Wireless Localization: Ranging (second part)

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Wireless Localization: Ranging Stefano Severi and Giuseppe Abreu [email protected] School of Engineering & Science - Jacobs University Bremen October 7, 2015

Transcript of Wireless Localization: Ranging (second part)

Wireless Localization: Ranging

Stefano Severi and Giuseppe [email protected]

School of Engineering & Science - Jacobs University Bremen

October 7, 2015

AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Phase-Difference RangingBasic Principle

x(t) = A0 cos (2πf1t+ ϕA).

y(t) = B0 cos (2πf1t+ ϕB).

ϕ1 = ϕB − ϕA.

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Phase-Difference RangingFrequency shifting

f1 = c/λ1,

ϕ1 = 2π

(2d

λ1−N1

)= 2π

(2f1d

c−N1

),

f2 = f1 + ∆f ,

ϕ2 = 2π

(2d

λ2−N2

)= 2π

(2f2d

c−N2

),

ϕ2 − ϕ1 = ∆ϕ =4πd∆f

c,

d =c

∆ϕ

∆f.

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Ranging for Indoor LocalizationSmart Ranging

Current localization solutions:

GNSS and cellular based,

otherwise fragile and underdeployed,

still suffering multipath and NLOS conditions.

Furthermore, mainly based on triangulation/trilateration:

requires point-to-point measurements,

pairwise communication,

overhead and redundancy.

They are far from being optimal!

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Ranging for Indoor LocalizationIncreasing the Efficiency

Superresolution Multipoint Ranging with Optimized Samplingvia Orthogonally Designed Golomb Rulers [1].

Exploit the differential nature of measurements,

avoid broadcast→ measurement toward anchor nodes,

orthogonal Golumb Ruler design,

optimized genetic algorithm for Golumb Ruler generation,

already implemented on 802.15.4-based commercialsolution (ToA and PDoA)!

[1], Oshiga O., Severi S., Abreug G.T.F, "Superresolution Multipoint Ranging with Optimized Sampling viaOrthogonally Designed Golomb Rulers", to appear on IEEE Transactions on Wireless Communications.

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Superresolution RangingUniform Set of adjacent Frequencies

Let us consider a uniform set of adjacent frequencies:

F , {f1, · · · , fn}, with fn = (n− 1)∆f + f1, (1)

The corresponding phase estimates at the respective frequencies are

ϕ = {ϕ1, · · · , ϕn}. (2)

and, in turn, the phase differences

ϕi+1 − ϕ1, with i = 1, · · · , n− 1 (3)

can be put in the vector

∆Φ = {∆ϕ1, · · · ,∆ϕn−1}. (4)

Matlab TipTo obtain the phase differences, use the command:phi = unwrap(phi);dphi = phi(2:end,:)-repmat(phi(1,:),length(freq)-1,1);Use every four columns to obtain the phase difference dphi.

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Superresolution RangingThe Steering Vector

Taking the vector ∆Φ as the argument of the element-wise complex

exponential function g(x) = exp(jx), we obtain

g(∆Φ) = [ej∆ϕ1 , · · · , ej∆ϕn−1 ]T = [ej4πd∆f

c , · · · , ej4πd(n−1)∆f

c ]T .(5)

One can immediately recognize from the above, the similarity between the

vector g(∆Φ) and the steering vector of a linear antenna array

[TUNCER09].

Steering Vector

A steering vector represents the set of phase delays a plane waveexperiences, evaluated at a set of array elements (antennas). The phasesare specified with respect to an arbitrary origin.

[TUNCER09 ] T. Tuncer and B. Friedlander, “Classical and Modern Direction-of-Arrival Estimation”. Elsevier

Science, 2009.

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Superresolution RangingThe Sample Array Covariance Matrix

The Sample Array Covariance Matrix Fundamental Property

The space spanned by its eigenvectors are partitioned into two orthogonalsubspaces, namely the signal plus noise subspace and the noise onlysubspace; the steering vectors corresponding to the direction of the signalare orthogonal to the noise subspace assuming they are uncorrelated.

Rx =1

K

K∑k=1

g(∆Φ(k))g(∆Φ(k))H (6)

where K is the number of snapshots and MH stands for the transpose of

the matrix M.

Matlab TipTo obtain the sample covariance matrix R, use the command:x = exp(1j*dphi.’);Rx = x’*x;

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Superresolution RangingSignal- and Noise-Subspace

Define M = n− 1 and make the following assumptions:

rank(g(∆Φ(k))

)= 1,

uniformity of the set of adjacent frequencies F .

