On the effect of random Snapshot timing jitter on the covariance Matrix for JADE estimation

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On the Effect of Random Snapshot Timing Jitter on the Covariance Matrix for JADE Estimation

A. BAZZI^1,2 D.T.M. SLOCK¹ L. MEILHAC²

1EURECOM-Mobile Communications Department

2CEVA-RivieraWaves

EUSIPCO, 2015

A. BAZZI, D.T.M. SLOCK, L. MEILHAC EUSIPCO, 2015 1 / 37

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Outline

1 Introduction

2 The System Model Snapshot

Analytical Formulation Assumptions

Problem Statement

3 Jitter Effect

Jitter Effect on Covariance Matrix

Consequence on Estimating number of Multipath Components

4 Estimation/Compensation of Jitter LS approach

Negative Eigenvalues approach

5 JADE (2D-MUSIC) No Jitter

Presence of Jitter Compensating for Jitter

6 Conclusion

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Intro

Indoor Localisation:

1 Radio-Based: Online estimation of AoA (Angle of Arrival) and/or ToA (Time of Arrival) , etc.

2 Fingerprinting: Offline training for Online use.

We focus on Radio-Based, and in particularJADE. Linear Model: x^(l) =Aγ^(l)+n^(l).

AoA andToAinformation found inA.

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Intro

We focus on Radio-Based, and in particularJADE. Linear Model: x^(l) =Aγ^(l)+n^(l).

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Intro

We focus on Radio-Based, and in particular JADE.

Linear Model: x^(l) =Aγ^(l)+n^(l). AoA andToAinformation found inA.

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Intro

Linear Model: x^(l)=Aγ^(l)+n^(l).

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Intro

Linear Model: x^(l)=Aγ^(l)+n^(l). AoA andToAinformation found inA.

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Intro

Various algorithms to estimate parameters inA:

1 Maximum Likelihood methods.

2 Subspace methods (2D-MUSIC/2D-ESPRIT/etc.)

It turns out that all existing algorithms are Model-Sensitive. In other words, such algorithms fail when model is

x^(l)=C^(l)Aγ^(l)+n^(l) with unknownC^(l).

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Intro

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Intro

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Intro

It turns out that all existing algorithms are Model-Sensitive.

In other words, such algorithms fail when model is x^(l)=C^(l)Aγ^(l)+n^(l) with unknownC^(l).

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Intro

It turns out that all existing algorithms are Model-Sensitive.

In other words, such algorithms fail when model is x^(l)=C^(l)Aγ^(l)+n^(l) with unknownC^(l).

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Outline

1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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Snapshot Illustration

OFDM Symbol

Transmi2er (TX) Mul;path Channel Receiver (RX)

. . . . . . . Antenna 1

Antenna N

s(t)= bm m=0 M−1

∑

^e^j2πmΔ^f^t

Channel'composed'of'q'paths.'' The'i^th'path'is'characterized'by:'

• Complex'amplitude'!_!^(!)'

• Time'of'Arrival'(ToA)'!!'

• Angle'of'Arrival'(AoA)'!!' '

!

!(!)!is!the!transmit!OFDM!symbol!

composed!of!!!subcarriers.!

!

(γ1 (l),τ1,θ1)

(γq (l),τq,θq)

. . . . .

r₁^(l)(t)

r_N^(l)(t) The$n^th$antenna$receives$a$sum$$

of$the$q#paths,$!_!^!(!).$

s(t)

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Snapshot Illustration (cont’d)

r_N^(l)(t) Collect

M

samples FFT

Sampler

. . . . δ^(l)

! x_MN−(M−1)^(l)

! x_MN^(l)

r₁^(l)(t) Collect

M

samples FFT

Sampler

. . . . δ^(l)

! x₁^(l)

! x_M^(l⁾

. . . . . .

. . . . . . .

. . . . .

. . . . . . .

. . . . .

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Outline

1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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Analytical Formulation

System Model

˜

x^(l)=C^(l)Aγ^(l)+n^(l), l= 1. . . L

where

A= [a(θ1)⊗c(τ1), . . . ,a(θq)⊗c(τq)]

a(θ) = [a1(θ), . . . ,aN(θ)]^T depends on array geometry (ex: ULA). c(τ) = [1, e^−j2πτ^M^f, . . . , e^{−j2πτ(M−1)}^M^f]^T

C^(l)=I_N ⊗diag{c(δ^(l))} γ^(l)= [γ₁^(l). . . γq^(l)]^T

n^(l) is anM N×1background noise vector.

