Transcription

1 修士論文の和文要旨 研究科 専攻大学院情報理工学研究科情報 通信工学専攻博士前期課程 氏名四ツ橋達彦学籍番号 論文題目 Lattice Reduction を用いた Sphere Detection の演算量削減法 要 旨 本研究は, Multiple-Input Multiple-Output(MIMO における無線通信の信号推定法を対象としている. MIMO とは複数の送受信アンテナを用いて独立なデータの送受信を行う技術である. MIMO は同一周波数帯を使用するため周波数利用効率をあげることができるが, 送信信号同士の干渉が起こり, 送信された信号が正しく推定できない場合がある. MIMO における従来の信号推定法として, ZF 法や MMSE 法や ML 法などがあり. それらの信号推定方法により信号を推定している. このうち, ML 法が最も良いビット誤り率 (BER 特性を示すが, 計算時間が膨大になってしまい, 実用的な方法とはいえない. そこで計算時間が短く,BER が ML 法に近い信号推定法が必要になる. 本研究では Lattice-Reduction(LR, Successive Interference Cancelation(SIC, Sphere Detection(SD を用い, BER の改善を行う. LR とは MIMO の技術の一つであり, 信号を準直交化させ信号の干渉を減らすことができる. さらに, LR は MMSE 法や ZF 法のような従来の信号推定方式と組み合わせることにより, ビット誤り率を良くすることができる. これらの理由により, LR は非常に有用性のあるものであると考えられる. SD とは, 通信路行列と 送信信号ベクトル候補から生成される複数レプリカが作る格子空間において, 受信信号点を中心とした初期半径の超球を仮定し, 超球内に格子点が見つかれば, 受信信号点からその格子点までの距離を新たな半径として, 新たな超球内で格子点が見つからなくなるまで探索を続ける方法である. この方法を用いることにより, ML 法の演算量を削減することが可能になる方法である. しかし, SD は送受信アンテナ数増加や, 変調方式が多値になるほど, 演算量が莫大なものとなってしまう欠点がある. 本稿ではまず始めに MIMO のシステムモデルと通信に用いる変調方式について紹介している. 次に MIMO 環境における従来の信号推定法について紹介している. その後, Lattice-Reduction(LR とその BER 特性について紹介している. その後, LR を用いて, 更に良好な BER 特性が得られる信号推定法を紹介している. そして最後に, 自身の提案法である信号推定法を紹介している. 提案法を用いることで, 送信アンテナ 8 本, 受信アンテナ 8 本での 8 8MIMO 通信において, QPSK 変調及び, 16QAM 変調, 64QAM 変調を用いた場合, ML 法と同様の BER 特性を維持しつつ, かつ SD よりも演算量を削減することが可能となる.

2 4 Lattice-Reduction Sphere detection

3 .,..,,.,, MIMO(Multi Input Multi Output. MIMO,,.,, (AWGN.,,,,,.,MIMO,ZF (Zero Forcing MMSE (Minimum Mean Square Error ML (Maximum Likelihood.., ML (BER,,., BER ML. LR SIC(Successive Interference Cancelation, BER. LR MIMO,., LR MMSE ZF,., LR. 1

4 1 MIMO (Multi Input Multi Output MIMO QPSK QAM QAM Offset-QPSK Offset-16QAM Offset-64QAM ZF MMSE ML Lattice-Reduction(LR LR Aided Detection (LRAD Forward&Backward LR (F&B LR F&B LR F&B LR

5 6 Seysen s Seysen s Seysen s Reciprocal Lattice-Reduction(LR LLL Reciprocal LR LLL SIC Ordering SIC SIC(Successive Interference Cancellation LR with SIC LR with Ordering SIC(OSIC Reciprocal Lattice Forward&Backward LR (F&B Reciprocal F&B Reciprocal BER Ordering SIC Forward&Backward LR(F&B OSIC F&B OSIC Reciprocal lattice F&B LR OSIC Reciprocal F&B LR OSIC(Reciprocal F&B OSIC M (N + M (N + M M M M M M M ŝ Shift and Scale Shift-Back and Scale-Back

6 LRAD F&B LR Seysen s LR Reciprocal LR LR with OSIC F&B Reciprocal F&B OSIC Recipeocal F&B OSIC Forward&Backward LR with OSIC ( F&B OSIC ( Sphere Detection (SD Sphere Detection(SD LR-with OSIC Sphere Detection(SD (

7 1 MIMO (Multi Input Multi Output 1.1 MIMO.MIMO(Multi Input Multi Output..,,. 1. MIMO MIMO 1.1. s. y H z, y = Hs + z. N, M, y. y N 1 = H N M s M 1 + z N 1 (1.1 H h h h h h1 h h3 h 4 = h h h h h h h h s s s = s s z 1 z z 3 z 4 y y y = y y : MIMO ( 4 4 5

8 y = Hs + z (1. : s M 1 = [s 1,..., s M ] T (1.3 : y N 1 = [y 1,..., y N ] T (1.4 AWGN : z N 1 = [z 1,..., z N ] T (1.5 h 1,1 h 1,M : H N M =..... (1.6 h N,1 h N,M MIMO,. y 1. = h 1,1 h 1,M..... s 1. + z 1. (1.7 y N h N,1 h N,M s M z N 1.3 MIMO.,., 4, 4 4 4MIMO, 6, 6 6 6MIMO, 8, 8 8 8MIMO. 6

9 .1,, 3, QPSK,16QAM,64QAM. LR. PHS. PSK(Phase Shift Keying :, QAM(Quadrature Amplitude Moduration : ASK(Amplitude Shift Keying :.,,,. 7

10 . QPSK QPSK.1. (01 + j ( ( 11 - j (10.1: QPSK QPSK. s m [±1 ± j], and C (sm [(00, (01, (11, (10] E s, N 0, γ E s = E [ s m ] = j j j + 1 j 4 = = (.1 E b = E s = = 1 (. E b N 0 = 1 N 0 : i.e., N 0 = 1 E b N 0 (.3 γ 1 = MN 0 E s = MN 0 E b = M E b N 0 (.4 8

11 .3 16QAM 16QAM.. (1000 ( 1100 (0100 ( j ( 1001 ( j (0101 ( ( 1011 ( 1111 j (0111 (0011 ( 1010 ( 1110 (0110 (0010 3j.: 16QAM 16QAM. s m = a + jb : [a b] [±1, ±3], and C (sm E s, N 0, γ E s = E [ s m ] (0000 (0001 (0011 (0010 (0100 (0101 (0111 (0110 (1100 (1101 (1111 (1110 (1000 (1001 (1011 (1010 = j j j j 4 = = 10 (.5 E b = E s 4 = 10 4 = 5 (.6 5 E b = : i.e., N 0 = 5 N 0 N 0 E b N 0 (.7 γ 1 = MN 0 E s = MN 0 4E b = M 4 E b N 0 (.8 9

12 .4 64QAM 64QAM.3. ( ( ( ( j ( ( ( ( ( ( ( ( j ( ( ( ( ( ( ( ( j ( ( ( ( ( ( ( j ( ( ( ( ( ( ( ( j ( ( ( ( ( ( ( ( ( j ( ( ( ( ( ( ( ( j ( ( ( ( ( ( ( ( j ( ( ( ( : 64QAM 64QAM. s m = a + jb : [a b] [±1, ±3, ±5, ±7], and C (sm ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

13 E s, N 0, γ E s = E [ s m ] = j j j j j j j j j j j j j j j j = 16 = 4 (.9 (.10 E b = E s 6 = 4 6 = 7 (.11 E b = 7 : i.e., N 0 = 7 N 0 N E 0 b N 0 (.1 γ 1 = MN 0 E s = MN 0 6E b = M 6 E b N 0 (.13 11

14 .5 Offset-QPSK Offset-QPSK.4. + j ( 01 (11 (00 ( : Offset-QPSK E s, N 0, γ E s = E [ s m (0.5 + j0.5 ] = 1 ( +1 + j + +1 j j + 1 j = 1 (.14 E b = E 5 s 4 = = 5 8 (.15 E b N 0 = 5 8 N 0 = 5 8N 0 : i.e., N 0 = 5 8 E b N 0 (.16 γ 1 = MN 0 E s = MN 0 8E b = M 16 E b N 0 (.17 1

15 .6 Offset-16QAM Offset-16QAM.5. (1000 ( 1100 (0100 ( j ( 1001 ( 1101 (0101 ( j ( 1011 ( 1111 (0111 ( j ( 1010 ( 1110 (0110 ( : Offset-16QAM E s, N 0, γ E s = E [ s m (1.5 + j1.5 ] = 1 ( +1 + j j j j = 5 (.18 E b = E 5 s 4 = 4 = 5 8 (.19 5 E b 8 = : i.e., N 0 = 5 N 0 N 0 8 E b N 0 (.0 γ 1 = MN 0 E s = M 16 E b N 0 (.1 13

