Joint Frequency-hopping Waveform Design for
MIMO Radar Estimation Using Game Theory
Keyong Han and Arye Nehorai
Preston M. Green Department of Electrical and Systems Engineering
Washington University in St. Louis
One Brookings Drive, St. Louis, MO 63130, USA
Email: {kyhannah,nehorai}@ese.wustl.edu
Phone: 314-935-7520 Fax: 314-935-7500
Abstract—Using a colocated Multiple Input Multiple Output
(MIMO) radar system, we consider the problem of joint design of
amplitudes and frequency-hopping codes for frequency-hopping
waveforms. First, we present the MIMO radar signal model and
the sparse representation. Then, we formulate a novel game
model and propose two joint design algorithms, one applying
a non-cooperative scheme, and the other applying a cooperative
scheme. Owing to the extremely large size of the feasible set of
the discrete code, we propose to use these algorithms to obtain
the -approximate Nash equilibrium (-ANE). We demonstrate
the improvement of the resulting codes and amplitudes through
numerical examples.
Index Terms—MIMO radar, frequency-hopping codes, sparse
modeling, optimal design, multiple objectives, game theory, approximate Nash equilibrium (-ANE)
I. I NTRODUCTION
Multiple Input Multiple Output (MIMO) radar is an active
technology that is attracting the attention of researchers due
to the improvement in performance it offers over conventional
single antenna systems. MIMO radar systems are commonly
used in two different antenna configurations: distributed [1]
and colocated [2]. Distributed MIMO radar improves performance by exploiting spatial diversity, whereas colocated
MIMO radar exploits waveform diversity. In this paper, we
refer to colocated MIMO radar. The authors in [3] and [4]
considered optimal frequency-hopping waveform designs for
colocated MIMO radar, but the frequency-hopping codes and
amplitudes were designed separately. Our goal in this paper is
to design them jointly, exploiting the advantages of both. Rao
[5] proposed a game based on Nash equilibrium to conduct
optimization problems and described the relationship between
Pareto-optimal solutions and game theory.
In this paper, we propose to employ game theory to solve
this joint optimization problem by viewing the two objective
functions, corresponding to frequency-hopping codes and amplitudes respectively, as two interacting players (see also [6]).
We will show that the joint design shows superiority over
separate designs.
This work was supported by the Department of Defense under the AFOSR
Grant FA9550-11-1-0210.
2013 IEEE Radar Conference (RadarCon13)
978-1-4673-5794-4/13/$31.00 ©2013 IEEE
II. S IGNAL M ODEL
In this section, we will describe the signal model for our
radar system. We first develop the measurement model for
the MIMO radar estimation problem. Then we transform the
measurement model to a sparse representation.
A. Measurement Model
We assume that there are MT transmitters and MR receivers
in the colocated MIMO radar system, arranged in uniform
linear arrays (ULA). The spacing between the transmitters is
dT , and the spacing between receivers is dR . Additionally, we
assume the targets are located in the far-field so that the angle
between them will be viewed as the same. Let these arrays
form the same angle θ with the targets. Let L pulses comprise
a waveform, and we write the frequency-hopping
waveform
L−1
from the mth transmitter as um (t) = l=0 φm (t−Tl ), where
Q−1
φm (t) = q=0 bm,q ej2πcm,q Δf t 1(0,Δt) (t − qΔt) and
1, if 0 < t < Δt,
1(0,Δt) (t) 0, otherwise.
Tl and Δt respectively denote the pulse repetition interval
and hopping interval duration. q and Q represent the hopping
index and the length of the code. We assume Δf Δt = 1. The
frequencies cm,q of the transmitted signal during each hopping
interval and the amplitudes bm,q of the transmitted sinusoid are
the waveform design parameters. The optimal design of the
frequency hopping waveform amounts to choosing cm,q and
bm,q for all the transmitters and all the hopping intervals. We
assume each cm,q ∈ {1, . . . , K}, where K is a positive integer.
