This document proposes a low complexity algorithm for jointly estimating the reflection coefficient, spatial location, and Doppler shift of a target for MIMO radar systems. It splits the estimation problem into two parts. The first part estimates the reflection coefficient in closed form. The second part jointly estimates the spatial location and Doppler shift using a 2D FFT approach. This allows significantly lower computational complexity compared to maximum likelihood estimation. Simulation results show the proposed estimator achieves the Cramér-Rao lower bound, providing optimal performance with low complexity.

1. Low Complexity Joint Estimation of Reflection
Coefficient, Spatial Location, and Doppler Shift for
MIMO-Radar by Exploiting 2D-FFT
Seifallah Jardak, Sajid Ahmed, and Mohamed-Slim Alouini
Computer, Electrical, and Mathematical Science and Engineering (CEMSE) Division
King Abdullah University of Science and Technology (KAUST)
Thuwal, Makkah Province, Saudi Arabia
Email: {seifallah.jardak, sajid.ahmed, slim.alouini}@kaust.edu.sa
Abstract—In multiple-input multiple-output (MIMO) radar, to
estimate the reflection coefficient, spatial location, and Doppler
shift of a target, maximum-likelihood (ML) estimation yields the
best performance. For this problem, the ML estimation requires
the joint estimation of spatial location and Doppler shift, which is
a two dimensional search problem. Therefore, the computational
complexity of ML estimation is prohibitively high. In this work,
to estimate the parameters of a target, a reduced complexity
optimum performance algorithm is proposed, which allow two
dimensional fast Fourier transform to jointly estimate the spatial
location and Doppler shift. To asses the performances of the
proposed estimators, the Cram´er-Rao-lower-bound (CRLB) is
derived. Simulation results show that the mean square estimation
error of the proposed estimators achieve the CRLB.
Keywords—MIMO-radar, Reflection coefficient, Doppler, Spa-
tial location, Cram´er-Rao lower bound.
I. INTRODUCTION
In contrast to the phased array radar where phase shifted
versions of the same waveform are transmitted from multiple
antennas, multiple-input multiple-output (MIMO) radar can
transmit independent or partially correlated waveforms. Such
waveforms in MIMO radar provide extra degrees-of-freedom
(DOF) that can be exploited for improved spatial resolution,
diversity, and better parametric identifiability. MIMO radars
can be divided into two categories, widely spaced [1] and
colocated [2]–[5] MIMO radars. The difference between these
two categories is the distance between the adjacent antennas.
The focus of this paper is on colocated MIMO radars.
For stationary targets, using colocated MIMO radar, param-
eters such as reflection coefficient and location are estimated
using adaptive techniques (see e.g., [6], [7] and the references
therein). To estimate the parameters of moving target, different
methods and models have been discussed in the literature.
In the bistatic MIMO-radar, the parameters of moving tar-
gets are estimated using maximum-likelihood (ML) estimator
in [8]. The ML estimator yields the optimal performance,
however, its computational complexity is very high, which
prevents its use in practice. Reduced computational com-
plexity algorithms, such as multiple-signal-classification (MU-
SIC) [9] and estimation-of-signal-parameters-via-rotational-
This work was funded by a CRG grant from the KAUST Office of
Competitive Research Fund (OCRF).
invariant-techniques (ESPRIT) [10], have been used to esti-
mate the parameters of moving targets in colocated MIMO-
radar. These algorithms are suboptimal and, if the signal matrix