16 .7 Offset-64QAM Offset-64QAM.6. ( ( ( ( ( ( ( ( j ( ( ( ( ( ( ( ( j ( ( ( ( ( ( ( ( j ( ( ( ( ( ( ( ( j ( ( ( ( ( ( ( ( j ( ( ( ( ( ( ( ( j ( ( ( ( ( ( ( ( j 0 ( ( ( ( ( ( ( ( : Offset-64QAM E s, N 0, γ E s = E [ s m (3.5 + j3.5 ] = 1 ( j j ( j j ( j j ( j j 4 = j j j j j j j j (. (.3 14

17 E b = E 1 s 6 = 6 = 7 4 (.4 7 E b 4 = = 7 : i.e., N 0 = 7 N 0 N 0 4N 0 4 E b N 0 (.5 γ 1 = MN 0 E s = M 4 E b N 0 (.6.8.,., MIMO N 0, γ 1 M. 15

18 3 3.1 MIMO Zero Forcing(ZF,Minimum Mean Square Error(MMSE,Maximum Likelifood(ML. 3. ZF ZF ZF H (pseudo inverse matrix H ZF ML. ZF. s (ZF = s (ZF 1. s (ZF M = WH y H y = H (Hs + z (3.1 ZF W H. W H H. H M M H = H 1, M N(N M H = ( H H H 1 H H. 3.3 MMSE MMSE MMSE γ H,. MMSE (3.. MMSE ZF,. W = arg min W E[ s WH y ] (3. 16

19 , H := [ H γ 1 I M ]. s (MMSE = s (MMSE 1. s (MMSE M = WH y = ( H H H + γ 1 I M 1 H H y = ( H H H + γ 1 I M 1 H H (Hs + z (3.3 γ 1 ( Es M AWGN 1 N 0 γ Es MN 0. (I M M M 3.4 ML ML ML (3.4 ŝ (ML = arg min s C (s [ y Hs ] (3.4 ML 3.5 M=N=4,M=N=8,, QPSK, 16QAM, 64QAM, ZF, MMSE, ML E b /N 0 BER. 17

20 3.1: QPSK (M=N=4 3.: QPSK (M=N=8 3.3: 16QAM (M=N=4 3.4: 16QAM (M=N=8 18

21 3.5: 64QAM (M=N=4 3.6: 64QAM (M=N=8 3.6 ZF MMSE BER,, M = N = 4 M = N = 8., MMSE., 16QAM,64QAM, ZF MMSE BER. ML,.,. 4 4MIMO 8 8MIMO,, BER. 19

22 4 Lattice-Reduction(LR Lattice-Reduction.,. Lattice Reduction,,. h s u h 黒丸で表した近傍点の位置に誤る確率が高い 送信点 (-1,1 (-1,1 (0,1 (-1,0 (0,0 (1,0 h 1 s 1 (-1,0 (0,0 (1,0 h u 1 1 (1,-1 (0,-1 (1,0,(-1,0 に誤る誤りシンボル数 1 (-1,1,(1,-1 に誤る誤りシンボル数 (1,-1 近傍点が (0,1(-1,0(0,-1(1,0 に変化する 4 つの近傍点いずれに誤った場合でも誤りシンボル数は 1, 1 BER(Bit Error Rate. Lattice-Reduction. 4. h h 1 h 1 h h 1 M =, H (= [h 1 h ] h 1 i 1 i 1 = h 1 h 1 (4.1 0

23 h i i 1 h i 1 h = h 1 h h 1 = ht 1 h h 1 (4. h h 1 (i 1 h i 1 = = = = ( h1 h h 1 ( h T 1 h h 1 ( h T 1 h h 1 ( h T 1 h h 1 i 1 i 1 h1 h 1 h 1 µ 1 h 1 (4.3 (4.3 µ 1 (4.4 µ 1 ht 1 h h 1 (4.4 µ 1 h h 1 h h 1 ( h = h µ 1 h 1 (4.5 h 1 h h h 1 h i i = h h (4.6 i h 1 i h 1 = h h 1 h = h T h 1 h (4.7 h 1 h (4.8 ( i h 1 i = ( h h 1 h i ( = h T h 1 h i ( = h T h 1 h h h ( = h T h 1 h h µ 1 h (4.8 1

24 (4.8 µ 1 (4.9 µ 1 h T h 1 h (4.9 h 1 µ 1 h h h 1 ( h 1 = h 1 µ 1 h ( LR LLL 4.1. h µ h 1 1 h 1 h 1 h h h 1 1 h 1 h 1 1 h 1 h 1 µ 1 h 4.1: LLL

25 Algorithm 1 4.1:LLL Algorithm 1: Input H = [h 1...h M ],T = I M = [t 1...t M ] : set p, δ = 3/4, ĥ1 = h 1 3: for p = 0 to M do 4: for q = p 1 to 1 do 5: Calculate µ p,q = ĥh q h p ĥ q 6: h p h p µ p,q h q (LR 7: t p t p µ p,q t q 8: end for 9: Let ĥp = h p 10: for q = p 1 to 1 do 11: Calculate µ p,q = ĥh q h p ĥq 1: ĥ p ĥp µ p,q ĥ q (GS 13: end for 14: if δ ĥp 1 > ĥp + µ p,p 1 ĥ p 1 then 15: swap the (p 1-th columns h p 1,t p 1 and the p-th columns h p,t p of H and 16: end if T,respectively. 17: p max(p 1, 18: if p = then 19: let ĥ1 h 0: go to 4: 1: else : p p + 1 3: go to 4: 4: end if 5: end for 6: Output H H = [h 1,..., h M ],T = [t 1,..., t M ] 3

26 4.3 LR Aided Detection (LRAD. LLL H.,, T. T. LR. y = Hs + z = (HT(T 1 s + z = H v + z (4.11, H HT, v T 1 s. (, T, H, H., LLL T. LRAD MMSE,., T 1 s (MMSE ṽ (MMSE. s (MMSE = H ȳ = ( H H H 1 H H ȳ (4.1 ṽ (MMSE = T 1 s (MMSE (4.13, ṽ (MMSE ˆv (MMSE. ˆv (MMSE = Q[ṽ (MMSE ] (4.14, ˆv (MMSE T ŝ (MMSE. 4.. ŝ = Tˆv (MMSE ( LR M=N=4, M=N=8 QPSK, 16QAM, 64QAM, ZF, MMSE, ML E b /N 0, BER. 4

27 送信信号 s y = Hs + z ~ s の推定 送信側 受信側 通信路 H L L Lアルゴリズム H, T を生成 AWGN T - 1 の vɶ T に変換 vɶ = 1 vɶ T sɶ 量子化 vˆ = Q [ ] ŝ に変換 sˆ = T vˆ vɶ 誤り判定 4.: LR 5

28 4.3: LRAD QPSK (M=N=4 4.4: LRAD QPSK (M=N=8 4.5: LRAD 16QAM (M=N=4 4.6: LRAD 16QAM (M=N=8 4.7: LRAD 64QAM (M=N=4 4.8: LRAD 64QAM (M=N=8 6

29 4.5 Lattice-Reduction BER., ML,.,ZF MMSE BER, ML Lattice Reduction MMSE,. 7

30 5 Forward&Backward LR (F&B LR 5.1 LLL LR BER. LLL, BER F&B LR. 5. F&B LR LLL H BER., 16QAM, M = N = 8 BER 5.1. BER. LLL H,,.F&B LR F&B LR F&B LR. s (MMSE = H ȳ = ( H H H 1 H H ȳ (5.1 ṽ = T (j 1 s (MMSE : j [1, ] (5. ˆv (j = Q[ṽ (j ] (5.3 ŝ (j = T (jˆv (j (5.4 ŝ = arg min j [1,] y Hŝ(j (5.5 (5.5, j, ŝ., LLL T T (j =1, LLL T T (j =. 8

31 LR LLL MMSE 16QAM 0dB (M=8 N=8 BER Transmitter Signal 5.1: BER(M = N = 8, 16QAM, E b /N 0 =17dB 入力 H, sɶ 昇順 -LR:T (1 (1 (1 1 vɶ = ( T sɶ (1 (1 ˆ [ vɶ ] v =Q = (1 (1 (1 sˆ T vˆ 準直交化空間のに vɶ を量子化する vˆ を s空間に戻す vɶ 降順 -LR:T ( ( ( 1 vɶ = ( T sɶ v =Q ( ( ˆ [ vɶ ] = ( ( ( sˆ T vˆ sˆ = argmin y Hsˆ j j {1,} ( j y Hs ( ˆ の中で最も小さい ŝ を判定信号とする 5.: F&B 9

