In
signal processingSignal processing is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals...
, a
sinc filter is an idealized
filterIn signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...
that removes all frequency components above a given bandwidth, leaves the low frequencies alone, and has
linear phaseLinear phase is a property of a filter, where the phase response of the filter is a linear function of frequency, excluding the possibility of wraps at . In a causal system, perfect linear phase can be achieved with a discrete-time FIR filter...
. The filter's
impulse responseIn signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...
is a
sinc functionIn mathematics, the sinc function, denoted by sinc and sometimes as Sa, has two definitions. In digital signal processing and information theory, the normalized sinc function is commonly defined by...
in the time domain, and its
frequency responseFrequency response is the measure of any system's output spectrum in response to an input signal. In the audible range it is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cables,...
is a
rectangular functionThe rectangular function is defined as:It is a simple step function....
.
It is an "ideal"
low-pass filterA low-pass filter is a filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in...
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 (aka
rectangular filter) has an infinite delay, but it is commonly found in conceptual demonstrations or proofs, such as the
sampling theoremThe Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...
and the
Whittaker–Shannon interpolation formulaThe Whittaker–Shannon interpolation formula is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples....
.
In mathematical terms, the desired frequency response is the
rectangular functionThe rectangular function is defined as:It is a simple step function....
:
where is an arbitrary cutoff frequency (aka
bandwidth) (in Hz).
In
signal processingSignal processing is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals...
, a
sinc filter is an idealized
filterIn signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...
that removes all frequency components above a given bandwidth, leaves the low frequencies alone, and has
linear phaseLinear phase is a property of a filter, where the phase response of the filter is a linear function of frequency, excluding the possibility of wraps at . In a causal system, perfect linear phase can be achieved with a discrete-time FIR filter...
. The filter's
impulse responseIn signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...
is a
sinc functionIn mathematics, the sinc function, denoted by sinc and sometimes as Sa, has two definitions. In digital signal processing and information theory, the normalized sinc function is commonly defined by...
in the time domain, and its
frequency responseFrequency response is the measure of any system's output spectrum in response to an input signal. In the audible range it is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cables,...
is a
rectangular functionThe rectangular function is defined as:It is a simple step function....
.
It is an "ideal"
low-pass filterA low-pass filter is a filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in...
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 (aka
rectangular filter) has an infinite delay, but it is commonly found in conceptual demonstrations or proofs, such as the
sampling theoremThe Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...
and the
Whittaker–Shannon interpolation formulaThe Whittaker–Shannon interpolation formula is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples....
.
In mathematical terms, the desired frequency response is the
rectangular functionThe rectangular function is defined as:It is a simple step function....
:
where is an arbitrary cutoff frequency (aka
bandwidth) (in Hz). The impulse response of such a filter is given by the inverse Fourier transform:
|
|
|
, in terms of the normalized sinc functionIn mathematics, the sinc function, denoted by sinc and sometimes as Sa, has two definitions. In digital signal processing and information theory, the normalized sinc function is commonly defined by...
. |
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
windowedIn signal processing, a window function is a function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation...
sinc filter is often used instead.
Brick-wall filters
An idealized
electronic filterElectronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
, 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 functionA transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
. The sinc filter is a brick-wall
low-pass filterA low-pass filter is a filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in...
, from which brick-wall
band-pass filterA band-pass filter is a device that passes frequencies within a certain range and rejects frequencies outside that range. An example of an analogue electronic band-pass filter is an RLC circuit...
s and
high-pass filterA high-pass filter is an LTI filter that passes high frequencies well but attenuates frequencies lower than the cutoff frequency. The actual amount of attenuation for each frequency is a design parameter of the filter...
s are easily constructed.
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 filters (since the filters are zero phase, their magnitude responses subtract directly):
The high-pass filter with lower band edge
BH is just a transparent filter minus a sinc filter, which makes it clear that the
Dirac delta functionThe Dirac delta or Dirac's delta is a mathematical construct introduced by theoretical physicist Paul Dirac. Informally, it is a generalized function representing an infinitely sharp peak bounding unit area: a 'function' δ that has the value zero everywhere except at x = 0 where its value is...
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
Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
forces its time response to be ever lasting) and infinite order (i.e., the response cannot be expressed as a
linear differential equationIn mathematics, a linear differential equation is of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
with a finite sum), but approximate implementations are sometimes used and they are frequently called brick-wall filters.
Frequency-domain sinc
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 filters, among many other applications, are almost universally used for
decimatingIn digital signal processing, decimation is a technique for reducing the number of samples in a discrete-time signal. The element which implements this technique is referred to as a decimator.Decimation is a two-step process:...
sigma–delta
ADCsAn analog-to-digital converter is a device which converts continuous signals to discrete digital numbers...
, as they are easy to implement and nearly optimum for this use.
See also
- Lanczos resampling
Lanczos resampling is a multivariate interpolation method used to compute new values for any digitally sampled data. It is often used for image scaling , but could be used for any other digital signal...
- Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...
- Anti-aliasing
In digital signal processing, anti-aliasing is the technique of minimizing the distortion artifacts known as aliasing when representing a high-resolution signal at a lower resolution...
- Whittaker–Shannon interpolation formula
The Whittaker–Shannon interpolation formula is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples....
External links