1509:
the effective width and to remove ringing effects of the resulting filter in the signal domain. This is done by choosing a narrow ideal filter impulse response function, e.g., an impulse, and a weighting function which grows fast with the distance from the origin, e.g., the distance squared. The optimal filter can still be calculated by solving a simple least squares problem and the resulting filter is then a "compromise" which has a total optimal fit to the ideal functions in both domains. An important parameter is the relative strength of the two weighting functions which determines in which domain it is more important to have a good fit relative to the ideal function.
637:, an uncertainty principle, the product of the width of the frequency function and the width of the impulse response cannot be smaller than a specific constant. This implies that if a specific frequency function is requested, corresponding to a specific frequency width, the minimum width of the filter in the signal domain is set. Vice versa, if the maximum width of the response is given, this determines the smallest possible width in the frequency. This is a typical example of contradictory requirements where the filter design process may try to find a useful compromise.
32:
397:
a significant reduction in computational complexity can be obtained if the filter can be separated as the convolution of one 1D filter in the horizontal direction and one 1D filter in the vertical direction. A result of the filter design process may, e.g., be to approximate some desired filter as a separable filter or as a sum of separable filters.
1508:
The previous method can be extended to include an additional error term related to a desired filter impulse response in the signal domain, with a corresponding weighting function. The ideal impulse response can be chosen independently of the ideal frequency function and is in practice used to limit
883:
and implies that the localization of various features such as pulses or steps in the filter response is limited by the filter width in the signal domain. If a precise localization is requested, we need a filter of small width in the signal domain and, via the uncertainty principle, its width in the
396:
Another issue related to computational complexity is separability, that is, if and how a filter can be written as a convolution of two or more simpler filters. In particular, this issue is of importance for multidimensional filters, e.g., 2D filter which are used in image processing. In this case,
362:
assures that every limited input signal produces a limited filter response. A filter which does not meet this requirement may in some situations prove useless or even harmful. Certain design approaches can guarantee stability, for example by using only feed-forward circuits such as an FIR filter.
392:
For discrete filters the computational complexity is more or less proportional to the number of filter coefficients. If the filter has many coefficients, for example in the case of multidimensional signals such as tomography data, it may be relevant to reduce the number of coefficients by removing
384:
A general desire in any design is that the number of operations (additions and multiplications) needed to compute the filter response is as low as possible. In certain applications, this desire is a strict requirement, for example due to limited computational resources, limited power resources, or
371:
In certain applications we have to deal with signals which contain components which can be described as local phenomena, for example pulses or steps, which have certain time duration. A consequence of applying a filter to a signal is, in intuitive terms, that the duration of the local phenomena is
322:
that the resulting filter have a small effective width in the signal domain as possible. The latter condition can be realized by considering a very narrow function as the wanted impulse response of the filter even though this function has no relation to the desired frequency function. The goal of
375:
According to the uncertainty relation of the
Fourier transform, the product of the width of the filter's impulse response function and the width of its frequency function must exceed a certain constant. This means that any requirement on the filter's locality also implies a bound on its frequency
613:
Parts of the design problem relate to the fact that certain requirements are described in the frequency domain while others are expressed in the time domain and that these may conflict. For example, it is not possible to obtain a filter which has both an arbitrary impulse response and arbitrary
388:
There are several ways in which a filter can have different computational complexity. For example, the order of a filter is more or less proportional to the number of operations. This means that by choosing a low order filter, the computation time can be reduced.
528:, which FIR filters normally do not). In theory, the impulse response of such a filter never dies out completely, hence the name IIR, though in practice, this is not true given the finite resolution of computer arithmetic. IIR filters normally require less
1499:
norm has to be approximated by means of a suitable sum over discrete points in the frequency domain. In general, however, these points should be significantly more than the number of coefficients in the signal domain to obtain a useful approximation.
363:
On the other hand, filters based on feedback circuits have other advantages and may therefore be preferred, even if this class of filters includes unstable filters. In this case, the filters must be carefully designed in order to avoid instability.