32 Algorithm 5.1:Backward LLL Algorithm 1: Input H = [h 1,..., h M ],T = I M = [t 1,..., t M ] : set p M 1, δ = 3/4, ĥm = h M 3: for p = M 1 to 1 do 4: for q = p + 1 to M do 5: Calculate µ p,q = ĥh q h p ĥ q 6: h p h p µ p,q h q (LR 7: t p t p µ p,q t q 8: end for 9: Let ĥp = h p 10: for q = p + 1 to M do 11: Calculate µ p,q = ĥh q h p ĥ q 1: ĥ p ĥp µ p,q ĥ q (GS 13: end for 14: if δ ĥp+1 ĥp + µ p,p+1 ĥ p+1 then 15: p p 1 16: else if δ ĥp+1 > ĥp + µ p,p+1 ĥ p+1 then 17: swap the (p + 1-th columns h p+1,t p+1 and the p-th columns h p,t p of H and T,respectively. 18: p min(p + 1, M 1 19: if p M then 0: goto (5 1: else if p > M then : ĥ M h M 3: end if 4: end if 5: H H = [h 1,..., h M ], T = [t 1,..., t M ] 6: end for 30

33 5.4 M=N=4, M=N=6,M=N=8,. QPSK, 16QAM, 64QAM, MMSE E b /N 0, BER. 5.5,F&B LLL LR BER 3[dB]BER. BER., LLL T (j=1 LLL T (j= ŝ,. F&B LR., ML BER BER. 31

34 5.3: F&B LR QPSK (M=N=4 5.4: F&B LR QPSK (M=N=6 5.5: F&B LR QPSK (M=N=8 5.6: F&B LR 16QAM (M=N=4 5.7: F&B LR 16QAM (M=N=6 5.8: F&B LR 16QAM (M=N=8 3

35 5.9: F&B LR 64QAM (M=N=4 5.10: F&B LR 64QAM (M=N=6 5.11: F&B LR 64QAM (M=N=8 33

36 6 Seysen s 6.1 Seysen s LLL H. Seysen s. 6. Seysen s Seysen s. Seysen s H, H. H. (1 Lattice H = [h 1 h M ] (6.1 [ ] { (H DualLattice H = h 1 h M = H H } 1 H H H = (H H = (H H (6. T := [t 1 t M ] = I M (6.3,, G G. ( (3 G H H H (6.4 G H H H = ( H H H 1 (6.5 (6.6 Set H := H (6.7 H := H (6.8 34

37 (4 λ i,j = x i,j (6.9 ( G j,i with x i,j 1 G i,i G j,i G j,j = 1 ( h H j h i hh h j hi i h j for all (i, j(i j (6.10 G i,j G i j., G i,j G i j. (5 ( ( i,j = S H S H i,j = h i h i + h j h j h i h i h j h j ( = G j,j G i,i R{λ i,jx i,j } λ i,j for all(i, j, i j (6.11 where h i = h i + λ i,j hj (6.1 h j = h j λ i,j h i (6.13 (6 T = I, λ i,j (i, j 0 (i.e.,λ i,j = 0 for all (i, j. 1. (6.15,(s, t. ( ( (s, t = arg max i,j with i,j = S H S H i,j (i,j (6.14 = G j,j G i,i ( R{λ i,j x i,j } λ i,j (6.15. G G. H ] s (i.e., H = [ h 1 h h s 1 h s h s+1 h M with h i = h i + λ i,j h j,, G. G s,j = G s,s = H ( hs + λ s,t ht hj = ( hh s hj + λ s,t hh t hj = G s,j + λ s,t G t,j : j s (6.16 h H s (6.17 G j,s = G s,j = h T s h j = h H j h s (

38 ,G. And G t,j = G t,t = G j,t = G H j,t H ( h t λ s,t h ( hh s h j = t h j λ s,t h H h s j = G t,j + λ s,tg a,j : j s (6.19 h H s (6.0 (6.1 h s = h s + λ s,t ht (6. T s = t s + λ s,t t t (6.3 h t = h t λ s,t h s (6.4 ] H s,t = [ h 1 h h s 1 h s h s+1 h M (6.5 T s,t = [ t 1 t t s 1 t ] st s+1, t M (6.6 [ H s,t = h 1 h h t 1h h t t+1 h ] M (6.7 (7 λ i,j i = j (i, j 0. (8 ((7,( Seysen s LLL H. M=N=4, M=N=6, M=N=8 QPSK,16QAM,64QAM, MMSE

39 6.1: Seysen s LR QPSK (M=N=4 6.: Seysen s LR QPSK (M=N=6 6.3: Seysen s LR QPSK (M=N=8 6.4: Seysen s LR 16QAM (M=N=4 6.5: Seysen s LR 16QAM (M=N=6 6.6: Seysen s LR 16QAM (M=N=8 37

40 6.7: Seysen s LR 64QAM (M=N=4 6.8: Seysen s LR 64QAM (M=N=6 6.9: Seysen s LR 64QAM (M=N=8 38

41 6.4 M=N=4 LLL LR Seysen s LR 0.[dB] BER. M=N=6 LLL LR Seysen s LR 1.3[dB] BER. M=N=8 LLL LR Seysen s LR.5[dB] BER. Seysen s LLL,,, Seysen s., Seysen s, Seysen s Measure. Seysen s Measure H H, LLL,., BER. 39

42 7 Reciprocal Lattice-Reduction(LR LLL 7.1 LLL H H = [h 1 h M ].,,, H H 1., H = (H H = (H H LLL,, H T [ ( ]., H H [ (T ] H Reciprocal LR LLL H Reciprocal Lattice. H ( H H = H ( H H H 1 (7.1 (1LLL (LLL H = [ ] h 1 h M, T := I M (7. ĥ 1 = h 1 (7.3 H = ] [ ] [h 1 h M, T = t 1 t M (7.4. H = H T (7.5 40

43 (7.5. H H = ( H T H = T H H H = T H { ( H H } H = T H H (7.6 (7.6. (H H = (T H H = H (T H = H (T H 1 (7.7,T H. ( T H = ( T H 1. (7.7. ( y = Hs + z = H T H 1 ( T H s + z = H H v + z (7.8 ( H H ( = H T H 1, v = T H s. (7.8 ZF,MMSE. s (ZF = H y = ( H H H 1 H H y (7.9 s (MMSE = { (H H H + γ 1 I M 1 H H } y (7.10 s ŝ. ṽ = T H s (7.11 ˆv = Q [ṽ] (7.1 ( ŝ = T H 1 ˆv ( H Reciprocal Lattice LLL. M=N=4, M=N=6, M=N=8 QPSK,16QAM,64QAM, MMSE

44 7.1: Reciprocal LR QPSK (M=N=4 7.: Reciprocal LR QPSK (M=N=6 7.3: Reciprocal LR QPSK (M=N=8 7.4: Reciprocal LR 16QAM (M=N=4 4

45 7.5: Reciprocal LR 16QAM (M=N=6 7.6: Reciprocal LR 16QAM (M=N=8 7.7: Reciprocal LR 64QAM (M=N=4 7.8: Reciprocal LR 64QAM (M=N=6 43

46 7.9: Reciprocal LR 64QAM (M=N=8 7.4, Reciprocal LR LR BER=10 5, M=N=4 BER 0.3[dB], M=N=8 BER 3[dB]. Reciprocal LR BER. Seysen s,., Reciprocal,. F&B. 44

47 8 SIC Ordering SIC 8.1 Successive Interference Cancellation(SIC, LR, Ordering SIC(OSIC. 8. SIC(Successive Interference Cancellation SIC, M N MIMO (M 1 N, (M N.,, BER. 45

48 , ZF SIC. M = N = , 4 4 MIMO. H h h h h h h h h = h h h h h h h h s = s s s s z 1 z z 3 y y 1 y = y 3 y4 z 4 8.1: 4 4 MIMO 4 s 4 ZF. s = (H H H 1 H H y = [s 1, s, s 3, s 4 ] T (8.1, ZF s 4, y H. y (1 = y h 4 s 4 (8. = [h 1, h, h 3 ] T [s (1 1, s(1, s(1 3 ]T (8.3 = H (1 s (1 + z (8.4 (8.5 H (1 [h 1, h, h 3 ] T s (1 [s (1 1, s(1, s(1 3 ]T (8.6 y y (1,. 46

49 8., 3 4 MIMO. H ( = h h h h h h h h h h h h s s (1 1 (1 (1 = s s (1 3 z 1 z z 3 z 4 y y (1 1 y (1 (1 = (1 y 3 y (1 4 8.: 3 4 MIMO, 3 4 MIMO, 3 s (1 3 ZF. s (1 = (H (1H H (1 1 H (1H y (1 = [s (1 1, s(1, s(1 3 ]T (8.7, ZF s (1 3, y (1 H (1. y ( = y (1 h 3 s (1 3 (8.8 = [h 1, h ] T [s ( 1, s( ]T (8.9 = H ( s ( + z (8.10 (8.11 H ( [h 1, h ] T s ( [s ( 1, s( ]T (8.1 (

