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Concept
Sinc filter
Summary
In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given cutoff frequency, without affecting lower frequencies, and has linear phase response. The filter's impulse response is a sinc function in the time domain \left(\tfrac{\sin(\pi t)}{\pi t}\right), and its frequency response is a rectangular function. It is an "ideal" low-pass filter in the frequency sense, perfectly passing low frequencies, perfectly cutting high frequencies; and thus may be considered to be a brick-wall filter.
Real-time filters can only approximate this ideal, since an ideal sinc filter (a.k.a. rectangular filter) is non-causal and has an infinite delay, but it is commonly found in conceptual demonstrations or proofs, such as the sampling theorem and the Whittaker–Shannon interpolation formula.
In mathematical terms, the desired frequency response is the rectangular function:
where B is an arbitrary cutoff frequency (a.k.a. bandwidth). The impulse response of such a filter is given by the inverse Fourier transform of the frequency response:
where sinc is the normalized sinc function.
As the sinc filter has infinite impulse response in both positive and negative time directions, it must be approximated for real-world (non-abstract) applications; a windowed sinc filter is often used instead. Windowing and truncating a sinc filter kernel in order to use it on any practical real world data set reduces its ideal properties.
An idealized electronic filter 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 transfer function). The sinc filter is a brick-wall low-pass filter, from which brick-wall band-pass filters and high-pass filters are easily constructed.
Official source
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Ontological neighbourhood
Related people (21)
Related concepts (11)
Anti-aliasing filter
An anti-aliasing filter (AAF) is a filter used before a signal sampler to restrict the bandwidth of a signal to satisfy the Nyquist–Shannon sampling theorem over the band of interest. Since the theorem states that unambiguous reconstruction of the signal from its samples is possible when the power of frequencies above the Nyquist frequency is zero, a brick wall filter is an idealized but impractical AAF. A practical AAF makes a trade off between reduced bandwidth and increased aliasing.
Sinc function
In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized. In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x). In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by In either case, the value at x = 0 is defined to be the limiting value for all real a ≠ 0 (the limit can be proven using the squeeze theorem).
Filter (signal processing)
In signal processing, a filter is a device or process that removes some unwanted components or features from a signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequencies or frequency bands. However, filters do not exclusively act in the frequency domain; especially in the field of many other targets for filtering exist.
Related MOOCs (4)
Digital Signal Processing I
Basic signal processing concepts, Fourier analysis and filters. This module can
be used as a starting point or a basic refresher in elementary DSP
Digital Signal Processing II
Adaptive signal processing, A/D and D/A. This module provides the basic
tools for adaptive filtering and a solid mathematical framework for sampling and
quantization
Digital Signal Processing III
Advanced topics: this module covers real-time audio processing (with
examples on a hardware board), image processing and communication system design.