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Power Allocation and User Assignment Scheme for Beyond 5G Heterogeneous Networks

The issue of spectrum scarcity in wireless networks is becoming prominent and critical with each passing year. Although several promising solutions have been proposed to provide a solution to spectrum scarcity, most of them have many associated tradeoffs. In this context, one of the emerging ideas relates to the utilization of cognitive radios (CR) for future heterogeneous networks (HetNets). This paper provides a marriage of two promising candidates (i.e., CR and HetNets) for beyond fifth generation (5G) wireless networks. More specifically, a joint power allocation and user assignment solution for the multi-user underlay CR-based HetNets has been proposed and evaluated. To counter the limiting factors in these networks, the individual power of transmitting nodes and interference temperature protection constraints of the primary networks have been considered. An efficient solution is designed from the dual decomposition approach, where the optimal user assignment is obtained for the optimized power allocation at each node. The simulation results validate the superiority of the proposed optimization scheme against conventional baseline techniques.

I. INTRODUCTION
In the last few years, the wireless systems have evolved to the point where a homogeneous cellular networks have achieved near optimal performance [1]. These advancements in the homogeneous cellular networks, though significant, may not be enough to support beyond fifth generation (5G) wireless networks [2]. To do so, dynamic and exhaustive improvements in spectral efficiency are needed. One of the proposed solutions is advanced networks which consist of multiple tiers of highpowered base stations (BSs) overlaid with low-powered BS both having different coverage areas, also known as heterogeneous networks (HetNets) [3]. Typically, a HetNet consists of main macro base stations (MBSs), a few pico base stations (PBSs), and several femto base stations (FBSs). An MBS in HetNets has a high transmission power and greater coverage area which is then overlaid with low-powered PBS and FBS [4], [5].
There are different purposes of PBS and FBS in the HetNets. Generally speaking, the PBS are laid in a dense traffic areas to improve coverage in hotspot areas, while FBS are overlaid in a manner to remove the coverage issues in homogeneous systems, thereby improving the overall performance. In conventional homogeneous cellular network, the mobile terminal is associated to a BS on the basis of downlink signal-to-interference-and-noise ratio (SINR). Specifically, a user is associated with a particular BS which offers SINR greater than the other BSs [6]. In HetNets, the SINR-based association principle leads to having load balancing issue among MBS and PBS. Hence, the network resources need to be intelligently distributed among BSs for higher spectral efficiency [5], [6].

A. Related Works
By the end of 2020, it is anticipated that up to 50 billion devices will exist in the world including static and mobile platforms [7]. Due to this reason, there has been an upsurge in the research on HetNets to provide efficient and long-term solutions [8]. Zhang et al. in [9] investigated the problem of user association in HetNets. They considered an optimization problem with different traffic capacity limits, quality of service (QoS) requirements and power budget constraints. The proposed scheme of user association improved the performance of the network. A bit-rate adoption method was proposed by the authors of [10] for body area HetNets. Their scheme showed the promise of increasing the efficiency of packet transmission regardless of placement of the node. The authors of [11] performed joint optimization of resource allocation and user association in HetNets.
They presented and compared three allocation strategies, i.e., orthogonal deployment, co-channel deployment, and partially shared deployment. Sequential quadratic programming (SQP)-based power optimization approach was presented in [12] to maximize the sum rate of small cell networks. Similar works have been done by the authors of [13], where a pricing-based approach for associating the user to a particular BS was proposed by the authors of [14] and the idea of a distributed price update strategy was presented for achieving the user fairness.
Liu et al. presented a fair user association scheme in [15]. They used the Nash bargaining solution (NBS), whereby, the optimization with fairness was achieved among competing BSs. In [16], the HetNets using orthogonal frequency division multiple access (OFDMA) network were presented by the authors. The aim was to manage radio resource by maximizing the throughput of user having minimum rate. From the perspective of non-orthogonal multiple access (NOMA), the author of [17] provided a novel idea to use interference-aided vehicular networks and highlighted some key challenges. Similarly, the authors of [18] managed the radio resources by a scheduling algorithm implemented by a central global resource controller (GRC). The algorithm optimized the attributes such as fairness among users, spectral efficiency and battery lifetime. Relay-based HetNets were considered by the authors of [19], wherein, they proposed a method to suppress inter-cell and intra-cell interference. Their proposed scheme was shown to outperform existing baseline methods in terms of sum-rate performance. Another similar and much recent work [20] provided optimal time switching and power splitting technique for improving the performance of wireless network. The authors in [21] jointly optimized power allocation and user association in ultra dense heterogeneous networks using non-cooperative game theory. Thus achieving the increase in system throughput as well as optimal power allocation.
Semov et al. proved that in HetNets by taking the geographical position into account the users throughput and fairness can be improved. [22]. A similar concept was employed in the form of relays by the authors of [23] to ensure proportional fairness among user equipment by taking into account backhaul links between the relays and BS. On the other hand, the authors of [24] considered distributed antenna system (DAS) and compared co-channel resource allocation schemes for energy-efficient communication. In [25], the authors studied a single cell HetNet having one macro and one picocell for efficient resource allocation such that the energy efficiency is maximized by proposing an iterative resource allocation algorithm. However, the same authors did not consider multi-cellular network for the evaluation. To achieve the spectral efficiency of the system more advanced dynamic spectrum access techniques (DSA) should be employed. The cognitive radio (CR) is an efficient DSA technique [26] that allows secondary (unlicensed) users (SU) to access the spectrum of primary (licensed) users (PU) in an opportunistic way [27]. In CR, spectrum sharing can be classified as spectrum overlay and spectrum underlay. In spectrum overlay, the SU can transmit simultaneously on the frequency band used by the PU through adjusting its transmit power such that it does not cause much interference for the PU [27].
Of late, the CR-based HetNets have been drawing a lot of research attention nowadays due to their dynamic resource allocation property. The power adjustment of BS and mobile users can be achieved by dynamic resource allocation [28]. In this regard, the author in [29] maximized the energy efficiency subject to power and interference constraint in OFDMA based CR networks. The authors applied the convex optimization theory and proposed an iterative algorithm. In [30], the authors considered a cognitive femtocell network using OFDMA and solved a sum-utility maximization and dynamic resource allocation problem with the help of dual decomposition method. In [31] the authors considered the stochastic optimization model for maximizing the long term energy efficiency in time varying heterogeneous networks. The proposed problem is a mixed integer problem and the Lagrange dual method has been used to solve it. In [32], the authors considered the resource allocation problem for rate maximization in multi-user cognitive heterogeneous networks. The authors considered the maximum transmit power of cognitive microcell base station and cross-tier interference constraints, simultaneously. They converted the non-convex optimization problem into a geometric programming problem and solved it in a distributed way using the Lagrange dual method.

