Sphere Decoding for Noncoherent Channels

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1 Sphere Decoding for Noncoherent Channels Lutz Lampe Deptartment of Electrical & Computer Engineering The University of British Columbia, Canada joint work with Volker Pauli, Robert Schober, and Christoph Windpassinger

2 Motivation 2 Wireless communication systems Operate in high frequency bands Offer high fidelity in various propagation environments Use cheap consumer devices

3 Motivation 3 E.g. low cost local oscillators no perfect carrier phase synchronization feasible noncoherent communication is a favorable choice

4 Motivation 3 E.g. low cost local oscillators no perfect carrier phase synchronization feasible noncoherent communication is a favorable choice

5 Motivation 3 E.g. low cost local oscillators no perfect carrier phase synchronization feasible noncoherent communication is a favorable choice

6 Motivation 4 Noncoherent channel: h[k]e jθ[k] n[k] s[k] r[k] Noncoherent multipath channel: noncoherent noncoherent reception without channel state information (CSI)

7 Motivation 4 Noncoherent channel: h[k]e jθ[k] n[k] s[k] r[k] Noncoherent multipath channel: h[k] {}}{ a[k]e jθ[k] n[k] s[k] r[k] noncoherent noncoherent reception without channel state information (CSI)

8 Motivation 4 Noncoherent channel: h[k]e jθ[k] n[k] s[k] r[k] Noncoherent multipath channel: h[k] {}}{ a[k]e jθ[k] n[k] s[k] r[k] noncoherent noncoherent reception without channel state information (CSI)

9 Outline 5 Transmitter: Differential Encoding Receiver: Multiple-Symbol Differential Detection Multiple-Symbol Differential Sphere Decoding Performance Results Summary

10 Outline 5 Transmitter: Differential Encoding Receiver: Multiple-Symbol Differential Detection Multiple-Symbol Differential Sphere Decoding Performance Results Summary

11 Differential Encoding 6 Consider phase-shift keying (PSK) signal set: s[k] = e j2π M m[k]

12 Differential Encoding 6 Consider phase-shift keying (PSK) signal set: s[k] = e j2π M m[k] s[k]? (θ?) Im r[k] = e jθ s[k] (h[k] + n[k]) Re

13 Differential Encoding 6 Consider phase-shift keying (PSK) signal set: s[k] = e j2π M m[k] s[k]? (θ?) Im r[k] = e jθ s[k] (h[k] + n[k]) Re

14 Differential Encoding 6 Consider phase-shift keying (PSK) signal set: s[k] = e j2π M m[k] s[k]? (θ?) Im r[k] = e jθ s[k] (h[k] + n[k]) Re

15 Differential Encoding 6 Consider phase-shift keying (PSK) signal set: s[k] = e j2π M m[k] s[k]? (θ?) Im r[k] = e jθ s[k] (h[k] + n[k]) Re Solution: encode data in phase difference arg{s[k]} arg{s[k 1]}

16 Differential Encoding 7 Differential phase-shift keying (DPSK): s[k] = v[k] s[k 1]

17 Differential Encoding 7 Differential phase-shift keying (DPSK): s[k] = v[k] s[k 1] Structure: v[k] s[k] s[k 1] Delay

18 Outline 7 Transmitter: Differential Encoding Receiver: Multiple-Symbol Differential Detection Multiple-Symbol Differential Sphere Decoding Performance Results Summary

19 Multiple-Symbol Differential Detection 8 Write: r[k] = s[k] h[k] + n[k] = v[k]s[k 1] h[k 1] + n[k] = v[k]r[k 1] v[k]n[k 1] + n[k]

20 Multiple-Symbol Differential Detection 8 Write: r[k] = s[k] h[k] + n[k] = v[k]s[k 1] h[k 1] + n[k] = v[k]r[k 1] v[k]n[k 1] + n[k]

21 Multiple-Symbol Differential Detection 8 Write: r[k] = s[k] h[k] + n[k] = v[k]s[k 1] h[k 1] + n[k] = v[k]r[k 1] v[k]n[k 1] + n[k]

