Noncooperative code design in radar networks: a gametheoretic approach
 Marco Piezzo^{1}Email author,
 Augusto Aubry^{2},
 Stefano Buzzi^{3},
 Antonio De Maio^{1} and
 Alfonso Farina^{4}
DOI: 10.1186/16876180201363
© Piezzo et al.; licensee Springer. 2013
Received: 10 July 2012
Accepted: 28 February 2013
Published: 28 March 2013
Abstract
A network of radars sharing the same frequency band, and using properly coded waveforms to improve features attractive from the radar point of view is considered in this article. Noncooperative games aimed at code design for maximization of the signaltointerference plus noise ratio (SINR) of each active radar are presented. Code update strategies are proposed, and, resorting to the theory of potential games, the existence of Nash equilibria is analytically proven. In particular, we propose noncooperative code update procedures for the cases in which a matched filter, a minimum integrated sidelobe level filter, and a minimum peak to sidelobe level filter are used at the receiver. The case in which the received data contain a nonnegligible Doppler shift is also analyzed. Experimental results confirm that the proposed procedures reach an equilibrium in few iterations, as well as that the SINR values at the equilibrium are largely superior to those in the case in which classical waveforms are used and no optimization of the radar code is performed.
Keywords
Game theory; Code design; Radar network; Interference mitigation; Nash equilibrium; Minimum peakto sidelobe level (PSL) filter; Minimum integratedtosidelobe level (ISL) filter1 Introduction
In the last decade, the importance of radar has grown progressively with the increasing dimension of the system: from a single colocated antenna to large sensor networks [1, 2]. The concept of heterogeneous radars working together has thoroughly been studied, opening the door to the ideas of multipleinput multipleoutput radar [3, 4], overthehorizon radar networks [5], and distributed aperture radar [6, 7]. These three scenarios are the examples of cooperative radar networks, in the sense that every single sensor contributes to the overall detection process. Unfortunately, in many practical situations, it is not possible to design the network a priori. As such, the sensors are just simply added to the already existing network (plug and fight), and each sensor exhibits its own detection scheme. This is the case of noncooperative radar networks[8, 9]; in this scenario, it is extremely important that each additional sensor interferes as little as possible with the preexisting elements, and, to this end, suitable techniques must be adopted. The usual approaches rely upon the employment of spatial and/or frequency diversity: the former resorts to forming multiple orthogonal beams, while the latter uses separated carrier frequencies to reduce interference [10, 11]. Another possibility is to exploit waveform diversity [12–14]; here, the basic concept is to suitably modulate the waveform of the new sensor so as to optimize the detection capabilities of the specific sensor, but, at the same time, controlling the interference introduced into the network. Notice that this is different from the approach employed in cooperative sensor network, where one must design waveforms so as to optimize the joint performance of the system [15–17].
With regard to the optimization of radar waveforms in a noncooperative scenario, we cite here the studies [18–21]. In [18], the design is based upon the maximization of the global signaltointerference plus noise ratio (SINR), and classic constraints such as phaseonly or finite energy are considered; in [19], instead, the problem of parameter estimation (e.g., direction of arrival) for a noncooperative radar is analyzed. In this article, we propose a different strategy, based upon a gametheoretic approach [22]; we thus deal with the active radars as if they were players of a properly modeled game, whose set of possible strategies is made up of a certain amount of prefixed transmit radar codes. We design utility functions, based on the framework of potential games [23], trying to improve the SINR of the active radars through a noncooperative game. Thus, we present several noncooperative games for radarcode optimization in a noncooperative environment, considering different types of receive filters [24] and accounting for the case of nonnegligible Doppler shifts too.
The remainder of this article is organized as follows. In Section 2, we give some background material on game theory and on potential games, which will be needed in the remaining part of the article. In Section 3, we present the considered radar network signal model and dwell on the proposed noncooperative games for radar code updating. Section 4 is devoted to the analysis of the performance of the proposed games, while, finally, Section 5 contains the conclusions.
2 Brief preliminaries on game theory
$\forall {s}_{k}^{\ast}\ne {s}_{k}.$ Otherwise stated, at an NE, no user can unilaterally improve its own utility by taking a different strategy. A quick reading of this definition might lead to think that at NE users’ utilities achieve their maximum values. Actually, this is not the case, since the existence of an NE point does not imply that no other strategy K tuple exists that can lead to an improvement of the utilities of some players while not decreasing the utilities of the remaining ones. These latter strategies are usually said to be Paretooptimal [22]. Otherwise stated, at an NE, each player, provided that the other players’ strategies do not change, is not interested in changing its own strategy. However, if some sort of cooperation would be available, players might agree to simultaneously switch to a different strategy K tuple, so as to improve the utility of some, if not all, players, while not decreasing the utility of the remaining ones. The gap existing between the achieved utilities at the NE and those achieved in correspondence of Paretooptimal points is sometime colorfully named “the price of anarchy”.