is ill conditioned, their performance degrades significantly.
In the proposed work, to estimate the parameters of a
moving target using MIMO-radar, a low complexity, non-
adaptive optimal algorithm is proposed. The problem of joint
estimation of the reflection coefficient, spatial location angle,
and Doppler shift is split into two estimation problems. The
first problem is a simple estimation problem, which yields the
closed-form solution for the reflection coefficient. While, the
second problem requires the joint estimation of spatial location
angle and Doppler shift, which is a two dimensional search
problem. Our manipulation of the cost function of the second
estimation problem allows us to exploit two-dimensional fast-
Fourier-transform (2D-FFT) to jointly estimate the spatial
location angle and Doppler shift of the target in a very low
computational complexity. To assess the performance of the
estimators, we derived the Cram´er-Rao-lower-bound (CRLB)
of the parameters and compared it with the mean-square-
estimation-error (MSEE) of the parameters.
The organization of the paper is as follows. In the following
section, the problem is formulated. The estimators of the
different parameters are derived in Section III. In Section
IV, the Fisher-information-matrix (FIM) of the parameters is
derived. Simulation results are presented in Section V. Finally,
conclusions are drawn in Section VI.
Notation: Bold upper case letters, X, and lower case let-
ters, x, respectively denote matrices and vectors. The identity
matrix of dimension N ×N is denoted by IN . Transpose, con-
jugate and conjugate transposition of a matrix are respectively
denoted by (·)T
, (·)∗
, and (·)H
. The statistical expectation is
denoted by E{.}. The real, imaginary, and absolute value of
a complex variable x are respectively represented by ℜ(x),
ℑ(x), and |x|.
II. PROBLEM FORMULATION
Consider a narrowband MIMO radar system with uniform-
linear-arrays (ULAs) at the transmitter and the receiver. Let dT
and dR respectively denote the inter-element spacing between
the nT transmitting and nR receiving antennas. A moving
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2. target of reflection coefficient βt is located at an angle θt,
which produces a normalized Doppler shift of fdt. Moreover,
the angles θ1 to θL denote the location of the L static
interferers, each of reflection coefficient βi. If xm(n) is the
baseband signal transmitted from antenna m, the received
signals after matched filter can be expressed in a vector form
as
y(n) = βtej2πfdtn
aR(θt)aT
T (θt)x(n) (1)
+
L∑
i=1
βiaR(θi)aT
T (θi)x(n) + v(n), n = 1, 2, . . . , N,
where N denotes the total number of symbols transmitted from
each antenna and
aT (θp) =
[
1 ej 2π
λ dT sin(θp)
· · · ej 2π
λ (nT −1)dT sin(θp)
]T
,
aR(θp) =
[
1 ej 2π
λ dR sin(θp)
· · · ej 2π
λ (nR−1)dR sin(θp)
]T
,
x(n) = [x1(n) x2(n) · · · xnT (n)]
T
,
and v(n) = [v1(n) v2(n) · · · vnR (n)]
T
,
are respectively the transmit and receive steering vectors cor-
responding to a location θp, the vector of transmitted symbols
at time index n, and the vector of complex white Gaussian
noise samples each of zero mean and σ2
n variance. Here, λ
denotes the wavelength of the transmitted signals.
III. PROPOSED PARAMETER ESTIMATION
In this work, the probing signals are linearly independent
waveforms. To maximize the signal-to-interference-plus-noise-
ratio (SINR), a beamformer weight vector, w, is used at the
receiver
wH
y(n) = βtej2πfdtn
wH
aR(θt)aT
T (θt)x(n)
+
L∑
i=1
βiwH
aR(θi)aT
T (θi)x(n) + wH
v(n). (2)
If the covariance matrix of the interference plus noise term is
denoted by Rin, the SINR can be defined as
SINR =
|βt|2
E
{
|ej2πfdtn
wH
aR(θt)aT
T (θt)x(n)|2
}
wHRinw
, (3)
where
Rin = E



