LMS NLMS LMS Least Mean Square LMS Normalized LMS NLMS AD 3 1 h(n) y(n) d(n) FIR w(n) n = 0, 1,, N 1 N N =

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1 LMS NLMS RLS FIR IIR c /(13)

2 LMS NLMS LMS Least Mean Square LMS Normalized LMS NLMS AD 3 1 h(n) y(n) d(n) FIR w(n) n = 0, 1,, N 1 N N = LMS y(n) x(n) h(n) J c /(13)

3 J = E[e 2 (n)] (3 1) = E[{d(n) y(n)} 2 ] = E[d 2 (n) 2d(n)h T x(n) + {h T x(n)}{x T (n)h}] = E[d 2 (n)] 2h T E[d(n)x(n)] + h T E[x(n)x T (n)]h = R dd (0) 2h T P dx + h T R xx h E h = [h 0 h 1...h N 1 ] T x = [x(n)x(n 1)...x(n N + 1)] T R dd (0) d(n) P dx R xx e(n) J E J h (3 1) h Wiener h = R 1 xx P dx (3 2) 2 LMS Wiener 1 2 h(n) Wiener J min Steepest decent LMS for n ++ (3 3) y(n) = h(n)tx(n) e(n) = y(n) d(n) h(n + 1) = h(n) + µe(n)x(n) end h(n + 1) n + 1 µ LMS 2N 3 LMS LMS Wiener (3 3) c /(13)

4 E[e(n)x(n)] = E[{d(n) y(n)}x(n)] (3 4) = E[{d(n) x T (n)h}x(n)] = P dx R xx h LMS (3 2) Wiener 1 Wiener NLMS 1 LMS 1) LMS µ 0 < µ < 2 λ max (3 5) λ max x(n) R xx µ R xx R xx R xx tr[r xx ] = M λ k k=1 (3 6) λ k 0 λ max M λ k k=1 (3 7) x(n) σ 2 x tr[r] = E[x 2 (n)] + E[x 2 (n 1)] + + E[x 2 (n N + 1)] = Nσ 2 x (3 8) (3 5) 0 < µ < 2 Nσ 2 x (3 9) µ (3 5) x(n) 2 NLMS (3 5) LMS h(n + 1) = h(n) + αe(n)x(n)/nσ 2 x (3 10) c /(13)

5 0 < α < 2 LMS NLMS LMS 1) S. Haykin,,,, c /(13)

6 recursive least-squares algorithm RLS 3-1 LMS RLS RLS J(n) n J(n) = λ n i {d(i) w T (n)x(i)} 2 i=1 (3 11) d(i) x(i) w(n) x(i) w(n) x T (i) = [ x(i), x(i 1),..., x(i N + 1) ] (3 12) w T (n) = [ w 0 (n), w 1 (n),..., w N 1 (n) ] (3 13) λ 1 1 (3 11) J(n) ŵ(n) (3 14) R(n)ŵ(n) = θ(n) (3 14) n R(n) = λ n i x(i)x T (i) i=1 (3 15) n θ(n) = λ n i x(i)d(i) i=1 (3 16) (3 15) (3 16) (3 14) ŵ(n) RLS 1) [ RLS ] P(0) = δ 1 I, ŵ(0) = O (3 17) n = 1, 2,... g(n) = P(n 1)x(n) λ + x T (n)p(n 1)x(n) (3 18) c /(13)

7 ν(n) = d(n) ŵ T (n 1)x(n) (3 19) ŵ(n) = ŵ(n 1) + g(n)ν(n) (3 20) P(n) = λ 1 {P(n 1) g(n)x T (n)p(n 1)} (3 21) δ P(n) (3 22) P 1 (n) = R(n) + δλ n I (3 22) RLS (3 11) (3 23) 1) n J(n) = δλ n w(n) λ n i {d(i) w T (n)x(i)} 2 i=1 (3 23) RLS RLS N O(N 2 ) O(N) 1) 2) R(n) RLS 3) RLS 4) R(n) leaky RLS 1) A.H. Sayed and T. Kailath, A state-space approach to adaptive RLS filtering, IEEE Signal Process. Mag., vol.11, no.3, pp.18-60, ) J. Benesty and T. Gänsler, New insights into the RLS algorithm, EURASIP J. Appl. Signal Processing, vol.2004, issue 3, pp ) S.H. Leung and C.F. So, Gradient-based variable forgetting factor RLS algorithm in time-varying environments, IEEE Trans. Signal Processing, vol.53, no.8, pp , ) E. Horita, K. Sumiya, H. Urakami, and S. Mitsuishi, A leaky RLS algorithm: its optimality and implementation, IEEE Trans. Signal Processing, vol.52, no.10, pp , c /(13)

8 FIR 3 2 1) x(i) 3 2 DFT X k (i)(k = 0, 1,, N 1) i k N DFT X k (i) W k (i) IDFT y(i) DFT IDFT FFT FFT (N) 1/N 2) DFT FFT W k (b) = W k (b 1) 1 X k (b) 2 E k(b) X k (b) (3 24) b D k (b) E k (b) D k (b) W k (b)x k (b) c /(13)

9 N N 3 3 3) N x(i n) (n = 0, 1,..., N 1) 3 3 DFT DFT FFT X k (i) W k (i) IDFT d(i) IDFT DFT FFT 1 W k (i) = W k (i 1) 1 X k (i) 2 e(i)x k(i) (3 25) e(i) c /(13)

10 DFT DCT 1) M. Dentino, J. McCool, and B. Widrow, Adaptive Filtering in the Frequency Domain, Proc. of the IEEE, vol.67, no.12, pp , ),,, pp , 3) S.S. Narayan, A.M. Peterson, and M.J. Narasimha, Transform Domain LMS Algorithm, IEEE Trans. Acoustics, Speech, and Signal Processing, vol.31, no.3, pp , ) S. Haykin, Adaptive Filter Theory, pp.18-67, Prentice-Hall, Inc., ) W.K. Jenkins, A.W. Hull, J.C. Strait, B.A. Schnaufer, and X. Li, Advanced Concepts in Adaptive Signal Processing, pp , c /(13)

11 , 2, 3) 2 4, 5) K(z) 3 4 H(z) d(i) s(i) x(i) e(i) c /(13)

12 x(i) K(z) 3 6 d(i) x(i) H(z) C(z) y(i) e(i) = d(i) + y(i) C(z) C(z) K(z)/C(z) 2) LMS C(z) Filtered-X 1) B. Widrow and S.D. Stearns, Adaptive Signal Processing, pp , Prentice-Hall, Inc., c /(13)

13 2),, pp ,, ) S. Haykin, Adaptive Filter Theory, pp.18-67, Prentice-Hall, Inc., ) W.K. Jenkins, A.W. Hull, J.C. Strait, B.A. Schnaufer, and X. Li, Advanced Concepts in Adaptive Signal Processing, pp , ),, 2, pp , c /(13)

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