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filters3

# filters3 - ECE-600 Phil Schniter Generalized linear phase...

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ECE-600 Phil Schniter November 14, 2010 Generalized linear phase: Recall that linear phase meant H ( e j ω ) = A ( ω ) e - j ω d for A ( ω ) R and d Z . We can relax this to generalized linear phase : H ( e j ω ) = A ( ω ) e - j ( ω d + φ 0 ) for 2 d Z and φ 0 { 0 , π 2 } while still retaining the desirable phase behavior. Interpretation: The group delay g ( ω 0 ) - d φ ( ω ) d ω ω = ω 0 is the e ff ective delay (in samples) caused by H ( e j ω ) at frequency ω 0 . For GLP, φ ( ω ) = - ω d - φ 0 , and so GLP H ( e j ω ) has g ( ω 0 ) = d for all ω 0 . 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

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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 ω ) = L - 1 n =0 h [ n ] e - j ω n = e - j ω L - 1 2 L - 1 n =0 h [ n ] e - j ω ( n - L - 1 2 ) = e - j ω L - 1 2 h [0] e j ω L - 1 2 + · · · + h [ L - 1] e - j ω L - 1 2 . Since e j θ = cos( θ )+ j sin( θ ) and e - j θ = cos( θ ) - j sin( θ ) , H ( e j ω ) = e - j ω L - 1 2 ( h [0] + h [ L - 1]) cos( ω L - 1 2 ) + j ( h [0] - h [ L - 1]) sin( ω L - 1 2 ) + ( h [1] + h [ L - 2]) cos( ω L - 3 2 ) + j ( h [1] - h [ L - 2]) sin( ω L - 3 2 ) + . . . . For H ( e j ω ) = A ( ω ) e j ( ω d + φ 0 ) with A ( ω ) R , we need: φ 0 = 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 . φ 0 = π 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 we’re 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 0 ) = L - 1 n =0 h [ n ] , HF gain: H ( e j π ) = L - 1 n =0 h [ n ]( - 1) n . As a consequence, we have four di ff erent filter “ types ”: I. odd-length symmetric ( φ 0 = 0 ) 0 5 0.5 0 0.5 1 0 2 40 20 0 20 h [ n ] H ( e j ω ) II. even-length symmetric ( φ 0 = 0 ) 0 5 0.5 0 0.5 1 0 2 40 20 0 20 h [ n ] H ( e j ω ) III. odd-length anti-symmetric ( φ 0 = π 2 ) 0 5 1 0 1 0 2 40 20 0 20 h [ n ] H ( e j ω ) IV. even-length anti-symmetric ( φ 0 = π 2 ) 0 5 1 0 1 0 2 40 20 0 20 h [ n ] H ( e j ω ) Note: group delay = L - 1 2 whether length L is even or odd. 11

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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 ( ω ) R , 2. Compute desired L - 1 2 -delayed impulse response: d [ n ] = F - 1 DTFT D ( ω ) e - j ( ω L - 1 2 + φ 0 ) 3. Apply causal length- L window w [ n ] to d [ n ] : h [ n ] = w [ n ] d [ n ] . Recall: time-domain windowing freq-domain convolution. 5 0 5 10 15 20 0.1 0 0.1 0.2 0.3 2 0 2 0 0.5 1 5 0 5 10 15 20 0 0.5 1 2 0 2 5 0 5 10 15 5 0 5 10 15 20 0.1 0 0.1 0.2 0.3 2 0 2 0 0.5 1 desired delayed impulse response d [ n ] ( L =17) rectangular window w [ n ] filter impulse response h [ n ] desired signed-magnitude response D ( ω ) window signed-magnitude response filter signed-magnitude response 12
ECE-600 Phil Schniter November 14, 2010 Window-based FIR design (cont.): Through choice of window, we can trade o ff between: 1. transition-band width Δ ω = | ω p - ω s | (in rad/sample) 2. passband/stopband deviation (Note: δ p = δ s here!) Examples for several windows: window Δ ω / 2 π

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filters3 - ECE-600 Phil Schniter Generalized linear phase...

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