Filter design and equalization - MIT OpenCourseWare

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Fourier transform of autocorrelation coefficients isn−1−iωτR(ω) = e r τ = r 0 + 2r t cos ωt = | H(ω)|τ t=12• always have R(ω)≥0 for all ω• can express magnitude specification asL(ω) 2 ≤ R(ω) ≤ U(ω) 2 ,ω ∈ [0,π]. . . convex in rFilter design 28
Spectral factorizationquestion: when is r ∈ R n the autocorrelation coefficients of some h ∈ R n ?answer: (spectral factorization theorem) if and only if R(ω) ≥ 0 for all ω• spectral factorization condition is convex in r• many algorithms for spectral factorization, i.e., finding an h s.t.R(ω) = |H(ω)| 2magnitude design via autocorrelation coefficients:• use r as variable (instead of h)• add spectral factorization condition R(ω) ≥ 0 for all ω• optimize over r• use spectral factorization to recover hFilter design 29
Page 2: FIR filtersfinite impulse response
Page 6 and 7: • h is optimization variableCheby
Page 8 and 9: Linear phase filterssuppose• n =
Page 10 and 11: Lowpass filter specificationsδ 11/
Page 12 and 13: Linear phase lowpass filter design
Page 14 and 15: example• linear phase filter, n =
Page 16 and 17: Equalizer designH(ω)G(ω)equalizat
Page 18 and 19: time-domain equalization: optimize
Page 20 and 21: Equalizer design exampleunequalized
Page 22 and 23: Chebychev equalizer design:minimize
Page 24 and 25: time-domain equalizer design:equali
Page 26 and 27: Filter magnitude specificationstran
Page 30 and 31: choose h to minimizelog-Chebychev m
Page 32: MIT OpenCourseWarehttp://ocw.mit.ed

Filter design and equalization - MIT OpenCourseWare

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