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 [1214]; 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 [1517].
With regard to the optimization of radar waveforms in a noncooperative scenario, we cite here the studies [1821]. 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
Formally speaking, a game
denote a certain strategy K tuple for the active players. Letting, as customary in the game theory literature, s_{k} denote the (K  1)dimensional vector whose entries are the strategies of all the players except the kth, the point (s_{1},s_{2},…,s_{K}) = (s_{k},s_{k}) is an NE if, for any player k, we have
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
2.1 Potential games
A potential game [23] is a normal form game wherein any change in the utility enjoyed by a given player
in reaction to a unilateral (i.e., assuming that the other players do not change their
strategies) change of strategy by that player is reflected by a similar change in
a global function, that is usually referred to as potential function. Formally speaking,
letting
for any
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
We consider a network of L noncooperative monostatic radar systems, where each sensor transmits a coded pulse composed of N subpulses. The signal backscattered toward the lth radar is filtered through a subpulse matched filter and then converted into digital. The vector r_{l},l = 1,…,L, containing the received sequence r_{k,l},k = 1,…,N, assumed temporally aligned with the returns from the range bin of interest, can be written as [26,27]
where c_{l} = [c_{l}(1) … c_{l}(N)]^{T} denotes the unitnorm N dimensional modulating sequence of the lth radar, α_{k,h} are complex parameters accounting for the radar cross section of the kth range bin illuminated by the hth radar (0 is conventionally chosen as the range bin of interest), n_{l} is the vector containing the filtered thermal noise samples at the lth radar (modeled as a zeromean complex circular white vector), the matrix
(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 lth 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 lth 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 lth one.
Now, the vector r_{l} is to suitably be processed in order to detect the possible presence of a target in the range cell of interest. We thus consider the following receiving structure: the vector r_{l} undergoes a linear transformation (projection over a suitable direction vector), and, then, its square modulus is compared with a threshold, i.e., we consider the detection rule
with (·)^{†} denoting conjugate transpose, · the modulus, d_{l}(c_{l}) an Ndimensional vector, function of the transmit code c_{l}, to be suitably designed (it could be a standard matched filter or a mismatched filter [2830] designed to optimize some performance metrics such as the integrated sidelobe level (ISL) or the peaktosidelobe level (PSL)—see more details in the sequel of the article), and η_{l} the detection threshold in the lth radar. Given the detection rule (3), we can define an SINR for the lth radar in the range cell of interest, γ_{l} say, as follows ^{a}
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 lth 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.
In the radar scenario, the beam pattern of the antennas is used as a means to improve the received SINR and to weaken interfering echoes. A simplified model for the beam pattern G(θ) is the one illustrated in Figure 1 where θ = 0 is the radar search direction ^{b}; for instance, such a shape can be approximated through an N element array [32]. Herein, we thus assume that the antenna gain may take two possible constant values, one for θ ∈ [θ_{∊},+θ_{∊}], and one (much lower than the former) outside the above interval: the side contributions are thus all equally weighted by the side beams. The effect of the antenna pattern can be therefore simply modeled as a proper L × L gain matrix G, whose (h,l)th element accounts for the effects of the hth radar on the lth system; the coefficients for l ≠ h are assumed to be a proper constant. The G(l,l) elements on the principal diagonal represent the main beam gain, weighting the useful signal for the lth radar.
Figure 1. Antenna beam pattern. Main beam: θ ∈ [θ_{∊},+θ_{∊}]. Side beam:
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
Given Equation (4), we begin with assuming that d_{l} = c_{l}, i.e., a conventional matched filter receiver is used, and consider minimization
of the denominator in (4), which is equivalent to optimizing γ_{l} since
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.
Unfortunately, when active radars update their own transmitted waveforms according to such a strategy, no sufficient condition has analytically been worked out for the existence of NE, and, moreover, numerical simulations have confirmed that when radars, in a roundrobin fashion, update their codes according to the strategy (5), an equilibrium is not always reached. The considered game has thus no pure strategy equilibrium. One possible way to circumvent such a problem is to properly modify the utility function to be considered so that the resulting game may have an NE point. In particular, if we choose to use the tool of potential games, the trick is to define a new utility function, strictly related to (4), but whose maximization by the competing radars leads to an NE. To this end, let us consider the opposite of the sum of the denominators of γ_{l}’s for the L active radars, i.e.,