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Analytical Formulation

System Model

˜

x^(l)=C^(l)Aγ^(l)+n^(l), l= 1. . . L

where

A= [a(θ1)⊗c(τ1), . . . ,a(θq)⊗c(τq)]

a(θ) = [a1(θ), . . . ,aN(θ)]^T depends on array geometry (ex: ULA).

c(τ) = [1, e^−j2πτ^M^f, . . . , e^{−j2πτ(M−1)}^M^f]^T C^(l)=I_N ⊗diag{c(δ^(l))}

γ^(l)= [γ₁^(l). . . γq^(l)]^T

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Analytical Formulation

System Model

˜

x^(l)=C^(l)Aγ^(l)+n^(l), l= 1. . . L

where

A= [a(θ1)⊗c(τ1), . . . ,a(θq)⊗c(τq)]

c(τ) = [1, e^−j2πτ^M^f, . . . , e^{−j2πτ(M−1)}^M^f]^T

C^(l)=I_N ⊗diag{c(δ^(l))} γ^(l)= [γ₁^(l). . . γq^(l)]^T

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Analytical Formulation

System Model

˜

x^(l)=C^(l)Aγ^(l)+n^(l), l= 1. . . L

where

A= [a(θ1)⊗c(τ1), . . . ,a(θq)⊗c(τq)]

γ^(l)= [γ₁^(l). . . γq^(l)]^T

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Analytical Formulation

System Model

˜

x^(l)=C^(l)Aγ^(l)+n^(l), l= 1. . . L

where

A= [a(θ1)⊗c(τ1), . . . ,a(θq)⊗c(τq)]

γ^(l)= [γ₁^(l). . . γq^(l)]^T

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Analytical Formulation

System Model

˜

x^(l)=C^(l)Aγ^(l)+n^(l), l= 1. . . L

where

A= [a(θ1)⊗c(τ1), . . . ,a(θq)⊗c(τq)]

γ^(l)= [γ₁^(l). . . γq^(l)]^T

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Outline

1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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Assumptions

1 q < M N

2 The matrix Ahas full column rank.

3 The entries of the multipath vectorγ^(l) are not fully correlated, i.e.

Rγγ is full rank.

4 n^(l) ∼ N(0, σ_n²I_{M N}) indep from signal partC^(l)Aγ^(l)

5 δ^(l)∼ N(0, σ_δ²) indep fromγ^(l) andn^(l).

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Outline

1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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Problem Statement

Given the data n x^(l)oL

l=1, estimate:

the jitter varianceσ²_δ.

the number of multipath components q. the times and angles of arrival

θ_i, τ_i ^q_i=1.

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Problem Statement

l=1, estimate:

the number of multipath components q.

the times and angles of arrival

θ_i, τ_i ^q_i=1.

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Problem Statement

l=1, estimate:

the number of multipath components q.

the times and angles of arrival

θ_i, τ_i ^q_i=1.

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Outline

1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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Jitter Effect on Covariance Matrix

R˜x˜x,El{˜x^(l)˜x^(l)^H}=ΥRxx

where

Rxx is the covariance matrix of datawithout jitter, i.e.

x^(l)=Aγ^(l)+n^(l)

Υ=JN ⊗T with

JN is the all-ones square matrix of sizeN. T^hm,ni=e⁻

2πMf(m−n)σδ²

2

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Jitter Effect on Covariance Matrix

R˜x˜x,El{˜x^(l)˜x^(l)^H}=ΥRxx

where

Rxx is the covariance matrix of datawithout jitter, i.e.

x^(l)=Aγ^(l)+n^(l)

Υ=JN ⊗T with

JN is the all-ones square matrix of sizeN. T^hm,ni=e⁻

2πMf(m−n)σδ²

2

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Outline

1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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Consequence on Estimating q

No Jitter

Assume no jitter, i.e. the model is:

x^(l)=Aγ^(l)+n^(l) ⇒R_xx=AR_γγA^H+σ_n²I_{M N} (Using Assumption 4).

Eigenvalues of R_xx

1 The matrix AR_γγA^H is of rank q (Under Assumptions 1-3).

2 Sorting the non-zero eigenvalues ofAR_γγA^H in decreasing order (λ1 ≥. . .≥λq), the q large eigenvalues ofRxx are {λ_i+σ²_n}^q_i=1 and its complementaryM N −q eigenvalues are equal to σ_n².

Conclusion

This means that, under high SNR, one could estimate the number of sources using the q largest eigenvalues of Rxx.

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Consequence on Estimating q (cont’d)

Theorem (C. S. Ballantine¹)

Let E andFbem×nmatrices. If Ais positive definite and Bis positive semi-definite with r positive diagonal elements, then rank(AB) =r.

1

1C. S. Ballantine;On the Hadamard product;Mathematische Zeitschrift 105;

1968;365-366.