B. Motivation and Contributions
Although the research works reported in recent years have considered the overall system's performance maximization for CR-based HetNets, the problem of fairness among different users has received little attention. The potential problem arises when the schemes proposed for the sum-rate maximization assign very few or no resources to some of the users with higher fading conditions. This uneven distribution of resources results in degrading the achievable performance for different users. This problem may become more serious for the CR-based HetNets due to the limiting factor of increased interference. Thus, optimization of the transmission for user fairness under more practical constraints becomes essential. To the best of our knowledge, the resource optimization and user assignment techniques for fair rate allocation have not been jointly investigated in the literature due to the higher level of complexity involved in finding the optimal solution.
To fill this gap in the literature and provide a comprehensive solution to the user fairness problem in CR-based HetNets, we provide a joint strategy for power allocation and user fairness. In particular, we consider the joint power optimization and user association problem in HetNets for achieving fairness among different users. We first formulate a joint optimization problem subject to power and interference constraints. Then, to provide a less complex and an efficient solution, we design an algorithm from the dual decomposition strategy. The presented numerical results indicate the importance and utility of our scheme in comparison to the baseline methodologies.

C. Organization
The remainder of this paper is organized as follows. The system model and problem formulation are presented in Section 2. In Section 3, the proposed solution has been described, while Section 4 discusses the simulation results. Finally, Section 5 presents some concluding remarks and future research directions. In addition, the list of acronyms used throughout this paper has been provided in Table I. II. SYSTEM MODEL AND PROBLEM FORMULATION This section describes the considered system model and provides the steps related to problem formulation.

A. System Model
A downlink CR transmission is considered, where the secondary HetNet system is reusing the spectrum of the primary network in an underlay mode, as shown in Fig. 1. The HetNet consists of a single macrocell overlaid with multiple pico BSs, intended to transmit data to multiple users. It is assumed that all the devices are equipped with a single antenna and experience independent and identically distributed (i.i.d) Rayleigh fading. The channel bandwidth is distributed among MBS and PBSs in such a way that MBS and PBSs are non-orthogonal to each other, while each PBS is orthogonal to other PBSs in the HetNet.
In this model, BSs are denoted as BS b , such that B = BS b |b = 1, 2, 3, ..., B, where BS 1 is modeled as macro and the remaining BSs act as pico BSs. Further, there are A users and C channels allocated to BS. The signal-to-interference-and-noise ratio (SINR) of a-th user from the b-th BS at the c-th channel is given as in [11] where P a,c,b is the transmit power for a-th user connected to b-th BS at c-th channel, α a,c,b denotes the association variable which belongs to set {0, 1}, σ 2 a is the power spectral density (PSD) of the noise while g a,c,b and f a,c,b are the channel gains from intended BS to user and from interfering BS to the user, respectively. Furthermore, P a,c,b denotes the transmit power of interfering BS.

B. Problem Formulation
One of our key objectives in this article is fair maximization of the data rate of each user in the network. To facilitate mathematical analysis, we introduce a binary variable α a,c,b , such that Based on the above expression, the channel allocation follows To protect PUs from the interference and to improve the performance of the system, resource allocation at BSs need to be performed such that where P a,c,b is the power allocated by b-th BS to c-th channel allocated to a-th user and h a,c,b represents the gain of interference channel c from b-th BS to primary BS. Finally, to ensure that the power consumed by each BS is less than or equal to the power budget, power allocation at each BS needs to be ensured as To achieve fairness in rates of different users, a max-min-based optimization framework is adopted such that the problem is stated, mathematically as s.t. (2), (3) and (4), where P b denote total power available at b-th BS and I th represents the sum interference threshold. Here, the first constraint ensures that the total power allocated by b-th BS must be within the available power budget. Similarly, the second constraint esures that the primary system is well protected from interference due to the underlay communication, and the third constraint makes sure that c-th channel is allocated to just one SU.