50 8.3, 4 MIMO. H ( h h h h = h h h h s ( ( s 1 = ( s z 1 z z 3 z 4 y y ( 1 y ( ( = ( y 3 y ( 4 8.3: 4 MIMO, 4 MIMO, s ( ZF. s ( = (H (H H ( 1 H (H y ( = [s ( 1, s( ]T (8.14, ZF s (, y ( H (. y (3 = y ( h s ( (8.15 = [h 1 ][s (3 1 ] (8.16 = H (3 s (3 + z (8.17 (8.18. H (3 [h 1 ] s (3 [s (3 1 ] (8.19 s = [s (3 1, s(, s(1 3, s 4] T (8.0, SIC. 48

51 8.3 LR with SIC SIC, s m y m 1 = y m h m s m. LR SIC. H 4 H 4. LR. y 4 = 1 y + ( P 1H 1 + j 1 + j 1 + j 1 + j (8.1 H 4 ṽ 4. ṽ 4 = (H H 4 H 4 1 H H 4 y 4 = W H 4 y 4 = [w 4,1, w 4, w 4,3 w 4,4 ] H y 4 (8. ṽ 4,4 = w4,4 H y 4. s v. v 4 ˆv 4,4., y 4, ˆv 4,4 y 3. y 3 = y 4 h 4,4ˆv 4,4 = [h 3,1, h 3,, h 3,3][v 3,1, v 3,, v 3,3 ] T + 1 T 1 z = H 3v T 1 z ( MIMO LR. ṽ 3 = (H H 3 H 3 1 H H 3 y 3 = W H 3 y 3 ṽ 3,3,. = [w 3,1, w 3,, w 3,3 ] H y 3 (8.4 ṽ 3,3 = w H 3,3y 3 (8.5, ṽ 3,3, ˆv 3,3. s v. v 3 ˆv 3,3., y 3, ˆv 3,3 y. 49

52 y = y 3 h 3,3ˆv 3,3 = [h,1, h,][v,1, v, ] T + 1 T 1 z = H v + 1 T 1 z (8.6 4 MIMO LR. ṽ = (H H H 1 H H y = W H y ṽ,,. = [w,1, w, ] H y (8.7 ṽ, = w H,y (8.8, ṽ,, ˆv,. s v. v ˆv,., y, ˆv, y 1. y 1 = y h,ˆv, = [h 1,1][v 1,1 ] + 1 T 1 z 1 4 MIMO LR. ṽ 1,1,. = H 1v T 1 z (8.9 ṽ 1,1 = w H 1,1y 1 (8.30, ṽ 1,1, ˆv 1,1. ˆv = [û 1,1, û,, û 3,3, û 4,4 ]. s = Tˆv ( P j 1 + j. 1 + j (8.31 ŝ, ŝ., LR with SIC. 8.5 SIC. 50

53 入力 H, y, m = M v H ɶ = w y,, v m m m m m m m = Q vɶ m m ˆ [ ],, if ( m > 1 NO m 1 m m, m m, m [,,, ] m 1 m 1,1 m 1, m 1, m 1 v v v m 1 ˆ [ ˆ,, ˆ,, ˆ ] m = y y h = H h h h v = = vɶ YES 1,1 m, m M, M 出力 = P + sˆ Tvˆ ( 1(1 1 j 8.4: SIC 51

54 8.4 LR with Ordering SIC(OSIC Ordering SIC, SIC BER BER., ˆv m.,.,. (H H H 1 ˆv m., (H H H 1 ˆv m SIC. ˆv.,,. y, H, s, z. s = [Re(s T Im(s T ] T (M 1 (8.3 y = [Re(y T Im(y T ] T (M 1 (8.33 z = [Re(z T Im(z T ] T (M 1 (8.34 [ ] H Re(H Im(H = (N M (8.35 Im(H Re(H M = N = 4, M = N = 6, M = N = 8 QPSK, 16QAM 64QAM E b /N 0 BER. 5

55 8.5: LR with OSIC QPSK (M=N=4 8.6: LR with OSIC QPSK (M=N=6 8.7: LR with OSIC QPSK (M=N=8 8.8: LR with OSIC 16QAM (M=N=4 53

56 8.9: LR with OSIC 16QAM (M=N=6 8.10: LR with OSIC 16QAM (M=N=8 8.11: LR with OSIC 64QAM (M=N=4 8.1: LR with OSIC 64QAM (M=N=6 54

57 8.13: LR with OSIC 64QAM (M=N=8 8.5 LR with OSIC, LRAD BER = 10 5, 5dB. 8 8MIMO 16QAM 64QAM, BER., SIC BER. SIC,, BER. 55

58 9 Reciprocal Lattice Forward&Backward LR (F&B Reciprocal Reciprocal Lattice 5 F&B LR F&B Reciprocal H ( H H = H ( H H H 1 (9.1 Reciprocal (9.1 H LLL., H LLL. LLL H (1, T (1, LLL H (, T (, (9., (9.3. LLL H (1,T (1, LLL H (, T (, (9.4, (9.5. H (1 = H ( = [h 1 h M ] (9. T (1 = T ( := I M (9.3 H (1 = [h 1 (1 h M (1 ] T (1 = [t 1 (1 t M (1 ] (9.4 H ( = [h 1 ( h M ( ] T ( = [t 1 ( t M ( ] (9.5 56

59 7 Reciprocal, (9.6. H (1 = H (1 T (1 H ( = H ( T ( (9.6 H H (1 = ( = T H (1 ( H H ( = = T H ( H (1 T (1 H = T H { ( H H } H H ( T ( H = T H { ( H H } H (9.7, (9.8 pseudo-inverse. ( H H (1 ( H H ( = ( = ( T H (1 H = H ( T H ( H = H ( T H (1 T H ( (1 H H 1 = T H (1 H (9.7 ( H H = T H ( H (9.8 = H ( = H ( (9.9, (9.10,. 1 T(1 H (9.9 T H ( 1 (9.10, y = Hs + z = H(T H (1 1 T H (1 s + z y = Hs + z = H(T H ( 1 T H ( s + z = (H H (1 v + z (9.11 = (H H ( v + z (9.1 (H H = H (T H 1 v = T H s (9.13 (9.11, (9.1. s (MMSE = H ȳ = ( H H H 1 Hȳ (9.14 ṽ (j = T (j H s (MMSE (9.15 ˆv (j = Q[ṽ (j ] (9.16 ŝ (j = (T (j H 1ˆv (j (9.17 ŝ = arg min j [1,] y Hŝ(j (

60 9.3 QPSK, 16QAM, 64QAM, 4 4MIMO, 6 6MIMO, 8 8MIMO F&B Reciprocal LRAD.,. CSI. 0, 1. 0, N 0 [W/Hz]. 9.4 BER F&B Reciprocal LRAD 4 4MIMO,6 6MIMO,8 8MIMO QPSK, 16QAM, 64QAM BER. 58

61 9.1: F&B Reciprocal QPSK (M=N=4 9.: F&B Reciprocal QPSK (M=N=6 9.3: F&B Reciprocal QPSK (M=N=8 9.4: F&B Reciprocal 16QAM (M=N=4 59

62 9.5: F&B Reciprocal 16QAM (M=N=6 9.6: F&B Reciprocal 16QAM (M=N=8 9.7: F&B Reciprocal 64QAM (M=N=4 9.8: F&B Reciprocal 64QAM (M=N=6 60

63 9.9: F&B Reciprocal 64QAM (M=N=8 9.5, F&B Reciprocal LRAD BER= MIMO 4dB BER. Reciprocal LR BER F&B Reciprocal LRAD BER BER= MIMO 1dB. LLL s (j=1 LLL s (j=,., Reciprocal LRAD F&B LRAD BER, BER. 61

64 10 Ordering SIC Forward&Backward LR(F&B OSIC Ordering-SIC 5 F&B LR F&B OSIC,., F&B LRAD, Lattice Reduction H (1, T (1, Lattice Reduction H (, T (,, (9.5. (9,1 (9,5, F&B LRAD. s (MMSE = H ȳ = ( H H H 1 H H ȳ (10.1 ṽ = T (j 1 s (MMSE : j 1, (10. ˆv = Q[ṽ (j ] (10.3 ŝ (j = T (jˆv (j (10.4 ŝ = arg min j 1, [ y Hŝ(j ] (10.5 F&B LRAD H (1,, H ( Ordering-SIC., (H (1H H (1 1, (H (H H ( 1,, ˆv m (1,ˆv m (,, SIC, [ˆv (1 m 1 ˆv(1 1 ],[ˆv( m 1 ˆv( 1 ] ˆv. 6