Upon some straightforward algebraic manipulations, we have
where the function T_{1}(c_{1},…,c_{j1},c_{j+1},…,c_{L}) does not depend on c_{j}. In Equation (7), we have isolated the terms depending on the jth radar code c_{j}; it thus readily follows that if we consider a game wherein the utility for the jth sensor is expressed as
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.
Therefore, viable alternatives to the matched filter may be sought. From this point
of view, relevant metrics to be considered are the ones related to the energies in
the sidelobes, which, with reference to the lth radar of (1), can be modeled as
Indeed, designing a filter with minimum ISL is tantamount to minimizing the total energy in the range sidelobes, see, for instance, [26,33]. In particular, with reference to the lth radar of model (1), the optimal ISL filter may be found as the solution to the following minimization problem:
where
and
in the sense that υ(11)
It is well known that problem (12) has a closed form solution
In particular, due to the direct connection between the radar code c_{l} and the optimal ISL filter, as well as the energy constraint in (12), maximizing the SINR related to the signal model reported in Equation (1) is equivalent to the minimization of its denominator, i.e., the quartic form
A solution to problem (13) can be again computed through an exhaustive search over the finite set Ω_{l}; however, there is no guarantee that an equilibrium is reached as all the radars iteratively update their codes in a sequential fashion. As for the previous section, we can resort to the potential games framework to obtain an utility function for the players, such that the resulting game admits an NE. We thus consider the following potential
which can be rewritten as
Therefore, in order to obtain an exact potential game with potential function T(·), we can consider the following expression for the utility of the jth user:
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
Besides the minimum ISL receive filter, another customary approach in radar applications is to constrain the level of the sidelobe peaks; the metric to be taken into account in this case is the PSL that, with reference to the lth radar, can be expressed as
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.
The computation of the minimum PSL filter is slightly more involved than the computation of the minimum ISL filter (which indeed was given in closed form), since it requires the detection of the range lobes with the highest peak level, and then their minimization; the problem can be thus formulated as the following fractional quadratic optimization problem:
where
Problem (19) can be restated into an equivalent form as
where the equivalence follows from the observation that υ((19))
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).
Based on the above assumptions, the maximization of the SINR for the pair
where, for each transmitted sequence c_{l} ∈ Ω_{l}, it is necessary to consider the corresponding filter d_{l}(c_{l})∈Σ_{l}. Again, for the purpose of correctly modeling the game among the L users, let us define the following potential:
where we assume that the correspondence between filters and transmitted sequences has already been defined. Specifically, we may resort to Table 1, that can be looked upon during the update procedure. After some algebraic transformations, we obtain
Table 1. Classes of phasecodesϕ
Given the above potential function, it is possible to define the utility for the jth user as
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
So far, we have implicitly assumed that the received signal is affected by either null or negligible Doppler shift. However, it is well known that if the targets illuminated by the network of radars rapidly change their position with unknown velocity and directions, then it is necessary to account for the effect (no more negligible) of the Doppler frequency shifts. To this end, we follow the same approach as in [27], extending it to the considered noncooperative scenario. Specifically, let us assume that ω_{l} = [ω_{N+1,l},…,ω_{N1,l}] is the Doppler shifts vector for the lth radar, with l = 1,…,L. Moreover, let
be the related Doppler shifted code sequence. The data model (1) can be thus modified as follows:
wherein ω_{0,l} is the Doppler shift associated to the range bin of interest. Now, should such a Doppler shift be known at the receiver, the following detection rule should be considered:
with d_{l}(c_{l}(ω_{0,l})) the Ndimensional detection vector, function of the (known) Doppler shifted code c_{l}(ω_{0,l}). Given Equation (29), the SINR equation (4) may easily be reformulated as follows:
In practical radar applications, however, the target Doppler shift is usually unknown, and the available knowledge is limited to the range [ω_{0},ω_{1}] of variability of the Doppler frequencies. The customary approach thus relies on a quantization of the said interval with a preassigned resolution (Δω) (a typical value of (Δω) is π/(10N) [27]) and, at the reception side, a bank of detection vectors, each one keyed to one of the quantized Doppler frequencies, is considered, followed by a maximum selector. Otherwise stated, denoting by ω(1),ω(2),…,ω(P) the P sample frequencies obtained by sampling with step (Δω) the interval [ω_{0},ω_{1}], the detection rule is actually expressed as
Now, in order to come up with a code update procedure, we should still focus on the
minimization of the denominator of Equation (20); note however that such a denominator
