This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ECE600 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 ECE600 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 antisymmetry around L 1 2 . Often were interested in realvalued h [ n ] . In this case, GLP implies symmetry or antisymmetry around the point L 1 2 . 10 ECE600 Phil Schniter November 14, 2010 Generalized linear phase (cont.): There are some implications to symmetry and antisymmetry. 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. oddlength symmetric ( = 0 ) 5 0.5 0.5 1 2 40 20 20 h [ n ] H ( e j ) II. evenlength symmetric ( = 0 ) 5 0.5 0.5 1 2 40 20 20 h [ n ] H ( e j ) III. oddlength antisymmetric ( = 2 ) 5 1 1 2 40 20 20 h [ n ] H ( e j ) IV. evenlength antisymmetric ( = 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 ECE600 Phil Schniter November 14, 2010 Windowbased FIR design: To design a (causal) length L GLP filter h [ n ] , 1. Specify desired signedmagnitude response D ( ) R , 2. Compute desired L 1 2delayed 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 ] ....
View
Full
Document
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.
 Fall '08
 Clymer,B
 Digital Signal Processing, Signal Processing

Click to edit the document details