Velocity Estimation of Target in MIMO Radar Environment with
Unknown Antenna Locations
P. PALANISAMY
Department of Electronics and Communication Engineering
National Institute of Technology
Tiruchirappalli-620015
INDIA
Abstract: - A new algorithm is formulated for MIMO Radar system where positions of target along with that of
the transmitter and receiver antennas are unknown. This algorithm considers a widely separated antenna MIMO
setup and can be used when transmitters and receivers are either stationary or moving at a very low velocity.
Here, the algorithm estimates the location of target with respect to the location of the first transmitter. The
TDOA and AOA available for LOS path between the transmitters and receivers along with the reflection path
from the target are utilized here. AOA is used only for initialization of antenna positions and target location.
Furthermore, accurate estimation using Davidon-Fletcher-Powell (DFP) Algorithm is performed. The paper
introduces a new algorithm to compute velocity of a target. Here, first the FDOA is estimated using a novel
approach and then velocity is estimated from the FDOA obtained. The velocity estimator for FDOA given
tracks CRLB and FDOA estimation tracks its corresponding CRLB upto -12dB. The algorithm uses target and
antenna locations to estimate velocity that can be found out from the algorithm introduced in paper.
Key-Words: - Target Localization, Velocity Estimation, MIMO Radars, DFP Algorithm, Antenna Positions
Estimation.
Received: September 12, 2021. Revised: April 17, 2022. Accepted: May 15, 2022. Published: June 28, 2022.
1 Introduction
MIMO radar have attracted significant attention in
last few years over conventional radar for the
superior performances in higher spatial resolution
[1], enhanced parameter identifiability [2], more
degrees of freedom (DOFs) [3], detection diversity
gain [4], better spatial coverage [5] and possibility
of direct application of adaptive array techniques
[6]. Two types of MIMO setup have been discussed,
MIMO with colocated antennas [7] and with widely
separated antennas [8].
MIMO radar with co-located antennas is better
for parameter estimation and beam-forming
performance as it has more effective spatial
degrees of freedom, since its transmitter and
receiver antennas are sufficiently close to observe
signals reflected from the target. MIMO radar with
widely separated antennas, also known as statistical
MIMO radar, exploits the diversity of the
propagation path, thus can be used where better
detection and estimation resolution is needed.
In a Radar system, the detection and
estimation of target parameter is the prime
application. The parameters of target into
consideration are location, velocity, acceleration,
Doppler frequency, Radar Cross
Section(RCS).etc.
Several methodologies have
been introduced for target detection [9] and
lo
calization. Detection techniques in clutter is
also discussed [10],[11]. Several
different
approaches have been adopted to estimate target
location and ve
locity, based on time of arrival
(TOA) [12], time difference of arrival (TDOA)
[13], angle of arrival (AOA) [14] or frequency
difference of arrival (FDOA) [15].
There is a rapid growth in literature on
MIMO radars. In [16], a method for estimating
target location when transmitter and receiver
location are
known
w
as prop
osed
which
trac
ks
Cram
e
´
r-Rao
l
o
w
er
bound
(CRLB).
In
[16],
the
problem of target localization is modeled in
MIMO radars using TDOA and AOA
measurements. This method solves the maximum
likelihood (ML) estimation problem of target
position with arena divided into grids and uses
steepest descent algorithm (SDA) to further
enhance accuracy while main
taing complexity
low. In [17] method for estimation of velocity is
introduced. It also discusses optimal antenna
placements. The paper [18] discuses improvement
in performance of estimators when number of
antennas are increased. In [19] spare
support
recovery is used to infer target properties both
position and velocity.
Optimal Energy allocation
is also discussed.
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All the available methods stand valid only
when the transmitter and receiver position are
accurately known beforehand. Thus, the
antennas have to be relatively stationary and the
positions have to found manually which could
consume considerable setup time and if at all by
any cause the antenna positions are changed, then
the system would have to be setup again. There fore
to conceptualize a portable MIMO Radar
system, an algorithm is vital which can elude the
setup time. We have come up with a new method
which estimates the antenna positions and target
positions at the same time. Here for example, if
we consider a MIMO radar setup where the
antennas are kept on a movable mount, then even
when the antennas change positions we do not have
to pause and obtain the coordinates of antennas, but
the algorithm estimates the target position with
respect to the first transmitter even when the
antennas are in motion.