376:
function's width. Consequently, it may not be possible to simultaneously meet requirements on the locality of the filter's impulse response function as well as on its frequency function. This is a typical example of contradicting requirements.
393:
those which are sufficiently close to zero. In multirate filters, the number of coefficients by taking advantage of its bandwidth limits, where the input signal is downsampled (e.g. to its critical frequency), and upsampled after filtering.
313:
However, in certain applications it may be the filter's impulse response that is explicit and the design process then aims at producing as close an approximation as possible to the requested impulse response given all other requirements.
317:
In some cases it may even be relevant to consider a frequency function and impulse response of the filter which are chosen independently from each other. For example, we may want both a specific frequency function of the filter
101:
The design of digital filters is a complex topic. Although filters are easily understood and calculated, the practical challenges of their design and implementation are significant and are the subject of advanced research.
1231:
272:
An all-pass filter passes through all frequencies unchanged, but changes the phase of the signal. Filters of this type can be used to equalize the group delay of recursive filters. This filter is also used in
489:
signals of all frequencies equally which is important in many applications. It is also straightforward to avoid overflow in an FIR filter. The main disadvantage is that they may require significantly more
1037:
described in
Knutsson et al., which minimizes the integral of the square of the error, instead of its maximum value. In its basic form this approach requires that an ideal frequency function of the filter
1030:
and returns the optimum coefficients. One possible drawback to filters designed this way is that they contain many small ripples in the passband(s), since such a filter minimizes the peak error.
548:
is inherently a non-linear function of frequency, the time delay through such a filter is frequency-dependent, which can be a problem in many situations. 2nd order IIR filters are often called '
569:
is fixed by some outside constraint, selecting a suitable sample rate is an important design decision. A high rate will require more in terms of computational resources, but less in terms of
98:
The filter design process can be described as an optimization problem. Certain parts of the design process can be automated, but an experienced designer may be needed to get a good result.
1022:
coefficients that minimize the maximum deviation from the ideal. Intuitively, this finds the filter that is as close as you can get to the desired response given that you can use only
881:
95:
that satisfies a set of requirements, some of which may be conflicting. The purpose is to find a realization of the filter that meets each of the requirements to an acceptable degree.
998:
A consequence of this theorem is that the frequency function of a filter should be as smooth as possible to allow its impulse response to have a fast decay, and thereby a short width.
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838:
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extended by the width of the filter. This implies that it is sometimes important to keep the width of the filter's impulse response function as short as possible.
1450:
1263:
1418:
520:, filters are the digital counterpart to analog filters. Such a filter contains internal state, and the output and the next internal state are determined by a
331:
signals (sum of sinusoids) is the primary filter requirement, while an unconstrained impulse response may cause unexpected degradation due to time spreading of
481:
are examples of FIR filters that are normally recursive (that use feedback). If the FIR coefficients are symmetrical (often the case), then such a filter is
1887:
1618:
Rabiner, Lawrence R., and Gold, Bernard, 1975: Theory and
Application of Digital Signal Processing (Englewood Cliffs, New Jersey: Prentice-Hall, Inc.)
347:. If the design process yields a noncausal filter, the resulting filter can be made causal by introducing an appropriate time-shift (or delay).
1007:
499:
323:
the design process is then to realize a filter which tries to meet both these contradicting design goals as much as possible. An example is for
350:
Filters that do not operate in real time (e.g. for image processing) can be non-causal. Noncausal filters may be designed to have zero delay.
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1662:
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effects. Often, this is done by adding analog anti-aliasing filters at the input and output, thus avoiding any frequency component above the
473:
is the order of the filter. FIR filters are normally non-recursive, meaning they do not use feedback and as such are inherently stable. A
53:
552:' and a common implementation of higher order filters is to cascade biquads. A useful reference for computing biquad coefficients is the
1026:
coefficients. This method is particularly easy in practice and at least one text includes a program that takes the desired filter and
194:
function, which describes, for each frequency, how important it is that the resulting frequency function approximates the desired one.
