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Unformatted text preview: EE236A (Fall 200708) 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 + h 1 e − jω + ··· + h n − 1 e − j ( n − 1) ω = n − 1 summationdisplay t =0 h t cos tω − j n − 1 summationdisplay t =0 h t sin tω ( j = √ − 1) periodic, conjugate symmetric, so only need to know/specify for ≤ ω ≤ π FIR filter design problem: choose h so H and h satisfy/optimize specs FIR filter design 6–2 example: (lowpass) FIR filter, order n = 21 impulse response h : 2 4 6 8 10 12 14 16 18 200.20.1 0.1 0.2 t h ( t ) frequency response magnitude  H ( ω )  and phase negationslash H ( ω ) : 0.5 1 1.5 2 2.5 3 103 102 101 10 10 1 ω  H ( ω )  0.5 1 1.5 2 2.5 3321 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 + h 1 e − jω + ··· + h n − 1 e − j ( n − 1) ω = e − jNω (2 h cos Nω + 2 h 1 cos( N − 1) ω + ··· + h N ) = e − jNω tildewide H ( ω ) • term e − jNω represents Nsample 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 Lowpass filter specifications ω δ 1 1 /δ 1 δ 2 ω p ω s π specifications: • maximum passband ripple ( ± 20 log 10 δ 1 in dB): 1 /δ 1 ≤  H ( ω )  ≤ δ 1 , ≤ ω ≤ ω p • minimum stopband attenuation ( − 20 log 10 δ 2 in dB):  H ( ω )  ≤ δ 2 , ω s ≤ ω ≤ π...
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This note was uploaded on 09/26/2009 for the course CAAM 236 taught by Professor Dr.vandenber during the Spring '07 term at Monmouth IL.
 Spring '07
 Dr.Vandenber

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