IEEE Signal Processing - May 2018 - 67
2
v v
v
G (~) e -ik· (R -R 0)
~!X
.
#0 2
r
P (i) e -ikd cos (i) di
= PJ 0 (kd ),
(9)
where the coordinate system in Figure 1(b) is assumed, J 0 (x)
is the zeroth-order Bessel function of the first kind, and d is
v 0 and R
v . Here, a continuthe Euclidean distance between R
ous integral is used to approximate the discrete sum, and
P (i) = P denotes the density of the energy of the multipaths
v =R
v 0, it degenerates to
coming from direction i. For R
d = 0, and thus s Rv [0] . P. Since the denominator of (4) is
the product of the energy received at two focal spots, it would
converge to P 2. Substituting (9) into the TRRS defined in (4)
leads to
TR (h T, Rv 0, h T, Rv ) . J 20 (kd ).
(10)
Since the theoretical approximation of the TRRS distribution
depends only on the distance between two points, in the following, we use TR (d ) = J 20 (kd ) to stand for the approximation of
the TRRS between two points with distance d between them.
This theoretical analysis can be verified using a mobile
channel probing platform equipped with stepping motors that
can control the granularity of the CIR measurements precisely
along any predefined direction. We collected extensive measurements of CIRs from different locations in a typical office
environment, and Figure 7 shows two representative results
measured at two locations approximately 20 m apart. In this
figure, the distance d away from each predefined focal spot
increases from 0 to 2m, with a resolution of 1 mm. It indicates
that the measured TRRS distributions agree with the theoretical analysis quite well, in that the positions of the peaks and valleys in the measured curves are almost the same as those of the
theoretical curves. Although the two locations are far apart, the
measured TRRS distributions exhibit similar damping patterns
when the distance d increases, which shows that the TRRS distribution is independent of the locations. We also see that the
measured TRRS distribution curves are above the theoretical
curve. This is due to the contribution of the direct path between
the transmitter and receiver, which adds an asymmetric component in the energy density function P (i) in (9). As a result,
the TRRS is a superposition of J 20 (kd ) and some unknown
function. Nevertheless, since the pattern J 20 (kd ) is dominant in
the TRRS distribution function and location independent, we
can exploit this feature for speed estimation and then estimate
the moving distance by integrating the speed over time.
Specifically, since the shape of the TRRS distribution function TR (d ) . J 20 (kd ) is determined only by the wave number k, which is dependent only on the carrier frequency and is
independent of location, it can be utilized as an intrinsic ruler
to measure distance. Consider that a receiver moves along a
straight line with a constant speed v, starting from location
v 0, and a transmitter keeps transmitting the TR waveform
R
v 0 at regular intervals. Then, the TRRS meacorresponding to R
sured at the receiver is just a sampled version of TR (d ), which
would also exhibit the Bessel-function-like pattern, as illustrated in Figure 8. The feature points, i.e., the local peaks and
valleys, on the Bessel-function-like TRRS distribution can be
used to estimate the instantaneous speed of a moving object.
For example, consider the first local peak of h (d) , in which
v 0 is approxthe theoretical distance d 1 from the starting point R
imately 0.61m. To estimate the moving speed, we need only estimate how much time t it takes for the receiver to reach the first
v 0 . Then, the speed estimation
local peak, starting from point R
becomes vt = (0.61m) /t. At a high sampling rate, we can assume
that the actual speed within t is constant, and then we can further
estimate the moving distance by integrating the instantaneous
speed over time. A more recent study [36] has shown that the
speed and distance of a moving object can be estimated based
on CSI from Wi-Fi with a similar accuracy, even without wearable devices. The moving direction can be estimated using IMU
measurements. Combining the estimation of the moving direction and moving distance, the moving object can be tracked.
Map-based position correction
Since the TRITS estimates the location of a moving object based
on its previous location and the current location displacement,
η (d)
/
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Theory
Experiment 1
Experiment 2
0
0.2 0.4 0.6 0.8
1 1.2 1.4 1.6 1.8
d (λ)
2
FIGURE 7. A comparison of the TRRS distribution between the experimental
results and the theoretical result [36].
1
0.8
TRRS
s Rv [0] =
0.6
0.4
0.2
0
d1 (ti,2, yi,2)
(ti,2, yi,2)
(ti,1, yi,1)
Estimation of the Time
Corresponding to the First Peak
0
0.05
0.1
Time Difference (s)
0.15
FIGURE 8. An illustration of the distance estimation method.
IEEE Signal Processing Magazine
|
May 2018
|
67
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