Slide set presented for the Wireless Communication module at Jacobs University Bremen, Fall 2015. Teacher: Dr. Stefano Severi, assistant: Andrei Stoica

1. Wireless Localization: Ranging
Stefano Severi and Giuseppe Abreu
s.severi@jacobs-university.de
School of Engineering & Science - Jacobs University Bremen
October 7, 2015

2. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Phase-Difference Ranging
Basic Principle
x(t) = A0 cos (2πf1t + ϕA).
y(t) = B0 cos (2πf1t + ϕB).
ϕ1 = ϕB − ϕA.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 2/18

3. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Phase-Difference Ranging
Frequency 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
4π
∆ϕ
∆f
.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 3/18

4. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Ranging for Indoor Localization
Smart 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!
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 4/18

5. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Ranging for Indoor Localization
Increasing the Efficiency
Superresolution Multipoint Ranging with Optimized Sampling
via 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 commercial
solution (ToA and PDoA)!
[1], Oshiga O., Severi S., Abreug G.T.F, "Superresolution Multipoint Ranging with Optimized Sampling via
Orthogonally Designed Golomb Rulers", to appear on IEEE Transactions on Wireless Communications.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 5/18

6. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Superresolution Ranging
Uniform 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 Tip
To 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.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 6/18

7. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Superresolution Ranging
The 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
= [e
j4πd∆f
c , · · · , e
j4πd(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 wave
experiences, evaluated at a set of array elements (antennas). The phases
are specified with respect to an arbitrary origin.
[TUNCER09 ] T. Tuncer and B. Friedlander, “Classical and Modern Direction-of-Arrival Estimation”. Elsevier
Science, 2009.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 7/18

8. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Superresolution Ranging
The Sample Array Covariance Matrix
The Sample Array Covariance Matrix Fundamental Property
The space spanned by its eigenvectors are partitioned into two orthogonal
subspaces, namely the signal plus noise subspace and the noise only
subspace; the steering vectors corresponding to the direction of the signal
are orthogonal to the noise subspace assuming they are uncorrelated.
ˆRx =
1
K
K
k=1
g(∆ ˆΦ(k))g(∆ ˆΦ(k))H
(6)
where K is the number of snapshots and MH
stands for the transpose of
the matrix M.
Matlab Tip
To obtain the sample covariance matrix R, use the command:
x = exp(1j*dphi.’);
Rx = x’*x;
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 8/18

9. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Superresolution Ranging
Signal- and Noise-Subspace
Define M = n − 1 and make the following assumptions:
rank g(∆ ˆΦ(k)) = 1,
uniformity of the set of adjacent frequencies F.
The eigendecomposition of the sample covariance matrix ˆRx gives:
ˆRx = ˆess
ˆΛssˆeH
ss
signal-subspace
+ ˆEns
ˆΛns
ˆEH
ns
noise-subspace
(7)
where the sample eigenvalues are sorted in descending order and the
matrices ˆess [ˆe1][M×1] and ˆEns [ˆe2, · · · , ˆeM ][M×M−1] contain in
their columns the signal- and noise-subspace eigenvectors of ˆRx
respectively.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 9/18

10. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Superresolution Ranging
Signal- and Noise-Subspace
The basis of noise-subspace ˆEns formed by the (M − 1) eigenvectors
associated with the (M − 1) smallest eigenvalues are orthogonal to the
complex exponential steering vector.
Therefore we can write this in mathematical terms:
g(∆Φ(k)) ⊥ Ens, (8)
or equivalently:
g(∆Φ(k))H
EnsEH
nsg(∆Φ(k)) = 0. (9)
Super-Resolution Goal
From g(∆ˆΦ(k)) we can construct ˆRx and, in turn, obtain Ens: the goal is
now to get a good estimate of the true steering vector g(∆Φ(k)).
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 10/18

11. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Golomb Ruler
A Nice but Challenging Tool
It’s a ruler without 2 pairs of marks at the same distance.
Example of perfect
Golomb ruler.
No perfect ruler exists beyond 4
marks,
optimal → no shorter of the same
order exists,
to find optimal high-order GR is a
NP-problem,
to find orthogonal high-order GRs
→ unsolved.
new genetic algorithm → proposed
approach.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 11/18

12. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Algorithm Evaluation
CRLB comparison
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 12/18

13. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Spectral Music
Multiple Signal Classification (MUSIC)
Recalling the full formulation of the measured steering vector
g(∆ˆΦ) = [e
j4π ˆd∆f
c , · · · , e
j4π ˆdM∆f
c ]T
,
the spectral MUSIC estimates the distance ˆd between the source s and
target t from the minimum of the function
f( ˆd) = g(∆ ˆΦ(k))H ˆEns
ˆEH
nsg(∆ ˆΦ(k)) (10)
by searching over ˆd using a fine grid as it exploits the orthogonality in eq.
(9).
Matlab Tip
To define the fine grid, use the command:
RangeD = 0:01:50;
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 13/18

14. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Root Music
Polynomial expansion of MUSIC
Root MUSIC replaces the search of the minimum of f( ˆd) by polynomial
rooting, and its only solution is the distance estimate ˆd between the source
s and target t.
The M × 1 complex exponential vector can be written as:
g(∆ ˆΦ(k)) = [e
j4π ˆd∆f
c , · · · , e
j4π ˆdM∆f
c ]T
(11)
= [z1
, · · · , zM
]T
,
where zi
ej4π ˆdi∆f/c
, and ∆f is the common uniform inter-frequency
spacing.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 14/18

15. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Root Music
Polynomial expansion of MUSIC
The function f( ˆd) is instead rewritten as:
f( ˆd) = g(z−1
)T ˆEns
ˆEH
nsg(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 ˆEns
ˆEH
ns.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 15/18

16. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Root Music
Polynomial expansion of MUSIC
The polynomial f(z) has (2M − 1) unitary modulus roots which form
conjugate reciprocal pairs
z = ej4π ˆd∆f/c
and ¯z = e−j4π ˆd∆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 ˆd by selecting the largest-magnitude root from those lying inside
the circle.
Matlab Tip
To 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);
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 16/18

17. Accurate
Ranging
Recap
Steering Vector
Golomb Ruler
Spectral Music
Root Music
Report 2/3
Ranging
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
s.severi@jacobs-university.de,
r.stoica@jacobs-university.de.
Specialization Lab - Fall 2015 Wireless Localization: Ranging October 7, 2015 17/18

18. Thank you!