Editing Linear phase (section)

== Examples ==
When a sinusoid<math>,\ \sin(\omega t + \theta),</math>&nbsp; passes through a filter with constant (frequency-independent) group delay <math>\tau,</math>&nbsp; the result is''':'''

:<math>A(\omega)\cdot \sin(\omega (t-\tau) + \theta) = A(\omega)\cdot \sin(\omega t + \theta - \omega \tau),</math>

where''':'''
*<math>A(\omega)</math> is a frequency-dependent amplitude multiplier.
*The phase shift <math>\omega \tau</math> is a linear function of angular frequency <math>\omega</math>, and <math>-\tau</math> is the slope.

It follows that a complex exponential function:

:<math>e^{i(\omega t + \theta)} = \cos(\omega t + \theta) + i\cdot \sin(\omega t + \theta), </math>

is transformed into:

:<math>A(\omega)\cdot e^{i(\omega (t-\tau) + \theta)} = e^{i(\omega t + \theta)}\cdot A(\omega) e^{-i\omega \tau}</math><ref group="note">
The multiplier <math>A(\omega) e^{-i\omega \tau}</math>, as a function of ω, is known as the filter's ''frequency response''.
</ref>

For approximately 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 <math>\omega \tau,</math>&nbsp; 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 [[Normalized frequency (digital signal processing)|frequency units]], the Nyquist frequency may be 0.5, 1.0, π, or ½ of the actual sample-rate.&nbsp; Some examples of linear and non-linear phase are shown below.

[[Image:Phase Plots.svg|thumb|400px|left|{{center|[[phase response]] vs [[Normalized frequency (digital signal processing)|normalized frequency]] (ω/π)}}]] {{multiple image
 | width = 300
 | footer = Two depictions of the frequency response of a simple FIR filter
 | footer_align = center
 | image1 = Frequency response of 3-term boxcar filter.svg
 | alt1 = 
 | caption1 = [[Bode plots]].  Phase discontinuities are π radians, indicating a sign reversal.
 | image2 = Amplitude & phase vs frequency for 3-term boxcar filter.svg
 | alt2 = 
 | caption2 = Phase discontinuities are removed by allowing negative amplitude.
 }}
{{clear}}
A discrete-time filter with linear phase may be achieved by an FIR filter which is either symmetric or anti-symmetric.<ref>{{cite web|last=Selesnick|first=Ivan|title=Four Types of Linear-Phase FIR Filters|url=http://cnx.org/content/m10706/latest/|work=Openstax CNX|publisher=Rice University|accessdate=27 April 2014}}</ref>&nbsp; A necessary but not sufficient condition is''':'''
:<math>\sum_{n =-\infty}^\infty h[n] \cdot \sin(\omega \cdot (n - \alpha) + \beta)=0</math>
for some <math>\alpha, \beta \in \mathbb{R} </math>.<ref>{{cite book|last1=Oppenheim|first1=Alan V|author2=Ronald W Schafer|title=Digital Signal Processing|date=1975|publisher=Prentice Hall|isbn=0-13-214635-5|edition=3}}</ref>