depends on the Doppler shifts {ω_{k,l}}, with l = 1,…,L and k = N + 1,…,N  1. In order to circumvent this drawback, a suitable technique is to consider the
statistical expectation of the denominator of (30), averaged with respect to the set
of Doppler shifts. Since in practice the detection vectors are considered only for
a finite number of Doppler frequencies, in performing the average we model the detection
vector as taking value in the discrete set
wherein the overline
Now, given the potential function (32), a noncooperative game can be obtained, similar to the case of negligible Doppler shift, by isolating the terms depending on a given code, say the jth. The utility function for the jth radar is thus written as
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
In this section, we assess the performance of the proposed noncooperative waveform
design techniques; to this end, we test the outlined algorithms in two distinct operational
scenarios, where the difference is mainly in the number of involved radars, as well
as their receive antenna pattern characterization. Precisely, we consider the following
two games
●
● Ω_{l} is a set of cardinality M = 653 which contains the sequences of length N = 16 available to the lth 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 lth 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
● respectively. ^{d}
With reference to the simulation setup of Figures 2a, 3a, and 4a, we choose four transmit sequences from Ω and consider them as the initial strategies for
Figure 2. SINR versus the number of iterations, for a set of (a) L = 4 players, Algorithm 1, (b)L = 6 players, Algorithm 1.
Figure 3. SINR versus the number of iterations, for a set of (a) L = 4 players, (b) L = 6 players, Algorithm 2.(c) Average ISL versus the number of active players, Algorithm 2. ISL at the NE points (solidcircle red line); ISL with a random choice (solidcross blue line).
Figure 4. SINR versus the number of iterations, for a set of L = 4 (a) and L =6 (b) players, Algorithm 3.(c) Average PSL versus the number of active players, Algorithm 3. PSL at the NE points (solidcircle red line); PSL with a random choice (solidcross blue line).
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
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
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.
Finally, in Figure 5, we analyze the average SINR among all the radars, at the NE, versus the number of active radars in the network, with respect to Algorithms 1, 2, and 3; in particular, for the latter two algorithms, the curves refer to the SINR values for the NE points of Figures 3c and 4c (as such random initial strategies have been considered for the radars operate according to a nogame approach and the results are averaged over 25 independent trials); for comparison purposes, we also report the average SINR obtained when the radars operate in a nogame scenario. The plots confirm that, at the Nash equilibria, the radar network actually may enjoy an increase in terms of SINR, with respect to the case in which nogame is allowed. Moreover, as expected, the performance gracefully degrades as the number of active radars increases. This is a pretty natural behavior, since the larger the number of radars the larger the power of the interfering signals.
Figure 5. Average SINR versus the number of active radars. Algorithm 1: SINR at the NE points (solidcircle blue line); SINR with a random choice (dottedcross blue line). Algorithm 2: SINR at the NE points (solidstar magenta line); SINR with a random choice (dottedpoint magenta line). Algorithm 3: SINR at the NE points (soliddiamond green line); SINR with a random choice (dottedplus green line).
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,
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
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
The aforementioned construction procedure does not provide sequences very attractive from the radar point of view; indeed, it can lead to signals with significant modulus variations, poor range resolution, high peak sidelobe levels, and more in general, to signals with an undesired ambiguity function behavior. These drawbacks can be circumvented imposing a control on the aforementioned performance metrics at the code design stage. Precisely, we can start from a good (in the sense of the ambiguity function properties) code c_{0} and devise some additional sequences which inherit some attractive properties of c_{0}. This goal can be achieved forcing the new sequences to lie in a suitable normball centered around c_{0}. In other words, we consider sequences which are solutions to the feasibility problem
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.
Solutions to problem (35) can be found according to the following algorithm.
1. Denote by a an N dimensional complex vector whose elements are continuous random variables.
2. Construct the unitnorm vector
3. Define the sequence
Exploiting the above procedure, we have updated the set Ω, so as to include additional 540 transmit sequences. In Table 2, we show the set of reference codes; for each sequence, we solve problem (35) K = 15 times (with 15 different feasible values of t), thus devising 9K possible codes. Finally, the procedure is implemented for δ_{∊} ∈ {0.41,0.63,0.75,0.9}.
Table 2. Set of similarity codes
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
Competing interests
The authors declare that they have no competing interests.
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.
References