The concept of best response dynamic is also worth being introduced. Given a certain strategy profile (s _{ k },s _{k }) for the active players, we say that a player implements a best response dynamic if he chooses as its new strategy ${\stackrel{~}{s}}_{k}=\text{arg}\phantom{\rule{1pt}{0ex}}\underset{x}{\mathrm{\text{max}}}{u}_{k}(x,{\mathit{s}}_{k})$. Given this definition, it descends that the set of chosen strategies at an NE is the best response for every active player.
2.1 Potential games
for any $k\in \mathcal{K}$, ${s}_{k},{\stackrel{~}{s}}_{k}\in {\mathcal{S}}_{k}$, and for any ${\mathit{s}}_{k}\in {\mathcal{S}}_{1}\times \cdots {\mathcal{S}}_{k1}\times {\mathcal{S}}_{k+1}\times \cdots {\mathcal{S}}_{K}$. Given a normal form game, a potential function subsumes the effects that any unilateral change of strategy may have on the utility enjoyed by that individual player. A moment of thought also reveals that every NE of an exact potential game must necessarily be a fixed point of the potential function, as well as that a best response dynamic in a potential game will converge to a NE in every game with continuous utility functions and compact strategy spaces [23].
Finally, it is also worth underlining that, if the potential function does represent a global performance measure for the considered system, potential games are an instance wherein users can serve the greater good while playing a noncooperative game and acting selfishly.
In the following, we will be using game theory concepts to model competition among a set of radars (the players) that tune their own transmitted code in order to maximize their SINR. Potential games, that have been used in recent years to obtain resource allocation schemes in wireless communication applications (see, e.g., [25] and references therein), will be used here in a radar scenario to come up with procedures convergent to an NE.
3 Problem formulation and code updating procedure
(k = 0,…N  1, (i,j) ∈ {1,…,N}^{2}) is the N × N shift matrix, and (·)^{ T } is the transpose operator. As to the modulating sequence c _{ l }, we suppose that it belongs to a finite set Ω _{ l } which containsall the possible sequences of length N that the l th radar can transmit.
It is interesting to provide an interpretation of the contributions appearing in the righthand side of (1). Indeed, the first term represents the signal component from the range bin of interest for the l th radar; the second contribution accounts for the selfinduced interference, while the third addend represents the interference caused by the other radars of the network on the l th one.
where the matrix G models the beampattern of the receive antenna.
The SINR γ _{ l } is indeed a measure of the detection capabilities of the l th radar in the range cell of interest. Note that at the denominator we have the contributions from the backscattered signals transmitted from the other (interfering) radars, weighted by the antenna pattern according to their direction of arrival; it thus follows that a proper design of the receive pattern helps to increase the detection capabilities.
3.1 Antenna beam pattern
The design of the receive antenna beam is of primary importance, especially in the case in which multiple radars operate in the same area. This problem is a classical one, and has deeply been analyzed in past years, especially with reference to wireless communications [31], where adaptive antennas are used in conjunction with power control and smart multiple access (MA) techniques. Obviously, it also plays a primary role in radar applications, where all the transmitting systems act as reciprocal sources of interference. Since we are considering here a noncooperative scenario, no MA or a priori coordination schemes can be applied. Similarly, since the ultimate goal of a radar is to maximize its detection capability, resorting to power control is unrealistic.
Given the outlined system model, our actual goal now is to design a noncooperative procedure for adapting the radar codes in order to maximize the individual detection performances.
3.2 Matched filter
for l = 1,…,L. The solution for c _{ l } to problem (5) exists and can be found through an exhaustive optimization over the finite set Ω _{ l }, with an acceptable computational complexity because in practice the quoted set contains a quite small number of elements.
we obtain an exact potential game with potential function T(·). Summing up, we propose the radar code update procedure reported in Algorithm 1.
Algorithm 1 Radar update procedure—matched filter
As already discussed, since at each iteration the potential function in (7) gets increased, and since it is upper bounded, it necessarily follows that the above iterative algorithm must reach a fixed point (NE). Notice however that there is in general no guarantee that such a fixed point is the global maximizer of the potential function, or just a local extremum [23].
