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Sinc filter

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Ideal low-pass filter or averaging filter

The normalizedsinc function, theimpulse response of a sinc-in-time filter and the frequency response of a sinc-in-frequency filter

Insignal processing, asinc filter can refer to either asinc-in-timefilter whoseimpulse response is asinc function and whosefrequency response is rectangular, or to asinc-in-frequency 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.

Sinc-in-time

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Sinc-in-time is an idealfilter that removes all frequency components above a givencutoff frequency, without attenuating lower frequencies, and haslinear phase response. It may thus be considered abrick-wall filter orrectangular filter.

Itsimpulse response is asinc function in thetime domain:

${\frac {\sin(\pi t)}{\pi t}}$

while itsfrequency response is arectangular function:

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}}

where $B {\displaystyle B}$ (representing itsbandwidth) is an arbitrary cutoff frequency.

Its impulse response is given by theinverse Fourier transform of its frequency response:

{\begin{aligned}h(t)={\mathcal {F}}^{-1}\{H(f)\}&=\int _{-B}^{B}\exp(2\pi ift)\,df\\&=2B\operatorname {sinc} (2Bt)\end{aligned}}

wheresinc is the normalizedsinc function.

Brick-wall filters

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An idealizedelectronic 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 thetransfer function). The sinc-in-time filter is a brick-walllow-pass filter, from which brick-wallband-pass filters andhigh-pass filters are easily constructed.

The lowpass filter with brick-wall cutoff at frequencyB_L has impulse response and transfer function given by:

h_{LPF}(t)=2B_{L}\operatorname {sinc} \left(2B_{L}t\right)

H_{LPF}(f)=\operatorname {rect} \left({\frac {f}{2B_{L}}}\right).

The band-pass filter with lower band edgeB_L and upper band edgeB_H is just the difference of two such sinc-in-time filters (since the filters are zero phase, their magnitude responses subtract directly):^[1]

h_{BPF}(t)=2B_{H}\operatorname {sinc} \left(2B_{H}t\right)-2B_{L}\operatorname {sinc} \left(2B_{L}t\right)

H_{BPF}(f)=\operatorname {rect} \left({\frac {f}{2B_{H}}}\right)-\operatorname {rect} \left({\frac {f}{2B_{L}}}\right).

The high-pass filter with lower band edgeB_H is just a transparent filter minus a sinc-in-time filter, which makes it clear that theDirac delta function is the limit of a narrow-in-time sinc-in-time filter:

h_{HPF}(t)=\delta (t)-2B_{H}\operatorname {sinc} \left(2B_{H}t\right)

H_{HPF}(f)=1-\operatorname {rect} \left({\frac {f}{2B_{H}}}\right).

Unrealizable

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As the sinc-in-time filter has infinite impulse response in both positive and negative time directions, it isnon-causal and has an infinite delay (i.e., itscompact support in thefrequency 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 alinear differential equation with a finite sum). However, it is used in conceptual demonstrations or proofs, such as thesampling theorem and theWhittaker–Shannon interpolation formula.

Sinc-in-time filters must be approximated for real-world (non-abstract) applications, typically bywindowing and truncating an ideal sinc-in-time filterkernel, but doing so reduces its ideal properties.^[2] This applies to other brick-wall filters built using sinc-in-time filters.

Stability

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The sinc filter is notbounded-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(2πBt)u(t), a sine wave starting at time 0, at the cutoff frequency.

Frequency-domain sinc

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Frequency response in dB of moving average filters. Frequency plotted relative to sampling frequency $f_{S}$ .

{\displaystyle f_{S}} — Frequency response in dB of moving average filters. Frequency plotted relative to sampling frequency $f_{S}$ .

Frequency response of a 16-sample sum using 1000 Hz sampling frequency, extended to 4x the Nyquist frequency. Because the transfer function is periodic, this repeated pattern continues forever.

The simplest implementation of asinc-in-frequency filter uses aboxcar impulse response to produce asimple moving average (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 finite impulse response (FIR) filter with all $N {\displaystyle N}$ coefficients equal. It is sometimes cascaded to produce higher-order moving averages (seeFinite impulse response § Moving average example andcascaded integrator–comb filter).

This filter can be used for crude but fast and easydownsampling (a.k.a. decimation) by a factor of $N . {\displaystyle N.}$ The simplicity of the filter is foiled by its mediocre low-pass capabilities. The stop-band contains periodic lobes with gradually decreasing height in between the nulls at multiples of ${\textstyle {\frac {f_{S}}{N}}}$ . The first lobe is -11.3 dB for a 4-sample moving average, or -12.8 dB for an 8-sample moving average, and -13.1 dB for a 16-sample moving average. An $N {\displaystyle N}$ -sample filter sampled at $f_{S}$ will alias all non-fully attenuated signal components lying above ${\textstyle {\frac {f_{S}}{2N}}}$ to thebaseband ranging fromDC to ${\textstyle {\frac {f_{S}}{2N}}.}$

A group averaging filter processing $N {\displaystyle N}$ samples has ${\tfrac {N}{2}}$ transmission zeroes evenly-spaced by ${\tfrac {f_{S}}{N}},$ with the lowest zero at ${\tfrac {f_{S}}{N}}$ and the highest zero at ${\tfrac {f_{S}}{2}}$ (theNyquist frequency). Above the Nyquist frequency, the frequency response is mirrored and then is repeated periodically above $f_{S}$ forever.

Themagnitude 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-degreephase shift.

Aninverse sinc filter may be used forequalization in the digital domain (e.g. aFIR filter) or analog domain (e.g.opamp filter) to counteract undesired attenuation in the frequency band of interest to provide a flat frequency response.^[3]

SeeWindow function § Rectangular window for application of the sinc kernel as the simplest windowing function.