Linear phase

<h2 id="definition">Definition</h2>
<p>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 <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of the filter's <a href="/facts/Impulse_response/l4vaE5er">impulse response</a>, and a linear phase version has the form:
</p>

H
        (
        ω
        )
        =
        A
        (
        ω
        )
         
        
          e
          
            −
            j
            ω
            τ
          
        
        ,
      
    
    {\displaystyle H(\omega )=A(\omega )\ e^{-j\omega \tau },}

<p>where:
</p>
<ul><li>A(ω) is a real-valued function.</li>
<li>
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 is the group delay.</li></ul>
<p>For a discrete-time application, the <a href="/facts/Discrete-time_Fourier_transform/Em8WprVs">discrete-time Fourier transform</a> of the linear phase impulse response has the form:
</p>

H
          
            2
            π
          
        
        (
        ω
        )
        =
        A
        (
        ω
        )
         
        
          e
          
            −
            j
            ω
            k
            
              /
            
            2
          
        
        ,
      
    
    {\displaystyle H_{2\pi }(\omega )=A(\omega )\ e^{-j\omega k/2},}

<p>where:
</p>
<ul><li>A(ω) is a real-valued function with 2π periodicity.</li>
<li>k is an integer, and k/2 is the group delay in units of samples.</li></ul>
<p>
  
    
      
        
          H
          
            2
            π
          
        
        (
        ω
        )
      
    
    {\displaystyle H_{2\pi }(\omega )}
  
 is a <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> that can also be expressed in terms of the <a href="/facts/Discrete-time_Fourier_transform/Em8WprVs">Z-transform</a> of the filter impulse response.  I.e.:
</p>

H
          
            2
            π
          
        
        (
        ω
        )
        =
        
          
            
            
              
                
                  
                    H
                    ^
                  
                
              
              (
              z
              )
              
            
            |
          
          
            z
            =
            
              e
              
                j
                ω
              
            
          
        
        =
        
          
            
              H
              ^
            
          
        
        (
        
          e
          
            j
            ω
          
        
        )
        ,
      
    
    {\displaystyle H_{2\pi }(\omega )=\left.{\widehat {H}}(z)\,\right|_{z=e^{j\omega }}={\widehat {H}}(e^{j\omega }),}

<p>where the 
  
    
      
        
          
            
              H
              ^
            
          
        
      
    
    {\displaystyle {\widehat {H}}}
  
 notation distinguishes the Z-transform from the Fourier transform.
</p>
<h2 id="examples">Examples</h2>
<p>When a sinusoid
  
    
      
        ,
         
        sin
        ⁡
        (
        ω
        t
        +
        θ
        )
        ,
      
    
    {\displaystyle ,\ \sin(\omega t+\theta ),}
  
  passes through a filter with constant (frequency-independent) group delay 
  
    
      
        τ
        ,
      
    
    {\displaystyle \tau ,}
  
  the result is:
</p>

A
        (
        ω
        )
        ⋅
        sin
        ⁡
        (
        ω
        (
        t
        −
        τ
        )
        +
        θ
        )
        =
        A
        (
        ω
        )
        ⋅
        sin
        ⁡
        (
        ω
        t
        +
        θ
        −
        ω
        τ
        )
        ,
      
    
    {\displaystyle A(\omega )\cdot \sin(\omega (t-\tau )+\theta )=A(\omega )\cdot \sin(\omega t+\theta -\omega \tau ),}

<p>where:
</p>
<ul><li>
  
    
      
        A
        (
        ω
        )
      
    
    {\displaystyle A(\omega )}
  
 is a frequency-dependent amplitude multiplier.</li>
<li>The phase shift 
  
    
      
        ω
        τ
      
    
    {\displaystyle \omega \tau }
  
 is a linear function of angular frequency 
  
    
      
        ω
      
    
    {\displaystyle \omega }
  
, and 
  
    
      
        −
        τ
      
    
    {\displaystyle -\tau }
  
 is the slope.</li></ul>
<p>It follows that a complex exponential function:
</p>

e
          
            i
            (
            ω
            t
            +
            θ
            )
          
        
        =
        cos
        ⁡
        (
        ω
        t
        +
        θ
        )
        +
        i
        ⋅
        sin
        ⁡
        (
        ω
        t
        +
        θ
        )
        ,
      
    
    {\displaystyle e^{i(\omega t+\theta )}=\cos(\omega t+\theta )+i\cdot \sin(\omega t+\theta ),}

<p>is transformed into:
</p>

A
        (
        ω
        )
        ⋅
        
          e
          
            i
            (
            ω
            (
            t
            −
            τ
            )
            +
            θ
            )
          
