
3244:
3192:
3154:
3026:
2937:
2908:
2893:{\displaystyle L=3,}
2875:
2847:
2824:
2533:
2519:{\displaystyle T/L.}
2499:
2384:
2355:
2327:
2307:
2283:
2263:
2043:
2005:
1997:representation of a
1968:
1935:
1912:
1880:
1865:{\displaystyle X(f)}
1847:
1777:
1751:
1731:
1711:
1685:
1665:
1629:
1609:
1582:
1562:
1542:
1518:
1498:
1474:
1463:{\displaystyle L=2,}
1445:
923:
894:
827:
801:
775:
733:
634:
575:
555:
535:
515:
473:
324:
300:interpolation filter
224:
194:
162:
123:
92:
3413:Harris, Frederic J.
2864:{\displaystyle L/T}
2344:{\displaystyle T/L}
1960:sequence. Then the
1792:{\displaystyle L-1}
1766:{\displaystyle L-1}
1700:{\displaystyle L-1}
1661: sequence at
790:{\displaystyle m-r}
469: and for any
209:{\displaystyle L-1}
119:Create a sequence,
3720:Integral transform
3715:Impulse invariance
3687:Bilinear transform
3323:Oppenheim, Alan V.
3268:
3230:
3218:
3178:
3041:
3039:
2955:
2950:
2923:
2890:
2861:
2836:{\displaystyle L.}
2833:
2801:
2749:
2593:
2516:
2459:
2367:{\displaystyle L:}
2364:
2341:
2313:
2289:
2269:
2230:
2156:
2149:
2023:
1999:periodic summation
1983:
1950:
1924:{\displaystyle T,}
1921:
1898:
1862:
1841:
1789:
1763:
1737:
1717:
1697:
1671:
1651:
1615:
1588:
1568:
1548:
1524:
1504:
1480:
1460:
1422:
1420:
1224:
1170:
1071:
1005:
906:{\displaystyle k,}
903:
880:
859:
813:{\displaystyle L.}
810:
787:
761:
716:
617:Derivation of Eq.1
599:
561:
541:
521:
485:{\displaystyle n,}
482:
459:
274:
206:
183:{\displaystyle x,}
180:
148:
98:
81:
3828:Signal processing
3810:
3809:
3735:Starred transform
3725:Laplace transform
3649:Speech processing
3618:Estimation theory
3356:Crochiere, R.E.;
3240: and
3217:
3038:
2949:
2926:{\displaystyle x}
2902:Nyquist frequency
2769:
2695:
2692:
2679:
2673:
2621:
2536:
2485:
2484:
2449:
2395:
2316:{\displaystyle L}
2292:{\displaystyle f}
2272:{\displaystyle T}
2256:
2255:
2218:
2168:
2154:
2116:
2113:
2097:
2048:
2046:
1986:{\displaystyle x}
1953:{\displaystyle x}
1876:of any function,
1874:Fourier transform
1835:at a low-rate of
1740:{\displaystyle y}
1720:{\displaystyle L}
1674:{\displaystyle L}
1618:{\displaystyle h}
1591:{\displaystyle L}
1571:{\displaystyle x}
1551:{\displaystyle L}
1527:{\displaystyle L}
1507:{\displaystyle L}
1483:{\displaystyle h}
1437:
1436:
1384:
1381:
1285:
1271:
1265:
1223:
1169:
1103:
1096:
1070:
1004:
858:
602:{\displaystyle h}
564:{\displaystyle k}
544:{\displaystyle K}
524:{\displaystyle h}
508:
507:
422:
419:
101:{\displaystyle L}
3835:
3608:Detection theory
3587:
3580:
3573:
3564:
3563:
3559:
3550:
3533:
3532:
3530:
3529:
3509:
3503:
3502:
3497:
3496:
3480:
3474:
3473:
3454:
3441:
3435:
3434:
3409:
3403:
3402:
3384:
3378:
3377:
3353:
3347:
3346:
3330:
3319:
3302:
3296:
3290:
3284:
3278:
3277:
3275:
3274:
3269:
3239:
3237:
3236:
3231:
3226:
3225:
3219:
3210:
3207:
3206:
3187:
3185:
3184:
3179:
3145:
3128:
3120:
3074:Half-band filter
3050:
3048:
3047:
3042:
3040:
3031:
3018: >
2989: >
2967:normalized units
2964:
2962:
2961:
2956:
2951:
2942:
2932:
2930:
2929:
2924:
2904:of the original
2899:
2897:
2896:
2891:
2870:
2868:
2867:
2862:
2857:
2842:
2840:
2839:
2834:
2810:
2808:
2807:
2802:
2797:
2796:
2767:
2751:
2748:
2694:
2693:
2691:
2687:
2677:
2671:
2666:
2661:
2657:
2656:
2652:
2619:
2612:
2595:
2592:
2525:
2523:
2522:
2517:
2509:
2479:
2468:
2466:
2465:
2460:
2455:
2451:
2450:
2442:
2419:
2414:
2396:
2388:
2377:
2373:
2371:
2370:
2365:
2350:
2348:
2347:
2342:
2337:
2322:
2320:
2319:
2314:
2298:
2296:
2295:
2290:
2278:
2276:
2275:
2270:
2250:
2239:
2237:
2236:
2231:
2226:
2225:
2219:
2211:
2203:
2202:
2192:
2187:
2169:
2161:
2155:
2152:
2150:
2145:
2144:
2143:
2114:
2112:
2098:
2093:
2076:
2074:
2072:
2067:
2036:
2032:
2030:
2029:
2024:
1993:sequence is the
1992:
1990:
1989:
1984:
1959:
1957:
1956:
1951:
1930:
1928:
1927:
1922:
1907:
1905:
1904:
1899:
1871:
1869:
1868:
1863:
1798:
1796:
1795:
1790:
1772:
1770:
1769:
1764:
1746:
1744:
1743:
1738:
1726:
1724:
1723:
1718:
1706:
1704:
1703:
1698:
1680:
1678:
1677:
1672:
1660:
1658:
1657:
1652:
1641:
1640:
1624:
1622:
1621:
1616:
1597:
1595:
1594:
1589:
1577:
1575:
1574:
1569:
1557:
1555:
1554:
1549:
1533:
1531:
1530:
1525:
1513:
1511:
1510:
1505:
1492:half-band filter
1489:
1487:
1486:
1481:
1470: function
1469:
1467:
1466:
1461:
1431:
1429:
1428:
1423:
1421:
1382:
1379:
1332:
1327:
1287:
1286:
1284:
1269:
1263:
1258:
1253:
1249:
1245:
1232:
1231:
1225:
1216:
1213:
1212:
1189:
1185:
1178:
1177:
1171:
1162:
1159:
1158:
1143:
1138:
1114:
1110:
1109:
1102:
1097:
1092:
1079:
1078:
1072:
1063:
1060:
1059:
1046:
1044:
1043:
1042:
1030:
1026:
1013:
1012:
1006:
997:
994:
993:
982:
981:
971:
966:
912:
910:
909:
904:
889:
887:
886:
881:
867:
866:
860:
851:
848:
847:
819:
817:
816:
811:
796:
794:
793:
788:
770:
768:
767:
762:
745:
744:
725:
723:
722:
717:
685:
684:
674:
669:
613:
612:
608:
606:
605:
600:
570:
568:
567:
562:
550:
548:
547:
542:
530:
528:
527:
522:
502:
491:
489:
488:
483:
468:
466:
465:
460:
420:
417:
370:
365:
317:
283:
281:
280:
275:
270:
269:
236:
235:
215:
213:
212:
207:
189:
187:
186:
181:
157:
155:
154:
149:
135:
134:
107:
105:
104:
99:
3843:
3842:
3838:
3837:
3836:
3834:
3833:
3832:
3813:
3812:
3811:
3806:
3744:
3658:
3627:
3613:Discrete signal
3596:
3591:
3554:
3545:
3542:
3540:Further reading
3537:
3536:
3527:
3525:
3510:
3506:
3494:
3492:
3481:
3477:
3467:
3445:Strang, Gilbert
3442:
3438:
3427:
3410:
3406:
3399:
3385:
3381:
3374:
3360:(1983). "2.3".