22 Multiple-Symbol Differential Detection 8 Write: r[k] = s[k] h[k] + n[k] Conventional differential detection: = v[k]s[k 1] h[k 1] + n[k] = v[k]r[k 1] v[k]n[k 1] + n[k] ˆv[k] = argmax{re{r [k]v r[k 1]}} v

23 Multiple-Symbol Differential Detection 8 Write: r[k] = s[k] h[k] + n[k] Conventional differential detection: = v[k]s[k 1] h[k 1] + n[k] = v[k]r[k 1] v[k]n[k 1] + n[k] ˆv[k] = argmax{re{r [k]v r[k 1]}} v r[k] ˆv[k] Delay ( ) r [k 1]

24 Multiple-Symbol Differential Detection 9 Idea: Use observation window of length N > 2 received samples Multiple-symbol differential detection (MSDD) Joint decision on N 1 data symbols based on observation of N received samples Performance Complexity exploit memory tree search ( memory increases capacity ) (complexity exponential in N)

25 Multiple-Symbol Differential Detection 9 Idea: Use observation window of length N > 2 received samples Multiple-symbol differential detection (MSDD) Joint decision on N 1 data symbols based on observation of N received samples Performance Complexity exploit memory tree search ( memory increases capacity ) (complexity exponential in N)

26 Multiple-Symbol Differential Detection 9 Idea: Use observation window of length N > 2 received samples Multiple-symbol differential detection (MSDD) Joint decision on N 1 data symbols based on observation of N received samples... r[k N + 1]... r[k 1] r[k]... observation window Performance Complexity exploit memory tree search ( memory increases capacity ) (complexity exponential in N)

27 Multiple-Symbol Differential Detection 9 Idea: Use observation window of length N > 2 received samples Multiple-symbol differential detection (MSDD) Joint decision on N 1 data symbols based on observation of N received samples... r[k N + 1]... r[k 1] r[k]... observation window Performance Complexity exploit memory tree search ( memory increases capacity ) (complexity exponential in N)

28 Multiple-Symbol Differential Detection 10 Maximum-likelihood (ML) MSDD Wilson et al., 1989 Divsalar & Simon, 1990 Leib & Pasupathy, ML MSDD for static channels Mackenthun, 1994 Suboptimum MSDD Linear-prediction based decision-feedback differential detection (DF-DD) Kam & Teh, 1983 Svensson, 1994 Schober et al., Two-step algorithms Xiaofu & Songgeng,

29 Multiple-Symbol Differential Detection 10 Maximum-likelihood (ML) MSDD Wilson et al., 1989 Divsalar & Simon, 1990 Leib & Pasupathy, ML MSDD for static channels Mackenthun, 1994 Suboptimum MSDD Linear-prediction based decision-feedback differential detection (DF-DD) Kam & Teh, 1983 Svensson, 1994 Schober et al., Two-step algorithms Xiaofu & Songgeng,

30 Multiple-Symbol Differential Detection 10 Maximum-likelihood (ML) MSDD Wilson et al., 1989 Divsalar & Simon, 1990 Leib & Pasupathy, ML MSDD for static channels Mackenthun, 1994 Suboptimum MSDD Linear-prediction based decision-feedback differential detection (DF-DD) Kam & Teh, 1983 Svensson, 1994 Schober et al., Two-step algorithms Xiaofu & Songgeng,

31 Multiple-Symbol Differential Detection 10 Maximum-likelihood (ML) MSDD Wilson et al., 1989 Divsalar & Simon, 1990 Leib & Pasupathy, ML MSDD for static channels Mackenthun, 1994 Suboptimum MSDD Linear-prediction based decision-feedback differential detection (DF-DD) Kam & Teh, 1983 Svensson, 1994 Schober et al., Two-step algorithms Xiaofu & Songgeng,

32 Outline 10 Transmitter: Differential Encoding Receiver: Multiple-Symbol Differential Detection Multiple-Symbol Differential Sphere Decoding Performance Results Summary

