Instrumentation & Measurement Magazine 24-4 - 78

Also, let A(fi) denote the signal amplitude at fi
j* as:
j argmin A f
* 
ii( ).
(2)
It is desirable that we do not use Dj*, since the signal received
at Sj* is not as strong as signals at other receivers. Thus,
Dj*
. We define
may contain NLOS errors.
Suppose that j* ∈ {1, 4}. Using the geometry in Fig. 1, the signal
delay at the center of the formation is estimated as:
Dc  23
2
DD

.
(3)
Suppose that j* ∈ {2, 3}. Using the geometry in Fig. 1, the signal
delay at the center of the formation is estimated as:
Dc  14
2
DD

.
(4)
There exists noise in signal delay measurements. Let C denote
the speed of signal. In simulations, we use C = 1400 m/s.
Using the geometry in Fig. 1, the signal delay at S1
is:
Fig. 1. The geometry of sensor configuration. S1 = [r, 0], S2
S3 = [0, -r], and S4
= [-r, 0].
the signal amplitude to estimate the DOA. The efficiency of the
proposed DOA estimation is verified by comparing it with the
MUSIC using simulations.
Definitions and Assumptions
Let Si
0], S2 = [0, r], S3
= [0, -r], and S4
(i ∈ {1, 2, 3, 4}) denote the coordinate of the ith sensor. The
geometry of sensor configuration is depicted in Fig. 1. S1
= [r,
= [-r, 0]. We consider a transmitter
which is sufficiently far from the sensors. T denotes the unit
vector from the transmitter to the origin.
Let ϕ denote the bearing angle of the signal, such that
- π ≤ ϕ < π. ϕ is measured counterclockwise direction from
the x-axis of the sensor configuration. We have:
Tc [ ( ), ( )].s
(1)
Here, s(.) = sin(.), and c(.) = cos(.). Our problem is to estimate
ϕ based on signal measurements at only four sensors.
Recall that each sensor is synchronized to every other sensor
and can measure the signal strength with respect to sampling
time.
Let Di (i ∈ {1, 2, 3, 4}) denote the signal delay measurement
using the ith sensor. Di can be measured using Discrete Fourier
Transform (DFT) and Maximum Likelihood (ML) estimation
of the signals' properties [10].
Using DFT, we can derive the relationship between the frequency
and its associated signal amplitude. Thus, we can find
a frequency where the signal amplitude is maximized. Let fi
denote the frequency where the signal amplitude received at
Si
nal delay Di
78
is maximized. Using the phase at fi
.
, we can derive the sigThe
signal delay at S4
is:
D D C D rc Ccc ( ()) .
44
    
TS
(8)
As a signal is reflected or scattered, its amplitude attenuates
significantly. Hence, as we estimate ϕ, we use receivers with
high signal amplitude, since they have more probability to receive
a direct signal (not a reflected signal) compared to other
receivers with low signal amplitude.
Let Δ > 0 denote a tuning parameter in this paper. Here, Δ
presents the estimated decrease of signal amplitude due to reflection
or scattering.
Suppose that min(A(f1
), A(f2
)) - max(A(f3
and D1 for estimation of ϕ. Let ϕe
Using (5) and (6), we get:
e
), A(f4
)) > Δ. This is
called the condition [C1]. If this condition holds, then we use
D2
denote the estimation of ϕ.
  atan 21D ccD D D )).
2( (  ), (
(9)
Here, atan2(y, x) is the phase (or angle) of the complex
number x + iy. Thus, it is possible that y or x in x + iy is zero.
However, it is not possible that both x and y in x + iy are zeros.
Suppose that min(A(f4
), A(f2
IEEE Instrumentation & Measurement Magazine
)) - max(A(f1
), A(f3
called the condition [C2]. If this condition holds, then we use:
June 2021
)) > Δ. This is
= [0, r],
D D C D rc Ccc ( ()) .
    
11
TS
Also, the signal delay at S2
22
is:
D D C D rs Ccc ( ()) .
33
    
TS
(7)
is:
D D TS C D rs Ccc ( ( )) .
   
The signal delay at S3
(6)
(5)

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