IEEE Signal Processing - May 2018 - 63

Distance (mm)

same time, as shown in Figure 4(c), where the CSIs on two
Wi-Fi channels and two receiving antennas are combined into
one fingerprint. In Wi-Fi, the frequency diversity is achieved
by performing frequency hopping on different Wi-Fi channels
[32], and the spatial diversity can be realized by collecting CSI
on multiple antenna links on multiple-input, multiple-output
(MIMO) Wi-Fi devices [33]. When we denote the maximum
frequency diversity by F, the maximum spatial diversity by S,
and the physical bandwidth for each Wi-Fi channel by B, the
effective bandwidth is given as S # F # B. Diversity exploitation increases the effective bandwidth to S # F times compared to the physical bandwidth B.

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2

-50
-40
-30
-20
-10
0
10
20
30
40
-40

Achieving centimeter accuracy via TRRS
For Wi-Fi devices with a spatial diversity of S and a frequency
diversity of F, the collected CSI values can be written as
Distance (mm)

g, F
r F = " h F(s, f ) ,sf ==11,,22,,g
H
,S ,
(s, f )

where h F stands for the CSI obtained on the sth antenna
link on the fth Wi-Fi channel. This expression provides a finegrained fingerprint with an effective bandwidth of S # F # B.
Thus, the TRRS in (1) could be extended to the fine-grained
r F, 1 and H
r F, 2 , with
fingerprint H
S

h=

S

F

-40

S =1 f =1

S

F

/ / K (2s,f ),

Distance (mm)

K2 =

s =1 f =

where
h

(s, f )

= max z

K

/ h (Fs,,1f ) [k] h F(s,,2f ) ) [k] e -jk

z

k -1

is the modified cross correlation on (s, f), and
(s, f )

K1

N sub

=

/

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2

20
30
40

F

and

2

(s , f )

( s, f )

h F, 1 [k] , K 2

N sub

=

/

(s, f )

(s, f )

-20
0
20
Distance (mm)
(b)

40

20
30
40

2

-40

h F, 2 [k]

0.1
0

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2

-50
-40
-30
-20
-10
0
10

k =1

k =1

40

-50
-40
-30
-20
-10
0
10

/ / h (s, f ), K 1 = / / K 1(s, f ),

s =1 f =1

0
-20
20
Distance (mm)
(a)

0.1
0

0
-20
20
Distance (mm)
(c)

40

0.1
0

(s, f )

are the CSI energies of h F, 1 and h F, 2 on (s, f), respectively.
r F, 1, H
r F, 2) calculated by (1)
Then, the fused TRRS TR (H
implies that it is a weighted average of CSI energies on different Wi-Fi channels and antenna links.
Treating the CSI as a location-specific fingerprint and the
TRRS as the classification metric, we discuss an IPS that consists
of an offline phase and an online one in the following. During
the offline phase, the positioning system collects CSI fingerprints
r F, , ,, = 1, 2, g, Nloc . In
from N loc locations of interest, denoted as " H
r F, test is obtained at a location
the online phase, a CSI fingerprint H

FIGURE 3. The ambiguity among nearby locations under (a) a 40-MHz effective bandwidth, (b) a 120-MHz effective bandwidth, and (c) a 360-MHz
effective bandwidth.

to be determined, which may be either one out of the N loc locations of interest mapped during the online phase or an unmapped
location. Then, the pairwise TRRS between the fingerprint
r F, test and each of the N loc fingerprints " H
r F, , ,, = 1, 2, g, Nloc is
H
r F, ,, H
r F, test). Finally, the location is detercomputed as TR (H
r F, ,, H
r F, test), as shown in (2). Here,
mined based upon TR (H

r F, ,, H
r F, test), If max TR (H
r F, ,, H
r F, test) $ C
argmax TR (H
, = 1, 2, g, N loc
4.
,t = * , = 1, 2, g, Nloc
0,
Otherwise

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May 2018

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