33 Multiple-Symbol Differential Sphere Decoding 11 Block diagram v[k] s[k] r[k] MSDSD ŝ[k] ˆv[k] T h[k] n[k] T ()

34 Multiple-Symbol Differential Sphere Decoding 11 Block diagram v[k] s[k] r[k] MSDSD ŝ[k] ˆv[k] T h[k] n[k] T () Rayleigh fading channel h[k]

35 Multiple-Symbol Differential Sphere Decoding 11 Block diagram v[k] s[k] r[k] MSDSD ŝ[k] ˆv[k] T h[k] n[k] T () Rayleigh fading channel h[k] ML MSDD r [r[k (N 1)], r[k (N 2)],..., r[k]] T = [r 1, r 2,..., r N ]] T s [s[k (N 1)], s[k (N 2)],..., s[k]] T = [s 1, s 2,..., s N ]] T h [h[k (N 1)], h[k (N 2)],..., h[k]] T = [h 1, h 2,..., h N ]] T

36 Multiple-Symbol Differential Sphere Decoding 12 Vector channel r = diag{s}h + n ML decision rule [Ho&Fung, 1992] ŝ = argmin{r H R 1 rr r} s where and R rr E{rr H s} = diag{s} ( E{hh H ) } + σ }{{ ni 2 N diag{s } } C (diag{s}) 1 = diag{s }, diag{s }r = diag{r}s ŝ = argmin{s T diag{r} H C 1 diag{r}s } s

37 Multiple-Symbol Differential Sphere Decoding 12 Vector channel r = diag{s}h + n ML decision rule [Ho&Fung, 1992] ŝ = argmin{r H R 1 rr r} s where and R rr E{rr H s} = diag{s} ( E{hh H ) } + σ }{{ ni 2 N diag{s } } C (diag{s}) 1 = diag{s }, diag{s }r = diag{r}s ŝ = argmin{s T diag{r} H C 1 diag{r}s } s

38 Multiple-Symbol Differential Sphere Decoding 12 Vector channel r = diag{s}h + n ML decision rule [Ho&Fung, 1992] ŝ = argmin{r H R 1 rr r} s where and R rr E{rr H s} = diag{s} ( E{hh H ) } + σ }{{ ni 2 N diag{s } } C (diag{s}) 1 = diag{s }, diag{s }r = diag{r}s ŝ = argmin{s T diag{r} H C 1 diag{r}s } s

39 Multiple-Symbol Differential Sphere Decoding 12 Vector channel r = diag{s}h + n ML decision rule [Ho&Fung, 1992] ŝ = argmin{r H R 1 rr r} s where and R rr E{rr H s} = diag{s} ( E{hh H ) } + σ }{{ ni 2 N diag{s } } C (diag{s}) 1 = diag{s }, diag{s }r = diag{r}s ŝ = argmin{s T diag{r} H C 1 diag{r}s } s

40 Multiple-Symbol Differential Sphere Decoding 12 Vector channel r = diag{s}h + n ML decision rule [Ho&Fung, 1992] ŝ = argmin{r H R 1 rr r} s where and R rr E{rr H s} = diag{s} ( E{hh H ) } + σ }{{ ni 2 N diag{s } } C (diag{s}) 1 = diag{s }, diag{s }r = diag{r}s ŝ = argmin{s T diag{r} H C 1 diag{r}s } s

41 Multiple-Symbol Differential Sphere Decoding 13 Cholesky factorization = LL H U (L H diag{r}) C 1 ŝ = argmin{ Us 2 } s M = 8, N = hypothetical vectors s

42 Multiple-Symbol Differential Sphere Decoding 13 Cholesky factorization = LL H U (L H diag{r}) C 1 ŝ = argmin{ Us 2 } s M = 8, N = hypothetical vectors s

43 Multiple-Symbol Differential Sphere Decoding 13 Cholesky factorization = LL H U (L H diag{r}) C 1 ŝ = argmin{ Us 2 } s M = 8, N = hypothetical vectors s

