Sinc filter

Calling them according to which domain the filter resembles a sinc avoids confusion.

Its impulse response is a sinc function in the time domain:

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 lowpass filter with brick-wall cutoff at frequency BL has impulse response and transfer function given by: The band-pass filter with lower band edge BL and upper band edge BH is just the difference of two such sinc-in-time filters (since the filters are zero phase, their magnitude responses subtract directly):[1] The high-pass filter with lower band edge BH is just a transparent filter minus a sinc-in-time filter, which makes it clear that the Dirac delta function is the limit of a narrow-in-time sinc-in-time filter: As the sinc-in-time filter has infinite impulse response in both positive and negative time directions, it is non-causal and has an infinite delay (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).

However, it is used in conceptual demonstrations or proofs, such as the sampling theorem and the Whittaker–Shannon interpolation formula.

Sinc-in-time filters must be approximated for real-world (non-abstract) applications, typically by windowing and truncating an ideal sinc-in-time filter kernel, but doing so reduces its ideal properties.

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.

The simplest implementation of a sinc-in-frequency filter uses a boxcar impulse response to produce a simple moving average (specifically if divide by the number of samples), also known as accumulate-and-dump filter (specifically if simply sum without a division).

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

will alias all non-fully attenuated signal components lying above

Though the sinc function really oscillates between negative and positive values, negative values of the frequency response simply correspond to a 180-degree phase shift.

An inverse sinc filter may be used for equalization in the digital domain (e.g. a FIR filter) or analog domain (e.g. opamp filter) to counteract undesired attenuation in the frequency band of interest to provide a flat frequency response.

The normalized sinc function , the impulse response of a sinc-in-time filter and the frequency response of a sinc-in-frequency filter.
The rectangular function , the frequency response of a sinc-in-time filter and the impulse response of a sinc-in-frequency filter.
Frequency response in dB of moving average filters. Frequency plotted relative to sampling frequency .
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.