65 ,. ŝ (1 = T (1ˆv (1 (10.6 ŝ ( = T (ˆv ( (10.7 arg min i (1, y Hŝ(i ( M = N = 4, M = N = 6, M = N = 8, QPSK, 16QAM 64QAM E b /N 0 BER. 入力 H, sɶ Forward-LR: H (1 (1, T Backward-LR: H ( (, T Ordering : (1 H o Ordering : ( H o QR 分解 : (1 R QR 分解 : ( R SIC : (1 ˆv SIC : ( ˆv = (1 (1 (1 sˆ T vˆ = ( ( ( sˆ T vˆ s y Hs ˆ = argmin ˆ j j {1,} ( 10.1: 63

66 10.: F&B OSIC QPSK (M=N=4 10.3: F&B OSIC QPSK (M=N=6 10.4: F&B OSIC QPSK (M=N=8 10.5: F&B OSIC 16QAM (M=N=4 64

67 10.6: F&B OSIC 16QAM (M=N=6 10.7: F&B OSIC 16QAM (M=N=8 10.8: F&B OSIC 64QAM (M=N=4 10.9: F&B OSIC 64QAM (M=N=6 65

68 10.10: F&B OSIC 64QAM (M=N= F&B OSIC 4 4MIMO BER 0.dB, MIMO BER. F&B LR MIMO. 8 8MIMO 1dB. F&B LR OSIC BER.,,,. 66

69 11 Reciprocal lattice F&B LR OSIC Reciprocal Lattice 5 F&B LR., 8 Ordering-SIC, BER Reciprocal F&B LR OSIC(Reciprocal F&B OSIC 7 Reciprocal. H ( H H = H ( H H H 1 (11.1 9, (11.1 H LLL,, LLL. LLL H (1, T (1, LLL H (, T (., (11., (11.3. H (1 = H ( = [h 1 h M ] (11. T (1 = T ( := I M (11.3 LLL H (1,T (1, LLL H (, T (., (11.4, (11.5. H (1 = [h 1 (1 h M (1 ] T (1 = [t 1 (1 t M (1 ] (11.4 H ( = [h 1 ( h M ( ] T ( = [t 1 ( t M ( ] (

70 7 Reciprocal, (11.6. H (1 = H (1 T (1 H ( = H ( T ( (11.6 H H (1 = ( = T H (1 ( H H ( = = T H ( H (1 T (1 H = T H { ( H H } H H ( T ( H = T H { ( H H } H (1 H H (1 = T H (1 H (11.7 ( H H ( (11.7, (11.8 pseudo-inverse. ( ( H H (1 = T H (1 H ( ( = H T H (1 = H ( ( H H ( = T H ( H ( = H T H ( (11.9, (11.10,. = T H ( H ( T(1 H (11.9 = H ( T H ( 1 (11.10, y = Hs + z = H(T H (1 1 T H (1 s + z y = Hs + z = H(T H ( 1 T H ( s + z (H H = H (T H 1 = (H H (1 v + z (11.11 = (H H ( v + z (11.1 v = T H s ( H (1, H ( Ordering-SIC., (H H (1 H (1 1, (H H ( H ( 1, ˆv m (1,ˆv m (, SIC, [ˆv (1 m 1 ˆv(1 1 ],[ˆv( m 1 ˆv( 1 ] ˆv. ŝ (1 = (T H (1 1ˆv (1 (11.14 ŝ ( = (T H ( 1ˆv ( (11.15,. arg min j (1, y Hŝ (j (

71 11.1. M = N = 4, M = N = 6, M = N = 8 QPSK, 16QAM 64QAM E b /N 0 BER. # 入力 H, sɶ H H H H # H 1 H H [( ] 昇順 -LR: H T # #, (1 (1 降順 -LR: H T # #, ( ( Ordering : H QR 分解 : R # (1 # (1 Ordering : H QR 分解 : R # ( # ( SIC : (1 ˆv SIC : ( ˆv sˆ (1 # (1 =T vˆ (1 ( # ( sˆ =T vˆ ( s y Hs ˆ = argmin ˆ j j {1,} ( 11.1: 69

72 11.: Reciprocal F&B OSIC QPSK 11.3: Reciprocal F&B OSIC QPSK (M=N=4 (M=N=6 11.4: Reciprocal F&B OSIC QPSK 11.5: Reciprocal F&B OSIC 16QAM (M=N=8 (M=N=4 70

73 11.6: Reciprocal F&B OSIC 16QAM 11.7: Reciprocal F&B OSIC 16QAM (M=N=6 (M=N=8 11.8: Reciprocal F&B OSIC 64QAM 11.9: Reciprocal F&B OSIC 64QAM (M=N=4 (M=N=6 71

74 11.10: Reciprocal F&B OSIC 64QAM (M=N= Reciprocal F&B OSIC, 10 F&B LR OSIC BER.,, BER. 7

75 1 1.1, BER. BER.., BER,.,.,, H (T Flop. Flop Floating point operations. x y x+y, x-y, xy, x/y 1[Flop] M (N + M (N + M 1 A B. a 1,1 a 1,(N +M A =..... a M,1 a M,(N +M, B = b 1. b (N +M, a 1,1 a 1,(N +M b 1 b (N +M, (1.1 73

76 (N + M (N + M 1, 4(N + M 1[Flops]., A. A M, 4(N + M 1 M [Flops]., 8M (N + M M [Flops]. ( s = H ȳ, ṽ = H ȳ, ȳ = Q T ȳ 1.. M M M 1 A = a 1,1 a 1,M..... a M,1 a M,M, B = b 1. b M (1., a 1,1 a 1,M b 1 b M, M M 1, 4M 1[Flops]., A, A M, (4M 1 M [Flops]., 8M M [Flops]. (ṽ = T 1 s, ŝ = Tˆv, ṽ = R 1 ȳ, s = R 1 ȳ 1..3 M M M 1 A B. A = a 1,1 a 1, a 1,M 0 a, a,m , B = b 1. b M ( a M,M, a 1,1 a 1,M b 1 b M, 74

77 M M 1, 4M 1[Flops].,, a, a,m b b M, M 1 M. 4M 3[Flops]. A,.. ( SIC M m=1 (m 1 = [ 1 M (M + 1] M = 4M [F lops] ( ŝ ŝ = arg min s C s[ y Hs ] (1.5 Hs 8MN N [Flops]., y Hs N [Flops]., y Hs. y Hs N,. (y hs (y hs N, N N 1, 4N 1[Flops]. (8MN + 4N 1[Flops] Shift and Scale Shift-Back and Scale-Back Shift and Scale, Shift-Back and Scale-Back.,. Shift and Scale, Shift-Back and Scale-Back. Shift and Scale ȳ S 1 [ȳ + ( K 1H1] (1.6 75

78 Shift back and Scale back Shift and Scale, Shift Back and Scale Back,, 8N [Flops]. s S s S ( K 1 (1.7 4N 4N 76

79 1.3, LRAD, 4 (LRAD. 1.1: LRAD 1.: ( :[Flops] 77

80 1.3. F&B LR, (F&B LR. 1.3: F&B LRAD 1.4: ( :[Flops] Seysen s LR, (Seysen s LR. 1.5: Seysen s LRAD Shift and Scale vɶ = H y S sˆ = 合計 S Tvˆ (Seysen s Algorithmにより準直交化した H Shift back and Scale back 8 M 8N 8 M ( N + M M 8N M M MN M N 1.6: ( :[Flops] M = N = 4 M = N = 6 M = N =

81 1.3.4 Reciprocal LR, (Reciprocal LR. 1.7: Reciprocal LR 1.8: ( :[Flops] LR with OSIC, (LR with OSIC. 1.9: LR with OSIC 1.10: ( :[Flops] 79

82 ɶ { F&B Reciprocal, 9 (F&B Reciprocal. 1.11: F&B Reciprocal s = H y M ( N + M M } Shift and Scale = j ( j ( j S : [1, ] ɶ v H y = S ˆ ( j ˆ ( j s Tv Shift back and Scale back j 合計 8 N {8 M ( N + M M} (8 M 8 N ( j sˆ = arg min [ y Hs ] (8MN + 4N 1 [1,] M M 4MN 10M 3N 1.1: ( :[Flops] M = N = 4 M = N = 6 M = N = F&B OSIC, 10 (F&B OSIC. 1.13: F&B OSIC 1.14: ( :[Flops] 80

83 ɶ Recipeocal F&B OSIC, 11 (Recipeocal F&B OSIC. 1.15: Recipeocal F&B OSIC s = H y M ( N + M M Shift and Scale S y = vɶ = R y ( j 1 ( j S S IC = T S Q y S j j j ( ( ( sˆ T vˆ Shift back and Scale back 8 N 8 M ( N + M M M ( 4 M (8 M M 8 N 1.16: ( :[Flops] M = N = 4 M = N = 6 M = N = ( j sˆ = arg min [ y Hs ] (8MN 4N 1 j 合計 [1, ] M + 3 M N 4 M + 4 N 81

