FACTOID # 84: 41% world's poor people live in India.
 
 Home   Encyclopedia   Statistics   Countries A-Z   Flags   Maps   Education   Forum   FAQ   About 
 
WHAT'S NEW
RECENT ARTICLES
More Recent Articles »
 

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

 

 

(* = Graphable)

 

 


Encyclopedia > Comb filter

In signal processing, a comb filter adds 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 spikes, giving the appearance of a comb. Signal processing is the processing, amplification and interpretation of signals and deals with the analysis and manipulation of signals. ... Signal processing is the processing, amplification and interpretation of signals and deals with the analysis and manipulation of signals. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ... A comb A comb for people with hair loss. ...


Comb filters exists in two different forms, feedforward and feedback; the names refer to the direction in which signals are delayed before they are added to the input. Feed-forward is a term describing a kind of system which reacts to changes in its environment, usually to maintain some desired state of the system. ... Feedback is (generally) information about actions. ...


Comb filters may be implemented in discrete time or continuous time; this article will focus on discrete-time implementations; the properties of the continuous-time comb filter are very similar. Discrete time is non-continuous time. ... Continuous time occurs when time is sampled continuously. ...

Contents

Feedforward form

Feedforward comb filter structure
Enlarge
Feedforward comb filter structure
Feedforward magnitude response for various positive values of α
Enlarge
Feedforward magnitude response for various positive values of α
Feedforward magnitude response for various negative values of α
Enlarge
Feedforward magnitude response for various negative values of α

The general structure of a feedforward comb filter is shown on the right. It may be described by the following difference equation: Image File history File links Comb_filter_feedforward. ... Image File history File links Comb_filter_feedforward. ... Image File history File links Comb_filter_response_ff_pos. ... Image File history File links Comb_filter_response_ff_pos. ... Image File history File links Comb_filter_response_ff_neg. ... Image File history File links Comb_filter_response_ff_neg. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

y[n] = x[n] + alpha x[n-K]

where K is the delay length (measured in samples), and α is a scaling factor applied to the delayed signal. If we take the Z transform of both sides of the equation, we obtain: In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...

Y(z) = (1 + alpha z^{-K}) X(z)

We define the transfer function as: In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...

H(z) = frac{Y(z)}{X(z)} = 1 + alpha z^{-K} = frac{z^K + alpha}{z^K}

Frequency response

To obtain the frequency response of a discrete-time system expressed in the Z domain, we make the substitution z = ejω. Therefore, for our feedforward comb filter, we get:

H(e^{j omega}) = 1 + alpha e^{-j omega K}

Often of interest is the magnitude response, which ignores phase. This is defined as:

| H(e^{j omega}) | = sqrt{Re{H(e^{j omega})}^2 + Im{H(e^{j omega})}^2}

In the case of the feedforward comb filter, this is:

| H(e^{j omega}) | = sqrt{(1 + alpha^2) + 2 alpha cos(omega K)}

Notice that the (1 + α2) term is constant, whereas the 2αcos(ωK) term varies periodically. Hence the magnitude response of the comb filter is periodic. In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ...


The graphs to the right show the magnitude response for various values of α, demonstrating this periodicity. Some important properties:

  • The response periodically drops to a local minimum (sometimes known as a notch), and periodically rises to a local maximum (sometimes known as a peak).
  • The levels of the maxima and minima are always equidistant from 1.
  • When alpha = pm 1, the minima have zero amplitude. In this case, the minima are sometimes known as nulls.
  • The maxima for positive values of α coincide with the minima for negative values of α, and vice versa.

A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some &#949; > 0 such that f(x*) &#8805; f(x) for all x with |x-x*| < &#949;. Stated less formally, a local maximum... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some &#949; > 0 such that f(x*) &#8805; f(x) for all x with |x-x*| < &#949;. Stated less formally, a local maximum...

Pole-zero interpretation

Looking again at the Z-domain transfer function of the feedforward comb filter:

H(z) = frac{z^K + alpha}{z^K}

we see that the numerator is equal to zero whenever zK = − α. This has K solutions, equally spaced around a circle in the complex plane; these are the zeros of the transfer function. The denominator is zero at zK = 0, giving K poles at z = 0. This leads to a pole-zero plot like the ones shown below. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ... In mathematics and signal processing, a pole/zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as: stability causal / anticausal region of convergence (ROC) minimum phase / non minimum phase In general, a rational transfer...

Pole-zero plot of feedfoward comb filter with K = 8 and α = 0.5
Enlarge
Pole-zero plot of feedfoward comb filter with K = 8 and α = 0.5
Pole-zero plot of feedfoward comb filter with K = 8 and α = − 0.5
Enlarge
Pole-zero plot of feedfoward comb filter with K = 8 and α = − 0.5

Image File history File links Comb_filter_pz_ff_pos. ... Image File history File links Comb_filter_pz_ff_pos. ... Image File history File links Comb_filter_pz_ff_neg. ... Image File history File links Comb_filter_pz_ff_neg. ...

