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filters - EE236A(Fall 2007-08 Lecture 6 FIR lter design FIR...

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EE236A (Fall 2007-08) Lecture 6 FIR filter design FIR filters linear phase filter design magnitude filter design equalizer design 6–1 FIR filters finite impulse response (FIR) filter: y ( t ) = n 1 summationdisplay τ =0 h τ u ( t τ ) , t Z u : Z R is input signal ; y : Z R is output signal h i R are called filter coefficients ; n is filter order or length filter frequency response: H : R C H ( ω ) = h 0 + h 1 e + · · · + h n 1 e j ( n 1) ω = n 1 summationdisplay t =0 h t cos j n 1 summationdisplay t =0 h t sin ( j = 1) periodic, conjugate symmetric, so only need to know/specify for 0 ω π FIR filter design problem: choose h so H and h satisfy/optimize specs FIR filter design 6–2
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example: (lowpass) FIR filter, order n = 21 impulse response h : 0 2 4 6 8 10 12 14 16 18 20 -0.2 -0.1 0 0.1 0.2 t h ( t ) frequency response magnitude | H ( ω ) | and phase negationslash H ( ω ) : 0 0.5 1 1.5 2 2.5 3 10 -3 10 -2 10 -1 10 0 10 1 ω | H ( ω ) | 0 0.5 1 1.5 2 2.5 3 -3 -2 -1 0 1 2 3 ω negationslash H ( ω ) FIR filter design 6–3 Linear phase filters suppose n = 2 N + 1 is odd and impulse response is symmetric about midpoint: h t = h n 1 t , t = 0 , . . . , n 1 then H ( ω ) = h 0 + h 1 e + · · · + h n 1 e j ( n 1) ω = e jNω (2 h 0 cos + 2 h 1 cos( N 1) ω + · · · + h N ) = e jNω tildewide H ( ω ) term e jNω represents N -sample delay tildewide H ( ω ) is real • | H ( ω ) | = | tildewide H ( ω ) | called linear phase filter ( negationslash H ( ω ) is linear except for jumps of ± π ) FIR filter design 6–4
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Lowpass filter specifications ω δ 1 1 1 δ 2 ω p ω s π specifications: maximum passband ripple ( ± 20log 10 δ 1 in dB): 1 1 ≤ | H ( ω ) | ≤ δ 1 , 0 ω ω p minimum stopband attenuation ( 20log 10 δ 2 in dB): | H ( ω ) | ≤ δ 2 , ω s ω π FIR filter design 6–5 Linear phase lowpass filter design sample frequency ( ω k = kπ/K , k = 1 , . . . , K ) can assume wlog tildewide H (0) > 0 , so ripple spec is 1 1 tildewide H ( ω k )
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