In the proposed algorithm, Cartesian plane is
fixed and the transmitter and receiver positions
are initialized used TDOA and AOA
measurements. Once transmitter and receiver
locations are initialized, the target position is
localized to grids by solving ML estimation,
inspired from [16], then for precise estimation of
all unknown positions Davidon-Fletcher-Powell
(DFP) from [20] is used.
This paper also introduces a new algorithm to
estimate velocity. Here first FDOA is estimated
and then using FDOA as input a new algorithm,
inspired from methodology in [16] is used to
estimate velocity. The velocity domain is discretized
to grids and the grid with nearest velocity values are
found using sparsity aware ML estimator.
In this paper, to estimate FDOA, a new
approach using iterative use of Non-Uniform
DFT is presented. It first finds the nearest
frequency with
respect to the resolution of the
current iteration bandwidth and for next iteration,
the bandwidth of interest is reduced and kept
around the frequency obtained in the last iteration.
This is repeated untill the bandwidth of the iteration
matches the required precision.
The paper is arranged as, Section 2 describes
system model, describes various parameters of
MIMO radar used in estimators. Section 3
elaborates the new method. In Section 3.1 a new
algorithm for estimating FDOA from signal is
elaborated. Section 3.2 a new approach for
approximating the velocity of target from FDOA
is presented. Input to velocity estimator is antenna
and target locations along with target doppler
signature (FDOA). Section 3.3 is new procedure
to estimate target position along antenna positions.
Section 3.3.1 discuses initialization procedure and
Section 3.3.2 elab
orates methodologies adopted for
more accurate estimation of target as well
as
antenna positions. Section 4 contains the
Numerical simulations results for testing the
proposed methods. Section 5 concludes the
paper.
2 System Model
Let us consider a MIMO setup with M
transmitters and N receivers distributed over a 2-
D surface. The surface is divided into K
grid
points for target localization. The positions of
transmitter and receiver are denoted b
y
x
m
=[x
m
,y
m
], m=1,2,..,M and x
n
=[x
n
,y
n
],
n=1,2,..,N
respectively. The position of a target is
denoted by x=[x, y] and its velocity v=[vx, vy]. This
is depicted in Fig.1
Fig.1 Schematic of MIMO Radar system Arrangement
It needs to be noted that, system considered is 2-D,
but can be extended to a 3-D localization.
Considering the wave propagation speed (that is,
speed of light) by c, the noisy TDOA and AOA
measurements due to LOS can be modeled as
(1)
(2)
where
and
respectively the measured
TDOA abd AOA which contains noise.
The target reflected path TDOA and AOA are
(3)
(4)
where
=
for m=1, 2, … ,M and
=
for n=1,2,…, N. TDOA and
AOA measurements are disturbed by independent
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zero mean Gaussian noises of
,
,
and
with the standard deviation of
,
,
and
. Here,
=
=
and
=
=
. Since all the measurements are obtained
in same environment.
The distance between transmitters and receiver
antennas can be found from TDOA as
and the bistatic range with target as .
The actual distance between transmitters and
receiver antennas are Rm,n=
+
for
m=1,2,…,M and n=1,2,…,N. The Frequency
difference of arrival (FDOA) can be modelled as
(5)
where
(6)
for m=1,2,…,M and n=1,2,…,N, is the actual
frequency difference due to Doppler shift and
where,
and
independent zero mean Gaussian random
variables with standard deviation
.
Let , 0tT, is signal
transmitted from the mth transmitter with total
energy as E and fc is the center frequency. Then, the
received signal at the nth receiver corresponding to
the signal transmitted by the mth transmitter after
reflection from target can be written as
rmn(t)=sr(t)+mn(t)
(7)
for m=,2,…,M and n=1,2,…,N
where
and where
is the unknown complex target
reflectivity and is the time delay of the path Rmn
and is the Doppler frequency detected by the
nth receiver due to the mth transmitter. The
observation interval T is assumed lengthy enough so
that all transmitted signals can be observed,
irrespective of their delay. That is, T>>max{}
and the parameter
(8)
for m=1,2…,M and n=1,2,…,N. In this,
. The noise at the nth receiver for the
signal from mth transmitter is denoted asmn(t) and
is a white Gaussian process with mean zero and
standard deviation .