Feedback form

Feedback comb filter structure
Enlarge
Feedback comb filter structure
Feedback magnitude response for various positive values of α
Enlarge
Feedback magnitude response for various positive values of α
Feedback magnitude response for various negative values of α
Enlarge
Feedback magnitude response for various negative values of α

Similarly, the general structure of a feedback comb filter is shown on the right. It may be described by the following difference equation: Image File history File links Comb_filter_feedback. ... Image File history File links Comb_filter_feedback. ... Image File history File links Comb_filter_response_fb_pos. ... Image File history File links Comb_filter_response_fb_pos. ... Image File history File links Comb_filter_response_fb_neg. ... Image File history File links Comb_filter_response_fb_neg. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

y[n] = x[n] + alpha y[n-K]

If we rearrange this equation so that all terms in y are on the left-hand side, and then take the Z transform, we obtain:

(1 - alpha z^{-K}) Y(z) = X(z)

The transfer function is therefore:

H(z) = frac{Y(z)}{X(z)} = frac{1}{1 - alpha z^{-K}} = frac{z^K}{z^K - alpha}

Frequency response

If we make the substitution z = ejω into the Z-domain expression for the feedback comb filter, we get:

H(e^{j omega}) = frac{1}{1 - alpha e^{-j omega K}}

The magnitude response is as follows:

| H(e^{j omega}) | = frac{1}{sqrt{(1 + alpha^2) - 2 alpha cos(omega K)}}

Again, the response is periodic, as the graphs to the right 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 α coincide with the minima for negative values of α, and vice versa.

However, there are also some important differences, due to the fact that the magnitude response has a term in the denominator: In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ...

  • The levels of the maxima and minima are no longer equidistant from 1.
  • The filter is only stable if | α | is strictly less than 1. As can be seen from the graphs, as | α | increases, the amplitude of the maxima rises increasingly rapidly.

In electrical engineering, specifically signal processing and control theory, BIBO Stability is a form of stability for signals and systems. ...

Pole-zero interpretation

Looking again at the Z-domain transfer function of the feedback comb filter:

H(z) = frac{z^K}{z^K - alpha}

This time, the numerator is zero at zK = 0, giving K zeros at z = 0. The denominator is equal to zero whenever zK = α. This has K 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. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

Pole-zero plot of feedback comb filter with K = 8 and α = 0.5
Enlarge
Pole-zero plot of feedback comb filter with K = 8 and α = 0.5
Pole-zero plot of feedback comb filter with K = 8 and α = − 0.5
Enlarge
Pole-zero plot of feedback comb filter with K = 8 and α = − 0.5

Image File history File links Comb_filter_pz_fb_pos. ... Image File history File links Comb_filter_pz_fb_pos. ... Image File history File links Comb_filter_pz_fb_neg. ... Image File history File links Comb_filter_pz_fb_neg. ...

Continuous-time comb filters

Comb filters may also be implemented in continuous time. The feedforward form may be described by the following equation: Continuous time occurs when time is sampled continuously. ...

y(t) = x(t) + alpha x(t - tau)

and the feedback form by:

y(t) = x(t) + alpha y(t - tau)

where τ is the delay (measured in seconds).


They have the following frequency responses, respectively:

H(omega) = 1 + alpha e^{-j omega tau}
H(omega) = frac{1}{1 - alpha e^{-j omega tau}}

Continuous-time implementations share all the properties of the respective discrete-time implementations.


Applications

Comb filters are used in a variety of signal processing applications. These include:

  • 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) for NTSC television decoders. The filters work to reduce artifacts such as dot crawl.

  Results from FactBites:
 
Flanging - Wikipedia, the free encyclopedia (1092 words)
This produces a swept 'comb filter' effect: peaks and notches are produced in the resultant frequency spectrum, related to each other in a linear harmonic series.
Extending the comb analogy, flanging uses a comb filter with regularly-spaced teeth (like a brand new hair comb), whereas phasing uses a comb filter with irregularly-spaced teeth (like a worn hair comb with teeth that have been bent or broken).
Thus, whilst the peaks and troughs of the comb filter are more-or-less in a linear harmonic series, there is a significant amount of non-linear behaviour too, causing the timbre of tape-flanging to sound more like a combination of what came to be known as flanging and phasing.
TV and Video Comb Filters (8893 words)
The output of a comb filter is S-Video.
For a "standard comb filter" it is OK for the dot of the lowercase I to be duller and elongated slightly upwards on the comb filter set while it fits sharp and clear on one or two lines (but maybe wider sideways) on a 260 line resolution set.
Comb filters are sometimes sold separately, as "composite to S-video converters".
  More results at FactBites »


 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments
Please enter the 5-letter protection code

Want to know more?
Search encyclopedia, statistics and forums:
 


Lesson Plans | Student Area | Student FAQ | Reviews | Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.