1957:
1809:
L.R. Rabiner; J.H. McClellan; T.W. Parks (April 1975). "FIR Digital Filter Design
Techniques Using Weighted Chebyshev Approximation".
1967:
1623:
1573:"A Suggested Explanation For (Some Of) The Audible Differences Between High Sample Rate And Conventional Sample Rate Audio Material"
310:
of the latter. That means that any requirement on the frequency function is a requirement on the impulse response, and vice versa.
75:
532:
resources than an FIR filter of similar performance. However, due to the feedback, high order IIR filters may have problems with
1880:
1014:. Here the user specifies a desired frequency response, a weighting function for errors from this response, and a filter order
1290:
574:
249:
A high-shelf filter passes all frequencies, but increases or reduces frequencies above the shelf frequency by specified amount.
246:
A low-shelf filter passes all frequencies, but increases or reduces frequencies below the shelf frequency by specified amount.
1579:
1597:
1792:
T.W. Parks; J.H. McClellan (March 1972). "Chebyshev
Approximation for Nonrecursive Digital Filters with Linear Phase".
1873:
266:
234:
1854:
306:
There is a direct correspondence between the filter's frequency function and its impulse response: the former is the
46:
40:
284:
is a specific all-pass filter that passes sinusoids with unchanged amplitude but shifts each sinusoid phase by ±90°.
1896:
92:
513:
498:
resources than cleverly designed IIR variants. FIR filters are generally easier to design than IIR filters - the
172:
20:
846:
57:
1011:
226:
passes frequencies above and below a certain range. A very narrow band-stop filter is known as a notch filter.
1528:
491:
458:
343:
Any filter operating in real time (the filter response only depends on the current and past inputs) must be
212:
passes high frequencies fairly well; it is helpful as a filter to cut any unwanted low-frequency components.
187:. For many purposes, this is not sufficient. To achieve steeper slopes, higher-order filters are required.
1962:
1707:
A.G. Deczky (October 1972). "Synthesis of
Recursive Digital Filters Using the Minimum p-Error Criterion".
1370:
598:
549:
414:
332:
324:
614:
frequency function. Other effects which refer to relations between the time and frequency domain are
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811:
779:
747:
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253:
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1041:
537:
168:. The steepness and complexity of the response curve determines the filter order and feasibility.
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1455:
1323:
1136:
1826:
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defined on the specified set of coordinates. The norm used here is, formally, the usual norm on
521:
288:
233:
passes all frequencies equally in gain. Only the phase shift is changed, which also affects the
161:
114:
1077:
899:
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This paper explains simply (between others topics) filters design theory and give some examples
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128:
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be its
Fourier transform. There is a theorem which states that if the first derivative of
503:
495:
451:
291:
is an all-pass that has a specified and constant group or phase delay for all frequencies.
274:
230:
202:
1472:. This can be done by solving the corresponding least squares problem. In practice, the
1689:
1423:
Once the error function is established, the optimal filter is given by the coefficients
1518:
1403:
474:
447:
419:
359:
240:
139:
1951:
1772:
602:
545:
436:
409:
344:
133:
1830:
506:) is one suitable method for designing quite good filters semi-automatically. (See
1724:
482:
328:
252:
A peak EQ filter makes a peak or a dip in the frequency response, commonly used in
175:
filter will only have a single frequency-dependent component. This means that the
1674:"Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function"
1572:
190:
In relation to the desired frequency function, there may also be an accompanying
634:
566:
541:
533:
486:
123:
119:
1340:
measures the deviation between the requested frequency function of the filter,
450:
are classified into one of two basic forms, according to how they respond to a
544:, and require careful design to avoid such pitfalls. Additionally, since the
478:
1801:
1743:
H. Knutsson; M. Andersson; J. Wiklund (June 1999). "Advanced Filter Design".
1716:
1865:
1549:
529:
165:
1822:
597:. The complexity (i.e., steepness) of such filters depends on the required
553:
1698:
1673:
624:
The asymptotic behaviour of one domain versus discontinuities in the other
1844:
590:
525:
385:
limited time. The last limitation is typical in real-time applications.
