Editing Linear phase

{{Short description|Filter whose phase response is proportional to frequency}}

In [[signal processing]], '''linear phase''' is a property of a [[filter (signal processing)|filter]] where the [[phase response]] of the filter is a [[linear function (calculus)|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 (the slope of the linear function), which is referred to as the [[group delay]]. 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 by having coefficients which are 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> Approximations can be achieved with [[infinite impulse response]] (IIR) designs, which are more computationally efficient. Several techniques are:
* a [[Bessel filter|Bessel]] transfer function which has a maximally flat group delay approximation function
* a [[Filter design#Phase and group delay|phase equalizer]]

== Definition ==
A filter is called a linear phase filter if the phase component of the frequency response is a linear function of frequency.  For a continuous-time application, the frequency response of the filter is the [[Fourier transform]] of the filter's [[impulse response]], and a linear phase version has the form:

:<math>H(\omega) = A(\omega)\ e^{-j \omega \tau},</math>

where:
*A(ω) is a real-valued function.
*<math>\tau</math> is the group delay.

For a discrete-time application, the [[discrete-time Fourier transform]] of the linear phase impulse response has the form:

:<math>H_{2\pi}(\omega) = A(\omega)\ e^{-j \omega k/2},</math>

where:
*A(ω) is a real-valued function with 2π periodicity.
*k is an integer, and k/2 is the group delay in units of samples.

<math>H_{2\pi}(\omega)</math> is a [[Fourier series]] that can also be expressed in terms of the [[Discrete-time_Fourier_transform#Relationship_to_the_Z-transform|Z-transform]] of the filter impulse response.  I.e.:

:<math>H_{2\pi}(\omega) = \left. \widehat H(z) \, \right|_{z = e^{j \omega}} = \widehat H(e^{j \omega}),</math>

where the <math>\widehat H</math> notation distinguishes the Z-transform from the Fourier transform.

== 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>

== Generalized linear phase ==

Systems with generalized linear phase have an additional frequency-independent constant <math>\beta</math> added to the phase.  In the discrete-time case, for example, the frequency response has the form:

:<math>H_{2\pi}(\omega) = A(\omega)\ e^{-j \omega k/2 + j \beta},</math>
:<math>\arg \left[ H_{2\pi}(\omega) \right] = \beta - \omega k/2 </math> for <math> -\pi < \omega < \pi </math>

Because of this constant, the phase of the system is not a strictly linear function of frequency, but it retains many of the useful properties of linear phase systems.<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=1}}</ref>

== See also ==
*[[Minimum phase]]

==Notes==
{{reflist|group=note}}

==Citations==
{{reflist}}

[[Category:Digital signal processing]]