Signal Processing - November 2016 - 131

Space

Space

f1

s1

f2

f3

f4

s4

Frequency

s2

Frequency
s2

f1

s1

t1

t2
t3

Hs 1,f 1

Frequency

s3

s1

t1

Space

Hs 1,f 2

t4

t1

Time

Hs 1,f 3

f2
f1

t2

Time

Hs 1,f 4

(a)

Hs 1,f 1

Hs 2,f 1

Hs 3,f 1

Time

Hs 4,f 1

Hs 1,f 1

(b)

Hs 2,f 1

Hs 1,f 2

Hs 2,f 2

(c)

figure 3. Leveraging frequency and spatial diversities in Wi-Fi to achieve large effective bandwidth: (a) using only the frequency diversity, (b) using only
the spatial diversity, and (c) using both of the frequency and spatial diversities.
f = 1, 2, f, F

H = {H s, f } s = 1, 2, f, S can provide finegrained fingerprint with an effective
bandwidth of S # F # W. Consequently,
TRRS in (1) can be extended to the finegrained fingerprint H and H, with h
and K, Kl modified as
S

h=

F

/ / h s, f ,

s=1 f=1
S
F

K=

/ / K s, f ,

s=1 f=1

Kl =

S

F

/ / Kl

s=1 f=1

,

s, f

(3)
where
h s, f = max
z

K

/ H s, f,k H sl*, f,k e -jk

z

(4)

k=1

represents the modified cross-correlation on the virtual link (s, f ), and
K
K s, f = / Kk = 1 | H s, f, k | 2, Kls, f = / k = 1
2
| H sl, f, k | are the channel energies of
H s, f and Hls, f on the virtual link (s, f ),
respectively.
Algorithm 1 elaborates on the calculation of c [ H, Hl ] . As shown in Algorithm 1, steps 4-9 are used to calculate

the channel energies on the virtual link
(s, f ), while steps 10-14 are targeted
to compute the modified cross-correlation of two CSIs on the virtual link
(s, f ). The channel energies and modified cross-correlation on each virtual
link are accumulated as shown in step 9
and step 15, respectively. Finally,
the TRRS is obtained by step 18. The
computation of h s, f is approximated
by
2 n (k - 1)
K
*
-j N
hr s, f = max n / k = 1 H s, f, k H sl , f, k e
that takes the same format of a discrete
Fourier transform of size N and thus
can be computed efficiently by fast
Fourier transform. Using a large N in
the computations leads to a more accurate approximation of h s, f .

phase. During the mapping phase, the
CSIs are collected from L locationsof-interest using Wi-Fi devices with
S antenna links and across F Wi-Fi
channels, denoted by " H , ,, = 1, 2, f,L . In
the localization phase, Hl is obtained
at a testing location, which may either
be one of the L locations-of-interest
or an unmapped location in the mapping phase. Then, the pairwise TRRS
c [ H ,, Hl ], is calculated for all locations-of-interest. Finally, the location
is determined based on c [ H ,, H], i.e,
(5), shown in the box at the bottom of
the page, where C is a threshold introduced to balance off the true positive
rate and false positive rate in location
determination. When c [ H ,, Hl ] falls
below C, the IPS cannot obtain a credible location estimation and returns 0 to
imply an unmapped location.

r

Localization using TRRS
There are two phases in the proposed
IPS: a mapping phase and a localization

,t = )

argmax c [ H ,, Hl ], If
, = 1, 2, f,L

0,

IEEE Signal Processing Magazine

max c [ H ,, Hl ] $ C

, = 1, 2, f,L

Otherwise

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November 2016

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131
Table of Contents for the Digital Edition of Signal Processing - November 2016
