μ μ μ

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

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

Time-Delay Estimation Bias / T c 0.4 0.3 0.2 0.1 0-0.1-0.2-0.3 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) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 Refl. Atten.: -3dB Refl. Atten.: -5dB Refl. Atten.: -10dB -0.4 0 0.5 1 1.5 2 2.5 3 Delay of the Reflection / T c 0 0 0.5 1 1.5 2 2.5 3 Delay of the Reflection / T c δ

δ

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

{ * } E e[ n] e [ l] = Qδ nl,

( 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) + ( ) *

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

{ } { * 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 2 0 2 d 1 e jω τ jω τ N 0 N d 1

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

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

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

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 )

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

± λ δ

RMS Time-Delay Estimation Error / T c 10 0 10-1 10-2 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) 0 2 4 6 8 10 12 14 16 18 20 Number of Pulses

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 10-2 -50-40 -30-20 -10 0 10 20 30 40 Signal to Interference Ratio (db), with respect to the first signal

RMS Time-Delay Estimation Error / c T 10 0 10-1 10-2 CRB Asymp. Efficient Estim., IQML, τ, W) g( Consistent Estim., ESPRIT,g( τ, I) w White-noise Estim., IQML, ( τ) f 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Delay of the Second Replica / T c

RMS Time-Delay Estimation Error / c T 10 0 10-1 10-2 CRB Asymp. Efficient Estim., IQML, τ, W) g( Consistent Estim., ESPRIT,g( τ, I) w White-noise Estim., IQML, ( τ) f 0 10 20 30 40 50 60 DOA of the Second Replica (degrees)

S τ = [ s τ s τ + T s τ + d T ] ( ) ( ) ( ) ( ( 1) ) T 0 0 1 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 1 1 0 0 g g 1 0 0 0 gd gd 1 1 d d 1 = [ 1 ] g g d 1 T ( z)

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

RMS Time-Delay Estimation Error / c T 10 0 10-1 CRB Proposed estimator (coupled iter.), τ, W) g( Exact ML estimator, τ) f( w White-noise estimator, (τ) f τ τ 1 0 10-2 0 0.5 1 1.5 Delay difference between the two rays / T c

α0 = α0a0 CRB 1/2 for the Time-Delay of the Direct Signal / T c 10-1 10-2 Without knowledge of the LOSS steering vector With knowledge of the LOSS steering vector 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Delay of the Reflection / c T

T Y = α a s ( τ ) + E 0 0 0 α Γ ( τ ) = ( τ ) 1 + ( τ ) ( τ ) ( τ ) rˆ ( τ ) Wˆ ( τ ) a * 1 1 * 1 * 0 0 0 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

( * 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 * 1 0 0 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

Asymptotic Cost Function 10 2 10 1 10 0 10-1 10-2 ML ML only with temporal reference MV beamformer 10-3 -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Trial Delay / T c

( pt ) ( p T ) s( τ ) (1 δ) s + δs ( + 1) 0 0 0 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

ˆ τ, ˆ α, w = arg min w Y α s ( τ ) * T 0 0 hyb 0 0 τ, α, w 0 0 2 2 * wa 0 = 1 τ, α 0 0 τ, α 0 0

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) * * 1 0 0 0 0 yy ys 0 w hyb ( ˆ τ, ˆ α ) 0 0 ˆ τ, ˆ α 0 0

RMS Time-Delay Estimation Error / c T 10-1 10-2 CRB ML MV beamformer ML, white-noise 10-3 0 2 4 6 8 10 12 14 16 18 20 Number of Pulses

-10-15 Hybrid beamformer MV beamformer Temporal ref. beamformer Temporal ref. beamformer (ML-TEE) Cancellation of the Interference (db) -20-25 -30-35 -40-45 -50-55 0 2 4 6 8 10 12 14 16 18 20 Number of Pulses

0.16 0.02 Time-Delay Estimation Bias / T c 0.14 0.12 0.1 0.08 0.06 0.04 0.02 ML ML only with temporal reference ML for white noise ML for white noise and only temporal reference Carrier-Phase Estimation Bias (cycles) 0.01 0-0.01-0.02-0.03-0.04 0-0.02 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Delay of the Reflection / T c -0.05-0.06 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Delay of the Reflection / T c ML ML for white noise

10-1 RMS Time-Delay Estimation Error / T c 10-1 10-2 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 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Delay of the Reflection / T c 10-3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Delay of the Reflection / T c

5 Cancellation of the Reflection (db) 0-5 -10-15 -20-25 -30 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 0 10 0 10-1 10-2 β(α 0,τ 0 ) α 0-35 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Delay of the Reflection / T c 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Delay of the Reflection / T c

0.1 10 0 0 Time-Delay Estimation Bias / T c -0.1-0.2-0.3-0.4 ML ML only with temporal reference MV beamformer ML for white noise -0.5-20 -10 0 10 20 30 40 50 Signal to Noise Ratio (SNR) of the direct signal (db) RMS Time-Delay Estimation Error / T c 10-1 10-2 10-3 CRB for the detailed model CRB for the simplified model ML ML only with temporal reference MV beamformer ML for white noise 10-4 -20-10 0 10 20 30 40 50 Signal to Noise Ratio (SNR) of the direct signal (db)

0.07 Time-Delay Estimation Bias / T c 0.06 0.05 0.04 0.03 0.02 0.01 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 0 10 0 10-1 10-2 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 0 0 20 40 60 80 100 120 Iterations 0 10 20 30 40 50 60 70 80 90 100 Iterations

σ 2 TE = ( * )( * 1 d τ ) 0 Ps ( τ ) d τ0 α0q α0 0 1 2 ( ) ( ) d( τ ) 0 d ( τ 0) = s dτ 0 σ 2 MV = ar QR a * 1 1 0 yy yy 0 ( τ ) s τ τ 2 d ( ) P d( ) a R α * * 1 0 ( ) 0 0 yy 0 0 2 σ aq a * 1 2 0 0 ML = * * 1 2 d ( 0 ) P ( ) d( 0) a0q a0 ( τ ) s τ τ 0 2

100 90 RMS Time-Delay Estimation Error / T c 10-1 10-2 2 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 (%) 80 70 60 50 40 30 20 ML estim. Λ ml (τ) MV beam. estim. Λ mv (τ) Robust ML estim. Ψ(τ) 10-20 -15-10 -5 0 5 10 15 20 Pointing Error (degrees) 0-20 -15-10 -5 0 5 10 15 20 Pointing Error (degrees)

τ 1 y() i

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

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

{ } 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

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

Wˆ = Wˆ + λi d # Ŵ

λ ft = 210 3 d

Timing RMSE (chips) 10-1 10-2 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 0 50 100 150 200 250 300 350 400 450 500 M, Length of the training sequence (bits)

Probability of acquisition 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 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-3 0.2 0 5 10 15 20 25 30 35 40 K, Number of users

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) 10-1 10-2 -40-30 -20-10 0 10 20 Signal to External Interference Ratio SIR (db)

1 0.9 Probability of acquisition 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 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-3 0-10 -5 0 5 10 15 20 25 30 35 Near-Far Ratio (db)

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

8 x 10-3 7.5 CRB ML ML with linear approximation RMS Time-Delay Estimation Error / T c 7 6.5 6 5.5 5 4.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Spacing of the linearization grid, T 0 / T c

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

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 1 2 0 1 x x 2 1+ x +