L∑
i=1
βiaR(θi)aT
T (θi)x(n)
2



+ σ2
nInR
.
Using (3), the Capon beamformer [11] that maximizes the
SINR is the solution of the following optimization problem
min
w
wH
Rinw
subject to wH
aR(θ) = 1, (4)
and can be derived as follows
w(θ) =
R−1
in aR(θ)
aH
R (θ)R−1
in aR(θ)
. (5)
To estimate the value of fdt, θt, and βt, the cost-function to
be minimized can be written as
{fdt, θt, βt} =argmin
fd,θ,β
J1(fd, θ, β)
=argmin
fd,θ,β
E
{
wH
(θ)y(n) − βej2πfdn
aT
T (θ)x(n)
2
}
.
(6)
The minimization of the above cost-function with respect to
β can be found as
∂J1
∂β∗
=E
{
βaT
T (θ)x(n)xH
(n)a∗
T (θ)
− e−j2πfdn
wH
(θ)y(n)xH
(n)a∗
T (θ)
}
= 0. (7)
Since all the transmitted waveforms are independent, i.e.,
E
{
x(n)xH
(n)
}
= InT
, (7) yields
ˆβ(fd, θ) =
1
nT
E
{
e−j2πfdn
wH
(θ)y(n)xH
(n)a∗
T (θ)
}
. (8)
Using (7) in (6), the cost function to be minimized in order to
estimate fdt and θt becomes
J2 =wH
(θ)Ryw(θ)
−
1
nT
E
{
e−j2πfdn
wH
(θ)y(n)xH
(n)a∗
T (θ)
} 2
. (9)
Using (5), the term inside the expectation operator in the above
equation can be written as
J2 = wH
(θ)Ryw(θ)
−
1
nT
E
{
e−j2πfdn
aH
R (θ)R−1
in y(n)xH
(n)a∗
T (θ)
} 2
aH
R (θ)R−1
in aR(θ)
2 .
(10)
In (10), it can be noticed that fd intervenes only in the
numerator of the second term of the right hand side expression.
Thus, to estimate the Doppler shift, the following cost function
should be maximized
J3 = E
{
e−j2πfdn
aH
R (θ)R−1
in y(n)xH
(n)a∗
T (θ)
} 2
. (11)
Assuming r(n) = R−1
in y(n), we can write
a(n) = e−j2πfdn
aH
R (θ)r(n)xH
(n)a∗
T (θ)
= e−j2πfdn
nR∑
q=1
rq(n)e−j 2π
λ
dR sin(θ)(q−1)
×
nT∑
p=1
x∗
p(n)e−j 2π
λ
dT sin(θ)(p−1)
= e−j2πfdn
nT∑
p=1
nR∑
q=1
rq(n)x∗
p(n)e−j2πfs(q−1+γ(p−1))
, (12)
where fs = dR
λ sin(θ) and γ = dT
dR
. By combining the same
frequency terms, we can write
a(n) =
(
x∗
1(n)r1(n) + x∗
1(n)r2(n)e−j2πfs
+ · · ·
+ x∗
1(n)rnR (n)e−j2π(nR−1)fs
+ x∗
2(n)r1(n)e−j2πγfs
+ x∗
2(n)r2(n)e−j2π(γ+1)fs
+ · · · + x∗
2(n)rnR (n)e−j2π(γ+nR−1)fs
+ . . . + x∗
nT
(n)rnR (n)e−j2π(nR−1+γ(nT −1)fs)
)
e−j2πfdn
.
(13)
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3. For example, if the distance between any two adjacent
antennas, at the transmitter and the receiver, is half of the
transmitted signal wavelength, i.e., dT = dR = λ
2 and γ = 1,
the expression of a(n) becomes
a(n) =
(
x1(n)r1(n) +
[
x1(n)r2(n) + x2(n)r1(n)
]
e−j2πfs
+
[
x1(n)r3(n) + x2(n)r2(n) + x3(n)r1(n)
]
e−j2πfs2
+ . . . + xnT (n)rnR (n)e−j2πfs(nT +nR−2)
)
e−j2πfdn
. (14)
Therefore, for any integer γ, we can write
E{a(n)} =
1
N
N−1∑
n=0
nR−1+γ(nT −1)
∑
m=0
f(n, m)e−j2πfdn
e−j2πfsm
,
(15)
where
f(n, m) =
nT∑
i=1
x∗
i (n)rm+1−γ(i−1)(n). (16)
Interestingly, the right hand side of (15) is similar to the
famous expression of the 2D-FFT. Therefore, using a 2D-FFT,
the Doppler frequency can be estimated as follows
ˆfd = argmax
fd,fs
N−1∑
n=0
γ(nT −1)
+nR−1∑
m=0
f(n, m)e−j2πfdn
e−j2πfsm
2
.
(17)
Next, to estimate the spatial location, two scenarios can be
considered. In the absence of interferers, i.e., Rin = σ2
nInR ,
it can be proved that minimizing the cost function in (10)
is equivalent to maximizing the cost function defined in (11)
(please see appendix). Thus, using a 2D-FFT, the spatial and
Doppler frequencies can be jointly estimated as follows
ˆfd, ˆfs = argmax
fd,fs
N−1∑
n=0
γ(nT −1)
+nR−1∑
m=0
f(n, m)e−j2πfdn
e−j2πfsm
2
.
(18)
The estimator of θt can be finally formulated as
ˆθt = sin−1
(
λ
dR
ˆfs
)
. (19)
For γ = 1, the number of samples is usually greater than the
number of transmit and receive antennas, i.e., N ≫ (nR+nT ).
Therefore, the resolution of the Doppler frequency, ˆfd, is ex-
pected to be higher than the resolution of the spacial frequency
ˆfs.
The second scenario takes into consideration the presence
of interferers. In this case, since the estimate ˆfd is known, a
search method should be applied to find the estimate ˆθt that
minimizes the cost function J2. To reduce the computational
cost, instead of evaluating the cost function J2 over all grid
points, we can restrict the search method over the region
centered around the maximum of J3.
IV. CRAM ´ER-RAO LOWER BOUND
In this section, the CRLB of the reflection coefficient βt,
the Doppler shift fdt, and the spatial location θt will be
derived. Let η = [ ℜ(βt) ℑ(βt) fdt θt ] be the vector
of unknown parameters and y be the vector of all received
samples from time n to (n+N −1) are stacked. Using similar
definitions for u and v, the problem in (1) can be reformulated
in a vector form as follows
y = u + v, (20)
where y =
[
yT
(n) yT
(n + 1) · · · yT
(n + N − 1)
]T
.
Under the assumption that the noise samples are spatially
uncorrelated, i.e., absence of interferers, the FIM for the
estimation of η can be found using the Slepian-Bangs formula
[12]
F(η) =
2
σ2
n
ℜ
(
∂uH
∂η
∂u
∂ηT
)
=
2
σ2
n
ℜ
(N−1∑
n=0
(
∂uH
(n)
∂η
∂u(n)
∂ηT
))
, (21)
where
∂uH
(n)
∂η
=






