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6 μ μ μ

7 s t j2 fct T () = a() t e π s t ka t e e j2π fct j2π fcτ0 R() = ( τ0)

9 xt () = α 0 dl () pt ( lt) + wt () l wt () N 2 (0, σ )

10 Time-Delay Estimation Bias / T c In-phase reflection Out-of-phase reflection Refl. Atten.: -3dB, δ=0.5 Refl. Atten.: -3dB, δ=0.05 Refl. Atten.: -10dB, δ=0.5 Refl. Atten.: -10dB, δ=0.05 Carrier-Phase Estimation Bias (cycles) Refl. Atten.: -3dB Refl. Atten.: -5dB Refl. Atten.: -10dB Delay of the Reflection / T c Delay of the Reflection / T c δ

12 δ

15 Y= AS() τ + E m N [ ] A = α α α m d 0 1 d 1 S() τ st ( s τ0) s(2 Ts τ0) snt ( s τ0) st ( τ ) s(2 T τ ) snt ( τ ) st ( s τd 1) s(2 Ts τd 1) snt ( s τd 1) s 1 s 1 s 1 = d N st () N

16 { * } E e[ n] e [ l] = Qδ nl,

17 ( τ ) { 1 } f (, τ AQ, ) = lnq+ Tr C, A Q ( τ ) = ˆ ˆ ( τ) ˆ ( τ) + ˆ ( τ) * * *, yy ys ys ss C A R AR R A AR A f 1 * ( ) ( ) ( ) (, τ A) = lnrˆ Rˆ τ Rˆ τ Rˆ τ yy ys ss ys ( ˆ ( ) ˆ 1( )) ˆ 1 ( ) ˆ ( ) ˆ A Rys τ Rss τ Rss τ A Rys τ Rss ( τ) + ( ) *

18 f () τ = ln Rˆ Rˆ () τ Rˆ () τ Rˆ () τ = ln Q () τ 1 * ˆ yy ys ss ys ML Qˆ () τ ML V () τ = ln I B() τ ˆ ˆ ˆ ˆ ˆ 1 B() τ = R R () τ R () τ R () τ R = Rˆ YP Y Rˆ S N 1/2 1 * 1/2 1/2 * 1/2 yy ys ss ys yy yy * ( τ) yy

19 { } { * Qˆ } ML YP Y w f ( τ) = Tr ( τ) = const-tr * S ( τ ) Qˆ () τ ML P S * ( τ ) jωτ 1 0 jωτ 1 d 1 S () τ = S * * ω e e e jω τ e e jω τ d 1 e jω τ jω τ N 0 N d 1

20 op { W } op B g (, W ) = Tr () τ τ ( ) 1 () i () i 1/2 g (, τ Wop ) = V () τ + op ( N ) g ( τ, W ) = V ( τ) + o (1) ( ij) ( ij) op p W = I B( τ ) ˆτ g(,) τ I w f () τ op

21 1 { } { 2 } 1 { 3 V () τ = ln I B() τ = Tr B() τ Tr B () τ Tr B () τ } 2 3 1/2 1/2 lim B( τ) = I Ryy QRyy 0 N V {( ˆ ˆ ) } () τ Tr I+ B() τ + B () τ + B () τ = g ( τ, W) () i 2 () i () i ˆ

22 V ( λ ) V () τ = ln I B() τ = ln 1 () τ k k = 1 m () i m () i () i λk τ λk τ () i () () () τ = g (, ˆ ) 1 λ ( ) τ τ 1 λ ( τˆ ) W k= 1 k k= 1 m () i () i { } λ g λ = = = g(,) τ I Tr B() τ () τ (,) τ I () τ m k k= 1 k= 1 k m k

23 V () τ = ln I B() τˆ + B() τˆ B() τ 1 ( ) ( ) = ln I B() τˆ + ln I+ I B() τˆ B() τˆ B() τ Ŵ { ˆ } { } V() τ ln I B() τˆ + Tr WB ˆ () τ Tr WB ˆ () τ = const + g(, τ Wˆ )