III. PROPOSED SOLUTION
To provide a viable solution to the aforementioned problem, we propose a solution based on duality theory. As is evident from the above expression, the allocation of channels and power loading are strongly coupled variables, thus, a joint optimization approach is needed. The problem P1 is a mixed binary integer programming and requires an exhaustive search. Fortunately, it has been shown by [33] that for a sufficiently large number of subchannels the gap between the dual solution and the primal solution reduces to zero, regardless of the non-convexity of the original problem. Thus, we exploit the duality theory to decompose and optimally solve the coupled problem.
To convert the complex max min problem into standard optimization form, we introduce intermediate variable t such that The introduction of the intermediate variable transforms the problem as s.t.
To obtain an immediate solution of auxiliary variables, from the structure of objective in (7), we utilize the fact that for any y ≥ 0, minimizing y is equivalent to minimization of y 2 . It is a known fact that maximizing y is equal to minimizing −y. Hence, to transform the problem into a standard minimization problem we replace the objective in (7) with its negative. After making these transformations P2 is written as (4) and (6). s.t.
The dual function associated with (9) is given by The expression in (10) can be written as s.t. For any given channel allocation, dual decomposition guides to solve the following sub-problems P5: min Using the KKT conditions to find the solution of the problems given in (12) and (13), we get where Φ a,c,b = λ a g a,c,b + σ 2 a η b + vh a,c,b , for all a, c, b, when (Ψ) + = max(0, Ψ). The detailed derivation steps have been provided in the Appendix. Now, to find the optimum value of α a,c,b , following optimization problem is considered s.t. Evidently, the optimal solution can be found, such that The dual problem is convex, hence, sub-gradient method can be adopted to find the solution. The dual variables are updated at each iteration as where δ is the step size. To obtain the joint optimization solution, power loading and channel allocation are updated in each iteration.