        
        =
        
          e
          
            i
            (
            ω
            t
            +
            θ
            )
          
        
        ⋅
        A
        (
        ω
        )
        
          e
          
            −
            i
            ω
            τ
          
        
      
    
    {\displaystyle A(\omega )\cdot e^{i(\omega (t-\tau )+\theta )}=e^{i(\omega t+\theta )}\cdot A(\omega )e^{-i\omega \tau }}
  
<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
<p>For approximately linear phase, it is sufficient to have that property only in the <a href="/facts/Passband/VebMrazm">passband</a>(s) of the filter, where |A(ω)| has relatively large values.  Therefore, both magnitude and phase graphs (<a href="/facts/Bode_plots/xcF1ej98">Bode plots</a>) 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 <a href="/facts/Principal_value/bCHR0LmD">principal value</a> of 
  
    
      
        ω
        τ
        ,
      
    
    {\displaystyle \omega \tau ,}
  
  instead of the actual value.
</p><p>In discrete-time applications, one only examines the region of frequencies between 0 and the <a href="/facts/Nyquist_frequency/zwFUKdVR">Nyquist frequency</a>, because of periodicity and symmetry.  Depending on the <a href="/facts/Normalized_frequency_(digital_signal_processing)/hsY0aY3E">frequency units</a>, 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.
</p>

<p>A discrete-time filter with linear phase may be achieved by an FIR filter which is either symmetric or anti-symmetric.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>  A necessary but not sufficient condition is:
</p>

∑
          
            n
            =
            −
            ∞
          
          
            ∞
          
        
        h
        [
        n
        ]
        ⋅
        sin
        ⁡
        (
        ω
        ⋅
        (
        n
        −
        α
        )
        +
        β
        )
        =
        0
      
    
    {\displaystyle \sum _{n=-\infty }^{\infty }h[n]\cdot \sin(\omega \cdot (n-\alpha )+\beta )=0}

<p>for some 
  
    
      
        α
        ,
        β
        ∈
        
          R
        
      
    
    {\displaystyle \alpha ,\beta \in \mathbb {R} }
  
.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="generalized-linear-phase">Generalized linear phase</h2>
<p>Systems with generalized linear phase have an additional frequency-independent constant 
  
    
      
        β
      
    
    {\displaystyle \beta }
  
 added to the phase.  In the discrete-time case, for example, the frequency response has the form:
</p>

H
          
            2
            π
          
        
        (
        ω
        )
        =
        A
        (
        ω
        )
         
        
          e
          
            −
            j
            ω
            k
            
              /
            
            2
            +
            j
            β
          
        
        ,
      
    
    {\displaystyle H_{2\pi }(\omega )=A(\omega )\ e^{-j\omega k/2+j\beta },}

arg
        ⁡
        
          [
          
            
              H
              
                2
                π
              
            
            (
            ω
            )
          
          ]
        
        =
        β
        −
        ω
        k
        
          /
        
        2
      
    
    {\displaystyle \arg \left[H_{2\pi }(\omega )\right]=\beta -\omega k/2}
  
 for 
  
    
      
        −
        π
        <
        ω
        <
        π
      
    
    {\displaystyle -\pi <\omega <\pi }

<p>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.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Minimum_phase/a9WcqXXc">Minimum phase</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="citations">Citations</h2>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Selesnick, Ivan. "Four Types of Linear-Phase FIR Filters". Openstax CNX. Rice University. Retrieved 27 April 2014. <a href="http://cnx.org/content/m10706/latest/" target="_blank">http://cnx.org/content/m10706/latest/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>The multiplier 
  
    
      
        A
        (
        ω
        )
        
          e
          
            −
            i
            ω
            τ
          
        
      
    
    {\displaystyle A(\omega )e^{-i\omega \tau }}
  
, as a function of ω, is known as the filter's frequency response. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Selesnick, Ivan. "Four Types of Linear-Phase FIR Filters". Openstax CNX. Rice University. Retrieved 27 April 2014. <a href="http://cnx.org/content/m10706/latest/" target="_blank">http://cnx.org/content/m10706/latest/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Oppenheim, Alan V; Ronald W Schafer (1975). Digital Signal Processing (3 ed.). Prentice Hall. ISBN 0-13-214635-5. <a href="0-13-214635-5" target="_blank">0-13-214635-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Oppenheim, Alan V; Ronald W Schafer (1975). Digital Signal Processing (1 ed.). Prentice Hall. ISBN 0-13-214635-5. <a href="0-13-214635-5" target="_blank">0-13-214635-5</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Linear phase open-in-new

Linear phase