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:

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:

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.

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?
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:

  • 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

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.
This article is issued from Wikipedia - version of the Sunday, September 06, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.