IV. SIMULATION RESULTS
In this section, we present the performance of the proposed scheme. The simulation environment for the proposed schemes is MATLAB. The noise PSD is σ 2 a = 0.1. For simulation we have considered the maximum number of users in the system is A = 30, the total channels available in the system is C = 20 and the total BS available in the system is B = 5. For the sake of the evaluation, we have compared our proposed optimal allocation and optimal power scheme (OAOP) with baseline schemes, i.e., fixed channel allocation and optimal power loading (FAOP) and fixed channel allocation and fixed power loading (FAFP).
In Fig. 2, peak to average rate ratio (PR) has been plotted against different parameters. Different values of PR show fairness among the users. The smaller values of PR indicates a more fair scheme. Moreover, the effect of changing picocell power P R p on PR has also been shown. The picocell power has been varied from 0.25 to 2. Considering P p = 0.25, the percentage gap between FAFP and FAOP is 49.74%, between FAFP and OAOP is 22.59% and between FAOP and OAOP is 45.42%. It can be seen that the value of PR decreases with an increase in the power. This indicates that high picocell power causes high interference which results in decreasing the PR. Moreover, a comparison of the three schemes clearly indicates that the OAOP scheme is the fairest among all. This is because optimization on channel allocation along with optimal power loading makes the problem more flexible compared to the cases when only power loading is optimized and if channel allocation and power loading are fixed. Also for P p = 0.5 the percentage gap between FAFP and FAOP is 49.83%, between FAFP and OAOP is 25.44% and between FAOP and OAOP is 51.05%. Finally, increasing the value to P p = 2, the percentage gap between FAFP and FAOP is 56.99%, between FAFP and OAOP is 33.12% and between FAOP and OAOP is 58.11%. Fig. 3 is showing the PR with changing interference constraint P R I th . The interference constraint has been varied from 20 to 40. Considering I th = 20, the percentage gap between FAFP and FAOP is 55.08%, between FAFP and OAOP is 31.83% and between FAOP and OAOP is 57.79%. Also for I th = 25 the percentage gap between FAFP and FAOP is 53.39%, between FAFP and OAOP is 30.57% and between FAOP and OAOP is 57.27%. Moreover, if I th = 30, then the percentage gap between FAFP and FAOP is 52.41%, between FAFP and OAOP is 29.95% and between FAOP and OAOP is 57.14%. Further increasing the value of I th = 35, the percentage gap between FAFP and FAOP is 52.00%, between FAFP and OAOP is 29.62% and between FAOP and OAOP is 57.11%. Finally, increasing the value to I th = 40, the percentage gap between FAFP and FAOP is 51.86%, between FAFP and OAOP is 28.91% and between FAOP and OAOP is 56.31%. Comparison of schemes in terms of the percentage gap shows that as the interference constraint I th increases the percentage gap is decreasing. Moreover, the average gap between the graphs of FAFP with FAOP and FAOP with OAOP scheme is near to each other. Graphs also show that OAOP outperforms all other schemes in terms of fairness. The reason behind this is the additional flexibility provided by optimum channel allocation. The effect of changing the power of macro base station P R m on the peak to average rate ratio is shown in Fig. 4. The macro power has been varied from 10 to 30. Considering P R m = 10, the percentage gap between FAFP and FAOP is 54.64%, between FAFP and OAOP is 31.83% and between FAOP and OAOP is 58.26%. Also for P R m = 15 the percentage gap between FAFP and FAOP is 56.05%, between FAFP and OAOP is 34.46% and between FAOP and OAOP is 61.48%. Moreover, if P R m = 20, then the percentage gap between FAFP and FAOP is 56.65%, between FAFP and OAOP is 36.27% and between FAOP and OAOP is 64.03%. Further increasing the value of P R m = 25, the percentage gap between FAFP and FAOP is 57.09%, between FAFP and OAOP is 37.59% and between FAOP and OAOP is 65.84%. Finally, increasing the value to P R m = 30, the percentage gap between FAFP and FAOP is 57.49%, between FAFP and OAOP is 38.61% and between FAOP and OAOP is 67.15%. Hence, comparing all the three schemes show that the OAOP is the superior scheme.
In the results presented here, the sum throughput has also been maximized along with achieving fairness among the users. Fig. 5 shows the sum throughput versus pico power P p . The P p has been varied from 0.25 to 2. The percentage gap of OAOP scheme with FAOP is 38%, between OAOP with FAFP is 7% and between FAOP with FAFP is 18.42% where the value of P p = 0.25. The results evidently show that the spectral efficiency of OAOP schemes is the most optimal among the other power allocation schemes. This is because when channel allocation is optimized and the channels are allocated to users in such a way that would maximize the total data rate of users. Then, increasing the value of P p = 0.5 percentage gap of OAOP with FAOP is 42.56%, between OAOP with FAFP is 7.825% and between FAOP with FAFP is 18.38%. Further increasing the value to P p = 2 OAOP with FAOP is 48.51%, between OAOP with FAOP is 10.12% and between FAOP with FAFP is 20.85%. Fig. 6 shows the sum throughput versus I th . The percentage gap of OAOP scheme with FAOP is 45.98%, between OAOP with FAFP is 9.73% and between FAOP with FAFP is 21.17% where the value of I th = 20. Further increasing the value of I th = 25 percentage gap of OAOP with FAOP is 46.09%, between OAOP with FAFP is 10.22% and between FAOP with FAFP is 22.17%. Moreover, if I th = 30, percentage gap between OAOP with FAOP is 46.21%, between OAOP with FAFP is 10.62% and between FAOP with FAFP is 23%. Considering I th = 35, OAOP with FAOP is 46.32%, between OAOP with FAFP is 10.98% and between FAOP with FAFP is 23.71%. Further increasing the value to I th = 40 OAOP with FAOP is 46.47%, between OAOP with FAOP is 11.01% and between FAOP with FAFP is 23.71%. Hence, the sum throughput of OAOP scheme is maximum.
The effect of varying macro power P m on the sum throughput is shown in Fig. 7. The P m has been varied from 10 to 30. The percentage gap of OAOP scheme with FAOP is 45.49%, between OAOP with FAFP is 9.73% and between FAOP with FAFP is 21.40% where the value of P m = 10. Further increasing the value of P m = 15 percentage gap of OAOP with FAOP is 46.69%, between OAOP with FAFP is 9.65% and between FAOP with FAFP is 20.67%. Moreover, if P m = 20, percentage gap between OAOP with FAOP is 47.27%, between OAOP with FAFP is 9.61% and between FAOP with FAFP is 20.32%. Considering P m = 25, OAOP with FAOP is 47.74%, between OAOP with FAFP is 9.58% and between FAOP with FAFP is 20.06%. Further increasing the value to P m = 30 OAOP with FAOP is 48.13%, between OAOP with FAOP is 9.56% and between FAOP with FAFP is 19.86%. The results show that OAOP scheme is the most optimal. Fig. 8 shows the values of sum throughput with a changing number of users U . From this plot, we identified that the performance of OAOP is far superior to other schemes which is due to the fact that adding new users to the system introduces new channel gains. This results in increasing the flexibility of the problem. In OAOP, both channel allocation and power loading are optimized, therefore, this scheme is better at taking advantage of the newly added users in the network. In FAOP only power loading is optimized so the addition of new users increases the data rate because of the reduced degree of freedom compared to OAOP. A very slight increase is observed in FAFP when new users are added, as, neither channel allocation nor   Fig. 9 shows the convergence of dual variable λ a , η and v with initial value as 0.6, η as 0.6 and v as 0.1. The smaller the value of step size more fine the convergence is, a bigger value leads to fast convergence.

V. CONCLUSION
This paper optimized the power allocation and user association to achieving fairness among different users in underlay CR-based HetNets. Specifically, we adopted a max-min-based fairness framework under various practical constraints. Problem was first transformed into a standard maximization optimization and then dual decomposition is used to solve the integer programming problem. The dual problem was solved through the sub-gradient method. The power allocation at each node is obtained from the convex optimization techniques for the known power allocation at all other nodes. The results are compared with the sub-optimal scenarios when the fixed channel allocation is assumed for the optimized/non-optimized power allocation. The results showed that the proposed scheme outperforms all other candidates and the spectral efficiency has been improved due to the simultaneous transmission with the primary network.

APPENDIX
The Lagrangian L associated with the optimization problem P5 is min P a,c,b (−λ a log 2 (1 + SIN R a,c,b ) + η b P a,c,b + vP a,c,b h a,c,b ) .
In the above expression, we have converted our problem from a constrained problem to an unconstrained problem.
By canceling the common terms and simplifying, we have After some mathematical simplifications, we get


Radar Waveform Strategy Based on Game Theory

In this paper, we proposed two waveform design methods based on game theory to address the problem of radar detection performance degradation in electronic warfare. Since radar and jammer are completely hostile, their interaction is modeled as two-person zero-sum game. Signal-to-Interference-plus-Noise Ratio (SINR) criterion is used in formulating the utility functions. The existence of Nash equilibrium in games is verified by mathematical derivation. Different game waveform strategies are designed for different information levels of radar and jammer. Iterative water-filling method and two-step waterfilling method are designed to achieve Cournot equilibrium and Stackelberg equilibrium, respectively. Simulation results reveal that game strategies can bring higher radar detection performance than No game signal, especially when jammer power is lower than radar power. Radar detection probability based on game theory can be increased by up to 10% without changing the power. This demonstrates game strategies have great potentials for radar waveform design in electronic warfare.