∂uH
(n)
∂ℜ(βt)
∂uH
(n)
∂ℑ(βt)
∂uH
(n)
∂fdt
∂uH
(n)
∂θt






∈ C4×nR
, (22)
and
∂u(n)
∂ηT
=
[
∂u(n)
∂ℜ(βt)
∂u(n)
∂ℑ(βt)
∂u(n)
∂fdt
∂u(n)
∂θt
]
∈ CnR×4
. (23)
Each entry of the vector in (23) can be expressed as follows
∂u(n)
∂ℜ(βt)
= ej2πfdtn
aR(θt)aT
T (θt)x(n) =
u(n)
βt
,
∂u(n)
∂ℑ(βt)
= jej2πfdtn
aR(θt)aT
T (θt)x(n) =
u(n)
−jβt
,
∂u(n)
∂fdt
= jβt2πnej2πfdtn
aR(θt)aT
T (θt)x(n) = (j2πn)u(n),
∂u(n)
∂θt
= jβtej2πfdtn 2π
λ
dR cos(θt)×
(
aT
T (θt)AT x(n) + aT
T (θt)x(n)AR
)
aR(θt),
where
AT =





0 0 · · · 0
0 γ
...
...
...
...
... 0
0 . . . 0 γ(nT − 1)





∈ RnT ×nT
, (24)
and
AR =





0 0 · · · 0
0 1
...
...
...
...
... 0
0 . . . 0 nR − 1





∈ RnR×nR
. (25)
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4. Using simple calculations, it can be noticed that the FIM is
independent of fdt. The CRLB of the parameters can be found
by simply inverting FIM. Since F(η) is independent of fdt,
the CRLB will also be independent of fdt. Due to the size
limitation, derivations of the close form solution of the FIM
and the CRLB are not given in this work but can be found in
[13].
V. SIMULATION RESULTS
In this section, two simulation results are presented to
demonstrate the performance of the proposed estimators. In
both of them, 10 transmit and 10 receive antennas with half-
wavelength inter-element spacing are used, i.e., dR = dT = λ
2
and γ = 1 . The results are averaged over 5, 000 Monte Carlo
realizations, the number of transmitted symbols is N = 32 and
the size of the 2D-FFT is 4096. The target of interest is located
at θt =10◦
and its reflection coefficient is βt =−1+2j. Since
the CRLB of the parameters is independent of the target’s
Doppler shift, for each realization, fdt is generated from a
Gaussian random variable with mean 0.25 and variance 0.001.
In the first simulation, we considered the case where there
are no interferers. Thus, the interference plus noise covariance
matrix, Rin, is diagonal and minimizing the cost function J2
becomes equivalent to maximizing J3. Fig. 1 shows that, at
very low SNR, the MSEE fails to meet the CRLB. However,
at low and high SNR, the proposed estimators achieve the
CRLB. It should be noted that at high SNR, as the CRLB
decreases, higher 2D-FFT orders should be used to increase
the resolution of the estimates. To overcome this issue without
increasing the resolution of the 2D-FFT, iterative algorithms
can be used to ameliorate the performance of the estimators
and match the CRLB.
In the second simulation, we considered the presence of
two interferers with reflection coefficients 100 times higher
than the SNR and located at −10◦
and 30◦
. Figure 2 presents
the performance of the estimates which minimize the cost
function J2. Similarly, at very low SNR, it is shown that the
MSEE does not attain the CRLB. Besides, a constant 4dB
gap can be noticed between the MSEE and the CRLB of the
parameter θt.
VI. CONCLUSION