44 Multiple-Symbol Differential Sphere Decoding 14 ML decoding in multiple-input multiple-output (MIMO) communications with CSI ŝ = argmin s r Hs 2 = argmin s with Cholesky factorization H H H = U H U and unconstrained least-squares solution s LS U(s s LS ) 2 Efficiently be solved by the application of sphere decoding (SD) Viterbo & Boutros, 1999 Damen et al., 2000 Agrell et al.,

45 Multiple-Symbol Differential Sphere Decoding 14 ML decoding in multiple-input multiple-output (MIMO) communications with CSI ŝ = argmin s r Hs 2 = argmin s with Cholesky factorization H H H = U H U and unconstrained least-squares solution s LS U(s s LS ) 2 Efficiently be solved by the application of sphere decoding (SD) Viterbo & Boutros, 1999 Damen et al., 2000 Agrell et al.,

46 Multiple-Symbol Differential Sphere Decoding 14 ML decoding in multiple-input multiple-output (MIMO) communications with CSI ŝ = argmin s r Hs 2 = argmin s with Cholesky factorization H H H = U H U and unconstrained least-squares solution s LS U(s s LS ) 2 Efficiently be solved by the application of sphere decoding (SD) Viterbo & Boutros, 1999 Damen et al., 2000 Agrell et al.,

47 Multiple-Symbol Differential Sphere Decoding 13 Cholesky factorization = LL H U (L H diag{r}) C 1 ŝ = argmin{ Us 2 } s Sphere decoding for noncoherent channels Multiple-Symbol Differential Sphere Decoding (MSDSD)

48 Multiple-Symbol Differential Sphere Decoding 13 Cholesky factorization = LL H U (L H diag{r}) C 1 ŝ = argmin{ Us 2 } s Sphere decoding for noncoherent channels Multiple-Symbol Differential Sphere Decoding (MSDSD)

49 Multiple-Symbol Differential Sphere Decoding 15 SD concept Us 2 R 2 s 1 s 2. s N 1 s N i = N u NN s N 2 d 2 N? R 2 ŝ N i = N 1. i = 1 u N 1N 1 s N 1 + u N 1N ŝ N 2 + d 2 N d 2 N 1 ŝ : update R := Uŝ? R 2 ŝ N 1

50 Multiple-Symbol Differential Sphere Decoding 15 SD concept Us 2 R 2 s 1 s 2. s N 1 s N i = N u NN s N 2 d 2 N? R 2 ŝ N i = N 1. i = 1 u N 1N 1 s N 1 + u N 1N ŝ N 2 + d 2 N d 2 N 1 ŝ : update R := Uŝ? R 2 ŝ N 1

51 Multiple-Symbol Differential Sphere Decoding 15 SD concept Us 2 R 2 s 1 s 2. s N 1 s N i = N u NN s N 2 d 2 N? R 2 ŝ N i = N 1. i = 1 u N 1N 1 s N 1 + u N 1N ŝ N 2 + d 2 N d 2 N 1 ŝ : update R := Uŝ? R 2 ŝ N 1

52 Multiple-Symbol Differential Sphere Decoding 15 SD concept Us 2 R 2 s 1 s 2. s N 1 s N i = N u NN s N 2 d 2 N? R 2 ŝ N i = N 1. i = 1 u N 1N 1 s N 1 + u N 1N ŝ N 2 + d 2 N d 2 N 1 ŝ : update R := Uŝ? R 2 ŝ N 1

53 Multiple-Symbol Differential Sphere Decoding 16 Phase ambiguities Fix s N = 1 and start sphere decoding with i = N 1 Schnorr-Euchner search strategy Ordering of hypothetical symbols s i according to d i a) Find the phase index m i (s i = e j2πm i/m ) of the best candidate point b) Zigzag Account for finite constellation size Initial radius Works without initial radius

54 Multiple-Symbol Differential Sphere Decoding 16 Phase ambiguities Fix s N = 1 and start sphere decoding with i = N 1 Schnorr-Euchner search strategy Ordering of hypothetical symbols s i according to d i a) Find the phase index m i (s i = e j2πm i/m ) of the best candidate point b) Zigzag Account for finite constellation size Initial radius Works without initial radius