Introduction
Optimizing radar transmit waveform is a key way to improve radar detection performance [1][2][3][4][5][6][7][8][9]. With the rapid development of electronic warfare technology, the confrontation between radar and jammer has become increasingly fierce. Cognitive radar can adaptively optimize transmit waveform based on prior knowledge [10]. But at the same time, enemy jammer intercepts radar waveform and releases jamming pertinently, resulting in a decline in radar performance. [11]. Therefore, optimizing the waveform to improve radar detection performance has become a significant research topic.
Game theory has long been the subject of many classic studies in economics, politics and biology. Recent research has validated that game theory can provide a useful account of improving radar system performance. In 2011, Gogineni and Nehorai first applied game theory to the radar field. They proposed a polarimetric waveform design scheme for distributed MIMO radar based on game theory [12]. In later work conducted by Panoui and Deligiannis, the power allocation schemes of Multistatic MIMO radar network based on different game models were investigated. Generalized Nash equilibrium, Stackelberg Nash equilibrium and Bayesian Nash equilibrium power strategies were solved to achieve high signal-to-noise ratio (SNR) gain and save power resources [13][14][15]. Considering the joint design of amplitudes and frequency-hopping codes for frequencyhopping waveforms, Han proposed a game theory framework to improve MIMO radar performance [16]. The distributed power control problem of radar network is formulated as cooperative game and non-cooperative game in [17], [18], which effectively improves the low probability of intercept (LPI) performance of radar network. Overall, these studies support the notion that game theory can improve radar system performance by optimizing signal or power. Such approaches, however, have failed to address the effects of jamming on radar. The degradation of radar performance caused by enemy jamming is a problem that must be considered in modern electronic warfare.
It is only since the work of Song et al. that the game theory in radar and jammer has gained momentum [19]. They studied the two-person zero-sum game model between MIMO radar and jammer and designed Stackelberg equilibrium solution based on mutual information (MI) criterion. Gao Hao studied the Bayesian game strategy of MIMO radar and jammer from the perspective of incomplete information [20]. But the effects of clutter are not taken into account. Actually, power allocation strategies are usually affected by clutter in electronic warfare. In our former work, Stackelberg game between MIMO radar and jammer in the presence of clutter was studied [21]. The optimization scheme based on MI criterion was explored and analyzed. To date, some published articles have begun to focus on the game phenomenon between radar and jam-mer, but such studies remain narrow in focus dealing only with MIMO radar. Unfortunately, the operational mechanism of SISO/SIMO radar is different from MIMO radar. Besides, the game scheme based on MI criterion cannot be well applied to non-Gaussian clutter environment. With the aim of improving SISO/SIMO radar detection performance in electronic warfare, this paper proposed waveform game strategies between radar and jammer in clutter based on SINR criterion.
Here, we propose radar waveform design methods from the perspective of game theory. Since jammer and radar are completely hostile, their interaction is modeled as two-person zero-sum game. Nash equilibriums in games are verified by mathematical derivation. To analyze the feasibility of the game waveform design method in different scenarios, two game models are considered as following: 1) Cournot Game Model (radar and jammer have equal status), and 2) Stackelberg Game Model (jammer has a late-mover advantage). To the best of our knowledge, this is the first time to propose SISO/SIMO radar waveform design strategies based on game theory in clutter. This method provides a stable solution for improving radar detection performance in electronic warfare. Both the theoretical analysis and numerical calculations demonstrate that radar performance can be improved through game strategies. Figure 1 shows a block diagram of radar signal model used for SINR-based waveform design. Here, s(t) is the complex-valued transmit waveform with finite-energy, duration T, and Fourier transform S(t). r(t) is the complexvalued receive filter impulse response. h(t) is the extended known target impulse response of finite duration T h and Fourier transform H(f). n(t) is a complex-valued, zeromean channel noise process with power spectral density (PSD) S nn (f), which is non-zero over the entire waveform bandwidth. c(t) is a complex-valued, random process representing a clutter component, e.g., ground clutter, and characterized by the PSD S cc (f). S nn (f) and S cc (f) are positive in the signal bandwidth W (i.e., the frequency band [-W/2, W/2]). Let jamming signal j(t) be the finite-energy waveform with finite duration T h and Fourier transform J(f). As seen in Fig. 1, the signal y(t) at the output of the receive filter is y s (t) and y j (t) are the signal and jamming components of the output y(t), respectively, which are defined by

Signal Model
The output SINR at time t 0 is where h(t) is a complex wide-sense stationary process. We define as the variance of H(f), called the energy spectral variance (ESV). Here The ESV describes the average energy of a finite-duration, zero-mean process in the same sense that a PSD describes the average power of an infinite-duration, wide-sense stationary process [4]. Put (5) into (4) and apply Schwarz's inequality yields the bound: The SINR achieves its maximum of if and only if the matched filter is of the form where k is an arbitrary constant. To visually represent the distributed power of the radar waveform on each sub-band, SINR may be converted from integral to discrete: where K is the frequency sample number, and Δf is the frequency sampling interval.