In this work, a 2D-FFT is used to estimate the reflection
coefficient, the Doppler shift, and the spatial location of a
target of interest. By deriving the Fisher information matrix of
the parameters, we concluded that the CRLB is independent of
the Doppler shift. Moreover, by comparing the MSEE with the
CRLB, we showed that the estimates have good performance at
low and high SNR. However, at very high SNR, the estimator’s
performance is limited by the 2D-FFT resolution. To overcome
this issue and enhance the performance of the estimators at
high SNR, iterative algorithms can be used and this is the
focus of our ongoing research [13].
APPENDIX
A two part proof will be derived to demonstrate that,
in absence of interferers, minimizing the cost function J2
defined in (10) is equivalent to maximizing the cost function
−30 −25 −20 −15 −10 −5 0 5 10
−100
−80
−60
−40
−20
0
20
MSEEandCRLB(dB)
SNR (dB)
ℜ{ˆβt}
ℑ{ˆβt}
ˆθt
ˆfd
Fig. 1. Comparison of the CRLB (dashed lines) with the MSEE (solid lines)
of βt, fd, and θt. Here, βt =−1+ 2j and θt =10◦.
−30 −25 −20 −15 −10 −5 0 5 10
−100
−80
−60
−40
−20
0
20
MSEEandCRLB(dB)
SNR (dB)
ℜ{ˆβt}
ℑ{ˆβt}
ˆθt
ˆfd
Fig. 2. Comparison of the CRLB (dashed lines) with the MSEE (solid lines)
of βt, fd, and θt determined by minimizing J2. Here, βt =−1+2j, θt =10◦,
and INR=20dB.
J3 introduced in (11). To this extent, we first need to prove
that
θt = argmax
θ
wH
(θ)Ryw(θ), (26)
where w(θ) is as defined in (5).
In absence of interferers, i.e., Rin = σ2
nInR , using (5), we
can write
wH
(θ)Ryw(θ) =
aH
R (θ)R−1
in RyR−1
in aR(θ)
(
aH
R (θ)R−1
in aR(θ)
)2
=
aH
R (θ)RyaR(θ)
n2
R
. (27)
Moreover, as the independent waveforms x(n) and the noise
v(n) are uncorrelated, the covariance matrix of the received
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5. signals Ry can be expressed as below
Ry = E
{
βtej2πfdtn
aR(θt)aT
T (θt)x(n) + v(n)
2
}
= |βt|
2
aR(θt)aT
T (θt)InT
a∗
T (θt)aR(θt)H
+ σ2
nInR
= |βt|
2
nT aR(θt)aR(θt)H
+ σ2
nInR
. (28)
Therefore, by combining (28) and (27), we can deduce that
wH
(θ)Ryw(θ) = |βt|
2 nT
n2
R
aH
R (θ)aR(θt)
2
+
σ2
n
nR
, (29)
which is clearly maximized at θ = θt.
In the second part of the demonstration, we will prove by
contradiction that if
θt = argmax
θ
wH
(θ)Ryw(θ), (30)
and
θt, fdt = argmin
fd,θ
wH
(θ)Ryw(θ)
−
1
nT
E
{
e−j2πfdn
wH
(θ)y(n)xH
(n)a∗
T (θ)
} 2
, (31)
then
fdt, θt = argmax
fd,θ
E
{
e−j2πfdn
wH
(θ)y(n)xH
(n)a∗
T (θ)
} 2
= argmax
fd,θ
|β (fd, θ)|
2
. (32)
Thus, let us assume that it exits a couple of variables (fde, θe)
such that
|β (fde, θe)| > |β (fdt, θt)| . (33)
Using (30), we can write
wH
(θe)Ryw(θe) ≤ wH
(θt)Ryw(θt), (34)
which leads to
wH
(θe)Ryw(θe) −
1
nT
|β (fde, θe)|
2
< wH
(θt)Ryw(θt) −
1
nT
|β (fdt, θt)|
2
. (35)
Because (35) contradicts (31), we can finally conclude that
solving (31) is equivalent to solving (32) under the noise only
assumption.
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