Further, to maintain the orthogonality of the waveforms, cm,q
satisfies cm,q = cm ,q , for m = m , ∀q.
For amplitudes, we constrain bmin ≤ |bm,q | ≤ bmax for all
transmitters and frequencies. This constraint provides control
over the power of the transmitted radar signals. Further, we
normalize
the transmitted energy for each waveform by asQ−1
suming q=0 |bm,q |2 = 1. For convenience, we arrange cm,q
into an MT × Q dimensional code matrix C = {cm,q }MT ×Q ,
which describes all the transmitted frequencies. Similarly, we
arrange bm,q into an MT × Q dimensional amplitude matrix
B = [bT1 ; . . . ; bTMT ], where bm = [bm,1 , . . . , bm,Q ]T .
Consider a target at (τ, ν, f ), where τ is the delay of
the target, ν is the Doppler frequency, and f is the spatial
frequency f = dR sin(θ)
, where λ is the wavelength of the
λ
carrier. We assume point targets in the surveillance scene, with
different scattering centers resonating at different frequencies.
Therefore, each target has a frequency-dependent radar cross
section (RCS).
The target response at the k th receive antenna is a summation of all reflected signals from all emitters. Considering
the situation of R targets, the received signal is a summation
of the reflected signals from all the targets. We sample the
received signal, obtaining
yk (n) =
MT R L−1
Q−1
arm,q bm,q ej2πcm,q Δf (nTS −Tl −τ
r
non-zero blocks, corresponding to different targets. Additionally, we stack the measurements and noise samples at all
receivers to obtain the measurement model:
y = Ψx + e.
)
m=1 r=1 l=0 q=0
×1(0,Δt) (nTS − qΔt − Tl − τ r )ej2πν
×ej2πf (γm+k) + ek (n),
r
(1)
In [3], the authors developed an objective function to optimally design the frequency-hopping
codes based on the ambi∞ 11
guity function: fp = −∞ 0 0 |Ω(τ, f, f )|p df df dτ, where
MT −1 MT −1 φ
j2π(f m,f m )γ
Ω(τ, f, f ) . The
m=0
m =0 rm,m (τ )e
cross correlation function is defined as (3), at the top of next
page, and the ambiguity function
Δt of the rectangular pulse
s(t) is given by χrect (τ, ν) 0 s(t)s(t + τ )ej2πτ dt. The
frequency hopping codes satisfy cm,q ∈ {0, 1, . . . , K − 1},
cm,q = cm ,q , for m = m . When we consider the sparse
recovery model, the delay range is [0, τ D ]. Therefore, the
optimization problem can be reduced to
τD 1 1
minC fp (B, C) =
|Ω(τ, f, f )|p df df dτ. (4)
Suppose we discretize the delay and Doppler space into D
and W uniformly separated grid points. Therefore we have a
total of DW grid points, among which only R correspond to
the real targets. Let τ d and ν w represent the delay and Doppler
of each grid point. For each grid point, we define the basis
function as
ej2πcm,q Δf (nTS −Tl −τ
d
)
l=0
× 10,Δt (nTS − qΔt − Tl − τ d )ej2πν
w
nTS j2πf (γm+k)
e
Stacking {ψ̃m,k,q (n, d, w)}N
n=1 according to samples, we
obtain an N × 1 vector ψ̃m,k,q (d, w). Then we stack
R
{ψm,k,q (d, w)}M
k=1 with respect to receivers into an N MR × 1
vector ψ̃m,q (d, w). Now, each ψ̃m,q (d, w) is the basis vector
for a different transmitter and frequency. For all transmitters
and frequencies, we further stack ψ̃m,q (d, w) into an N MR ×
MT Q matrix Ψ(d, w). Considering all the combinations of
(di , wj ), i = 1, . . . , D, j = 1, . . . , W, we can finally stack
Ψ(d, w) into an N MR × DW MT Q matrix Ψ. This is the
dictionary matrix, defining the basis elements of the sparse
representation. Considering all the different transmitters and
frequency intervals, we stack arm,q bm,q into an MT Q vector
xr . Now, we define the sparse vector as
r
x , if (τ d , ν w ) = (τ r , ν r ),
x(d, w) =
0,
otherwise.