3 Proposed Method
3.1 FDOA Estimation
A novel approach to estimate frequency shift
is introduced. Let W be the bandwidth of interest.
For radar case, W must be chosen such that it can
contain the Doppler shift caused by maximum
velocity of the target under consideration. Now, the
sampling period Ts must be chosen such that
Ts1/2W so as fulfill Nyquist criteria. Let Ns be
the number of samples collected by sampling
at sampling period . It is to be noted that
. We can write
(9)
For the iterative algorithm to begin the following
parameters are to be considered.
(10)
(11)
Now let us define vector w of size such
that
(12)
where
(13)
Let n be another vector
(14)
Now to form DFT basis ,
then E can be written as
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(15)
Algorithm 1
Pseudo-code for FDOA Estimation
while do
Generate w rom (13) and take n from (14)
Generate E from (15)
Y=RE
end while
Let be the instantaneous bandwidth for each
iteration. Initially
Let δ be the
resolution required for estimation. The iterations are
performed till reaches. In each iteration the
frequency band of interest i.e.
is
discretized into frequencies using (12) and peak
magnitude in frequency response is found out. The
bandwidth around this peak value becomes our new
bandwidth of interest. After required number of
iterations is performed, the value gives the FDOA
estimation with resolution of .
3.2 Velocity Estimation
Considering target at position x and moving with a
velocity v as described in Section 2 and FDOA
measurements obtained from 3.1. The actual FDOA
without the noise is defined by (5).
Let us form a vector F of measured doppler
shifts (FDOA), i.e.
F is noised added
version of actual Doppler shift,
Here the assumption
is that we know the target location x and antenna
locations
. This can be found using
Section 3.3. Using FDOA instead of TDOA in [16],
also not using AOA and making necessary changes,
a new algorithm for estimating velocity is
formulated.
Select K number of grid points
in
velocity domain to compute the objective function of
the ML estimation for all grid points and select the
minimum one. Now we can form the matrix A by
finding f (v) in different grid points.
A1=
(16)
Now, in order to obtain the velocity of a target,
the values of f(v) is compared with the received
measurements in all grid points. Thus, the target
velocity estimation problem can be written in the
sparse representation framework as
where is a MN×1 vector containing the FDOA
measurement noise, thus , where
. Vector z is a K×1 vector with
(K-1) zeros and a one element which is
corresponding to the index of the grid point where
the target velocity is closest.
Since has an in-deterministic nature,
thus the conventional maximum likelihood (ML)
estimation is not viable. Therefore, a simple solution
for this problem is to compute the objective function
of the ML estimation for all grid points and select
the minimum one (brute force). This trivial method
is of high complexity and limited positioning
accuracy according to grid size in velocity domain.
Instead, a compressed sensing technique can be
considered taking the sparsity in target's velocity.
Thus, the target velocity estimation problem can be
expressed using the minimization.
=argmin(A1z-F)
(A1z-F)T + ||z||1
(17)
where is a regularization parameter that controls
the sparsity of and is the covariance matrix of ε.
Also consider a matrix W such that
. By applying this, (17) can be
expressed as
=argmin||
||2+||z||1
(18)
where
We can find a
nearest grid index of the target velocity and
initialize that as target velocity as .
In grid-based localization, the target velocity
which are not located on the grids i.e, off-grid are
not accurately localized. In order to resolve this
problem, an algorithm based on dictionary learning
(DL) techniques can be used which is designed to
minimize the following cost function.
(v)=[F-f(v)]T
[F-f(v)]=||
||2
(19)
where
F
˜
is
the
column
of
A
˜
1
which
corresp
onds
to
the
estimated
target
velocity on the grids in velocity
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domain. It can be seen that he cost function is convex
with respect to v. Thus, simple steepest decent
(SDA) can iteratively estimate the true target’s
velocity from the matrix A1. Note that, in the
following method we just employ FDOAs in order to
satisfy convexity condition. SDA iteration
equation is written as
v(i+1)=v(i)-µ(i)v(v(i))
(20)
In Appendix 5, the derivation for final recursion
equation for updating the estimated velocity vector
is explained. Thus, the velocity vector v at the
iteration can be written as
(21)
where, eT=F-f(v(i)) and
(22)
for m=1,2,..,M and n=1,2,…,N.