1845:
An extensive list of filter design articles and software at
Circuit Sage
1745:
Proc. Scandinavian
Symposium on Image Analysis, Kangerlussuaq, Greenland
435:
The design of linear analog filters is for the most part covered in the
110:
Typical requirements which may be considered in the design process are:
1226:{\displaystyle \varepsilon =\|W\cdot (F_{I}-{\mathcal {F}}\{f\})\|^{2}}
180:
19:"Filter theory" redirects here. For the theory on mate selection, see
709:
be the variance of the filter. The variance of the filter response,
184:
1400:. However, the deviation is also subject to the weighting function
465:, filters express each output sample as a weighted sum of the last
405:
It must also be decided how the filter is going to be implemented:
589:
For any digital filter design, it is crucial to analyze and avoid
176:
1130:
in the signal domain where the filter coefficients are located.
618:
The uncertainty principle between the time and frequency domains
1869:
1859:
25:
524:
of the previous inputs and outputs (in other words, they use
1860:
Yehar's digital sound processing tutorial for the braindead!
1376:
1367:, and the actual frequency function of the realized filter,
1274:
1196:
1074:
is specified together with a frequency weighting function
327:
in which the frequency response (magnitude and phase) for
1655:
Digital Signal
Processing: Signals, Systems, and Filters
243:
has an amplitude response proportional to the frequency.
581:
with other signals in the system may also be an issue.
1727:(1974). "Nonrecursive Digital Filter Design Using the
1598:"MQA Time-domain Accuracy & Digital Audio Quality"
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1080:
1044:
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935:
902:
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814:
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750:
715:
683:
651:
1754:
Digital Signal Processing: A Computer-Based Approach
1850:
A list of digital filter design software at dspGuru
1736:
Proc. 1974 IEEE Int. Symp. Circuit Theory (ISCAS74)
1636:
Digital Filters: Analysis, Design, and Applications
1033:Another method to finding a discrete FIR filter is
1006:One common method for designing FIR filters is the
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1464:
1444:
1412:
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1359:
1332:
1312:
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917:
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832:
800:
768:
733:
701:
669:
1771:A.V. Oppenheim; R.W. Schafer; J.R. Buck (1999).
205:is used to cut unwanted high-frequency signals.
145:Finite (in duration) impulse response required?
1881:
8:
1678:EURASIP Journal on Applied Signal Processing
1387:
1381:
1214:
1207:
1201:
1169:
884:frequency domain cannot be arbitrary small.
677:be the variance of the input signal and let
197:Typical examples of frequency function are:
888:Discontinuities versus asymptotic behaviour
1888:
1874:
1866:
876:{\displaystyle \sigma _{r}>\sigma _{f}}
179:of the frequency response is limited to 6
1777:. Prentice-Hall, Upper Saddle River, NJ.
1697:
1638:(2 ed.). McGraw-Hill, New York, NY.
1504:Simultaneous optimization in both domains
1483:
1477:
1457:
1428:
1405:
1375:
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1372:
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1325:
1304:
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1273:
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1195:
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964:
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781:
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749:
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714:
693:
688:
682:
661:
656:
650:
605:and the highest frequency of the signal.
76:Learn how and when to remove this message
39:This article includes a list of general
1540:
1420:before the error function is computed.
1018:. The algorithm then finds the set of
1008:Parks-McClellan filter design algorithm
500:Parks-McClellan filter design algorithm
1585:from the original on 28 November 2009.
219:passes a limited range of frequencies.
1529:Finite impulse response#Filter design
7:
1738:. San Francisco, CA. pp. 20–23.
1672:S.W.A. Bergen; A. Antoniou (2005).
1608:from the original on 10 March 2023.
