filters3 - ECE-600 Phil Schniter November 14, 2010...

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Unformatted text preview: ECE-600 Phil Schniter November 14, 2010 Generalized linear phase: Recall that linear phase meant H ( e j ) = A ( ) e- jd for A ( ) R and d Z . We can relax this to generalized linear phase : H ( e j ) = A ( ) e- j ( d + ) for 2 d Z and { , 2 } while still retaining the desirable phase behavior. Interpretation: The group delay g ( )- d ( ) d = is the effective delay (in samples) caused by H ( e j ) at frequency . For GLP, ( ) =- d- , and so GLP H ( e j ) has g ( ) = d for all . Thus, GLP filters delay all frequency components equally! As we will now see, the GLP property implies a form of symmetry in the FIR impulse response h [ n ] . . . 9 ECE-600 Phil Schniter November 14, 2010 Generalized linear phase (cont.): Notice that, for any length- L (order L- 1 ) FIR filter, H ( e j ) = L- 1 n =0 h [ n ] e- jn = e- j L- 1 2 L- 1 n =0 h [ n ] e- j ( n- L- 1 2 ) = e- j L- 1 2 h [0] e j L- 1 2 + + h [ L- 1] e- j L- 1 2 . Since e j = cos( ) + j sin( ) and e- j = cos( )- j sin( ) , H ( e j ) = e- j L- 1 2 ( h [0] + h [ L- 1]) cos( L- 1 2 ) + j ( h [0]- h [ L- 1]) sin( L- 1 2 ) + ( h [1] + h [ L- 2]) cos( L- 3 2 ) + j ( h [1]- h [ L- 2]) sin( L- 3 2 ) + ... . For H ( e j ) = A ( ) e j ( d + ) with A ( ) R , we need: = 0 : n : h [ n ] + h [ L- 1- n ] R h [ n ]- h [ L- 1- n ] I h [ n ] = h * [ L- 1- n ] conjugate symmetry around L- 1 2 . = 2 : n : h [ n ] + h [ L- 1- n ] I h [ n ]- h [ L- 1- n ] R h [ n ] =- h * [ L- 1- n ] conjugate anti-symmetry around L- 1 2 . Often were interested in real-valued h [ n ] . In this case, GLP implies symmetry or anti-symmetry around the point L- 1 2 . 10 ECE-600 Phil Schniter November 14, 2010 Generalized linear phase (cont.): There are some implications to symmetry and anti-symmetry. Consider that DC gain: H ( e j ) = L- 1 n =0 h [ n ] , HF gain: H ( e j ) = L- 1 n =0 h [ n ](- 1) n . As a consequence, we have four different filter types : I. odd-length symmetric ( = 0 ) 5 0.5 0.5 1 2 40 20 20 h [ n ] H ( e j ) II. even-length symmetric ( = 0 ) 5 0.5 0.5 1 2 40 20 20 h [ n ] H ( e j ) III. odd-length anti-symmetric ( = 2 ) 5 1 1 2 40 20 20 h [ n ] H ( e j ) IV. even-length anti-symmetric ( = 2 ) 5 1 1 2 40 20 20 h [ n ] H ( e j ) Note: group delay = L- 1 2 whether length L is even or odd. 11 ECE-600 Phil Schniter November 14, 2010 Window-based FIR design: To design a (causal) length- L GLP filter h [ n ] , 1. Specify desired signed-magnitude response D ( ) R , 2. Compute desired L- 1 2-delayed impulse response: d [ n ] = F- 1 DTFT D ( ) e- j ( L- 1 2 + ) 3. Apply causal length- L window w [ n ] to d [ n ] : h [ n ] = w [ n ] d [ n ] ....
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This note was uploaded on 03/05/2011 for the course ECE 600 taught by Professor Clymer,b during the Fall '08 term at Ohio State.

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filters3 - ECE-600 Phil Schniter November 14, 2010...

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