84 13 Forward&Backward LR with OSIC ( ,, BER Forward & Backward LR with OSIC(F&B OSIC,. 13. F&B OSIC F&B OSIC Lattice Reduction BER,,.. F&B OSIC,,. (13.1. arg min i (1, [ y Hŝ (i ] (13.1 y Hŝ (i = y HTT 1 ŝ (i = y H ˆv (i = y Q R ˆv (i = Q H y R ˆv (i = y R ˆv (i (13., (13.1 (8MN + 4N 1[Flops], (13. (4M + M[Flops]., y R ˆv (i. 8

85 13.3 1(,..,. LR with OSIC, LR with OSIC MIMO,6 6MIMO,8 8MIMO QPSK, 16QAM, 64QAM. 83

86 16 LR with OSIC correct y -R v (M=4 N=4 LR with OSIC error y -R v (M=4 N=4 35 LR with OSIC correct y -R v (M=6 N=6 LR with OSIC error y -R v (M=6 N= Average y -R v Average y -R v : QPSK (M=N=4 13.: QPSK (M=N=6 60 LR with OSIC correct y -R v (M=8 N=8 LR with OSIC error y -R v (M=8 N= LR with OSIC correct y -R v (M=4 N=4 LR with OSIC error y -R v (M=4 N= Average y -R v Average y -R v : 16QAM 13.3: QPSK (M=N=8 (M=N=4 84

87 30 LR with OSIC correct y -R v (M=6 N=6 LR with OSIC error y -R v (M=6 N= LR with OSIC correct y -R v (M=8 N=8 LR with OSIC error y -R v (M=8 N= Average y -R v 15 Average y -R v : 16QAM 13.6: 16QAM (M=N=6 (M=N=8 1 LR with OSIC correct y -R v (M=4 N=4 LR with OSIC error y -R v (M=4 N=4 5 LR with OSIC correct y -R v (M=6 N=6 LR with OSIC error y -R v (M=6 N= Average y -R v 6 Average y -R v : 64QAM 13.8: 64QAM (M=N=4 (M=N=6 85

88 45 40 LR with OSIC correct y -R v (M=8 N=8 LR with OSIC error y -R v (M=8 N= Average y -R v : 64QAM (M=N=8 9, LR with OSIC,., LR with OSIC X(E b /N 0., Forward LR with OSIC, Backward LR with OSIC ,4 4MIMO, 6 6MIMO, 8 8MIMO QPSK, 16QAM, 64QAM X(E b /N 0. (1Forward-LR-OSIC から推定値 If y Else go to ( (1 R v < ˆ X ( E N b 0 (Backward-LR-OSIC から推定値 ˆv (1 を導出 end ( ˆv を導出 13.10: 86

89 16 14 LR with OSIC correct y -R v (M=4 N=4 LR with OSIC error y -R v (M=4 N=4 X[E b /N 0 ] (M=4 N= LR with OSIC correct y -R v (M=6 N=6 LR with OSIC error y -R v (M=6 N=6 X[E b /N 0 ] (M=6 N=6 1 5 Average y -R v Average y -R v : (M=N=4 X(E b /N 0 QPSK 13.1: (M=N=6 X(E b /N 0 QPSK LR with OSIC correct y -R v (M=8 N=8 LR with OSIC error y -R v (M=8 N=8 X[E b /N 0 ] (M=8 N= LR with OSIC correct y -R v (M=4 N=4 LR with OSIC error y -R v (M=4 N=4 X(E b /N 0 (M=4 N= Average y -R v 30 Average y -R v : (M=N=8 X(E b /N 0 QPSK 13.14: X(E b /N 0 16QAM (M=N=4 87

90 30 LR with OSIC correct y -R v (M=6 N=6 LR with OSIC error y -R v (M=6 N=6 X[E b /N 0 ] (M=6 N= LR with OSIC correct y -R v (M=8 N=8 LR with OSIC error y -R v (M=8 N=8 X[E b /N 0 ] (M=8 N= Average y -R v 15 Average y -R v : X(E b /N 0 16QAM 13.16: X(E b /N 0 16QAM (M=N=6 (M=N=8 1 LR with OSIC correct y -R v (M=4 N=4 LR with OSIC error y -R v (M=4 N=4 X(E b /N 0 (M=4 N=4 5 LR with OSIC correct y -R v (M=6 N=6 LR with OSIC error y -R v (M=6 N=6 X[E b /N 0 ] (M=6 N= Average y -R v 6 Average y -R v : X(E b /N 0 64QAM 13.18: X(E b /N 0 64QAM (M=N=4 (M=N=6 88

91 45 40 LR with OSIC correct y -R v (M=8 N=8 LR with OSIC error y -R v (M=8 N=8 X[E b /N 0 ] (M=8 N= Average y -R v : X(E b /N 0 64QAM (M=N=8 9, MIMO, X(E b /N 0. F&B LR with OSIC, Forward LR with OSIC ˆv (1 Backward LR with OSIC ˆv (. X(E b /N 0, Forward LR with OSIC ˆv (1, Backward LR with OSIC ˆv (, X(E b /N 0,BER ,8 4 4MIMO, 6 6MIMO, 8 8MIMO QPSK, 16QAM, 64QAM BER Proposed method [1] QPSK (M=4 N=4 F&B LR with OSIC QPSK (M=4 N=4 LR with OSIC QPSK (M=4 N=4 ML QPSK (M=4 N=4 BER : 1 BER QPSK (M=N=4 89

92 10 0 Proposed method [1] QPSK (M=6 N=6 F&B LR with OSIC QPSK (M=6 N=6 LR with OSIC QPSK (M=6 N=6 ML QPSK (M=6 N= Proposed method [1] QPSK (M=8 N=8 F&B LR with OSIC QPSK (M=8 N=8 LR with OSIC QPSK (M=8 N=8 ML QPSK (M=8 N= BER BER : 1 BER QPSK 13.: 1 BER QPSK (M=N=6 (M=N= Proposed method [1] 16QAM (M=4 N=4 F&B LR with OSIC 16QAM (M=4 N=4 LR with OSIC 16QAM (M=4 N=4 ML 16QAM (M=4 N= Proposed method [1] 16QAM (M=6 N=6 F&B LR with OSIC 16QAM (M=6 N=6 LR with OSIC 16QAM (M=6 N=6 ML 16QAM (M=6 N= BER BER : 1 BER 16QAM 13.4: 1 BER 16QAM (M=N=4 (M=N=6 90

93 10 0 Proposed method [1] 16QAM (M=8 N=8 F&B LR with OSIC 16QAM (M=8 N=8 LR with OSIC 16QAM (M=8 N=8 ML 16QAM (M=8 N= Proposed method [1] 64QAM (M=4 N=4 F&B LR with OSIC 64QAM (M=4 N=4 LR with OSIC 64QAM (M=4 N=4 ML 64QAM (M=4 N= BER BER : 1 BER 16QAM 13.6: 1 BER 64QAM (M=N=8 (M=N= Proposed method [1] 64QAM (M=6 N=6 F&B LR with OSIC 64QAM (M=6 N=6 LR with OSIC 64QAM (M=6 N=6 ML 64QAM (M=6 N= Proposed method [1] 64QAM (M=8 N=8 F&B LR with OSIC 64QAM (M=8 N=8 LR with OSIC 64QAM (M=8 N=8 ML 64QAM (M=8 N= BER BER : 1 BER 64QAM 13.8: 1 BER 64QAM (M=N=6 (M=N=8 91

94 , X(E b /N 0, E b /N 0,, (1 (.,(1,, ( ,38.., 13.9, (1Forward Ordering-SIC が使われる確率 70% (Backward Ordering-SIC が使われる確率 30% とすると LR-MMSE LR-OSIC F&B-OSIC 8 8MIMO 163 [Flops] 1904 [Flops] 3760[Flops] (LR-OSIC+ ノルム計算 (F&B-OSIC = 606[Flops] となる (1 ( 13.9: 1 (M=N=8 9

95 1 0.9 Forward LR with OSIC BackwardLR with OSIC Forward LR with OSIC BackwardLR with OSIC Probability Probability : : 1 QPSK (M=N=4 QPSK (M=N= Forward LR with OSIC BackwardLR with OSIC Forward LR with OSIC BackwardLR with OSIC Probability Probability : : 1 QPSK (M=N=8 16QAM (M=N=4 93

96 1 0.9 Forward LR with OSIC BackwardLR with OSIC Forward LR with OSIC BackwardLR with OSIC Probability Probability : : 1 16QAM (M=N=6 16QAM (M=N= Forward LR with OSIC BackwardLR with OSIC Forward LR with OSIC BackwardLR with OSIC Probability Probability : : 1 64QAM (M=N=4 64QAM (M=N=6 94