1393:{\displaystyle {\mathcal {F}}\{f\}}
1709:IEEE Trans. Audio Electroacoustics
45:it lacks sufficient corresponding
14:
929:which is discontinuous has order
1855:Analog Filter Design Demystified
507:
30:
16:Signal processing design process
1774:Discrete-Time Signal Processing
1291:discrete-time Fourier transform
833:{\displaystyle \sigma _{f}^{2}}
801:{\displaystyle \sigma _{s}^{2}}
769:{\displaystyle \sigma _{r}^{2}}
734:{\displaystyle \sigma _{r}^{2}}
702:{\displaystyle \sigma _{f}^{2}}
670:{\displaystyle \sigma _{s}^{2}}
1596:Robjohns, Hugh (August 2016).
1571:Story, Mike (September 1997).
1439:
1433:
1282:{\displaystyle {\mathcal {F}}}
1252:
1246:
1210:
1178:
1090:
1084:
1067:{\displaystyle F_{I}(\omega )}
1061:
1055:
912:
906:
641:The variance extension theorem
621:The variance extension theorem
91:is the process of designing a
1:
1756:. McGraw-Hill, New York, NY.
1657:. McGraw-Hill, New York, NY.
959:has an asymptotic decay like
1465:{\displaystyle \varepsilon }
1333:{\displaystyle \varepsilon }
1146:{\displaystyle \varepsilon }
1265:is the discrete filter and
601:and the ratio between the
267:Group delay and phase delay
106:Typical design requirements
1984:
1794:IEEE Trans. Circuit Theory
1096:{\displaystyle W(\omega )}
918:{\displaystyle F(\omega )}
299:
264:
18:
1958:Digital signal processing
1936:
1903:
1897:Signal-processing filters
1320:spaces. This means that
629:The uncertainty principle
514:Infinite impulse response
21:Filter theory (sociology)
1968:Signal processing filter
1802:10.1109/TCT.1972.1083419
1734:-sinh Window Function".
1717:10.1109/TAU.1972.1162392
1012:Remez exchange algorithm
988:{\displaystyle t^{-n-1}}
380:Computational complexity
148:Computational complexity
93:signal processing filter
1103:and set of coordinates
948:{\displaystyle n\geq 0}
459:Finite impulse response
289:fractional delay filter
60:more precise citations.
1913:High-pass filter (HPF)
1823:10.1109/PROC.1975.9794
1796:. CT-19 (2): 189–194.
1711:. AU-20 (4): 257–263.
1493:
1466:
1446:
1414:
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1314:
1283:
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1227:
1147:
1124:
1097:
1068:
989:
949:
919:
896:be a function and let
877:
834:
802:
770:
735:
703:
671:
156:The frequency function
1908:Low-pass filter (LPF)
1699:10.1155/ASP.2005.1910
1494:
1492:{\displaystyle L^{2}}
1467:
1447:
1415:
1395:
1362:
1360:{\displaystyle F_{I}}
1335:
1315:
1313:{\displaystyle L^{2}}
1284:
1260:
1228:
1148:
1125:
1123:{\displaystyle x_{k}}
1098:
1069:
990:
950:
920:
878:
835:
803:
771:
736:
704:
672:
599:signal-to-noise ratio
571:anti-aliasing filters
554:RBJ Audio EQ Cookbook
469:input samples, where
415:Analog sampled filter
325:high-resolution audio
261:Phase and group delay
254:parametric equalizers
1653:A. Antoniou (2006).
1634:A. Antoniou (1993).
1476:
1456:
1445:{\displaystyle f(x)}
1427:
1404:
1371:
1344:
1324:
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1269:
1258:{\displaystyle f(x)}
1240:
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1137:
1107:
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1042:
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933:
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748:
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649:
401:Other considerations
296:The impulse response
1752:S.K. Mitra (1998).
1690:2005EJASP2005...44B
1035:filter optimization
829:
797:
765:
741:, is then given by
730:
698:
666:
538:arithmetic overflow
282:Hilbert transformer
1604:. Sound On Sound.