3.3 Minimum ISL filter
The matched filter, considered in the previous section, is obviously the classical receiving structure used in detection problems. However, it does not allow to completely control the sidelobe energies, a feature that may be critical in radar applications. Indeed, this limitation may strongly affect the target detection capabilities of the radar system, especially in scenarios where multiple radars have to coexist in the same area, thus becoming themselves the main source of reciprocal interference.
in the sense that υ(11)$=\frac{\upsilon (12)}{{b}^{2}}\phantom{\rule{1em}{0ex}}(\upsilon (\xb7)$ stands for the optimal value of problem (·)), for any b > 0.
It is well known that problem (12) has a closed form solution ${\mathit{x}}^{\star}({\mathit{c}}_{l})=b{\mathit{Q}}_{l}{\mathit{c}}_{l}$, for any constant b > 0, with ${\mathit{Q}}_{l}\triangleq \frac{{\mathit{R}}_{l}^{1}}{{\mathit{c}}_{l}^{\u2020}{\mathit{R}}_{l}^{1}{\mathit{c}}_{l}}$ (indeed, it is possible to prove that R is strictly positive definite and thus invertible, provided c _{ l }(1) ≠ 0 and c _{ l }(N) ≠ 0 [34]); as a consequence, in order to solve (12), it suffices to focus on (13) with b = 1 and ψ = 0.
We summarize the steps for the radar code update procedure in the Algorithm 2.
Algorithm 2 Radar update procedure—minimum ISL filter
3.4 Minimum PSL filter
Note that designing a filter minimizing the PSL is equivalent to cutting all the sidelobes in the filter response, and constraining the mainlobe peak to a desired level.
which belongs to the class of the LP [29, 30] or SOCP [27] problems for the case of real or complex transmitted code sequence and optimization variable, respectively.
Obviously, an optimal solution x ^{⋆} for problem (21) is a function of the radar code c _{ l } used by the player; therefore, the finite set Ω _{ l } of the possible radar sequences and the set, say Σ _{ l }, of the possible optimal PSL filters are related by a onetoone correspondence. Otherwise stated, specifying Ω _{ l } also leads to specify Σ _{ l }, in the sense that the set of the filters can be computed directly offline, and populated by the possible solutions for the problem (21).
Classes of phasecodes ϕ
Parameters  ϕ _{ l }  Parameters  ϕ _{ l } 

r = 15  GolombZhang  –  Palindronic P4 
–  MPS (Minimum Peak Sidelobe)  r = 3  Chu 
–  ZadoffChu  r = 13  GolombZhang 
r = 5,q = 10  Zadoff  r = 17  Chu 
r = 27,q = 8  Zadoff  r = 3,q = 16  Zadoff 
–  P3  r = 21,q = 0  Zadoff 
r = 3  GolombZhang  –  – 
whose iterative maximization by the active radars leads to a new potential game admitting NE points. Algorithm 3 summarizes the radar code update iterations for the case at hand.^{c}
Algorithm 3 Radar update procedure—minimum PSL filter
3.5 Nonnegligible Doppler shift
In writing the above equation, we have made explicit the functional dependence of the matrix F _{ j } on the code c _{ j }, which has to properly be accounted for in the utility maximization. Summing up, for nonnegligible Doppler shifts and matched filter reception, each radar should update its code to maximize the utility in (34), and the detection rule to be considered should be the one reported in Equation (31).
Similar considerations can be done for the cases in which a minimum ISL or PSL filters are used. For the sake of brevity, however, we avoid providing more details on this, since it would not add conceptual value to this study.
4 Performance analysis

${\mathcal{\mathcal{L}}}_{1}=\{1,2,3,4\}$ is the set of 4 players (i.e., the set of four radars actually transmitting), while ${\mathcal{\mathcal{L}}}_{2}=\{1,2,3,4,5,6\}$ is the set of six players (i.e., the set of six radars actually transmitting);

Ω _{ l } is a set of cardinality M = 653 which contains the sequences of length N = 16 available to the l th player. The same set is considered for each radar, i.e., Ω _{ l } is actually independent of the index l (and indeed we will be denoting it by Ω in the following). The full details on the sequences of the set Ω are reported in Appendix.

