CDMA 1 CDMA ( ) CDMA CDMA CDMA 1 ( ) Hopfield [1] Hopfield 1 E-mail: okada@brain.riken.go.jp 1
1: 1 [] Hopfield Sourlas Hopfield [3] Sourlas 1? CDMA.1 DS/BPSK CDMA (Direct Sequence; DS) (Binary Phase-Shift-Keying; BPSK) [4, 5]
...... i η i = ±1 {ξ µ }{ξ µ } 1... {ξ µ } {n µ } η 1 η. η {y µ } : DS/BPSK Prob[η i = ±1] = 1 (i =1,,,). (1) ( ) p p p α = p O(1), () µ ξ µ i i Prob[ξ µ i = ±1] = 1, (i =1,,,, µ=1,,,p). (3) DS/BPSK µ i η i ξ µ i ξµ i η i 0 1/β 0 ) n µ (0, 1β0 (4) y µ = 1 ξ µ i η i + n µ, (5) n µ (5) 1/ y µ (5) 1 β 0 ( P (y µ β0 {η i })= π exp β 0 y µ 1 ) ξ µ i η i. (6) 3
{y µ } (µ =1,,,p) {ξ µ i } (i =1,,,, µ = 1,,,p) {η i } (i =1,,,) s i = ±1 (6) p({s i } {y µ 1 }) = Z(β 0 ) exp( β 0H(s)) (7) H(s) = 1 J ij s i s j h 0 i s i (8) Z(β 0 ) = s J ij = 1 h 0 i = i,j 1 exp( β 0 H(s)) (9) ξ µ i ξµ j (10) µ=1 ξ µ i yµ, µ=1 = αη i + 1 ξ µ i ξµ j η j + 1 ξ µ i nµ (11) µ=1 3 CDMA CDMA CDMA 3.1 n µ ξ µ 1 η 1 ξ µ. ξ µ. η η y µ 3: Ising 3 [6] µ ξ µ (3) ±1 Prob[ξ µ i = ±1] = 1 (1) 4
η i ξ µ (η i ξ µ i ) (4) 0 1/β 0 y µ (5) y µ = 1 ξ µ i η i + n µ. (13) p p (ξ µ,y µ ) η η η p (ξ µ,y µ ) η η η i ±1 DS/BPSK CDMA [7] 3. Hopfield CDMA Hopfield [1] Hopfield Hopfield 4 i s i 4: Hopfield s i =1 s i = 1 j i J ij J ij (10) Hopfield i i h i h i = J ij s j + h 0 i, (14) j i h 0 i i T β = 1 T, (15) 5
Glauber s i = ±1 Prob[s i = ±1] = 1 ± tanh(βh i), (16) CDMA (7) (11) 1/β 0 Prob[s i = ±1] = 1 ± tanh(β 0h i ), (17) h i = J ij s j + h 0 i, (18) j i (18) J ij h 0 i (10) (11) (14) (16) (17) (18) J ij Hopfield Hopfield h 0 i DS/BPSK CDMA ξ µ n µ h 0 i 4 {y µ } {ξ µ i } {η i} 4.1 ˆη CD i = sgn(h 0 i ) (19) h 0 i (11) h 0 i = αη i + 1 ξ µ i ξµ j η j + 1 ξ µ i nµ (0) µ=1 [8] (0) (19) ˆη CD i = η i (0) 6
4. {ξ µ i } MAP( ) (7) (11) p({s i } {y µ }) {s i } {η i } MAP ˆη MAP = arg min H(s), (1) s Tanaka MAP [9] (1)? (17) (18) 1/β 0 =0 (MCMC ) H(s) 0 H(s) (18) h i h i = αη i + 1 ξ µ i nµ + 1 µ=1 ξ µ i ξµ j η j 1 ξ µ i ξµ j s j () {η i } {s i } {η i } () 3 4 η i 4.3 ( ) y µ ^ 1 η 1 M.F ^ 1 η M.F ^ 1 η 3 M.F. ^ 1 η M.F s t a g e ^ η 1 ^ η ^ η 3. ^ η s t a g e 3..... s t a g e L ^ L η 1 ^ L η ^ L η 3. ^ L η M.F: matched filter 5: 7
MAP P [10] CDMA 5 [11, 1] 5 L ( ) 1 4.1 ˆη t+1 ˆη i 1 = sgn(h0 i ). (3) i = sgn(u t i) (4) u t i = h 0 i J ij ˆη j t (5) j i = αη i + 1 ξ µ i ξµ j ηj 1 ξ µ i ξµ j ˆηt j + 1 ξ µ i nµ (6) (6) t ˆη t i η i MCMC ˆη t i η i (17) (18) 0 (4) (6) t t ˆη t η ( ) M t = 1 µ=1 η iˆη i t, (7) t 1 P t b Pb t = 1 M t. (8) 5 ( ) 5.1 = 4000 6 α = P b E b / 0 E b 1 0 E b 0 = αβ 0, (9) 8
1 0.1 0.01 Pb 0.001 0.0001 1e-05 0 4 6 8 10 t 6: α = ( E b / 0 =4, 5, 6, 7, 8, 9[dB]) [9] 1 5. p α = p/ O(1) [9] (7) (8) t M t Pb t CDMA Hopfield CDMA [13, 14, 15] [16] (6) t M t 0 (3) M 0 M 1 = 1 η i ˆη i = 1 η i sgn(h 0 i ) (30) h 0 i = αη i + zi 0 (31) zi 0 = 1 ξ µ i ξµ j η j + 1 ξ µ i nµ (3) (3) z 0 i i 0 α(1+1/β 0) 9
( M 1 = erf α (1 + 1/β 0 ) ), (33) t>1 (4) (6) ˆη t+1 i = F (h 0 i j i J ij ˆη t j) = sgn(αη i + zi 0 zi t ) t 1 (34) zi t = 1 ξ µ i ξµ j ˆηt j (35) (35) ˆη t j ξµ i zt i [13, 14, 15] Amari-Maginu [13] Amari-Maginu z t i 1 {ξµ i } M t [17] 6 Amari-Maginu ˆη t i [14, 15] 1 [14, 15] 6 CDMA Amari-Magninu [13] CDMA 10
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[15] M. Okada, otions of associative memory and sparse coding, eural etworks, vol.9, no.9, pp. 149-1458 1996. [16] T. J. Richardson and R. L. Urbanke The capacity of low-density parity-check codes under message-passing decoding, IEEE Trans.Inform. Theory, vol.47, no., pp.599-618, Feb. 001. [17] CDMA, C001-174, 00. T. Tanaka, Density evolution for multistage CDMA multiuser detector, Proc. 00 Int. Symp. Inform. Theory, Lausanne, Switzerland, pp. 3, 00. 1