3354:
3350:
3343:
3320:
3316:
3311:
3306:
3305:
3297:
3293:
3285:
3281:
3245:
3242:
3241:
3221:
3220:
3208:
3202:
3201:
3193:
3190:
3189:
3155:
3152:
3151:
3146:
3142:
3137:
3132:
3131:
3124:transition band
3121:
3117:
3112:
3094:Data conversion
3060:
3029:
3027:
3024:
3023:
3006:by a factor of
2975:
2940:
2938:
2935:
2934:
2909:
2906:
2905:
2876:
2873:
2872:
2853:
2848:
2845:
2844:
2825:
2822:
2821:
2774:
2770:
2750:
2699:
2683:
2667:
2662:
2660:
2659:
2648:
2626:
2622:
2608:
2594:
2540:
2534:
2531:
2530:
2505:
2500:
2497:
2496:
2486:
2441:
2428:
2424:
2415:
2401:
2387:
2385:
2382:
2381:
2356:
2353:
2352:
2333:
2328:
2325:
2324:
2308:
2305:
2304:
2284:
2281:
2280:
2264:
2261:
2260:
2257:
2221:
2220:
2210:
2198:
2197:
2188:
2174:
2160:
2151:
2121:
2117:
2099:
2077:
2075:
2068:
2054:
2049:
2047:
2044:
2041:
2040:
2006:
2003:
2002:
1969:
1966:
1965:
1936:
1933:
1932:
1913:
1910:
1909:
1881:
1878:
1877:
1848:
1845:
1844:
1809:
1778:
1775:
1774:
1752:
1749:
1748:
1732:
1729:
1728:
1712:
1709:
1708:
1686:
1683:
1682:
1666:
1663:
1662:
1636:
1632:
1630:
1627:
1626:
1610:
1607:
1606:
1583:
1580:
1579:
1563:
1560:
1559:
1543:
1540:
1539:
1519:
1516:
1515:
1499:
1496:
1495:
1475:
1472:
1471:
1446:
1443:
1442:
1440:
1438:
1419:
1418:
1328:
1317:
1259:
1254:
1252:
1251:
1227:
1226:
1214:
1208:
1207:
1200:
1196:
1173:
1172:
1160:
1154:
1153:
1152:
1148:
1139:
1125:
1112:
1111:
1105:
1104:
1098:
1074:
1073:
1061:
1055:
1054:
1047:
1045:
1038:
1037:
1008:
1007:
995:
989:
988:
987:
983:
977:
973:
967:
953:
942:
926:
924:
921:
920:
895:
892:
891:
862:
861:
849:
843:
842:
828:
825:
824:
802:
799:
798:
776:
773:
772:
740:
736:
734:
731:
730:
680:
676:
670:
656:
635:
632:
631:
618:
576:
573:
572:
556:
553:
552:
536:
533:
532:
516:
513:
512:
509:
474:
471:
470:
366:
355:
325:
322:
321:
262:
258:
231:
227:
225:
222:
221:
195:
192:
191:
163:
160:
159:
130:
126:
124:
121:
120:
93:
90:
89:
86:
55:and filtering (
17:
12:
11:
5:
3841:
3831:
3830:
3825:
3808:
3807:
3805:
3804:
3799:
3794:
3789:
3784:
3779:
3770:
3765:
3760:
3754:
3752:
3746:
3745:
3743:
3742:
3737:
3732:
3727:
3722:
3717:
3712:
3706:
3700:
3694:
3689:
3684:
3683:
3682:
3677:
3666:
3664:
3660:
3659:
3657:
3656:
3651:
3646:
3641:
3635:
3633:
3629:
3628:
3626:
3625:
3620:
3615:
3610:
3604:
3602:
3598:
3597:
3590:
3589:
3582:
3575:
3567:
3561:
3560:
3552:
3541:
3538:
3535:
3534:
3518:dsprelated.com
3504:
3475:
3465:
3436:
3425:
3404:
3397:
3379:
3372:
3348:
3341:
3313:
3312:
3310:
3307:
3304:
3303:
3291:
3279:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3229:
3224:
3216:
3213:
3205:
3200:
3197:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3139:
3138:
3136:
3135:Page citations
3133:
3130:
3129:
3114:
3113:
3111:
3108:
3107:
3106:
3101:
3096:
3091:
3086:
3081:
3076:
3071:
3066:
3059:
3056:
3037:
3034:
3011:
3010:
3001:
2974:
2971:
2954:
2948:
2945:
2922:
2919:
2916:
2913:
2889:
2886:
2883:
2880:
2860:
2856:
2852:
2832:
2829:
2812:
2811:
2800:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2773:
2766:
2763:
2760:
2757:
2754:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2698:
2690:
2686:
2682:
2676:
2670:
2665:
2655:
2651:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2625:
2618:
2615:
2611:
2607:
2604:
2601:
2598:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2539:
2515:
2512:
2508:
2504:
2483:
2482:
2473:
2471:
2469:
2458:
2454:
2448:
2445:
2440:
2437:
2434:
2431:
2427:
2423:
2418:
2413:
2410:
2407:
2404:
2400:
2394:
2391:
2375:
2363:
2360:
2340:
2336:
2332:
2312:
2288:
2268:
2254:
2253:
2244:
2242:
2240:
2229:
2224:
2217:
2214:
2209:
2206:
2201:
2196:
2191:
2186:
2183:
2180:
2177:
2173:
2167:
2164:
2159:
2148:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2120:
2111:
2108:
2105:
2102:
2096:
2092:
2089:
2086:
2083:
2080:
2071:
2066:
2063:
2060:
2057:
2053:
2034:
2022:
2019:
2016:
2013:
2010:
1995:Fourier series
1982:
1979:
1976:
1973:
1964:(DTFT) of the
1949:
1946:
1943:
1940:
1920:
1917:
1897:
1894:
1891:
1888:
1885:
1861:
1858:
1855:
1852:
1808:
1805:
1788:
1785:
1782:
1762:
1759:
1756:
1736:
1716:
1696:
1693:
1690:
1670:
1650:
1647:
1644:
1639:
1635:
1614:
1587:
1567:
1547:
1523:
1503:
1479:
1459:
1456:
1453:
1450:
1435:
1434:
1433:
1432:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1331:
1326:
1323:
1320:
1316:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1283:
1280:
1277:
1274:
1268:
1262:
1257:
1248:
1244:
1241:
1238:
1235:
1230:
1222:
1219:
1211:
1206:
1203:
1199:
1195:
1192:
1188:
1184:
1181:
1176:
1168:
1165:
1157:
1151:
1147:
1142:
1137:
1134:
1131:
1128:
1124:
1120:
1117:
1115:
1113:
1108:
1101:
1095:
1091:
1088:
1085:
1082:
1077:
1069:
1066:
1058:
1053:
1050:
1041:
1036:
1033:
1029:
1025:
1022:
1019:
1016:
1011:
1003:
1000:
992:
986:
980:
976:
970:
965:
962:
959:
956:
952:
948:
945:
943:
941:
938:
935:
932:
929:
928:
902:
899:
879:
876:
873:
870:
865:
857:
854:
846:
841:
838:
835:
832:
809:
806:
786:
783:
780:
760:
757:
754:
751:
748:
743:
739:
727:
726:
715:
712:
709:
706:
703:
700:
697:
694:
691:
688:
683:
679:
673:
668:
665:
662:
659:
655:
651:
648:
645:
642:
639:
620:
619:
616:
611:
609:is non-zero.
598:
595:
592:
589:
586:
583:
580:
560:
540:
520:
506:
505:
496:
494:
492:
481:
478:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
416:
413:
410:
407:
404:
401:
398:
395:
392:
389:
386:
383:
380:
377:
374:
369:
364:
361:
358:
354:
350:
347:
344:
341:
338:
335:
332:
329:
315:
296:
295:
292:lowpass filter
284:
273:
268:
265:
261:
257:
254:
251:
248:
245:
242:
239:
234:
230:
205:
202:
199:
179:
176:
173:
170:
167:
147:
144:
141:
138:
133:
129:
97:
85:
82:
15:
9:
6:
4:
3:
2:
3840:
3829:
3826:
3824:
3821:
3820:
3818:
3803:
3800:
3798:
3797:Undersampling
3795:
3793:
3792:Sampling rate
3790:
3788:
3785:
3783:
3780:
3778:
3774:
3771:
3769:
3766:
3764:
3761:
3759:
3756:
3755:
3753:
3751:
3747:
3741:
3740:Zak transform
3738:
3736:
3733:
3731:
3728:
3726:
3723:
3721:
3718:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3693:
3690:
3688:
3685:
3681:
3678:
3676:
3673:
3672:
3671:
3668:
3667:
3665:
3661:
3655:
3652:
3650:
3647:
3645:
3642:
3640:
3637:
3636:
3634:
3630:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3605:
3603:
3599:
3595:
3588:
3583:
3581:
3576:
3574:
3569:
3568:
3565:
3557:
3553:
3548:
3544:
3543:
3523:
3519:
3515:
3508:
3501:
3490:
3486:
3479:
3472:
3468:
3462:
3458:
3453:
3452:
3446:
3440:
3433:
3428:
3422:
3418:
3414:
3408:
3400:
3394:
3390:
3383:
3375:
3369:
3365:
3364:
3359:
3358:Rabiner, L.R.
