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filter whose impulse response is rectangular and whose frequency response is a sinc function. Calling them according to which domain the filter resembles a sinc avoids confusion. If the domain is unspecified, sinc-in-time is often assumed, or context hopefully can infer the correct domain.
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of the frequency response (plotted in these graphs) is useful when one wants to know how much frequencies are attenuated. Though the sinc function really oscillates between negative and positive values, negative values of the frequency response simply correspond to a 180-degree
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1398:. That is, a bounded input can produce an unbounded output, because the integral of the absolute value of the sinc function is infinite. A bounded input that produces an unbounded output is sgn(sinc(
392:{\displaystyle H(f)=\operatorname {rect} \left({\frac {f}{2B}}\right)={\begin{cases}0,&{\text{if }}|f|>B,\\{\frac {1}{2}},&{\text{if }}|f|=B,\\1,&{\text{if }}|f|<B,\end{cases}}}
444:
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1488:(specifically if divide by the number of samples), also known as accumulate-and-dump filter (specifically if simply sum without a division). It can be modeled as a FIR filter with all
630:
with full transmission in the pass band, complete attenuation in the stop band, and abrupt transitions is known colloquially as a "brick-wall filter" (in reference to the shape of the
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1801:
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data samples, output the accumulator result, zero the accumulator, and repeat) is foiled by its mediocre low-pass capabilities. Its poorest attenuation in the stop-band is -13.3
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600:{\displaystyle {\begin{aligned}h(t)={\mathcal {F}}^{-1}\{H(f)\}&=\int _{-B}^{B}\exp(2\pi ift)\,df\\&=2B\operatorname {sinc} (2Bt)\end{aligned}}}
757:
1469:
16-sample averaging (above), extended to 4x the
Nyquist frequency. Because the transfer function is periodic, this repeated pattern continues forever.
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forces its time response not to have compact support meaning that it is ever-lasting) and infinite order (i.e., the response cannot be expressed as a
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is just the difference of two such sinc-in-time filters (since the filters are zero phase, their magnitude responses subtract directly):
1119:{\displaystyle H_{BPF}(f)=\operatorname {rect} \left({\frac {f}{2B_{H}}}\right)-\operatorname {rect} \left({\frac {f}{2B_{L}}}\right).}
141:
1994:
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993:{\displaystyle h_{BPF}(t)=2B_{H}\operatorname {sinc} \left(2B_{H}t\right)-2B_{L}\operatorname {sinc} \left(2B_{L}t\right)}
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1386:, but doing so reduces its ideal properties. This applies to other brick-wall filters built using sinc-in-time filters.
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1934:
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2011:
1937:) to counteract undesired attenuation in the frequency band of interest to provide a flat frequency response.
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As the sinc-in-time filter has infinite impulse response in both positive and negative time directions, it is
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Sinc-in-time filters must be approximated for real-world (non-abstract) applications, typically by
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with a finite sum). However, it is used in conceptual demonstrations or proofs, such as the
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coefficients equal. It is sometimes cascaded to produce higher-order moving averages (see
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is just a transparent filter minus a sinc-in-time filter, which makes it clear that the
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1243:{\displaystyle h_{HPF}(t)=\delta (t)-2B_{H}\operatorname {sinc} \left(2B_{H}t\right)}
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and most high frequency components are only slightly more attenuated than that. An
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1336:{\displaystyle H_{HPF}(f)=1-\operatorname {rect} \left({\frac {f}{2B_{H}}}\right).}
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of a sinc-in-time filter and the frequency response of a sinc-in-frequency filter.
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831:{\displaystyle H_{LPF}(f)=\operatorname {rect} \left({\frac {f}{2B_{L}}}\right).}
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of a sinc-in-time filter and the impulse response of a sinc-in-frequency filter.
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2041:"APPLICATION NOTE 3853: Equalizing Techniques Flatten DAC Frequency Response"
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Transmission plots for group averaging filters using 1000 Hz sampling rate:
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744:{\displaystyle h_{LPF}(t)=2B_{L}\operatorname {sinc} \left(2B_{L}t\right)}
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for application of the sinc kernel as the simplest windowing function.