{u _{ l }} represents the utility function for the l th player, as defined in the discussed Algorithms 1, 2, and 3, for l = 1,…,4 or l = 1,…,6, respectively for the first and the second games;

G is the L _{ i } × L _{ i } matrix describing the antenna gain pattern of the L _{ i } players, for i = {1,2}. We consider a general scenario wherein each radar may have its own antenna beam pattern, but we normalize, without loss of generality, to 0 dB the maximum gain of each antenna. Indeed, we consider the following pattern models for the games ${\mathcal{G}}_{1}$ and ${\mathcal{G}}_{2}$:$\begin{array}{l}{\mathit{G}}_{{{\mathcal{G}}_{1}}_{\text{dB}}}=\left[\begin{array}{llll}0& 30& 19& 20\\ 20& 0& 19& 20\\ 20& 30& 0& 20\\ 20& 30& 19& 0\end{array}\right],\\ {\mathit{G}}_{{{\mathcal{G}}_{2}}_{\text{dB}}}=\left[\begin{array}{llllll}0& 30& 19& 20& 15& 23\\ 20& 0& 19& 20& 15& 23\\ 20& 30& 0& 20& 15& 23\\ 20& 30& 19& 0& 15& 23\\ 20& 30& 19& 20& 0& 23\\ 20& 30& 19& 20& 15& 0\end{array}\right],\end{array}$
respectively. ^{d}
In Figure 2a,b, we plot the SINR of each player versus the number of iterations required by Algorithm 1 to converge to an NE, for the games ${\mathcal{G}}_{1}$ and ${\mathcal{G}}_{2}$, respectively; these plots show the impact of the chosen code (strategy) on the SINR of the set of players, as they pick up different codes from the set Ω. Note that the starting codes (strategies) do not provide satisfactory values of γ _{ l } for all the set of players; indeed, in both the games the majority of the sensors experiment quite a low level of SINR, with the exception of the first and the last players. The curves highlight that, as the players change their transmitting codes according to Algorithm 1, the SINR of each player converges to a fixed value: after a certain amount of iterations, the iterative algorithm thus reaches a fixed code (strategy). In particular, both the sets of players share an average increase in their respective performances, quantifiable in about 2.10 dB for the first game and 1.51 dB for the second game, and no particular loss is observed due to the growth of the number of transmitting radars. Moreover, convergence is reached after a few iterations.
In Figure 3a,b, the same analysis is conducted for Algorithm 2. Again, the starting strategy seems to be quite disadvantageous for both the sets of active radars, and in particular for the second game (specifically, we experience unpleasant performances in the cases of radars 1 and 4, with reference to the first game, and radars 3–4 for the second game). Resorting to the coding procedure of Algorithm 2, however, all the radars increase the respective performances; in particular, we observe an average increase, in the achieved SINR values, of 1.56 dB for game ${\mathcal{G}}_{1}$ and 2.66 dB for game ${\mathcal{G}}_{2}$. The analysis also shows a gain in terms of ISL values, due to the game approach. Specifically, in Figure 3c, we provide a comparison between the average ISL, with respect to the increasing number of active radars (for the case at hand, we assume a maximum of ten radars), obtained with the Algorithm 2 and the nogame strategy, respectively. In the setup of this simulation, random initial strategies have been selected for the radars and the results have been averaged over 25 independent trials. The plots highlight that the nogame approach is very sensitive to the number of sensors composing the network; in fact increasing values of ISL can be observed when the number of active radars increases. On the contrary, the updating procedure of Algorithm 2 is capable of ensuring a quite flat ISL behavior.
In Figure 4a,b, we focus on the performance of Algorithm 3, and similar comments as for the previous two algorithms can be made. The average increase, in terms of SINR, can be quantified in 3.60 dB for the first game, and 1.19 dB for the second one.
In Figure 4c, we consider the average PSL versus the number of active radars, for both the nogame approach and the noncooperative game technique of Algorithm 3. The same simulation conditions as in Figure 3c have been considered concerning the initial choice. Notice that the average PSL for the nogame approach appears quite unpleasant, as worse and worse PSL values are obtained increasing the number of active sensors. On the contrary, Algorithm 3 seems quite robust in terms of PSL with respect to the number of active radars.
Overall, the results of this section confirm the effectiveness of the proposed algorithms, as well as that all the considered games converge to an equilibrium.
5 Conclusion
In this article, we have considered a network of radars sharing the same frequency band, and tuning their transmitted waveforms in order to improve their SINR.
We have assumed that each radar can select the waveform to be transmitted from a finite set. Hence, we have proposed code updating strategies according to some noncooperative games, based on the potential games framework, to account for the cases of matched filter detection, minimum ISL and minimum PSL detection. Finally, we have discussed the situation where a nonnegligible Doppler shift exists in the received data. In all the considered scenarios, the existence of NE is analytically proven.