Further, we stack the vectors x(d, w), corresponding
to different delay and Doppler grid points, to obtain a DW MT Q dimensional block-sparse vector x =
[x(1, 1)T , . . . , x(D, W )T ]T . In this vector, there are only R
In this section, we will formulate the game model for the
joint design of the frequency-hopping waveforms, and then
jointly design the frequency-hopping codes and amplitudes
using game theory.
A. Game Model
B. Sparse Model
L−1
where
T
T
T
T
• y = [y1 , y2 , . . . , yM ] , an N MR × 1 vector;
R
T
• yk = [yk (1), yk (2), . . . , yk (N )] , an N × 1 vector;
T
T
T
T
• e = [e1 , e2 , . . . , eM ] , an N MR × 1 vector;
R
T
• ek = [ek (1), ek (2), . . . , ek (N )] , an N × 1 vector;
We will use the block matching pursuit (BMP) [7] method to
recover the unknown target parameters.
III. J OINTLY O PTIMAL D ESIGN U SING G AME T HEORY
nTS
∀n = 1, . . . , N , where N is the total number of samples at
each receiver during one processing interval, and TS denotes
the corresponding sampling interval. γ dT /dR , arm,q represents the target’s RCS, and ek (n) is the additive noise received
at the k th receiver, assumed to follow Gaussian distribution.
Equation (1) gives the measurement model.
ψ̃m,k,q (n, d, w) =
(2)
.
0
0
0
For amplitudes, we apply the optimization problem proposed in [4]. We want to maximize the smallest recovered
estimate in x̃m (bm ) = [x̂1m (bm ), . . . , x̂R
m (bm )] for each
transmitter: maxbm minr∈1,...,R {x̃rm (bm )}, where x̂rm Q−1 2
r
2
r
q=0 bm,q |âm,q | and âm,q are the estimates of target attenuations, given initial bm,q = 1. The amplitude constraints
Q−1
2
are |bm,q | ≥ bmin , |bm,q | ≤ bmax , and
q=0 |bm,q | = 1.
Equivalently, the optimization problem can be converted to
minbm f˜m (bm , C) = maxr∈1,...,R {Z − x̃rm (bm )},
(5)
− x̃rm (bm )
where Z is a large constant, guaranteeing that Z
is
non-negative.
A normal strategic game is a model of interacting decision makers, consisting of players, their strategies, and the
players’ cost functions [8]. In our joint design problem, we
have MT + 1 objective functions. As for the MT objective
functions for amplitude design, their variable choices are
independent of each other. Accordingly, we write these MT
cost functions into one summation function: fs (B, C) =
MT ˜
m=1 fm (bm , C). Therefore, the optimization problem (5)
is equivalent to minB fs (B, C). We assume that each player
φ
rm,m
(τ ) =
Q−1
Q−1
bm,q bm ,q χrect (τ − (q − q)Δt, (cm,q − cm ,q )Δf ) × ej2πΔf ((cm,q −cm ,q ))qΔt ej2πΔf cm ,q τ .
(3)
q=0 q =0
where j wi,j = 1, wi,j = wj,i . In our problem, we choose
the second player’s cost function to remain the same, without
considering the first player’s influence. All that changes is the
first player’s objective function. Additionally, we replace the
absolute cost function with a relative cost function. Details are
shown as follows:
Q−1
2
• Initialize C0 randomly, and initialize bm0 , m
=
B = {B||bm,q | ≥ bmin , |bm,q | ≤ bmax ,
|bm,q | = 1}, and
1, . . . , MT , to be equal.
q=0
• At step k, fix Ck , then conduct the single objective
C = {C|cm,q ∈ {1, 2, . . . , K}MT Q , cm,q = cm ,q , for m = m }.