The initial value v(0) of v is chosen from the estimate
of v from previously. The value of µ(i)) is selected
according such that 0 < µ(i) <
in which
is
the maximum eigen value of . To
further clarify the velocity estimation procedure,
following pseudo-code describes the estimation
procedure in step by step. The Cramer-Rao lower
bound (CRLB)) on the estimation error is
summarized in Appendix B.1 .
Algorithm 2
Pseudo-code for Velocity Estimation
for : Number of blocks do
block divided into K sub-blocks.
Solve ML estimation for (18) and find the
nearest block.
end for
Set from the previous estimated grid point
and set = inf (very large value)
while do
Compute from (21)
end while
3.2 Target and Antenna Position Estimation
3.3.1 Initialization
Antenna Position
Now, in order to fix a coordinate system, location
i.e. position of the first transmitter is considered to
be origin and the LOS path connecting
(first receiver) as x-axs. That is,
. and
are target TDOA and AOA
respectively. If M and N are number of transmitters
and receivers considering only single target, then
number of unknown quantities for 2-D case will be
We use AOA information along with TDOA.
Let us denote,
(23)
(24)
Let us define M vectors by rearranging D' and
(25)
(26)
where m = 1, 2,….,M Now let us initialize the
receiver locations as
(27)
It should be noted that
must be forced to zero.
Now for transmitters
[0,0].
(28)
for
Target Position
For initializing target position x, the ML-Estimation
concept from [16] is used. The antenna positions
considered are
. K number of grid points
in spatial domain. Now we can write the
Bistatic Range
and by taking TDOA received due to
target reflection path , where
., We can represent
,
.
Here, and the measure
Matrix A2 can be
computed as,
(29)
The problem of target localization can be
expressed in the sparse representation framework
given by where ε is
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vector containing the noises in measurement of
TDOA and AOA, i.e.,
, where,
T
and
. Vector z is a K x 1
vector with (K-1) zeros and a one at to the index
which corresponds to the grid point where the target
is located.
Since has an un-deterministic nature,
thus the conventional maximum likelihood (ML)
estimation is not viable. Therefore, a simple solution
for this problem is to compute the objective function
of the ML estimation for all grid points and select
the minimum one (brute force). This trivial method
is of high complexity and limited positioning
accuracy according to grid size in velocity domain.
Instead, a compressed sensing technique can be
considered taking the sparsity in target's location.
Thus, the target localization problem can be
expressed using the l1 minimization procedure.
(A2z-B)
(A2z-B)T+
(30)
where
is a regularization parameter that controls
the sparsity of z and is the covariance matrix of
. A matrix W is so introduced such that
. Now, (30) can be rewritten as
(31)
where =WA2 and
= WB. We canfind a nearest
grid location of the target and initialize that as target
location as x0.
3.3.2 Precise Estimation
For precise estimation of the required parameters we
are using DFP algorithm. Let N' be the number of
parameters to be estimated. Let F(X) be the cost
function and the gradient of the cost
function. Here Cγ is the covariance matrix of γ,
where
.
(32)
where,
X=[x,y,x2t,y2t,x3t,y3t,…xMt,yMt,x1r,y1r,x2r,y2r,…xNr,yNr]
computed in Appendix 5
Let us set a small value δ for limiting the
convergence. Let
. The
iterations are carried out till . Initially
Dj is set as an Identity matrix of order N'. Now, i is
varied from 1 to N’ computing equation (33), (34),
(35), (36) and (37). Thus N' iteration is performed to
converge X.
(33)
(34)
(35)
(36)
(37)
Once the limiting condition is satisfied the
parameters constituting vector X is precisely
estimated.
Fig. 2: Transmitters, receivers and target positions
with respect to the in spatial domain
Algorithm 3
DFP Algorithm for Precise Estimation
while do
Set D0 = Identity(N’)
for j<N’ do
Solve for
end for
end while
4 Simulation and Results
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m,n
In this section, the proposed algorithms are tested
using MATLAB. Fig.9 shows the performance of
algorithm presented in Section 3.2. Here first
FDOA is estimated as per Section 3.1 and then
velocity is estimated. Fig.5 is a plot between MSE
of estimated FDOA with respect to different SNR.