Numerical results have confirmed that the proposed games are effective in improving the system performance, in the sense that at the NE each radar may enjoy an SINR that is larger than that corresponding to the case of a random choice of the coded waveform to transmit. Moreover, it has also been verified that there is a graceful performance degradation as the number of active radars increases.
Possible future research tracks might account for the possibility of some form of cooperation between the radars of the network as well as the extension of the procedure to the case where more advanced decision strategies (in place of the linear filter followed by an envelope detector) are used. By doing so, we can also confer to the system additional desired robust features such as for instance the constant falsealarm rate property.
Appendix
Code design procedure
We choose our Ndimensional radar codes so that c = 1, $c\in {\u2102}^{N}$; otherwise stated, we fill the set Ω with sequences lying on the unitnorm sphere. To this end, we consider both standard codes available in open literature and adhoc coding procedure.
As to the former class, we refer to some wellknown phasecoding techniques [35] to design the first 13 possible transmit sequences of the set Ω. Specifically, we assume that ${\mathit{c}}_{l}=\frac{1}{\sqrt{N}}{e}^{\sqrt{1}{\mathit{\varphi}}_{l}}$, where ϕ _{ l }= [ϕ _{ l }(1),…,ϕ _{ l }(N)]^{ T }is the phase sequence of the l th code, and l = 1,…,13. In Table 1, we summarize the classes of phase codes herein used, as well as the values of the parameters applied in the respective design procedures.^{e}
In addition, to properly test our noncooperative procedures, we increase the number of possible strategies enriching with other suitable codes the set Ω. We resort to the following construction procedure. First of all, we force the coefficients c _{ l }(i),i = 1,…,N, to belong to a welldefined finite set Ω _{∗} with cardinality M. Then, we obtain the transmit sequences picking up randomly the codes from the set ${\Omega}_{\ast}^{N}$ with cardinality M ^{ N }. Finally, we normalize the selected sequences so as to get unitnorm codes. For the specific case at hand, we set ${c}_{l}(i)\triangleq \{{a}_{i}+\sqrt{1}{b}_{i}\}/\sqrt{2N}$ for l = 14,…,113, with $\{{a}_{i},{b}_{i}\}\in {\{1,+1\}}^{2}$. With such a choice we can produce up to 2^{2N } possible codes. Thus, we randomly choose 100 codes from such a set, and use them in our simulations.
where the parameter ∊ ∈ [0,2] quantifies the desired similarity level; the smaller ∊, the higher the degree of similarity among the ambiguity functions of the designed radar code and the reference sequence.
 1.
Denote by a an N dimensional complex vector whose elements are continuous random variables.
 2.
Construct the unitnorm vector ${{c}_{0}}^{\perp}=(\mathit{I}{c}_{0}{{c}_{0}}^{\u2020})\mathit{a}/\parallel (\mathit{I}{c}_{0}{{c}_{0}}^{\u2020})\mathit{a}\parallel $.
 3.
Define the sequence ${\mathit{c}}^{t}=\sqrt{t}{c}_{0}+\sqrt{1t}{{c}_{0}}^{\perp}$, where the parameter t complies with t ≥ (1  ∊/2)^{2} = δ _{ ∊ } and t ≤ 1.
Set of similarity codes
Parameters  ϕ _{0}  Parameters  ϕ _{0} 

r = 3  Chu  –  Px 
–  Frank  r = 17  GolombZhang 
–  MPS (Minimum Peak Sidelobe)  r = 6,q = 6  Zadoff 
–  P4  –  Polyphase Barker 
–  P1  –  – 
Endnotes
^{a}Actually, the SINR definition should include also the coefficients α _{·,·}; however, no prior knowledge of these coefficients may be reasonably assumed, and we are thus omitting them in the SINR definition reported in (4).^{b}We are considering a bidimensional scenario where G(θ) is the azimuth beam pattern. However, the extension to a threedimensional situation accounting for both azimuth and elevation is quite easy. ^{c}With Ω _{ j } we are denoting the cardinality of the set Ω _{ j }, whereas with ${c}_{j}^{i}$ we are indicating the i th element of Ω _{ j }. ^{d}Recall that in the above gain matrices the (m,n)th element is a coefficient weighting the interference from the m th radar on the n th receiver. ^{e}The reader might refer to [35], which is an exhaustive compendium of the classic radar coding techniques.
Declarations
Acknowledgments
The effort of A. De Maio and M. Piezzo is sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655–091–3006. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright notation thereon.
Authors’ Affiliations
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