optimization design for Player 2: minB f2 (B, Ck )),
resulting in the optimized Bk+1 . Using the newly
(B, C) denotes a profile of actions by the two players.
generated
Bk+1 , we optimize Player 1’s new objecA Nash equilibrium is a profile of strategies such that the
M T
f˜m (bm,k+1 ,C)
tive function minC f1∗ =
strategy for each player is an optimal response to the strategy
m=1 wm f˜m (bm,k ,Ck ) +
of the other player. Owing to the numerous choices of C, up
k+1 ,C)
wMT +1 ff11(B
(Bk ,Ck ) , obtaining Ck+1 .
to K MT Q , it is almost impossible for us to find the global
• Replace Bk , Ck with Bk+1 , Ck+1 , and repeat the above
minimal point for the cost function f1 , making it difficult to
step until an -ANE is satisfied or k is large enough.
find the Nash equilibrium. To circumvent this problem, we
prefer to use the simulated annealing algorithm [9] to achieve
IV. N UMERICAL SIMULATIONS
an acceptable solution (B ∗ , C ∗ ), yielding the -ANE1 :
We consider a uniform linear transmitting array with
∗
∗
∗
transmitter
and receiver numbers MT = 3 and MR = 3
f1 (B , C ) ≤ f1 (B, C ) + , ∀B ∈ B,
respectively.
Choose θ = 300 and let the array spacing
f2 (B ∗ , C ∗ ) ≤ f2 (B ∗ , C) + , ∀C ∈ C,
dT = dR = λ2 . The length of the frequency-hopping
code Q equals 5. The number of frequencies K equals
where is a non-negative parameter.
7. Suppose L = 10 pulses comprise a waveform. The
B. Optimization Algorithm Using Game Theory
chip duration is t = 1μS, whereas the time interval
First, we consider the non-cooperative case, with each between pulses is 3mS. The minimum hopping frequency
player having no knowledge of the other player’s information. interval is f = 1 MHz. Since we chose K = 7, the
They know only their own information. The algorithm is maximum hopping-frequency is Kf = 7 MHz. Therefore,
we sample at a Nyquist rate of 14 × 106 samples per
expressed as follows:
• Initialize C0 randomly, and initialize bm0 , m
= second. Further, we suppose three targets are present in the
scenario. We specify the amplitudes of K attenuations
1, . . . , MT to be equal.
for each target a1 = [0.4, 0.2, 0.5, 0.8, 0.1, 0.4, 0.3],
• At step k, fix Ck , then conduct the single objective
2
=
[0.6, 0.2, 0.8, 0.9, 0.1, 0.3, 0.5],
a3
=
optimization design for Player 2: minB f2 (B, Ck ), re- a
[0.2,
0.4,
0.3,
0.7,
0.4,
0.1,
0.9].
We
can
use
this
attenuation
sulting in the optimized Bk+1 . Using the newly generated Bk+1 , we optimize Player 1’s objective function information to estimate the RCS corresponding to different
hopping-frequencies and transmitters.
minC f1 (Bk+1 , C), obtaining Ck+1 .
When considering sparse recovery, we discretize the target
• Replace Bk , Ck with Bk+1 , Ck+1 , and repeat the above
delay-Doppler
space. The grid size in delay is t = 1μS, and
step until an -ANE is satisfied or k is large enough.
the Doppler dimension is 10 Hz. The delay space is uniformly
We apply the convex programming software CVX [11] [12]
divided in the interval [0, 10] μS, and the Doppler grid points
to solve the single amplitude optimization problem.
lie uniformly in the interval [0, 100] Hz. Therefore, we have
In the cooperative2 case, we suppose each player knows
a total of V = 11 × 11 = 121 grid points, out of which only
part of the other player’s information by incorporating their
three correspond to the true targets. We
1 assume
the true delays
cost functions in his or her own cost function. The
resulting
2 3
and
Doppler
shifts
of
the
targets
are
τ
,
τ
,
τ
= [4, 9, 1] μs,
cost function for each player is fi∗ = wi,i fi + j=i wi,j fj , 1 2 3 ν , ν , ν = [80, 60, 40] Hz. We define the
signal-to-noise
Ψx2
1 The -approximate Nash equilibrium [10] here is for a pure strategy game,
dB, where E{·}
ratio (SNR) as SNR = 10log E(e
2)
which is different from the one concerning mixed strategies defined in most
denotes the expected value of {·}.
existing literature.
