In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, a comb filter is a
filter implemented by adding a delayed version of a
signal to itself, causing constructive and destructive
interference. The
frequency response of a comb filter consists of a series of regularly spaced notches in between regularly spaced ''peaks'' (sometimes called ''teeth'') giving the appearance of a
comb
A comb is a tool consisting of a shaft that holds a row of teeth for pulling through the hair to clean, untangle, or style it. Combs have been used since prehistoric times, having been discovered in very refined forms from settlements dating ba ...
.
Applications
Comb filters are employed in a variety of signal processing applications, including:
*
Cascaded integrator–comb (CIC) filters, commonly used for
anti-aliasing during
interpolation and
decimation operations that change the
sample rate of a discrete-time system.
* 2D and 3D comb filters implemented in hardware (and occasionally software) in
PAL and
NTSC
The first American standard for analog television broadcast was developed by National Television System Committee (NTSC)National Television System Committee (1951–1953), Report and Reports of Panel No. 11, 11-A, 12–19, with Some supplement ...
analog television decoders, reduce artifacts such as
dot crawl.
*
Audio signal processing, including
delay
Delay (from Latin: dilatio) may refer to:
Arts, entertainment, and media
* ''Delay 1968'', a 1981 album by German experimental rock band Can
* '' The Delay'', a 2012 Uruguayan film
People
* B. H. DeLay (1891–1923), American aviator and ac ...
,
flanging,
physical modelling synthesis
Physical modelling synthesis refers to sound synthesis methods in which the waveform of the sound to be generated is computed using a mathematical model, a set of equations and algorithms to simulate a physical source of sound, usually a musi ...
and
digital waveguide synthesis Digital waveguide synthesis is the synthesis of audio using a digital waveguide. Digital waveguides are efficient computational models for physical media through which acoustic waves propagate. For this reason, digital waveguides constitute a maj ...
. If the delay is set to a few milliseconds, a comb filter can model the effect of
acoustic standing waves in a cylindrical cavity or
in a vibrating string.
* In astronomy the
astro-comb promises to increase the precision of existing
spectrographs by nearly a hundredfold.
In
acoustics, comb filtering can arise as an unwanted artifact. For instance, two
loudspeakers playing the same signal at different distances from the listener, create a comb filtering effect on the audio. In any enclosed space, listeners hear a mixture of direct sound and reflected sound. The reflected sound takes a longer, delayed path compared to the direct sound, and a comb filter is created where the two mix at the listener.
Implementation
Comb filters exist in two forms, ''feedforward'' and ''
feedback''; which refer to the direction in which signals are delayed before they are added to the input.
Comb filters may be implemented in
discrete time or
continuous time forms which are very similar.
Feedforward form
The general structure of a feedforward comb filter is described by the
difference equation:
:
where
is the delay length (measured in samples), and is a scaling factor applied to the delayed signal. The
transform of both sides of the equation yields:
:
The
transfer function is defined as:
:
Frequency response
The frequency response of a discrete-time system expressed in the -domain, is obtained by substitution . Therefore, for the feedforward comb filter:
:
Using
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
, the frequency response is also given by
:
Often of interest is the ''magnitude'' response, which ignores phase. This is defined as:
:
In the case of the feedforward comb filter, this is:
:
The term is constant, whereas the term varies
periodically. Hence the magnitude response of the comb filter is periodic.
The graphs show the magnitude response for various values of , demonstrating this periodicity. Some important properties:
*The response periodically drops to a
local minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
(sometimes known as a ''notch''), and periodically rises to a
local maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
(sometimes known as a ''peak'' or a ''tooth'').
*For positive values of , the first minimum occurs at half the delay period and repeat at even multiples of the delay frequency thereafter:
::
.
*The levels of the maxima and minima are always equidistant from 1.
*When , the minima have zero amplitude. In this case, the minima are sometimes known as ''nulls''.
*The maxima for positive values of coincide with the minima for negative values of
, and vice versa.
Impulse response
The feedforward comb filter is one of the simplest
finite impulse response filters. Its response is simply the initial impulse with a second impulse after the delay.
Pole–zero interpretation
Looking again at the -domain transfer function of the feedforward comb filter:
:
the numerator is equal to zero whenever . This has solutions, equally spaced around a circle in the
complex plane; these are the
zeros of the transfer function. The denominator is zero at , giving
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in ...
at . This leads to a
pole–zero plot like the ones shown.
Feedback form
Similarly, the general structure of a feedback comb filter is described by the
difference equation:
:
This equation can be rearranged so that all terms in
are on the left-hand side, and then taking the transform:
:
The transfer function is therefore:
:
Frequency response
Substituting into the -domain expression for the feedback comb filter:
:
The magnitude response is as follows:
:
Again, the response is periodic, as the graphs demonstrate. The feedback comb filter has some properties in common with the feedforward form:
*The response periodically drops to a local minimum and rises to a local maximum.
*The maxima for positive values of coincide with the minima for negative values of
, and vice versa.
*For positive values of , the first maximum occurs at 0 and repeats at even multiples of the delay frequency thereafter:
::
.
However, there are also some important differences because the magnitude response has a term in the
denominator:
*The levels of the maxima and minima are no longer equidistant from 1. The maxima have an amplitude of .
*The filter is only
stable
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
if is strictly less than 1. As can be seen from the graphs, as increases, the amplitude of the maxima rises increasingly rapidly.
Impulse response
The feedback comb filter is a simple type of
infinite impulse response filter.
If stable, the response simply consists of a repeating series of impulses decreasing in amplitude over time.
Pole–zero interpretation
Looking again at the -domain transfer function of the feedback comb filter:
:
This time, the numerator is zero at , giving zeros at . The denominator is equal to zero whenever . This has solutions, equally spaced around a circle in the
complex plane; these are the poles of the transfer function. This leads to a pole–zero plot like the ones shown below.
Continuous-time comb filters
Comb filters may also be implemented in
continuous time. The feedforward form may be described by the equation:
:
where is the delay (measured in seconds). This has the following transfer function:
:
The feedforward form consists of an infinite number of zeros spaced along the jω axis.
The feedback form has the equation:
:
and the following transfer function:
:
The feedback form consists of an infinite number of poles spaced along the jω axis.
Continuous-time implementations share all the properties of the respective discrete-time implementations.
See also
*
Dirac comb
*
Fabry–Pérot interferometer
References
External links
*
{{DEFAULTSORT:Comb Filter
Signal processing
Filter theory