Fig.6 shows MSE of estimated velocity with
respect to noise variance in FDOA (σ
FDOA
). Here
a 2×2 MIMO is tested. The transmitters are at
[(2.5712, 3.0642), (1.2968, 8.1879)] and receivers
at [(3.1231, 6.6976), (0.8682, 4.9240)] in Kms.
The signal energy is taken as 400 and M and N are
both 2. Target position is considered [0, 0]. The
velocity of target is considered [0.568,
0.081]km/s. Thus, it is tested for higher velocity
value.
Fig.7 and Fig.8 shows estimated velocity for 1000
Monte-Carlo trials for SNR
=
0dB and SNR
=
-
12dB respectively. The carrier signal is
considered to be of frequency
f
c
=
1GHz.
Bandwidth BW
=
50KHz and sampling period
5µs. Number of samples collected is N
s
=
400 and
reflection coefficient ζ
mn
=
1 is considered.
Testing is done with two MIMO setups, 5×5
and 3×3. In MIMO 5×5,
transmitters are located
at (0,0), (1,9), (5,5), (6,1) and (6,9), and receivers
are
placed at (5,0) , (2,5), (5,8), (7,7) and (7,3). For
MIMO 3×
3, transmitters
are at (0, 0), (5, 8) and
(9, 9) and receivers at (4, 0), (1, 9) and (5, 5).
Let us consider a target positioned at (3.54,6.23) in
both cases. All the distances are to be considered
in Kms.
Fig.2 shows antenna and target arrangement in
spatial domain. The estimate of target’s position is
shown in Figure 3 and Figure 4 for MIMO 5×5
and MIMO 3×3respectively at σα= 5 deg.
Fig.3: Plot between MSE of target position vs
TDOA noise standard deviation
w
ith MIMO
5×5
Fig.4: Plot between MSE of target position vs
TDOA noise standard deviation
w
ith MIMO
3×3
Performance of the proposed
method is evaluated
in the presence of TDOA and AOA noises. The
MSE of the target’s position versus the standard
deviation of TDOA noises was calculated using
1000 - trial Monte-Carlo runs. It can be observed
that the MMSE of y is less than that of x in Figure
3. This is because the spatial distribution of
antenna are better for y-coordinate than x-
coordinate (Figure 2).
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Fig.5: MSE Vs SNR for estimated FDOA
Fig.6:
MSE of velocity estimation Vs noise
with CRLB
5 Conclusion
In this paper, we formulated the problem of target
localization in MIMO radars in widely separated
framework with unknown antenna locations. The
target localization is done considering that the
transmitters and receivers are stationary or
moving with a very low velocity such that its
positions do not change much with in the
estimation interval. Further a new approach for
estimating velocity is introduced. Here FDOA is
estimated first and then the estimated FDOA
with noisy is used to estimate velocity.
Fig.7: Estimated velocity for 1000 iterations at
SNR
=
0.
Figure 8: Estimated velocity for 1000 iterations at
SNR
=
−12
Appendix A.
Appendix A.1. Cost Function
Simplification of 3.2
By some mathematical manipulations, the cost
function Γ(
v
) can be formed as:
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Figure 9: MSE (m2/s2) Vs SNR (dB) for a MIMO
2 × 2 Setup
The derivation of the cost function with
respect to vk , k = 1, 2 is as follows:
For k = 1, 2 it can be written as
for m=1,2,…,M and n=1,2,…,N.
Appendix A.2. Simplification of Gradient
of F (X)
The cost function
. If X is a vector of
N number of variables then
. Then Gradient of
can be written as
Now,
Here,
Then,
Computing each partial derivative
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Appendix B. CRLB
Appendix B.1. CRLB Derivation For
Velocity Estimation
We aim to derive the
CRLB
for velocity
estimation. The velocity v is estimated from the
FDOA
observations. The
CRLB
can be
calculated using the trace of the inverse of Fisher
information matrix, denoted by
I.
For a
Gaussian observations, with mean vector µ and
covariance matrix C
є
, then
I
can be written as,
In the present study, µ
= f
(
v
) and C
є
is
independent of
v
. Thus the second term in above
equation is equal to zero and the first term
yields:
CRLB can be plotted using
([
]
−1
).
Appendix B.2. CRLB Derivation for
Combined Velocity Estimation from
Signal Using Algorithm
For retrieving noise variance from SNR
Now the CRLB can be obtained as
Let C
є
be the covariance matrix of noise in
r
m,n
(t). Then,
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