We compare the performance of the proposed design method
2 The cooperative case here is different from a cooperative game. We
and the existing separate design methods, using the empirical
consider the coalition between different players.
is associated with one objective function. Therefore, we have
two players. The cost function for the first player is chosen
as f1 (B, C) = fp (B, C). The cost function for the second
player is f2 (B, C) = fs (B, C). The strategy spaces for the
two players are respectively
15
14
0.98
Empirical CDF
0.94
0.92
Without Optimal Design
Optimize Only C
Non−cooperative Design
Cooperative Design
Optimize Only B
0.9
0.88
−6
−5
−4
−3
Magnitude (dB)
−2
−1
Fig. 1: Empirical CDF of |Ω(τ, f, f )|.
TABLE I: Target estimates at an SNR of -21.58 dB
(τ 1 , ν 1 )
(4μ, 90Hz)
(4μ, 80Hz)
(4μ, 80Hz)
(τ 2 , ν 2 )
(6μ, 70Hz)
(9μ, 50Hz)
(9μ, 60Hz)
12
11
10
9
8
7
0
Design Method
Only Optimize C
Only Optimize B
Cooperative Design
Performance Metric Δ
13
0.96
Without Optimal Design
Optimize Only C
Non−cooperative Design
Cooperative Design
Optimize Only B
(τ 3 , ν 3 )
(2μ, 30Hz)
(2μ, 50Hz)
(1μ, 40Hz)
cumulative distribution function (CDF) of |Ω(τ, f, f )| applied
in [3]. We use Monte Carlo method by sampling from the
function |Ω(τ, f, f )|, and plot the percentage of samples of
|Ω(τ, f, f )| less than different magnitudes.
Fig. 1 shows the results of five different design methods.
Since the CDF criterion is mainly designed to measure the performance improvement resulting from designing code matrix
C, it is not surprising to see that the method designing only C
achieves the best performance. When designing B only, the
performance becomes worse than even the case without any
optimal design. This is explainable from game theory point of
view. Each player tries to choose strategies that minimize his
or her own cost function, which will not necessarily contribute
to the other player’s benefits. In this example, it degrades the
other player’s performance. Game theory based joint design
performs much better than that designing only B.
For amplitudes, we use the performance metric [4]: Δ =
minx̂S
maxx̂S̄ , where S and S̄ denote the recovered support sets of
the correct and incorrect target indices, respectively. We want
this metric to be as large as possible. Fig. 2 shows the result,
which is similar to that of Fig. 1. Note that the cooperative
design achieves the best performance.
As we assumed, three targets are present. We used 30
iterations for the BMP algorithm. The reconstructed target
parameters at an SNR of -21.58 dB are shown in Table I,
corresponding to three different designs. Only the cooperative
design recovered all the three targets correctly.
V. C ONCLUDING REMARKS
We proposed joint optimization designs using game theory
to compute the adaptive code matrix and amplitude ma-
6
−3
−2
−1
0
SNR (in dB)
1
2
3
Fig. 2: Performance metric Δ comparison.
trix for frequency-hopping waveforms. For this purpose, we
considered a colocated MIMO radar. Both the cooperative
case and non-cooperative case were considered to optimize
the frequency-hopping codes and amplitudes. We numerically
demonstrated the advantage of jointly optimal design methods
over separate design methods. In future work, we will consider
non-uniform grid spacing to improve the estimation accuracy,
and will validate our results using real radar data.
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