CJ Baker, AL Hume, Netted radar sensing. IEEE Aerosp. Electron. Syst. Mag 18(2), 3–6 (2003). Publisher Full Text

VS Chernyak, Fundamentals of Multisite Radar System: Multistatic Radars and Multiradar Systems (Gordon and Breach Science Publishers, New York, 1998)

E Fishler, A Haimovich, R Blum, L Cimini, D Chizhik, R Valenzuela, Spatial diversity in radars: models and detection performance. IEEE Trans. Signal Process 54(3), 823–838 (2006)

J Li, P Stoica, MIMO Radar Signal Processing (Wiley, New York, 2008)

A De Maio, GA Fabrizio, A Farina, WL Melvin, L Timmoneri, Challenging issues in multichannel radar array processing. Proceedings of the IEEE International Radar Conference, 2007 (IEEE (Edinburgh, UK, April 2007), pp. 856–862

L Landi, R Adve, Timeorthogonalwaveformspacetime adaptive processing for distributed aperture radars, Sect. B—part III, ch. 19. Principles of Waveform Diversity and Design (SciTech Publishing, Raleigh, NC, 2011)

RS Adve, RA Schneible, G Genello, P Antonik, Waveformspacetime adaptive processing for distributed aperture radars. Proceedings of the IEEE International Radar Conference 2005 (IEEE (Arlington, USA, May 2005), pp. 93–97

KH Berthke, B Röde, M Schneider, A Schroth, A novel noncooperative nearrange radar network for traffic guidance and control on airport surfaces. IEEE Trans. Control Syst. Technol. 1(3), 168–178 (1993). Publisher Full Text

H Huang, D Lang, The comparison of attitude and antenna pointing design strategies of noncooperative spaceborne bistatic radar. Proceedings of the IEEE International Radar Conference 2005 (IEEE (Arlington, USA, May 2005), pp. 568–571

HD Ly, Q Liang, Spatialtemporalfrequency diversity in radar sensor networks. Proceedings of the IEEE Military Communications Conference 2006 (IEEE Washington, DC, USA, October 2006), pp. 1–7

N Levanon, Multifrequency complementary phasecoded radar signal. Proc. IEE Radar Sonar Navigat 17(6), 276–284 (2000)

A Farina, Waveform diversity: past, present, and future. Proceedings of the Third International Waveform Diversity & Design Conference 2007 (Plenary Talk, Pisa, Italy, June 2007)

A Aubry, A De Maio, A Farina, M Wicks, Knowledgeaided (potentially cognitive) transmit signal and receive filter design in signaldependent clutter. IEEE Trans. Aerosp. Electron. Syst 49(1), 93–117 (2013)

