MIT6_079F09_lec09

MIT6_079F09_lec09 - Filter design FIR filters Chebychev...

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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 u ( t ) , t Z =0 (sequence) u : Z R is input signal (sequence) y : Z R is output signal h i are called filter coecients n is filter order or length Filter design 2 filter frequency response: H : R C H ( ) = h 0 + h 1 e i + + h n 1 e i ( n 1) n 1 n 1 = h t cos t + i h t sin t 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 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 ( ) ): | | 1 10 10 0 0.5 1 1.5 2 2.5 3 | H ( ) | 1 10 2 10 3 10 frequency response phase ( i.e. , H ( ) ): 3 2 H ( ) 0 0.5 1 1.5 2 2.5 3 3 2 1 0 1 Filter design 5 Chebychev design minimize max H ( ) H des ( ) [0 , ] | | 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 ( k ) H des ( k ) | sample points 1 < < m are fixed ( e.g. , k = k/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 k cos( n 1) k A ( k ) = sin k sin( n 1) k H des ( k ) b ( k ) = H des ( k ) h 0 ....
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MIT6_079F09_lec09 - Filter design FIR filters Chebychev...

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