24 1 W = W Rˆ YP Y Rˆ W S ( τ) N { 1/2 1/2 * 1/2 1/2 } g( τ, ) Tr ýy * ýy 1 ( * * 1 ) 1 * * * = 1 = S ( τ) S ω ω ω ω ω G P P S G G S S G G S

25 ± λ δ

26 RMS Time-Delay Estimation Error / T c CRB Asymp. Efficient Estim., IQML, g( τ, W) Consistent Estim., ESPRIT,g(τ, I) w White-noise Estim., IQML, f ( τ) Asymp. Efficient Estim., IQML ("2 iter"), g( τ, W) Number of Pulses

27 10 0 RMS Time-Delay Estimation Error / c T 10-1 CRB Asymp. Efficient Estim., IQML, τ, W) g( Consistent Estim., ESPRIT,g( τ, I) w White-noise Estim., IQML, ( τ) f Signal to Interference Ratio (db), with respect to the first signal

28 RMS Time-Delay Estimation Error / c T CRB Asymp. Efficient Estim., IQML, τ, W) g( Consistent Estim., ESPRIT,g( τ, I) w White-noise Estim., IQML, ( τ) f Delay of the Second Replica / T c

29 RMS Time-Delay Estimation Error / c T CRB Asymp. Efficient Estim., IQML, τ, W) g( Consistent Estim., ESPRIT,g( τ, I) w White-noise Estim., IQML, ( τ) f DOA of the Second Replica (degrees)

30 S τ = [ s τ s τ + T s τ + d T ] ( ) ( ) ( ) ( ( 1) ) T ˆ ˆ * ( τ ) N { 1/2 1/2 * g( τ, W) TrW R 1/2 1/2 } yy YP Y Ryy W g G = * = S 1 ( * * 1 ) 1 * * * = 1 = S ( τ ) S ω ω ω ω ω G P P S G G S S G G S gd gd g g gd gd 1 1 d d 1 = [ 1 ] g g d 1 T ( z)

31 d 1 d 1 1 d 0 n= 0 ( π ) d ( z) = z + g z + + g = z x exp( j2 T n/ NTs) x( τ) = exp( j2 πτ/ NT s ) ( z) g= Kt() x d t( x) = 1 x x. ( * * 1 ) GS S G 1 ω ω ( * * 1 ) GS 1 ω SωG gx x x T * (, W) = t (1/ ) K CKt( ) T

32 RMS Time-Delay Estimation Error / c T CRB Proposed estimator (coupled iter.), τ, W) g( Exact ML estimator, τ) f( w White-noise estimator, (τ) f τ τ Delay difference between the two rays / T c

34 α0 = α0a0 CRB 1/2 for the Time-Delay of the Direct Signal / T c Without knowledge of the LOSS steering vector With knowledge of the LOSS steering vector Delay of the Reflection / c T

35 T Y = α a s ( τ ) + E α Γ ( τ ) = ( τ ) 1 + ( τ ) ( τ ) ( τ ) rˆ ( τ ) Wˆ ( τ ) a * 1 1 * 1 * ˆ ˆ ˆ ys 0 W 0 P ˆ ˆ s rys 0 W 0 r ys 0 ˆ H ˆ 1 Ps a0 W ( τ 0) a0 2 Wˆ ( τ ) = Rˆ rˆ ( τ ) rˆ ( τ ) Pˆ * 1 0 yy ys 0 ys 0 s

36 ( * ˆ ˆ ˆ 1 ) Γ ( τ ) = Rˆ 1 α ( τ ) r ( τ ) R a 0 yy 0, ML 0 ys 0 yy 0 ˆ α ˆ ( τ ) ˆ ( τ ) = aw r * ys 0 0, ML ˆ * ˆ 1 P s aw 0 ( τ 0) a0 τ = ˆ τ 0 0, ML * ˆ 1ˆ 0 yy ys ( τ 0) MV ( τ 0) ˆ τ Λ ( τ ) = ar r Λ ˆ ( ) ˆ ( ) = 1 Λ ( τ ) 0, ML ML 0 * 1 P s rˆ ys τ ˆ 0 R yy r ys τ0 2 TE 0