A Aubry, A De Maio, M Piezzo, A Farina, M Wicks, Cognitive design of the receive filter and transmitted phase code in reverberating environment. IET Radar Sonar Navigat 6(9), 822–833 (2012). Publisher Full Text

J Li, L Xu, P Stoica, KW Forsythe, DW Bliss, Range compression and waveform optimization for MIMO radar: a CramérRao bound based study. IEEE Trans. Signal Process 56(1), 218–232 (2008)

A De Maio, M Lops, Design principles of MIMO radar detectors. IEEE Trans. Aerosp. Electron. Syst 43(3), 886–898 (2007)

A Aubry, M Lops, AM Tulino, L Venturino, On MIMO detection under nongaussian target scattering. IEEE Trans. Inf. Theory 56(11), 5822–5838 (2010)

N Subotic, K Cooper, P Zulch, Conditional and constrained joint optimization of radar waveforms. Proceedings of the International Waveform Diversity and Design Conference, 2007 (IEEE (Pisa, Italy, June 2007), pp. 387–394

M Greco, F Gini, P Stinco, A Farina, L Verrazzani, Adaptive waveform diversity for crosschannel interference mitigation

G Scutari, DP Palomar, S Barbarossa, Competitive design of multiuser MIMO systems based on game theory: a unified view. IEEE J. Sel. Areas Commun 26(7), 1089–1103 (2008)

A De Maio, F Gini, L Patton, Waveform design for noncooperative radar networks, ch. 10. Waveform Design and Diversity for Advanced Radar Systems ((Institution of Engineering and Technology, London, UK, 2012)

D Fudenberg, J Tirole, Game Theory (MIT Press, Cambridge, MA, 1991)

D Monderer, LS Shapley, Potential games. Games Econ. Behav 14(44), 124–143 (1996)

M Piezzo, S Buzzi, A De Maio, A Farina, Noncooperative code design in radar networks: a gametheoretic approach. Evolutionary and Deterministic Methods for Design, Optimization and Control (EUROGEN) ((Capua, Italy, September 2011), pp. 14–16

S Buzzi, G Colavolpe, D Saturnino, A Zappone, Potential games for energyefficient power control and subcarrier allocation in uplink multicell OFDMA systems. IEEE J. Sel. Topics Signal Process 6(2), 89–103 (2012)

KR Griep, JA Ritcey, JJ Burlingame, Polyphase codes and optimal filters for multiple user ranging. IEEE Trans. Aereosp. Electron. Syst 31(2), 752–767 (1995)

P Stoica, J Li, M Xue, Transmit codes and receive filters for radar: a look at the design process. IEEE Signal Process. Mag 25(6), 94–109 (2008)

YI Abramovich, MB Sverdlik, Synthesis of a filter which maximizes the signaltonoise ratio under additional quadratic constraints. Radio Eng. Electron. Phys. 15(11), 1977–1984 (1970)

YI Abramovich, MB Sverdlik, Synthesis of filters maximizing the signalto noise ratio in the case of a minimax constraint on the sidelobes of the crossambiguity function. Radio Eng. Electron. Phys 16, 253–258 (1971)

S Zoraster, Minimum peak range sidelobe filters for binary phasecoded waveforms. IEEE Trans. Aereosp. Electron. Syst 16(1), 112–115 (1980)

JC Liberti Jr, TS Rappaport, Analytical results for capacity improvements in CDMA. IEEE Trans. Veh. Technol 43(3), 680–690 (1994). Publisher Full Text

S Anderson, M Millnert, M Viberg, B Wahlberg, An adaptive array for mobile communication systems. IEEE Trans. Veh. Technol 40(1), 230–236 (1991). Publisher Full Text

VT Dolgochub, MB Sverdlik, Generalized γfilters. Radio Eng. Electron. Phys 15, 147–150 (1970)

P Stoica, J Li, M Xue, On binary probing signals and instrumental variables receivers for radar. IEEE Trans. Inf. Theory 54(8), 3820–3825 (2008)