55 Multiple-Symbol Differential Sphere Decoding 16 Phase ambiguities Fix s N = 1 and start sphere decoding with i = N 1 Schnorr-Euchner search strategy Ordering of hypothetical symbols s i according to d i a) Find the phase index m i (s i = e j2πm i/m ) of the best candidate point b) Zigzag Account for finite constellation size Initial radius Works without initial radius

56 Multiple-Symbol Differential Sphere Decoding 16 Phase ambiguities Fix s N = 1 and start sphere decoding with i = N 1 Schnorr-Euchner search strategy Ordering of hypothetical symbols s i according to d i a) Find the phase index m i (s i = e j2πm i/m ) of the best candidate point b) Zigzag Account for finite constellation size Initial radius Works without initial radius

57 Multiple-Symbol Differential Sphere Decoding 16 Phase ambiguities Fix s N = 1 and start sphere decoding with i = N 1 Schnorr-Euchner search strategy Ordering of hypothetical symbols s i according to d i a) Find the phase index m i (s i = e j2πm i/m ) of the best candidate point b) Zigzag Account for finite constellation size Initial radius Works without initial radius

58 function MSDSD(U, M, N, R) input: N N upper triangular matrix U, constellation size M, dimension N, initial radius R output: Maximum-likelihood decision ŝ 1 d N := u NN // initialize length 2 s N := 1 // fix last component of s 3 i := N 1 // start with component i = N 1 4 k i := u (N 1)N // sum of last N i components 5 [m i, step i, n i ] = findbest (k i, u ii, M) // find best candidate point 6 loop 7 d 2 i := ki + u ii e j 2π M m i + d 2 i+1 // update squared length 8 if d i < R and n i M { // check radius and constellation size 9 s i := e j 2π M m i // store candidate component 10 if i 1 { // component 1 not reached yet 11 i := i 1 // move down 12 k i := N ι=i+1 u iιs ι // add last N i components 13 [m i, step i, n i ] := findbest (k i, u ii, M) // find best candidate point 14 } 15 else { // first component reached 16 ŝ := s // best point so far 17 R := d i // update sphere radius 18 i := i + 1 // move up 19 [m i, step i, n i ] := findnext (m i, step i, n i ) // next point examined for component i 20 } 21 } 22 else { 23 do { 24 if i == N 1 return ŝ and exit // outside sphere and no component left 25 i := i + 1 // move up 26 } while n i == M // while all constellation points examined 27 [m i, step i, n i ] := findnext (m i, step i, n i ) // next point examined for component i 28 } 29 goto loop subfunction ([m i, step ( i, n i ])) = findbest (k i, u ii, M) // Finds best MPSK signal point F1-1 p := M 2π angle k i u ii // unconstrained phase index (p IR) F1-2 m i := p // constrained phase index (m i Z) F1-3 n i := 1 // counter of examined candidates F1-4 step i := sign(p m i ) // step size for phase index subfunction [m i, step i, n i ] = findnext (m i, step i, n i ) // Selects next MPSK signal point // according to Schnorr-Euchner strategy F2-1 m i := m i + step i // zig-zag through MPSK constellation F2-2 step i = step i sign(step i ) // update step size F2-3 n i := n i + 1 // count examined candidates

59 Multiple-Symbol Differential Sphere Decoding 18 Example

60 Multiple-Symbol Differential Sphere Decoding 19 MSDSD vs. DF-DD u iι = r ι p N i ι i /σ e,n i s 1 s 2. p i ι : σe,i 2 : ιth coefficient, p i 0 1 error variance of an ith order linear backward minimum mean-squared error (MMSE) predictor for the discrete-time random process h[k] + n[k] s N 1 s N Estimation of s i based on (tentative) decisions ŝ l, i + 1 l N, can be interpreted as linear-prediction based DF-DD with window size N i + 1

61 Outline 19 Transmitter: Differential Encoding Receiver: Multiple-Symbol Differential Detection Multiple-Symbol Differential Sphere Decoding Performance Results Summary