97 1 0.9 Forward LR with OSIC BackwardLR with OSIC Probability : 1 64QAM (M=N=8 95

98 1400 Proposed method [1] QPSK (M=4 N=4 F&B LR with OSIC QPSK (M=4 N=4 LR with OSIC QPSK (M=4 N= Proposed method [1] QPSK (M=6 N=6 F&B LR with OSIC QPSK (M=6 N=6 LR with OSIC QPSK (M=6 N= FLOPS FLOPS : 1 QPSK 13.40: 1 QPSK (M=N=4 (M=N= Proposed method [1] QPSK (M=8 N=8 F&B LR with OSIC QPSK (M=8 N=8 LR with OSIC QPSK (M=8 N= Proposed method [1] 16QAM (M=4 N=4 F&B LR with OSIC 16QAM (M=4 N=4 LR with OSIC 16QAM (M=4 N= FLOPS 3000 FLOPS : 1 QPSK 13.4: (M=N=8 (M=N=4 1 16QAM 96

99 3000 Proposed method [1] 16QAM (M=6 N=6 F&B LR with OSIC 16QAM (M=6 N=6 LR with OSIC 16QAM (M=6 N= Proposed method [1] 16QAM (M=8 N=8 F&B LR with OSIC 16QAM (M=8 N=8 LR with OSIC 16QAM (M=8 N= FLOPS FLOPS : (M=N=6 1 16QAM 13.44: (M=N=8 1 16QAM 1400 Proposed method [1] 64QAM (M=4 N=4 F&B LR with OSIC 64QAM (M=4 N=4 LR with OSIC 64QAM (M=4 N= Proposed method [1] 64QAM (M=6 N=6 F&B LR with OSIC 64QAM (M=6 N=6 LR with OSIC 64QAM (M=6 N= FLOPS FLOPS : (M=N=4 1 64QAM 13.46: (M=N=6 1 64QAM 97

100 4500 Proposed method [1] 64QAM (M=8 N=8 F&B LR with OSIC 64QAM (M=8 N=8 LR with OSIC 64QAM (M=8 N= FLOPS : 1 64QAM (M=N=8 13.5,LR with OSIC, X(E b /N 0,., F&B LR with OSIC. BER F&B LR with OSIC, 1 BER., E b /N 0,, (1., E b /N 0, LR with OSIC,. 98

101 14 Sphere Detection (SD 14.1 ML Sphere Detection(SD. 14. Sphere Detection(SD SD, H s H s, y C 0,,,. y H s C 0 (14.1, H QR [ H N M = Q N M Q N (N M ] [ R M M 0 (N M M ] (14., Q Q R,. y = Q H y, C 0 = C 0 (Q H y, (14.1, y R s C 0 (14.3., R, M 1 y R s s.( 14.1,

102 y ɶ y h h h s M 1 y h h h s 1 M Hs = = y h h h sɶ N N1 N NM M ɶ ɶ ɶ ɶ ɶ ɶ y h s h s h s M y h s h s h s 1 1 M y h s h s h s ɶ ɶ ɶ N N1 1 N NM M M M ɶ ɶ y y 1 r r r s M 1 y r r sɶ M Rsɶ = ɶ y r s M 0 MM M ɶ ɶ ɶ y r s r s r s M M y r sɶ r sɶ M M = y r s ɶ M MM M ɶ 最下行に M 個の未知数 最下行に 1 個の未知数 14.1: ML 14.: SD y R s : y i r ii s i r im s M (i [1, M], y R s., s QPSK. SD,. y im r MM s M s M, y im r MM s M, s M (QPSK 4. 4 C 0 1, sm., 1 y M 1 r M 1M 1 s M 1 r M 1M s M, s M, s M., y M 1 r M 1M 1 s M 1 r M 1M s M s M 1, 4. 4, y M r MM s M + y M 1 r M 1M 1 s M 1 r M 1M s M C 0, s M 1., y M r MM s M + y M 1 r M 1M 1 s M 1 r M 1M s M C 0 sm 1, 1, s M,., C 0 1, C,.., s M s 1 s. 100

103 1 行目 : S 1 最小メトリック = 4 : 計算されたブランチメトリック 行目 : S 行目 : S 最小メトリック 4 を探索し終えるまでに要したブランチメトリックの演算回数は 0 回となる 14.3: M=3, QPSK, SD SD,. 14.3, M = 3, QPSK, SD.,,, SD.,, (., C 0., ,, C 0 C = 5., 1, 3, , 7 5,. 1, 4, 5, 4., 3 1,, 1, =4, 4. 1, 4,. 3, 101

104 ,., 3, 1, ,,. 1,, 14.3., 4, s 3, s, s , 0. ML,, 4 3 = 64,, 64 3 = 19, SD ML. SD ML,,,.,., SD. bestdist.,,,., P. y = 1 (y + ( P 1H[1,..., 1] T (

105 Algorithm :SD Algorithm 1: Input w = Q T y,g = R 1 : bestdist = 3: k K 4: dist k 0 5: e k = GW 6: x k = e kk 7: x k = max(x k, 0; x k = min(x k, Q max ; 8: y = (e kk x k /g kk 9: step k = sgn(v 10: while ( do 11: newdist = dist k + v 1: if (newdist < bestdist (x k 0 x k Q max then 13: if (k 1 then 14: for i = 1,..., k 1 do 15: e k 1,i = e ki vg ik 16: end for(14 17: k k 1 18: dist k = newdist 19: x k = e kk 0: x k = max(x k, 0; x k = min(x k, Q max ; 1: y = (e kk x k /g kk : step k = sgn(v 3: else 4: ˆx = x 5: bestdist = newdist 6: k k + 1 7: x k = x k + step k 8: step k = step k sgn(step k 9: if (x k < 0 x k > Q max then 30: x k = x k + step k 31: step k = step k sgn(step k 3: end if(9 33: y = (e kk x k /g kk 34: end if(13 35: else 36: if k = K then 37: {return ˆx} 38: else 39: k k : x k = x k + step k 41: step k = step k sgn(step k 4: if (x k < 0 x k > Q max then 43: x k = x k + step k 44: step k = step k sgn(step k 45: end if(4 46: y = (e kk x k /g kk 47: end if(36 48: end if(1 49: end while(10 103

106 14.3 QPSK MIMO M = N = 4 M = N = 6, M = N = 8 BER 14.4, 14.5, M = N = 4, M = N = 6 SD, C = 5, C = 10, C = 15, C =. M = N = 8 SD, C = 5, C = 10, C = 15, C = 0, C =., 16QAM M = N = 4 M = N = 6, M = N = 8 BER 14.7, 14.8, M = N = 4, M = N = 6 SD, C = 5, C = 10, C = 15, C =. M = N = 8 SD, C = 5, C = 10, C = 15, C = 0, C =., 64QAM M = N = 4 M = N = 6, M = N = 8 BER 14.7, 14.8, M = N = 4, M = N = 6 SD, C = 5, C = 10, C = 15, C =. M = N = 8 SD, C = 5, C = 10, C = 15, C = 0, C =., QPSK MIMO M = N = 4 M = N = 6, M = N = , 14.11, 14.1., 16QAM MIMO M = N = 4 M = N = 6, M = N = , 14.14, , 64QAM MIMO M = N = 4 M = N = 6, M = N = , 14.17,

107 10 0 SD QPSK (C=5 SD QPSK (C=10 SD QPSK (C=15 SD QPSK (C=INF 10 0 SD QPSK (C=5 SD QPSK (C=10 SD QPSK (C=15 SD QPSK (C=INF BER BER : SD QPSK BER (M=N=4 14.5: SD QPSK BER (M=N= SD QPSK (C=5 SD QPSK (C=10 SD QPSK (C=15 SD QPSK (C=0 SD QPSK (C=INF SD 16QAM (C=5 SD 16QAM (C=10 SD 16QAM (C=15 SD 16QAM (C=INF BER BER : SD QPSK BER (M=N=8 14.7: SD 16QAM BER (M=N=4 105

108 SD 16QAM (C=5 SD 16QAM (C=10 SD 16QAM (C=15 SD 16QAM (C=INF SD 16QAM (C=5 SD 16QAM (C=10 SD 16QAM (C=15 SD 16QAM (C=0 SD 16QAM (C=INF BER BER : SD 16QAM BER (M=N=6 14.9: SD 16QAM BER (M=N= SD 64QAM (C=5 SD 64QAM (C=10 SD 64QAM (C=15 SD 64QAM (C=INF 10 0 SD 64QAM (C=5 SD 64QAM (C=10 SD 64QAM (C=15 SD 64QAM (C=INF BER BER : SD 64QAM BER (M=N= : SD 64QAM BER (M=N=6 106