The eigendecomposition of the sample covariance matrix Rx gives:

Rx = essΛsseHss︸ ︷︷ ︸

signal-subspace

+ EnsΛnsEHns︸ ︷︷ ︸

noise-subspace

(7)

where the sample eigenvalues are sorted in descending order and the

matrices ess , [e1][M×1] and Ens , [e2, · · · , eM ][M×M−1] contain in

their columns the signal- and noise-subspace eigenvectors of Rx

respectively.

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Superresolution RangingSignal- and Noise-Subspace

The basis of noise-subspace Ens formed by the (M − 1) eigenvectorsassociated with the (M − 1) smallest eigenvalues are orthogonal to thecomplex exponential steering vector.

Therefore we can write this in mathematical terms:

g(∆Φ(k)) ⊥ Ens, (8)

or equivalently:

g(∆Φ(k))HEnsEHnsg(∆Φ(k)) = 0. (9)

Super-Resolution GoalFrom g(∆Φ(k)) we can construct Rx and, in turn, obtain Ens: the goal isnow to get a good estimate of the true steering vector g(∆Φ(k)).

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Golomb RulerA Nice but Challenging Tool

It’s a ruler without 2 pairs of marks at the same distance.

Example of perfectGolomb ruler.

No perfect ruler exists beyond 4marks,

optimal→ no shorter of the sameorder exists,

to find optimal high-order GR is aNP-problem,

to find orthogonal high-order GRs→ unsolved.

new genetic algorithm→ proposedapproach.

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Algorithm EvaluationCRLB comparison

Example of perfect Golomb ruler.Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 12/18

AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Spectral MusicMultiple Signal Classification (MUSIC)

Recalling the full formulation of the measured steering vector

g(∆Φ) = [ej4πd∆f

c , · · · , ej4πdM∆f

c ]T ,

the spectral MUSIC estimates the distance d between the source s andtarget t from the minimum of the function

f(d) = g(∆Φ(k))HEnsEHnsg(∆Φ(k)) (10)

by searching over d using a fine grid as it exploits the orthogonality in eq.

(9).

Matlab TipTo define the fine grid, use the command:

RangeD = 0:01:50;

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Root MusicPolynomial expansion of MUSIC

Root MUSIC replaces the search of the minimum of f(d) by polynomial

rooting, and its only solution is the distance estimate d between the source

s and target t.

The M × 1 complex exponential vector can be written as:

g(∆Φ(k)) = [ej4πd∆f

c , · · · , ej4πdM∆f

c ]T (11)

= [z1, · · · , zM ]T ,

where zi , ej4πdi∆f/c, and ∆f is the common uniform inter-frequency

spacing.

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Root MusicPolynomial expansion of MUSIC

The function f(d) is instead rewritten as:

f(d) = g(z−1)T EnsEHnsg(z) , f(z)

= [z−1, · · · , z−M ]

e11 · · · e1M... · · ·

...

eM1 · · · eMM

z1

...

zM

,f(z) =

(M−1)∑l=−(M−1)

alzl, (12)

where al is the sum of the lth diagonal entries of EnsEHns.

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Root MusicPolynomial expansion of MUSIC

The polynomial f(z) has (2M − 1) unitary modulus roots which form

conjugate reciprocal pairs

z = ej4πd∆f/c and z = e−j4πd∆f/c. (13)

Due to the presence of noise, the root locations are distorted and the root

corresponding to the true distance d does not lie on the unit circle.

Therefore, the Root-Music computes all roots of f(z) and estimates the

distance d by selecting the largest-magnitude root from those lying inside

the circle.

Matlab TipTo obtain the distances for Music and Root-Music and the function in Eq.(10), use the command:[EstDistMusic f] = MusicSpectrum(Rx,1,1,RangeD);EstDistRMusic = RootMUSIC(Rx,1);

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AccurateRangingRecap

Steering Vector

Golomb Ruler

Spectral Music

Root Music

Report 2/3Ranging

Complete the lab experience writing a report with:

1 plot 4 different phase measurements ϕ (dphi(:,[a:b])) of all

frequencies for both cable and wireless measurements.

2 compute the distance estimates using Music and Root-Music algorithms

using every four phase measurements (400/4 = 100 distance estimates).

3 plot the f from the Music algorithm for one distance estimate.

Please print and deliver the report within the aforementioned deadline to

[email protected],

[email protected].

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Thank you!