Waveform Strategy
This section assumes that both radar and jammer are cognitive. Some prior knowledge can be used to adaptively optimize waveforms. Signal-to-Interference-plus-Noise Ratio (SINR) criterion is used in formulating the utility functions.

No Game
Firstly, radar and jammer unilateral optimal schemes are given. Assume that the power of radar and jammer is P S and P J , respectively; the waveform power constraint may be written as

Radar Unilateral Strategy
In the radar unilateral model, when radar makes a decision, it does not know the jammer's decision. For reasons of conservation and rationality, radar assumes that jammer releases Gaussian white noise jamming in signal band, which does not change the target characteristics. For this case, to maximize SINR, radar can choose the following strategy to optimize the transmit waveform: Notice that the objective function is a convex function of |S(f k )| 2 , and the power constraint is linear. Therefore, an optimal scheme can be obtained by the water-filling method [1]. Considering the objective function and constraints, the waveform optimization problem is transformed into a convex programming problem, which is solved by Lagrange multiplier method: Take L(|S(f k )| 2 , λ)'s derivative and make it equal to zero to maximize SINR. We can get the radar optimized signal:

Jammer Unilateral Strategy
Suppose the jammer knows radar's income function (maximizing SINR). In order to degrade radar performance, jammer needs to optimize the jamming signal based on the detected radar waveform parameters. Assuming that radar waveform spectrum is stable, the jamming waveform design strategy can be: The kernel in (14) is monotonically decreasing and concave, and power constraint of P J is linear, so the optimization solution is unique. The jamming waveform can be obtained by the water-filling method: Solve the optimized jamming waveform Without considering the game, the final strategy of radar and jammer is as follows:

Cournot Game
In the electronic warfare, the relationship between radar and jammer is detection and anti-detection. Both action strategies can be explained by game model.
The Cournot model was proposed by the French economist Antoine Augustin Cournot in 1838. The model illustrates how the production decisions of firms that compete with each other and do not coordinate with each other produce a result that lies between the competitive equilibrium and the monopoly equilibrium [22]. Ref. [23] analyzed the game equilibria always exist if two agents have symmetric information. If a game is competitive and has a unique pure-strategy Nash equilibrium, all the players prefer to stay at Nash equilibrium under the assumptions of conservativeness and rationality. In the Cournot game, radar and jammer have symmetric information and equal status. Therefore, it is significant to analyze Nash equilibrium strategy between radar and jammer.
In order to solve the optimization problem in Cournot game, we first need to find and prove the game equilibrium solution between radar and jammer. The optimization scheme has to satisfy the following four characteristics to ensure equilibrium [19]:   Proof: When the game reaches the equilibrium point, for reasonable reasons, if there is no signal power in a subband f k , the jammer will not allocate any jamming power in the sub-band, otherwise it will waste a limited amount of jamming power. In order to optimize the interference performance, the jammer will distribute jamming power in the sub-band which has radar signal power. Therefore, the characteristic (1) is verified.
When J(f k ) > 0, |S(f k )| 2 > 0 can be obtained from characteristic (1). So   x Obviously, x m  x n cannot be the optimal solutions, which contradicts the assumptions. Therefore, the characteristic A) is proved. The same reason can be proved that characteristics B), C) are established.
Characteristics (3) and (4) can also be proved by the counter-evidence method. Therefore, the Cournot Nash equilibrium exists, and we can achieve it by iterative waterfilling method. For a given jammer's strategy, we obtain the radar's best response strategy via maximizing the radar's utility. Substituting this radar's strategy into the jammer's utility function, we obtain jammer's best response to this radar's strategy via maximizing the jammer's utility. Repeat this process until both radar and jammer strategies remain unchanged. This stable result is Nash equilibrium in game theory. At this equilibrium point, radar and jammer have no incentive to deviate from this point. The algorithm can be summarized as shown in Tab. 1.

Stackelberg Game
In Sec. 3.2, the status of radar and jammer are equal. Considering the status between radar and jammer during the game may be asymmetrical. In that case, the Cournot model is difficult to apply. Game model established by German economist Heinrich von Stackelberg reflects this asymmetrical competition [22]. As two participants in game Iterative water-filling algorithm 1 Initialize two players' actions in Section 3.1: Maximize the radar's utility: Update the radar's strategy: Maximize the jammer's utility:

Jammer Power Fixed
First, we investigate Nash equilibrium between radar and jammer. Assume that the jammer can accurately estimate the radar signal frequency band and the jamming power is 20 dBw. Figure 2 illustrates the game equilibrium exists and is achieved at point E. Figure 3 shows the dynamic iterative process of Nash equilibrium. Point A is the initial position. The SINR versus iteration numbers is shown in Fig. 4. After 4 iterations, all players cannot improve their own benefits by changing their strategy individually. The Cournot game eventually reached equilibrium with the SINR of 9.782 dB. This shows that the equilibrium strategies are the best choice for conservative and rational radar and jammer. At the equilibrium point, both radar and jammer have no incentive to deviate this point.   Next, power distribution in frequency domain is carried out for different waveforms. Based on the Neyman-Pearson theory, assuming the radar false alarm probability is 10 -4 , we can derive the radar detection probability by SINR. As can be seen from Fig. 5 and Fig. 6, different signal waveforms result in different power allocation strategies. When the radar does not consider the game with the jammer, the radar designs the optimal waveform according to prior knowledge. It can be seen that compared with chirp signal, the radar signal optimized based on SINR criterion allocates more energy in the frequency band of high target impulse response, low clutter and noise.
In Cournot game, two players take into account each other's strategies when formulating the waveform strategy. First, radar optimizes signal to maximize SINR. Then, to minimize SINR, the jammer designs jamming waveform according to the radar signal. After several iterations, the   game finally reached equilibrium. It can be seen that the radar will allocate more power to the frequency band with high target-clutter ratio (TCR) to suppress jamming signal, so as to obtain high income. At this point, if jammer changes the strategy, radar will have a better choice, so the game remains equilibrium.
In Stackelberg game, different game sequences result in different information levels. Radar moves first, and jammer optimizes its waveform after observing the radar's strategy. Since the jamming strategy is not known, radar can only predict the jamming waveform and optimize the signal accordingly. Jammer has more posteriori information and a higher game level. It can be seen from Tab. 2 that the output SINR of Stackelberg game is higher than No game and Chirp signal, but lower than Cournot game. Therefore, in radar game model, followers have more battlefield information and late-mover advantage.

Jammer Power Variable
Next, the radar game strategy performance is tested when jamming power changes. Assume that the power jammer varies from 10-30 dBw. The radar optimization scheme with jammer's power changes is shown in Fig. 7. With the increase of jamming power in the game process,  the high TCR frequency band has been completely covered by jamming, which makes it difficult for radar to obtain target information. In Cournot game, the radar distributes more energy to the band where TCR is low but σ h 2 (f k ) is high to avoid jamming, such as sub-band 3. In Stackelberg game, radar distributes more energy to high TCR band to counter against jamming, such as sub-band 4. The radar detection probability curves of different optimized waveforms are shown in Fig. 8.
The most striking result to emerge from Fig. 8 is that compared with No game and Chirp signal, the optimized signal by game theory can achieve higher detection performance. Further analysis shows that the degree of information obtained can affect radar performance in games. In other words, the game has a late-mover advantage in the electronic warfare. It is noteworthy that, for radar, the lower the jamming signal power is, the more obvious the performance improvement by game strategy. At constant transmit power, radar can use game theory to improve the detection performance by optimizing the waveform power spectrum. Unfortunately, if the jamming power is too high, it is difficult for radar to obtain valid results. When the power of jammer is 10 times higher than that of radar (30 dBw), it is difficult to improve the radar performance by optimizing radar waveform. This implies that the optimization of radar performance by game strategy is limited. Overall, radar waveform optimization design based on game theory is significant to improve radar detection performance in electronic warfare. The power difference between radar and jammer will affect the optimization effect.

Conclusion
Radar waveform power spectrum design is critical to radar system performance. In electronic warfare, the interaction between radar and jammer will affect the radar performance. This paper proposes waveform optimization strategies based on SINR from game theory perspective. Since jammer and radar are completely hostile, their interaction is modeled as two-person zero-sum game. SINR criterion is used in formulating the utility functions. Firstly, the SINR-based radar and jammer unilateral optimization strategy is studied. Secondly, Cournot game and Stackelberg game strategies are designed for different information levels of radar and jammer. In Cournot game, radar and jammer have symmetrical information. We find that Nash equilibrium can be achieved by the iterative water-filling method. In Stackelberg game, the information of radar and jammer is asymmetrical. The two-step water-filling method is used to achieve Stackelberg equilibrium. Finally, the performance of different optimization schemes is compared via simulation. Simulation results reveal that game strategies can bring higher detection probability to radar, and the power difference between radar and jammer will affect the optimization effect. Furthermore, the follower has a latemover advantage in radar and jammer Stackelberg game. This study has shown that without changing transmit power, radar can use game theory to improve the detection performance by optimizing the waveform power spectrum.


Human face detection and recognition using contour generation and matching algorithm

Received Jan 26, 2019 Revised Apr 30, 2019 Accepted May 25, 2019 Face detection, monitoring of human face and recognition is a high-quality venture in safety, surveillance and regulation enforcement systems with greater accuracy. There are many motives at the back of in this trouble. Which include function of the human face; pose version, negative environmental conditions, terrible lightning and picture tilt etc,. So we must locate a few feasible solutions to clear up those issues by means of imposing a brand new system. Our proposed venture might be improving the issues on Face detection, monitoring of human face and recognition issues.

INTRODUCTION
Face detection, monitoring of human face and recognition is an amazing project in safety, surveillance and law enforcement structures with greater accuracy [1]. There are many reasons at the back of on this problem. Inclusive of role of the human face, pose variant, negative environmental conditions, terrible lightning and picture tilt and so forth. So we ought to find a few possible answers to clear up these troubles through imposing a brand new gadget. First we can get the frames camera or web cam. At the same time, a detector analyzes the frames to find a face and highlight it within the panel. Face Tracker is shown in motion within the pictures on the top of the web page [2,3].
The detection algorithm is executed speedily while there is a face present in the image (it will be around 40ms), however may additionally take substantially longer to decide there may be no face (as plenty as 200ms). Essential aspects of the coding are locating ways to speed up the detection, and making sure that lengthy detection processing don't sluggish down the relaxation of this system (in particular, the rapid rendering of successive images onto the panel) [4]. Detection is finished with the aid of a contours matching algorithm, pre-educated to find facial features (whilst regarded the front-on). The classifier's education calls for a remarkable deal of time.
The subsequent processing will increase to understand the tracked face. The difference between face detection and reputation is that popularity returns a call for the face. The control software program for the launcher ought to then decide whether the individual is a chum or for before firing a missile. Reorganization is applied by connected thing Operators [5]. Contour matching set of rules used for identification of faces. The feasibility of the usage of contour matching for human face has been identified and it is presented through experimental research. The benefit of the use of contour matching is that the structure of the face is