1489:
1462:
1442:
1410:
1390:
1357:
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1310:
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1255:
1223:
1143:
1133:An error function
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1093:
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985:
945:
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830:
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731:
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522:linear combination
162:frequency response
115:Frequency response
1945:
1944:
1939:Electronic filter
1784:978-0-13-754920-7
1763:978-0-07-286546-2
1664:978-0-07-145424-7
1645:978-0-07-002117-4
1550:"Digital Filters"
1413:{\displaystyle W}
633:As stated by the
609:Theoretical basis
595:Nyquist frequency
425:Mechanical filter
308:Fourier transform
86:
85:
78:
1975:
1928:Band-stop filter
1923:Band-pass filter
1890:
1883:
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1602:soundonsound.com
1593:
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1524:Prototype filter
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843:This means that
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660:
302:Impulse response
224:band-stop filter
217:band-pass filter
210:high-pass filter
164:is an important
129:impulse response
81:
74:
70:
67:
61:
56:this article by
47:inline citations
34:
33:
26:
1983:
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1978:
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1918:All-pass filter
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504:Remez algorithm
448:Digital filters
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443:Digital filters
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231:all-pass filter
203:low-pass filter
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71:
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52:Please help to
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1839:External links
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1817:(4): 595–610.
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1552:. GRM Networks
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1153:is defined as
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502:(based on the
475:moving average
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431:Analog filters
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420:Digital filter
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300:Main article:
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265:Main article:
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241:differentiator
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171:A first-order
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603:sampling rate
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585:Anti-aliasing
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437:linear filter
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410:Analog filter
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360:stable filter
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160:The required
155:
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140:Stable filter
138:
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134:Causal filter
132:
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105:
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89:Filter design
80:
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66:December 2012
59:
55:
49:
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28:
27:
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1684:(12): 1910.
1681:
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1554:. Retrieved
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1023:
1019:
1015:
1005:
997:
956:
926:
893:
891:
842:
644:
632:
612:
588:
575:Interference
564:
542:limit cycles
517:
483:linear phase
470:
466:
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452:unit impulse
446:
434:
404:
395:
391:
387:
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349:
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329:steady state
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235:group delay.
196:
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170:
159:
109:
100:
97:
88:
87:
72:
63:
44:
1725:J.K. Kaiser
1578:. dCS Ltd.
1002:Methodology
635:Gabor limit
567:sample rate
565:Unless the
561:Sample rate
546:phase shift
534:instability
508:Methodology
124:group delay
120:Phase shift
58:introducing
1952:Categories
1811:Proc. IEEE
1535:References
492:processing
479:CIC filter
477:filter or
151:Technology
41:references
1460:ε
1328:ε
1215:‖
1192:−
1176:⋅
1170:‖
1164:ε
1141:ε
1088:ω
1059:ω
978:−
972:−
940:≥
910:ω
865:σ
852:σ
817:σ
785:σ
753:σ
718:σ
686:σ
654:σ
530:computing
439:section.
354:Stability
339:Causality
335:signals.
333:transient
192:weighting
173:recursive
166:parameter
142:required?
136:required?
1831:12579115
1606:Archived
1580:Archived
1513:See also
591:aliasing
526:feedback
485:, so it
367:Locality
1686:Bibcode
1556:13 July
1289:is the
955:, then
579:beating
550:biquads
54:improve
1829:
1781:
1760:
1661:
1642:
1622:
1236:where
540:, and
496:memory
487:delays
345:causal
185:octave
43:, but
1827:S2CID
1583:(PDF)
1576:(PDF)
516:, or
461:, or
177:slope
1779:ISBN
1758:ISBN
1682:2005
1659:ISBN
1640:ISBN
1620:ISBN
1558:2020
894:f(t)
892:Let
861:>
645:Let
577:and
494:and
183:per
1819:doi
1798:doi
1713:doi
1694:doi
518:IIR
463:FIR
320:and
229:An
122:or
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287:A
280:A
239:A
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181:dB
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471:N
467:N
277:.
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73:(
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23:.
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