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HO3_315aSP09_filter_approx

HO3_315aSP09_filter_approx - Filter Approximations Boris...

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Filter Approximations Boris Murmann Stanford University [email protected] Copyright © 2009 by Boris Murmann B. Murmann 1 EE315A HO #3 Filter Approximation Problem Approximate ideal response by one that implements a reasonable compromise between filter complexity (number of poles and zeros) and approximation error B. Murmann 2 EE315A HO #3

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Second Order Lowpass Filter Q p =0.5 -10 0 e [dB] Q p =1/sqrt(2) Q p =1 1 H(s) = -20 Magnitude 2 2 1 P P P s s Q + + ω ω 10 4 10 5 10 6 -40 -30 Magnitude response is “maximally flat” (no peaking) for Q = 1/sqrt(2) f [Hz] B. Murmann 3 Q P = 1/sqrt(2) EE315A HO #3 Pole Positions 6 x 10 5 6 x 10 5 6 x 10 5 2 4 nary 2 4 nary 2 4 nary 1 Q -4 -2 0 Imagin -4 -2 0 Imagin 4 -2 0 Imagin 0 5 P Q . = 2 P = 1 P Q = -6 -4 -2 0 x 10 5 -6 Real -6 -4 -2 0 x 10 5 -6 Real -6 -4 -2 0 x 10 5 -6 -4 Real 1 1 0 2 P cos Q α = = D 1 1 45 2 P cos Q α = = D 1 1 60 2 P cos Q α = = D B. Murmann 4 EE315A HO #3