Image Processing and Normalization Process
From the Figure 1, the captured images are divided into two jointly one of a kind sets: the trained set and the test set. The trained set is used to start and prepare the device to identify subjective pix and to highquality composition the algorithm parameters. The take a look at set is the set of photographs that is used to evaluate the performance of the machine after schooling is completed [13]. The captured images are preprocessed to improve the popularity overall performance. After the pre-processing stage, all of the new images ought to an identical in dimension, so after cropping the face from the whole image, those new snap shots are resized to 150 x 112 pixels. Any other difficulty particular to faces, is that of facial features, complexion and whether or not the concern is carrying glasses or has a moustache and/or a beard. The device should be able to identify faces whether or not they may be smiling, sad, sporting glasses, and not wearing glasses. For this reason its miles essential to apply a trained face database that includes different pictures of identical challenge. The photographs should be preprocessed to lead those to suitable for reputation purposes. This usually consists of the following responsibilities:

Histogram Equalization
Histogram equalization is achieved which enhances the assessment of photographs with the aid of transforming the values in a depth photograph in order that the histogram of the output picture about matches a specified histogram.

Noise removal
This step is used to estimate and remove any noise if gift within the image.

Normalization
This is used to compensate for any illumination versions or relative sizes among two sets of faces. That is accomplished the usage of pixel cost normalization. In this stage the imply pixel fee of all of the pixels within the image is calculated. Also the usual deviation of pixel values is calculated. The usage of those values normalize the pixel values of the picture by the usage of the subsequent components Pixel price normalized image = (Blurred photograph pixel -Pixel suggest) / fashionable deviation.

Contour generation
Middle of the system in which the contour of a face is generated from the picture. The whole human face is considered as a contour map, with the areas of steady gray stage brightness together with this by means of the contour traces. For this reason contour lines for a given face may be generated. a. Getting to know face look fashions and models for realtime observable motion estimation & clustering b. Studying Gaussian mixture primarily based shade fashions for equally monitoring pores and skin tone items and multi-color based forefront and historical past segmentation and tracking. c. Gaining knowledge of an adaptive temporal coloration version to address extreme lights adjustments.
The primary difficulty in modeling shade robustly is the shade fidelity problem which arises because of version in shade values delivered about via lighting adjustments. Information fusion in object detection and monitoring the usage of both motion and color cues was used to result in the specified consistency in face tracking .The device is able to perform face detection and tracking in the following way. Related to face detection and tracking, we also addressed the problem of actual-time head pose estimation which is important in monitoring transferring faces across perspectives [14]. We introduced a composite Gabor wavelet transform as a representation scheme for capturing pose changes. We derived a pose Eigen-area primarily based at the essential additives evaluation to represent and interpret the distribution of pose modifications from continuous sequences of face rotation intensive [15].

Different Image Matching Algorithms
Four image matching strategies are very generally used. A) Blob Detection Technique B) Template Matching C) SURF feature extraction and D) SIFT. In our work preferably use Template Matching strategies because Template matching is a 'brute-force' algorithm for object recognition.

RESULT AND DISCUSSIONS 4.1. Case i: Image Position Detection
The exclusive role of a human face had been captured and presented to the algorithm for role detection. Each role turned into modulated with a selected constellation, noise changed into added and filtered for actual time consequences and the detected role became as compared with the real. The modulation of position facts turned into accomplished the usage of QAM-256 scheme [16]. The captured input positions, matching algorithm all through runtime and the categorized output is proven in Figure 2 to determine 3.

Case ii: Privacy Preservation in Video Images
The textual content turned into hidden inside the video and to protect the data privateness, slicing of video picture has been done. The input frame, sliced frames and the matched photos (i.e. where the question text matched with the hidden text) are proven in Figure 3(a) to parent 3(c). The video frames available for public storage corresponds to the sliced frames and as a result privacy protection is finished. A pattern of the three snap shots, the equal textual content turned into hidden and the corresponding pictures on my own had been correctly recovered. An image-based tracking system can detect the correct position of the moving human face images even poor lightning. Tracking accuracy, processing time and applicability to poor environment is the identical in this method. Consequently, this research improves the performance of the image tracking system to be simplified and not required a separate tracking system. The new algorithm has been implemented is also able to successfully detect the fall event within 255ms. Once the corresponding feature is verified by the current state, it can proceed to next state; otherwise, the system resets to the initial state and waiting for the appearance of another feature sequence. Moreover, a distinguished performance up to 92% on the sensitivity and 99.75% on the specificity is obtained with a set of 450 test activities.

CONCLUSION
Use of our proposed method we can detect and track the human face with higher efficiency is 10% better than the existing methods of image detection and tracking techniques. In terms of face recognition also it will provide better response over the available methods. Also this technique reduces the delay of response considerably. Because of these advantages it will be useful in Security systems, Classification of images, Image enhancement, Automatic image rotation, Portrait / passport photography, Law Enforcement Surveillance etc.This work is not a location specific and it can be used by everywhere.