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will alias all non-fully attenuated signal components lying above
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The lowpass filter with brick-wall cutoff at frequency
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is the limit of a narrow-in-time sinc-in-time filter:
2012:"An Intuitive Look at Moving Average and CIC Filters"
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Finite impulse response § Moving average example
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has impulse response and transfer function given by:
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This filter can be used for crude but fast and easy
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that removes all frequency components above a given
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2075:Brick Wall Digital Filters and Phase Deviations
1382:and truncating an ideal sinc-in-time filter
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180:{\displaystyle {\frac {\sin(\pi t)}{\pi t}}}
634:). The sinc-in-time filter is a brick-wall
1989:. Cambridge University Press. p. 81.
1396:bounded-input–bounded-output (BIBO) stable
1129:The high-pass filter with lower band edge
841:The band-pass filter with lower band edge
1982:
1942:Window function § Rectangular window
1548:The simplicity of the filter (accumulate
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27:Ideal low-pass filter or averaging filter
2009:
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1373:Whittaker–Shannon interpolation formula
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114:response. It may thus be considered a
1355:and has an infinite delay (i.e., its
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429:Its impulse response is given by the
1796:{\displaystyle {\tfrac {f_{S}}{N}},}
1712:A group averaging filter processing
426:) is an arbitrary cutoff frequency.
1868:{\displaystyle {\tfrac {f_{S}}{2}}}
1832:{\displaystyle {\tfrac {f_{S}}{N}}}
1523:(a.k.a. decimation) by a factor of
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1703:{\textstyle {\frac {f_{S}}{2N}}.}
1476:The simplest implementation of a
1653:{\textstyle {\frac {f_{S}}{2N}}}
1462:
1448:
1434:
2053:from the original on 2023-09-18
2022:from the original on 2023-04-02
1754:{\displaystyle {\tfrac {N}{2}}}
1514:cascaded integrator–comb filter
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1929:in the digital domain (e.g. a
1484:impulse response to produce a
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2010:Verbeure, Tom (2020-09-30).
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1365:linear differential equation
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1986:Practical signal processing
1947:
1455:32-sample averaging (above)
433:of its frequency response:
10:
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1594:-sample filter sampled at
1441:4-sample averaging (above)
2110:Filter frequency response
2100:Digital signal processing
1933:) or analog domain (e.g.
431:inverse Fourier transform
102:Sinc-in-time is an ideal
1839:and the highest zero at
1803:with the lowest zero at
646:are easily constructed.
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89:is rectangular, or to a
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1614:{\displaystyle f_{S}}
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1486:simple moving average
1421:Frequency-domain sinc
1402:)). Another is sin(2
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1964:Anti-aliasing filter
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195:rectangular function
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52:rectangular function
1923:inverse sinc filter
1762:transmission zeroes
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120:rectangular filter.
2080:Brick-wall filters
1983:Mark Owen (2007).
1954:Lanczos resampling
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1540:{\displaystyle N.}
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2095:Signal processing
1996:978-0-521-85478-8
1877:Nyquist frequency
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1764:evenly-spaced by
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1725:{\displaystyle N}
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1587:{\displaystyle N}
1562:{\displaystyle N}
1501:{\displaystyle N}
1478:sinc-in-frequency
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644:high-pass filters
640:band-pass filters
632:transfer function
628:electronic filter
415:{\displaystyle B}
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91:sinc-in-frequency
64:signal processing
16:(Redirected from
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2069:External links
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2049:. 2012-08-20.
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1927:equalization
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134:time domain
68:sinc filter
2089:Categories
2057:2024-01-02
2026:2023-08-24
1970:References
1931:FIR filter
1353:non-causal
189:while its
85:and whose
1911:magnitude
1906:forever.
1390:Stability
1380:windowing
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2020:Archived
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318:if
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1359:in the
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