37 Asymptotic Cost Function ML ML only with temporal reference MV beamformer Trial Delay / T c

38 ( pt ) ( p T ) s( τ ) (1 δ) s + δs ( + 1) ΛML ( τ 0) δ N 1 s( τ 0 ) = Sω u( z) z= exp( j2 πτ0 / NT s ) u() z = 1 z z T Λ ( τ ) = ML 0 u( z ) S Y Rˆ a a Rˆ YS u( z) 1 * 1 * 1 * ω yy 0 0 yy ω ( ˆ ) yy 1 * 1 * ( z ) ω N ω ( z) u S I Y R Y S u

39 ˆ τ, ˆ α, w = arg min w Y α s ( τ ) * T 0 0 hyb 0 0 τ, α, w * wa 0 = 1 τ, α 0 0 τ, α 0 0

Rˆ a βτ (, α) = 1 α arˆ r ( τ) * * 1 0 0 0

40 1 * ˆ 1 yy 0 hyb ( τ0, α0) = α0 yy ys ( τ0) + β( τ0, α0 ) * ˆ 1 ar 0 yy a0 w R r Rˆ a βτ (, α) = 1 α arˆ r ( τ) * * yy ys 0 w hyb ( ˆ τ, ˆ α ) 0 0 ˆ τ, ˆ α 0 0

41 RMS Time-Delay Estimation Error / c T CRB ML MV beamformer ML, white-noise Number of Pulses

42 Hybrid beamformer MV beamformer Temporal ref. beamformer Temporal ref. beamformer (ML-TEE) Cancellation of the Interference (db) Number of Pulses

43 Time-Delay Estimation Bias / T c ML ML only with temporal reference ML for white noise ML for white noise and only temporal reference Carrier-Phase Estimation Bias (cycles) Delay of the Reflection / T c Delay of the Reflection / T c ML ML for white noise

44 10-1 RMS Time-Delay Estimation Error / T c CRB for the detailed model CRB for the simplfied model ML ML only with temporal reference ML for white noise RMS Carrier-Phase Estimation Error (cycles) 10-2 CRB for the detailed model CRB for the simplified model ML ML for white noise Delay of the Reflection / T c Delay of the Reflection / T c

45 5 Cancellation of the Reflection (db) Hybrid beamfomer MV beamformer Temporal ref. beamformer Temporal ref. beamformer (ML-TEE) Mean and STD of the coefficients β(α 0,τ 0 ) and α β(α 0,τ 0 ) α Delay of the Reflection / T c Delay of the Reflection / T c

46 Time-Delay Estimation Bias / T c ML ML only with temporal reference MV beamformer ML for white noise Signal to Noise Ratio (SNR) of the direct signal (db) RMS Time-Delay Estimation Error / T c CRB for the detailed model CRB for the simplified model ML ML only with temporal reference MV beamformer ML for white noise Signal to Noise Ratio (SNR) of the direct signal (db)

47 0.07 Time-Delay Estimation Bias / T c Hybrid Beamformer (reflection delay ) 0.4T c Hybrid Beamformer (reflection delay ) T c ML (reflection delay 0.4T and T) c c Reflection delay 0.4 T c Mean and STD of the coefficients β(α 0,τ 0 ) and α β(α 0,τ 0 ), Reflection delay: 0.4T c α 0, Reflection delay: 0.4T c β(α 0,τ 0 ), Reflection delay: T c α 0, Reflection delay: T c Reflection delay T c Iterations Iterations

48 σ 2 TE = ( * )( * 1 d τ ) 0 Ps ( τ ) d τ0 α0q α ( ) ( ) d( τ ) 0 d ( τ 0) = s dτ 0 σ 2 MV = ar QR a * yy yy 0 ( τ ) s τ τ 2 d ( ) P d( ) a R α * * 1 0 ( ) 0 0 yy σ aq a * ML = * * 1 2 d ( 0 ) P ( ) d( 0) a0q a0 ( τ ) s τ τ 0 2

