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, 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 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, one that has full transmission in the pass band, and complete attenuation in the stop band, with 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.

The lowpass filter with brick-wall cutoff at frequency *B*_{L} has impulse response and transfer function given by:

The band-pass filter with lower band edge *B*_{L} and upper band edge *B*_{H} is just the difference of two such sinc filters (since the filters are zero phase, their magnitude responses subtract directly):^{ [1] }

The high-pass filter with lower band edge *B*_{H} is just a transparent filter minus a sinc filter, which makes it clear that the Dirac delta function is the limit of a narrow-in-time sinc filter:

Brick-wall filters that run in realtime are not physically realizable as they have infinite latency (i.e., its compact support in the frequency domain 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 linear differential equation with a finite sum), but approximate implementations are sometimes used and they are frequently called brick-wall filters.^{[ citation needed ]}

The name "sinc filter" is applied also to the filter shape that is rectangular in time and a sinc function in frequency, as opposed to the ideal low-pass sinc filter, which is sinc in time and rectangular in frequency. In case of confusion, one may refer to these as **sinc-in-frequency** and **sinc-in-time,** according to which domain the filter is sinc in.

Sinc-in-frequency CIC filters, among many other applications, are almost universally used for decimating delta-sigma ADCs, as they are easy to implement and nearly optimal for this use.^{ [2] }

The simplest implementation of a Sinc-in-frequency filter is a group-averaging filter, also known as accumulate-and-dump filter. This filter also performs a data rate reduction.

It collects *N* data samples, accumulates them and provides the accumulator value as output. Thus, the decimation factor of this filter is *N*. It can be modeled as a FIR filter with all *N* coefficients equal, followed by a N-time downsampling block. The simplicity of the filter, requiring just an accumulator as central data processing block, is foiled with strong aliasing effects: an N sample filter aliases all attenuated and unattenuated signal components lying above to the baseband ranging from 0 to (*f _{S}* is the input sample rate).

A group averaging filter processing *N* samples has *N*/2 transmission zeroes.

The picture "transmission function of a 16sample group averaging filter" shows how the transmission function looks above the Nyquist frequency.

The sinc filter is not bounded-input–bounded-output (BIBO) stable. 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(*t*)). Another is sin(2π*Bt*)*u*(*t*), a sine wave starting at time 0, at the cutoff frequency.

The **Nyquist–Shannon sampling theorem** is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that permits a discrete sequence of *samples* to capture all the information from a continuous-time signal of finite bandwidth.

In physics and electrical engineering, a **cutoff frequency**, **corner frequency**, or **break frequency** is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.

A **wavelet** is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one recorded by a seismograph or heart monitor. Generally, wavelets are intentionally crafted to have specific properties that make them useful for signal processing.

A **low-pass filter** (**LPF**) is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter design. The filter is sometimes called a **high-cut filter**, or **treble-cut filter** in audio applications. A low-pass filter is the complement of a high-pass filter.

A **high-pass filter** (**HPF**) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency depends on the filter design. A high-pass filter is usually modeled as a linear time-invariant system. It is sometimes called a **low-cut filter** or **bass-cut filter** in the context of audio engineering. High-pass filters have many uses, such as blocking DC from circuitry sensitive to non-zero average voltages or radio frequency devices. They can also be used in conjunction with a low-pass filter to produce a bandpass filter.

The **Whittaker–Shannon interpolation formula** or **sinc interpolation** is a method to construct a continuous-time bandlimited function from a sequence of real numbers. The formula dates back to the works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935, and in the formulation of the Nyquist–Shannon sampling theorem by Claude Shannon in 1949. It is also commonly called **Shannon's interpolation formula** and **Whittaker's interpolation formula**. E. T. Whittaker, who published it in 1915, called it the **Cardinal series**.

In signal processing and statistics, a **window function** is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions.

In signal processing, **undersampling** or **bandpass sampling** is a technique where one samples a bandpass-filtered signal at a sample rate below its Nyquist rate, but is still able to reconstruct the signal.

**Chebyshev filters** are analog or digital filters having a steeper roll-off than Butterworth filters, and have passband ripple or stopband ripple. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. Type I Chebyshev filters are usually referred to as "Chebyshev filters", while type II filters are usually called "inverse Chebyshev filters".

In mathematics, the **Gibbs phenomenon,** discovered by Henry Wilbraham (1848) and rediscovered by J. Willard Gibbs (1899), is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The *n*th partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as *n* increases, but approaches a finite limit. This sort of behavior was also observed by experimental physicists, but was believed to be due to imperfections in the measuring apparatus.

In signal processing, a **finite impulse response** (**FIR**) **filter** is a filter whose impulse response is of *finite* duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely.

In mathematics, physics and engineering, the **sinc function**, denoted by sinc(*x*), has two slightly different definitions.

The **rectangular function** is defined as

In mathematics, a **Dirac comb** is a periodic tempered distribution constructed from Dirac delta functions

The **raised-cosine filter** is a filter frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference (ISI). Its name stems from the fact that the non-zero portion of the frequency spectrum of its simplest form is a cosine function, 'raised' up to sit above the (horizontal) axis.

A **triangular function** is a function whose graph takes the shape of a triangle. Often this is an isosceles triangle of height 1 and base 2 in which case it is referred to as *the* triangular function. Triangular functions are useful in signal processing and *communication systems engineering* as representations of idealized signals, and the triangular function specifically as an integral transform kernel function from which more realistic signals can be derived, for example in kernel density estimation. It also has applications in pulse-code modulation as a pulse shape for transmitting digital signals and as a matched filter for receiving the signals. It is also used to define the **triangular window** sometimes called the Bartlett window.

The **zero-order hold** (**ZOH**) is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter (DAC). That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval. It has several applications in electrical communication.

**First-order hold** (**FOH**) is a mathematical model of the practical reconstruction of sampled signals that could be done by a conventional digital-to-analog converter (DAC) and an analog circuit called an integrator. For FOH, the signal is reconstructed as a piecewise linear approximation to the original signal that was sampled. A mathematical model such as FOH is necessary because, in the sampling and reconstruction theorem, a sequence of Dirac impulses, *x*_{s}(*t*), representing the discrete samples, *x*(*nT*), is low-pass filtered to recover the original signal that was sampled, *x*(*t*). However, outputting a sequence of Dirac impulses is impractical. Devices can be implemented, using a conventional DAC and some linear analog circuitry, to reconstruct the piecewise linear output for either predictive or delayed FOH.

In signal processing, particularly digital image processing, **ringing artifacts** are artifacts that appear as spurious signals near sharp transitions in a signal. Visually, they appear as bands or "ghosts" near edges; audibly, they appear as "echos" near transients, particularly sounds from percussion instruments; most noticeable are the pre-echos. The term "ringing" is because the output signal oscillates at a fading rate around a sharp transition in the input, similar to a bell after being struck. As with other artifacts, their minimization is a criterion in filter design.

In optics, the **Fraunhofer diffraction equation** is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.

- ↑ Mark Owen (2007).
*Practical signal processing*. Cambridge University Press. p. 81. ISBN 978-0-521-85478-8. - ↑ Chou, W.; Meng, T.H.; Gray, R.M. (1990). "Time domain analysis of sigma delta modulation".
*Acoustics, Speech, and Signal Processing*.**3**: 1751–1754. doi:10.1109/ICASSP.1990.115820.

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