Linear phase

Linear phase is a property of a filter, where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time (usually delayed) by the same constant amount, which is referred to as the phase delay. And consequently, there is no phase distortion due to the time delay of frequencies relative to one another.

For discrete-time signals, perfect linear phase is easily achieved with a finite impulse response (FIR) filter.  Approximations can be achieved with infinite impulse response (IIR) designs, which are more computationally efficient.  Several techniques are:

• a Bessel transfer function which has a maximally flat group delay
• a maximally flat group delay approximation function
• a phase equalizer

Examples

When a sinusoid$,\ \sin(\omega t + \theta),$  passes through a filter with group delay $\tau,$  the result is:

$A(\omega)\cdot \sin(\omega (t-\tau) + \theta) = A(\omega)\cdot \sin(\omega t + \theta - \omega \tau),$

where:

• $A(\omega)$ is a frequency-dependent amplitude multiplier.
• The phase shift $\omega \tau$ is a linear function of frequency (ω), and $-\tau$ is the slope.

For linear phase, it is sufficient to have that property only in the passband(s) of the filter, where |A(ω)| has relatively large values. Therefore both magnitude and phase graphs (Bode plots) are customarily used to examine a filter's linearity. A "linear" phase graph may contain discontinuities of π and/or 2π radians. The smaller ones happen where A(ω) changes sign. Since |A(ω)| cannot be negative, the changes are reflected in the phase plot. The 2π discontinuities happen because of plotting the principal value of $\omega \tau,$  instead of the actual value.

In discrete-time applications, one only examines the region of frequencies between 0 and the Nyquist frequency, because of periodicity and symmetry. Depending on the frequency units, the Nyquist frequency may be 0.5, 1.0, π, or ½ of the actual sample-rate.  Some examples of linear and non-linear phase are shown below.

phase response vs normalized frequency (ω/π)

Bode plots. Phase discontinuities are π radians, indicating a sign reversal.

Phase discontinuities are removed by allowing negative amplitude.

Two depictions of the frequency response of a simple FIR filter

A filter with linear phase may be achieved by an FIR filter which is either symmetric or anti-symmetric.^[1]  A necessary but not sufficient condition is:

$\sum_{n =-\infty}^\infty h[n] \cdot \sin(\omega \cdot (n - \alpha) + \beta)=0$

for some $\alpha, \beta$.^[2]

Generalized linear phase

Investigation of potential copyright issue Please note this is about the text of this Wikipedia article; it should not be taken to reflect on the subject of this article. Do not restore or edit the blanked content on this page until the issue is resolved by an administrator, copyright clerk or OTRS agent.
If you have just labeled this page as a potential copyright issue, please follow the instructions for filing at the bottom of the box.
The previous content of this page or section has been identified as posing a potential copyright issue, as a copy or modification of the text from the source(s) below, and is now listed on Wikipedia:Copyright problems (listing): http://www.testbank007.com/wp-content/uploads/2014/08/013139407X_ism01.pdf (Duplication Detector report)   Unless the copyright status of the text on this page is clarified, the problematic text or the entire page may be deleted one week after the time of its listing. Temporarily, the original posting is still accessible for viewing in the page history.
Can you help resolve this issue?  If you hold the copyright to this text, you can license it in a manner that allows its use on Wikipedia. Click "Show" to see how.  You must permit the use of your material under the terms of the Creative Commons Attribution-Sharealike 3.0 Unported License (CC-BY-SA) and the GNU Free Documentation License (GFDL) (unversioned, with no invariant sections, front-cover texts, or back-cover texts). Explain your intent to license the content on this article's discussion page To confirm your permission, you can either display a notice to this effect at the site of original publication or send an e-mail from an address associated with the original publication to permissions-en at wikimedia dot org or a postal letter to the Wikimedia Foundation. These messages must explicitly permit use under CC-BY-SA and the GFDL. See Wikipedia:Donating copyrighted materials. Note that articles on Wikipedia must be written from a neutral point of view and must be verifiable in published third-party sources; consider whether, copyright issues aside, your text is appropriate for inclusion in Wikipedia.  To demonstrate that this text is in the public domain, or is already under a license suitable for Wikipedia, click "Show".  Explain this on this article's discussion page, with reference to evidence. Wikipedia:Public domain and Wikipedia:Compatibly licensed may assist in determining the status.  Otherwise, you may write a new article without copyright-infringing material. Click "Show" to read where and how.  Your rewrite should be placed on this page, where it will be available for an administrator or clerk to review it at the end of the listing period. Follow this link to create the temporary subpage. Simply modifying copyrighted text is not sufficient to avoid copyright infringement—if the original copyright violation cannot be cleanly removed or the article reverted to a prior version, it is best to write the article from scratch. (See Wikipedia:Close paraphrasing.) For license compliance, any content used from the original article must be properly attributed; if you use content from the original, please leave a note at the top of your rewrite saying as much. You may duplicate non-infringing text that you had contributed yourself. It is always a good idea, if rewriting, to identify the point where the copyrighted content was imported to Wikipedia and to check to make sure that the contributor did not add content imported from other sources. When closing investigations, clerks and administrators may find other copyright problems than the one identified. If this material is in the proposed rewrite and cannot be easily removed, the rewrite may not be usable. State that you have created a rewrite on this article's discussion page.
About importing text to Wikipedia Posting copyrighted material without the express permission of the copyright holder is unlawful and against Wikipedia policy. If you have express permission, this must be verified either by explicit release at the source or by e-mail or letter to the Wikimedia Foundation. See Wikipedia:Declaration of consent for all enquiries. Policy requires that we block those who repeatedly post copyrighted material without express permission.	Instructions for filing If you have tagged the article for investigation, please complete the following steps: Add the following to the bottom of Wikipedia:Copyright_problems/2015 April 23 * {{subst:article-cv|:Linear phase}} from http://www.testbank007.com/wp-content/uploads/2014/08/013139407X_ism01.pdf. ~~~~ Place this notice on the talk page of the contributor of the copyrighted material: {{subst:Nothanks-web|pg=Linear phase|url=http://www.testbank007.com/wp-content/uploads/2014/08/013139407X_ism01.pdf}} ~~~~ To blank a section instead of an entire article, add the template to the beginning of the section and </div> at the end of the portion you intend to blank.

See also

• Minimum phase

Citations

1. Selesnick, Ivan. "Four Types of Linear-Phase FIR Filters". Openstax CNX. Rice University. Retrieved 27 April 2014.
2. Oppenheim, Alan V; Ronald W Schafer (1975). Digital Signal Processing (3 ed.). Prentice Hall. ISBN 0-13-214635-5.
3. Oppenheim, Alan V; Ronald W Schafer (1975). Digital Signal Processing (1 ed.). Prentice Hall. ISBN 0-13-214635-5.