MIT6_079F09_lec09

MIT6_079F09_lec09 - Filter design FIR filters • •...

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Unformatted text preview: Filter design FIR filters • • Chebychev design • linear phase filter design • equalizer design • filter magnitude specifications 1 FIR filters finite impulse response (FIR) filter: n − 1 y ( t ) = h τ u ( t − τ ) , t ∈ Z τ =0 (sequence) u : Z R is input signal • → (sequence) y : Z R is output signal • → • h i are called filter coeﬃcients • n is filter order or length Filter design 2 filter frequency response: H : R C → H ( ω ) = h 0 + h 1 e − iω + + h n − 1 e − i ( n − 1) ω · · · n − 1 n − 1 = h t cos tω + i h t sin tω t =0 t =0 (EE tradition uses j = √ − 1 instead of i ) • • H is periodic and conjugate symmetric, so only need to know/specify for ≤ ω ≤ π FIR filter design problem: choose h so it and H satisfy/optimize specs Filter design 3 example: (lowpass) FIR filter, order n = 21 impulse response h : h ( t ) 0.2 0.1 0 −0.1 −0.2 0 2 4 6 8 10 12 14 16 18 20 t Filter design 4 frequency response magnitude ( i.e. , H ( ω ) ): | | 1 10 10 0 0.5 1 1.5 2 2.5 3 | H ( ω ) | −1 10 −2 10 −3 10 ω frequency response phase ( i.e. , H ( ω ) ): 3 2 H ( ω ) 0 0.5 1 1.5 2 2.5 3 −3 −2 −1 0 1 ω Filter design 5 Chebychev design minimize max H ( ω ) − H des ( ω ) ω ∈ [0 ,π ] | | • h is optimization variable H des : R C is (given) desired transfer function • → • convex problem • can add constraints, e.g. , | h i | ≤ 1 sample (discretize) frequency: minimize max k =1 ,...,m | H ( ω k ) − H des ( ω k ) | • sample points ≤ ω 1 < · · · < ω m ≤ π are fixed ( e.g. , ω k = kπ/m ) • m ≫ n (common rule-of-thumb: m = 15 n ) • yields approximation (relaxation) of problem above Filter design 6 Chebychev design via SOCP: minimize t subject to A ( k ) h − b ( k ) ≤ t, k = 1 , . . . , m where 1 cos ω k cos( n − 1) ω k A ( k ) = · · · − sin ω k − sin( n − 1) ω k · · · ℜ H des ( ω k ) b ( k ) = ℑ H des ( ω k ) h 0 ....
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MIT6_079F09_lec09 - Filter design FIR filters • •...

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