3352:
3344:
3342:0-13-754920-2
3338:
3334:
3329:
3324:
3318:
3314:
3300:
3295:
3288:
3283:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3227:
3214:
3211:
3198:
3195:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3149:
3144:
3140:
3125:
3119:
3115:
3105:
3102:
3100:
3099:Interpolation
3097:
3095:
3092:
3090:
3087:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3064:Downsampling
3062:
3061:
3055:
3053:
3035:
3032:
3021:
3017:
3009:
3005:
3002:
3000:
2996:
2995:
2994:
2992:
2988:
2984:
2980:
2970:
2968:
2952:
2946:
2943:
2917:
2911:
2903:
2887:
2884:
2881:
2878:
2858:
2854:
2850:
2830:
2827:
2819:
2818:
2798:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2771:
2761:
2758:
2752:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2696:
2688:
2684:
2680:
2674:
2668:
2653:
2649:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2623:
2613:
2609:
2605:
2602:
2596:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2537:
2529:
2528:
2527:
2513:
2510:
2506:
2502:
2493:
2491:
2481:
2474:
2472:
2470:
2456:
2452:
2446:
2443:
2438:
2435:
2432:
2429:
2425:
2421:
2408:
2405:
2402:
2398:
2392:
2389:
2379:
2378:
2374:
2361:
2358:
2338:
2334:
2330:
2310:
2302:
2299:has units of
2286:
2266:
2252:
2245:
2243:
2241:
2227:
2215:
2212:
2207:
2204:
2194:
2181:
2178:
2175:
2171:
2165:
2162:
2157:
2146:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2118:
2106:
2100:
2094:
2087:
2084:
2078:
2061:
2058:
2055:
2051:
2038:
2037:
2033:
2020:
2014:
2008:
2000:
1996:
1977:
1971:
1963:
1944:
1938:
1918:
1915:
1895:
1889:
1883:
1875:
1856:
1850:
1838:
1834:
1830:
1826:
1822:
1818:
1813:
1804:
1802:
1786:
1783:
1780:
1760:
1757:
1754:
1734:
1714:
1694:
1691:
1688:
1668:
1645:
1637:
1633:
1612:
1603:
1601:
1585:
1565:
1545:
1537:
1521:
1501:
1493:
1477:
1457:
1454:
1451:
1448:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1376:
1370:
1367:
1364:
1361:
1355:
1352:
1346:
1343:
1340:
1334:
1329:
1324:
1321:
1318:
1314:
1310:
1304:
1301:
1298:
1295:
1289:
1281:
1278:
1275:
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1266:
1260:
1246:
1242:
1239:
1236:
1233:
1220:
1217:
1204:
1201:
1197:
1193:
1190:
1186:
1182:
1179:
1166:
1163:
1149:
1145:
1132:
1129:
1126:
1122:
1118:
1116:
1099:
1093:
1089:
1086:
1083:
1080:
1067:
1064:
1051:
1048:
1034:
1031:
1027:
1023:
1020:
1017:
1014:
1001:
998:
984:
978:
974:
960:
957:
954:
950:
946:
944:
936:
930:
919:
918:
917:
916:
900:
897:
877:
874:
871:
868:
855:
852:
839:
836:
833:
830:
823:
807:
804:
784:
781:
778:
755:
752:
749:
741:
737:
710:
704:
701:
695:
692:
689:
681:
677:
663:
660:
657:
653:
649:
643:
637:
630:
629:
628:
627:
622:
621:
615:
614:
610:
593:
590:
587:
584:
578:
558:
538:
518:
504:
497:
495:
493:
479:
476:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
414:
408:
405:
402:
399:
393:
390:
384:
381:
378:
372:
367:
362:
359:
356:
352:
348:
342:
339:
336:
333:
327:
319:
318:
314:
313:
309:
305:
301:
293:
289:
286:Interpolation
285:
271:
266:
255:
249:
246:
240:
232:
228:
219:
203:
200:
197:
190:separated by
177:
171:
165:
145:
139:
131:
127:
118:
114:
113:
112:
111:
95:
76:
72:
70:
66:
62:
58:
57:interpolation
54:
50:
46:
42:
38:
34:
33:interpolation
30:
26:
22:
3801:
3787:Quantization
3782:Oversampling
3773:Nyquist rate
3768:Downsampling
3526:. Retrieved
3517:
3507:
3499:
3493:. Retrieved
3488:
3478:
3470:
3450:
3439:
3430:
3416:
3407:
3388:
3382:
3362:
3351:
3332:
3317:
3294:
3282:
3143:
3118:
3079:Oversampling
3051:
3019:
3015:
3012:
3007:
2998:
2990:
2986:
2982:
2978:
2976:
2815:
2813:
2494:
2489:
2487:
2475:
2380:
2258:
2246:
2039:
1842:
1836:
1832:
1828:
1824:
1820:
1816:
1800:
1604:
1599:
1535:
1441:In the case
1439:
914:
821:
728:
625:
623:
510:
498:
320:
311:
299:
297:
287:
217:
116:
109:
87:
69:compact disc
60:
56:
52:
48:
44:
32:
28:
24:
18:
3670:Z-transform
3489:eetimes.com
3299:Harris 2004
3287:Harris 2004
2303:. Sampling
820: Thus
308:dot product
3817:Categories
3802:Upsampling
3663:Techniques
3632:Sub-fields
3528:2024-01-31
3495:2024-06-27
3491:. EE Times
3466:0961408871
3426:0131465112
3398:0849385792
3373:0136051626
3309:References
3004:Downsample
2301:hertz (Hz)
1931:equal the
571:for which
511:where the
45:Upsampling
37:resampling
25:upsampling
3777:frequency
3257:−
3161:≜
2785:π
2776:−
2746:∞
2743:±
2722:±
2713:±
2697:∑
2675:≜
2664:⟶
2637:π
2628:−
2590:∞
2587:±
2563:±
2554:±
2538:∑
2439:⋅
2433:−
2417:∞
2412:∞
2409:−
2399:∑
2208:−
2190:∞
2185:∞
2182:−
2172:∑
2147:⏟
2132:π
2123:−
2095:⏞
2070:∞
2065:∞
2062:−
2052:∑
1784:−
1758:−
1707:of every
1692:−
1600:polyphase
1413:−
1404:…
1353:⋅
1344:−
1315:∑
1267:≜
1256:⟶
1205:−
1191:⋅
1180:−
1141:∞
1136:∞
1133:−
1123:∑
1094:⏞
1052:−
1032:⋅
1018:−
969:∞
964:∞
961:−
951:∑
872:−
834:−
782:−
753:−
702:⋅
693:−
672:∞
667:∞
664:−
654:∑
451:−
442:…
391:⋅
382:−
353:∑
264:↑
201:−
115:Expansion
53:expansion
49:expansion
29:expansion
3758:Aliasing
3750:Sampling
3522:Archived
3223:⌋
3204:⌊
3058:See also
1602:filter.
1229:⌋
1210:⌊
1175:⌋
1156:⌊
1076:⌋
1057:⌊
1010:⌋
991:⌊
864:⌋
845:⌊
43:system.
3328:"4.6.2"
3127:cutoff.
1872:be the
220:
79:filter.
65:density
3711:(DTFT)
3601:Theory
3463:
3423:
3395:
3370:
3339:
2768:
2678:
2672:
2620:
2490:result
2115:
1536:phases
1383:
1380:
1270:
1264:
421:
418:
61:signal
31:, and
3705:(DFT)
3699:(DCT)
3110:Notes
2817:Eq.2,
2259:When
39:in a
3461:ISBN
3421:ISBN
3393:ISBN
3368:ISBN
3337:ISBN
2977:Let
2478:Eq.3
2249:Eq.2
2153:DTFT
1843:Let
1833:x(t)
1825:x(t)
501:Eq.1
3457:101
3033:0.5
2944:0.5
2001:of
1837:1/T
304:FIR
19:In
3819::
3775:/
3520:.
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3498:.
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1613:h
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1452:=
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