49 RMS Time-Delay Estimation Error / T c Theoretical ML σ ml 2 Theoretical MV σ mv 2 Simulated ML σ ml 2 Simulated MV σ mv Robust ML (Ψ(τ)) Percentage of outlayers (%) ML estim. Λ ml (τ) MV beam. estim. Λ mv (τ) Robust ML estim. Ψ(τ) Pointing Error (degrees) Pointing Error (degrees)

53 τ 1 y() i

54 [ α τ ] y() i = A( ) d () i + e() i i= 0 M α A( τ 1 ) : d [ ] () i = d () i d ( i 1) : T

55 E e( i) e ( l) = Qδil { * }, Q = Q I sp Q= I Q m P te

56 { } ln I+ B Tr B, B 0 α ˆ τ = arg max f ( τ ) 1 1 τ 1 Wˆ = Rˆ Rˆ Rˆ Rˆ 1 * yy yd dd yd

57 Ŵ ˆ s W Q ˆ s W Ŵ Wˆ s = P+ σ 2 I Ŵ min P, σ 2 ˆ 2 W P σ I rank { P} F = d

58 Wˆ = Wˆ + λi d # Ŵ

59 λ ft = d

60 Timing RMSE (chips) Space-Time div. fdt=0 Time div. fdt=0 Space div. fdt=0 CRB Space-Time div. fdt=2e-3 Time div. fdt=2e-3 Space div. fdt=2e M, Length of the training sequence (bits)

61 Probability of acquisition Space-Time div. fdt=0 Time div. fdt=0 Space div. fdt=0 Space-Time div. fdt=2e-3 Time div. fdt=2e-3 Space div. fdt=2e K, Number of users

62 Space-Time div. fdt=0 Time div. fdt=0 Space div. fdt=0 CRB Space-Time div. fdt=2e-3 Time div. fdt=2e-3 Space div. fdt=2e-3 Timing RMSE (chips) Signal to External Interference Ratio SIR (db)

63 1 0.9 Probability of acquisition Space-Time div. fdt=0 Time div. fdt=0 Space div. fdt=0 Space-Time div. fdt=2e-3 Time div. fdt=2e-3 Space div. fdt=2e Near-Far Ratio (db)

69 xt () = α 0 dl () pt ( lt) + wt () l wt N σ 2 () (0, ) ˆ ϕ * 0 = ylt ( + τ0) d( l) ˆ L ˆ c τ0 = d l e y lt + τ0 l L * j ˆ0 ϕ ( ) Re ( ) ( ) nc ( τ ) = y( lt + τ ) 0 0 l 2

70 8 x CRB ML ML with linear approximation RMS Time-Delay Estimation Error / T c Spacing of the linearization grid, T 0 / T c

71 σ 2 τ = 1 N { 2 ( ) } N N Λ τ 0 2 ( lim Λ ( τ )) lim E ( ) N 0 * ˆ 1ˆ Λ 0 yy ys ( τ 0) MV ( τ 0) ML ( τ ) = ar r Λ ˆ ( ) ˆ ( ) = 1 Λ ( τ ) 0 * 1 P s rˆ ys τ ˆ 0 R yy r ys τ0 2 TE 0 α 0 a 0

72 a0 = a( ρ) Ψ ( τ ) = E { Λ ( τ ; ρ) } 0 ρ ML 0 ( ρ) ˆ ( τ ) ( ρ) * 0 Λ ( τ0 ; ρ) = a W a * ˆ 1 a ρ Ryy a ( ) ( ρ) 2 ρ a( ρ) a0 + ρb0 + h x x 2 1+ x +

Σχετικά έγγραφα

l 0 l 2 l 1 l 1 l 1 l 2 l 2 l 1 l p λ λ µ R N l 2 R N l 2 2 = N x i l p p R N l p N p = ( x i p ) 1 p i=1 l 2 l p p = 2 l p l 1 R N l 1 i=1 x 2 i 1 = N x i i=1 l p p p R N l 0 0 = {i x i 0} R