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Consequence on Estimating q (cont’d)

Presence of Jitter

˜

x^(l)=C^(l)Aγ^(l)+n^(l)

⇒R_x˜_˜_x=ΥRxx

⇒R_x˜_˜_x=ΥAR_γγA^H +σ_n²I_{M N}

Conclusion

Assuming one antenna (N = 1) and using the previous theorem, the rank of ΥARγγA^H is M for anyq.

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Outline

1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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LS approach

Definition

Let B be anM N×M N matrix, i.e. B=







B_1,1 · · · B_1,N ... . .. ... B_N,1 · · · B_N,N





, where Bi,j is the(i, j)^th block matrix of sizeM ×M.

We define the following function:

F^B(k) = _2(M−k)N¹ 2

P

|m−n|=k i=1...N j=1...N

|B^hm,ni_i,j |, k= 1. . . M−1

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LS approach (cont’d)

Recall that

Rx˜˜x=ΥRxx = (JN ⊗T)Rxx

where

JN is the all-ones square matrix of size N

T^hm,ni=e⁻

2πMf(m−n)σδ

2 2

It is easy to verify that F^R^˜^x˜^x(k) =F^R^xx(k).e⁻

(2πMf kσδ)2

2 , k= 1. . . M −1

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LS approach (cont’d)

LS Estimate

For large M, we can say that F^R^xx(k) is almost invariant compared to e⁻

(2πMf kσδ)2

2 , i.e. F^R^xx(k) =F^R^xx. We have:

f =Pg where:

f =

log F^R^˜^x˜^x(1)

· · ·log F^R^˜^x˜^x(M−1)T

P=







−^(2πM₂^f⁾² 1 ... ...

−^(2π(M^−1)M₂ ^f⁾² 1







g=

σ²_δ log F^R^xx

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LS approach (cont’d)

LS Estimate

The LS estimate is obtained by solving:

ˆ

g^LS = arg min

g

kf−Pgk²₂ = (P^HP)⁻¹P^Hf =

σˆ_δ² log Fˆ^R^xx

Compensate

1 Form the matrixΥˆ = (J_N⊗T), whereˆ

Tˆ^hm,ni=e⁻

2πMf(m−n)ˆσδ

2 2

2 Compensate by:

Rˆ_xx =Υˆ^HR_˜_x˜_x

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LS approach (cont’d)

k

0 10 20 30 40 50 60

FR(k) in (dB)

2 4 6 8 10 12 14 16 18 20 22

F^R^xx(k) F^R^~^x~^x(k) F^R^{^}^xx(k) H(,k)

Simulation parameters q = 15paths.

M = 52subcarriers and N = 3 antennas.

SNR = 20 dB andL= 100Snapshots.

Mf= 0.3125MHz

σ_δ = 25nsec, we getσˆ_δ=27.2 nsec

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LS approach (cont’d)

EigenValue Index

0 20 40 60 80 100 120 140 160

Magnitude

-1000 0 1000 2000 3000 4000 5000 6000

6_R_xx 6R~x~x

6_R_^

xx

The Over-Compensated Case σ_δ = 25nsec<σˆδ =27.2 nsec.

The ”over-compensated” covariance matrix Rˆxx admits negative eigenvalues.

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Outline

1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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Negative Eigenvalues approach

Idea

Search for the smallest value of σ˘_δ∈[0, S]that yields

λ˘min=λmin R˘xx

<0

where

R˘_xx =Υ˘^H R_˜_x˜_x

Υ˘ =J_N⊗T,˘ T˘^hm,ni =e⁻

2πMf(m−n)˘σδ

2 2

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Negative Eigenvalues approach (cont’d)

6</in nsec

0 20 40 60 80 100 120

66min

-400 -350 -300 -250 -200 -150 -100 -50 0

7

</= 20nsec 7

</= 40nsec 7

</= 60nsec 7

</= 80nsec

Simulation Parameters

Same as before, but differentσ_δ for each curve.

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Outline

1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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JADE with No Jitter

Simulation Parameters

M = 52,N = 3, SNR= 20dB, L= 100 q = 3paths:

1 (θ1, τ1) = (−40^◦,0nsec)

2 (θ2, τ2) = (0^◦,30nsec)

3 (θ3, τ3) = (70^◦,95nsec)

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Outline

1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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JADE in the Presence of Jitter

Same parameters as previous slide but σδ= 95nsec

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Outline

1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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JADE after Compensating for Jitter

Compensate then JADE

True jitter variance: σ_δ= 95nsec Estimated jitter variance: σˆδ= 99nsec

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1 Introduction

Problem Statement

3 Jitter Effect

6 Conclusion

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QUESTIONS ?