109 BER SD 64QAM (C=5 SD 64QAM (C=10 SD 64QAM (C=15 SD 64QAM (C=0 SD 64QAM (C=INF Number of Operations 10 3 SD QPSK (C=5 SD QPSK (C=10 SD QPSK (C=15 SD QPSK (C=INF : SD 64QAM BER (M=N= : SD QPSK (M=N=4 Number of Operations SD QPSK (C=5 SD QPSK (C=10 SD QPSK (C=15 SD QPSK (C=INF Number of Operations SD QPSK (C=5 SD QPSK (C=10 SD QPSK (C=15 SD QPSK (C=INF : SD QPSK (M=N= : SD QPSK (M=N=8 107

110 Number of Operations SD 16QAM (C=5 SD 16QAM (C=10 SD 16QAM (C=15 SD 16QAM (C=INF Number of Operations SD 16QAM (C=5 SD 16QAM (C=10 SD 16QAM (C=15 SD 16QAM (C=INF : SD 16QAM (M=N= : SD 16QAM (M=N=6 Number of Operations SD 16QAM (C=5 SD 16QAM (C=10 SD 16QAM (C=15 SD 16QAM (C=INF Number of Operations SD 64QAM (C=5 SD 64QAM (C=10 SD 64QAM (C=15 SD 64QAM (C=INF : SD 16QAM (M=N= : SD 64QAM (M=N=4 108

111 10 6 SD 64QAM (C=5 SD 64QAM (C=10 SD 64QAM (C=15 SD 64QAM (C=INF 10 8 SD 64QAM (C=5 SD 64QAM (C=10 SD 64QAM (C=15 SD 64QAM (C=INF Number of Operations Number of Operations : SD 64QAM (M=N=6 14.1: SD 64QAM (M=N= SD, C ML BER., ML., ML BER., C, 4 4 C = 5, 6 6 C = 10, 8 8 C = 15 E b /N 0 ML, E b /N 0 ML BER., BER=10 3 ML BER, BER. 109

112 15 LR-with OSIC Sphere Detection(SD ( ML SD LR with OSIC, ML,, SD. 15., LR with OSIC ṽ,, ˆv.. ȳ = Hs + z (15.1 ȳ = HTT 1 s + z (15. ȳ = H v + z (15.3 ȳ = Q R v + z (15.4 Q Tȳ = R v + Q T z (15.5 y = R v + z (15.6,(R T R 1, SIC ˆv m, ˆv m 1, ˆv 1. ˆv, y R ˆv. (15.6 y R ˆv z., y R ˆv, LR with OSIC. y = R v + z y R ˆv = z (

113 , X(E b /N 0. y Rˆv < X (E b /N 0 (15.8 (15.8,, LR with OSIC ˆv. ( 13 (15.8 ŝ. ŝ = Tˆv (15.9 y Rˆv > X (E b /N 0 (15.10, (15.10,, LR with OSIC, SD, SD ŝ.,, E b /N 0 SD BER, LR with OSIC, E b /N 0 ML BER, SD QPSK MIMO M = N = 4 M = N = 6, M = N = 8 BER 15.1, 15., M = N = 4 SD, C = 5. M = N = 6 SD, C = 10. M = N = 8 SD, C = 15., 16QAM M = N = 4 M = N = 6, M = N = 8 BER 15.4, 15.5, M = N = 4 SD, C = 5. M = N = 6 SD, C = 10. M = N = 8 SD, C = 15., 64QAM M = N = 4 M = N = 6, M = N = 8 BER 15.7, 15.8, M = N = 4 SD, C = 5. M = N = 6 SD, C = 10. M = N = 8 SD, C = 15., QPSK MIMO M = N = 4 M = N = 6, M = N = , 15.11, 15.1., 16QAM MIMO M = N = 4 M = N = 6, M = N = , 15.14, , 64QAM MIMO M = N = 4 M = N = 6, M = N = , 15.17,

114 10 0 Proposed method[] QPSK (C=5 SD QPSK (C=5 LR with OSIC QPSK ML QPSK 10 0 Proposed method[] QPSK (C=10 SD QPSK (C=10 LR with OSIC QPSK ML QPSK BER BER : QPSK BER 15.: QPSK BER (M=N=4 (M=N= Proposed method[] QPSK (C=15 SD QPSK (C=15 LR with OSIC QPSK ML QPSK 10 0 Proposed method[] 16QAM (C=5 SD 16QAM (C=5 LR with OSIC 16QAM ML 16QAM BER BER : QPSK BER 15.4: 16QAM BER (M=N=8 (M=N=4 11

115 10 0 Proposed method[] 16QAM (C=10 SD 16QAM (C=10 LR with OSIC 16QAM ML 16QAM 10 0 Proposed method[] 16QAM (C=15 SD 16QAM (C=15 LR with OSIC 16QAM ML 16QAM BER BER : 16QAM BER 15.6: 16QAM BER (M=N=6 (M=N= Proposed method[] 64QAM (C=5 SD 64QAM (C=5 LR with OSIC 64QAM ML 64QAM 10 0 Proposed method[] 64QAM (C=10 SD 64QAM (C=10 LR with OSIC 64QAM ML 64QAM BER BER : 64QAM BER 15.8: 64QAM BER (M=N=4 (M=N=6 113

116 10 0 Proposed method[] 64QAM (C=15 SD 64QAM (C=15 LR with OSIC 64QAM ML 64QAM Proposed method[] QPSK (C=5 QRM-MLD QPSK LR with OSIC QPSK SD QPSK(C=5 BER Number of Operations : 64QAM BER 15.10: QPSK (M=N=8 (M=N= Proposed method[] QPSK (C=10 QRM-MLD QPSK LR with OSIC QPSK SD QPSK (C= Proposed method[] QPSK (C=15 QRM-MLD QPSK LR with OSIC QPSK SD QPSK (C=15 Number of Operations 10 3 Number of Operations : QPSK 15.1: QPSK (M=N=6 (M=N=8 114

117 Number of Operations Proposed method[] (C=5 16QAM QRM-MLD 16QAM LR with OSIC 16QAM SD (C=5 16QAM Number of Operations Proposed method[] (C=10 16QAM QRM-MLD 16QAM LR with OSIC 16QAM SD(C=10 16QAM : 16QAM 15.14: 16QAM (M=N=4 (M=N= Proposed method[] 16QAM (C=15 QRM-MLD 16QAM LR with OSIC 16QAM SD 16QAM (C=15 Proposed method[] (C=5 64QAM QRM-MLD 64QAM LR with OSIC 64QAM SD (C=5 64QAM Number of Operations Number of Operations : 16QAM 15.16: 64QAM (M=N=8 (M=N=4 115

118 10 5 Proposed method[] (C=10 64QAM QRM-MLD 64QAM LR with OSIC 64QAM SD (C=10 64QAM 10 6 Proposed method[] 64QAM (C=15 QRM-MLD 64QAM LR with OSIC 64QAM SD 64QAM (C=15 Number of Operations 10 4 Number of Operations : 64QAM 15.18: 64QAM (M=N=6 (M=N= LR with OSIC ˆv y R ˆv, X(E b /N 0 SD., LR with OSIC., ML BER., MIMO, SD. 8 8MIMO 64QAM, BER

119 16 MIMO,, BER,. MIMO.,., BER., 1, MIMO, 3 ZF, MMSE, ML. 4 Lattice-Reduction(LR, 5 8 LR. 9 11,,. 1,, , 1, 10 (F&B OSIC LRAD,. 14, ML, Sphere Detection(SD,. 15,,. BER, LR with OSIC,MIMO 4 4 QPSK, ML BER., MIMO QAM, ML BER., SD, E b /N 0 QPSK, 16QAM, 64QAM,. E b /N 0, MIMO,., LR with OSIC, SD. 117

120 ,,.,. Hou Wei(,,,, 118

121 [1] Tadashi Fujino. Introduction to Lattice-Reduction Aided Detection(Rev.10 [] A.K.Lenstra, H.W.Lenstra Jr. and L.Lovasz, Factioring polynomials with rational coefficients, pp [3] T.Fujino and T.Shimokawa, Combined forward and backward lattice-reduction aided MMSE detection in MIMO systems, Proc.68thIEEEVehicularTechnol.Conf.(VTC 08 Fall,Calgary, Canada, Sep [4] Luis G.Barbero. A COMPARISON OF COMPLEX LATTICE REDUCTION ALGO- RITHM FOR MIMO DETECTION [5] LOW-COMPLEXITY MIMO DATA DETECTION USING SEYSEN S LATTICE REDUC- TION SLGORITHM [6] A Performance Study of MIMO Detectors [7] Tetsuyoshi Shimokawa. A combined forward and backward lattice reduction aided MMSE list detection [8] Kyungchun Lee, Joohwan Chun, and Lajos Hanzo. Optimal Lattice-Reduction Aided Successive Interference Cancellation for MIMO Systems 119