62 Performance Results 20 Figures of merit Power efficiency Computational complexity System parameters Fading according to Clarke s model with normalized bandwidth B f T = and 8DPSK Comparison with Mackenthun s algorithm (MA) Prediction-based DF-DD Xiaofu and Songgeng s algorithm (X&S-A) (only 4PSK)

63 Performance Results 20 Figures of merit Power efficiency Computational complexity System parameters Fading according to Clarke s model with normalized bandwidth B f T = and 8DPSK Comparison with Mackenthun s algorithm (MA) Prediction-based DF-DD Xiaofu and Songgeng s algorithm (X&S-A) (only 4PSK)

64 Performance Results 20 Figures of merit Power efficiency Computational complexity System parameters Fading according to Clarke s model with normalized bandwidth B f T = and 8DPSK Comparison with Mackenthun s algorithm (MA) Prediction-based DF-DD Xiaofu and Songgeng s algorithm (X&S-A) (only 4PSK)

65 Performance Results MA 10 2 BER > N=2 DF-DD 10 3 coherent DPSK N=2 N=6 N=10 coherent MSDSD log 10 ( E b /N 0 ) [db] >

66 Performance Results DPSK 42 N = 6 N = 8 required 10log 10 ( E b /N 0 ) [db] > N = 10 N = 16 N = numerical results simulation results coherent position i > (see paper for details of subset MSDSD and error-rate analysis)

67 Performance Results DPSK 8DPSK average # of multiplications > 10 3 lower bound for MSDSD MSDSD DF-DD MA log 10 ( E b /N 0 ) [db] >

68 Performance Results DPSK 10log 10 ( E b /N 0 ) = 10 db 10log 10 ( E b /N 0 ) = 20 db 10log 10 ( E b /N 0 ) = 30 db 10log 10 ( E b /N 0 ) = 40 db log N 1 (average # of multiplications) MSDSD DF-DD X&S-A 2.5 MA lower bound for MSDSD Observation window length N

69 Performance Results 25 4DPSK 10log 10 (E b /N 0 )=20dB 10log 10 (E b /N 0 )=30dB 10log 10 (E b /N 0 )=40dB 10 2 BER > 10 3 BER and # of multiplications for MSDSD without limitation average # maximum # maximum allowed # of multiplications >

70 Outline 26 Transmitter: Differential Encoding Receiver: Multiple-Symbol Differential Detection Multiple-Symbol Differential Sphere Decoding Performance Results Summary

71 Summary 27 Low-complexity ML MSDD has been a long-standing problem Solution via application of sphere decoding Multiple-symbol differential sphere decoding (MSDSD) Expected complexity is orders of magnitudes below that of brute-force search Excellent performance versus complexity trade-off Gains in power efficiency almost for free Variations of MSDSD Subset MSDSD MSDSD with limited maximum complexity MSDSD with different initialization

72 Summary 27 Low-complexity ML MSDD has been a long-standing problem Solution via application of sphere decoding Multiple-symbol differential sphere decoding (MSDSD) Expected complexity is orders of magnitudes below that of brute-force search Excellent performance versus complexity trade-off Gains in power efficiency almost for free Variations of MSDSD Subset MSDSD MSDSD with limited maximum complexity MSDSD with different initialization

73 Summary 27 Low-complexity ML MSDD has been a long-standing problem Solution via application of sphere decoding Multiple-symbol differential sphere decoding (MSDSD) Expected complexity is orders of magnitudes below that of brute-force search Excellent performance versus complexity trade-off Gains in power efficiency almost for free Variations of MSDSD Subset MSDSD MSDSD with limited maximum complexity MSDSD with different initialization

74 Summary 27 Low-complexity ML MSDD has been a long-standing problem Solution via application of sphere decoding Multiple-symbol differential sphere decoding (MSDSD) Expected complexity is orders of magnitudes below that of brute-force search Excellent performance versus complexity trade-off Gains in power efficiency almost for free Variations of MSDSD Subset MSDSD MSDSD with limited maximum complexity MSDSD with different initialization

ML Detection with Blind Linear Prediction for Differential Space-Time Block Code Systems

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