Multi-Carrier Jamming Mitigation: A Proactive Game Theoretic Approach

Abstract

Wireless communications systems are highly vulnerable to jamming. There is a large body of research demonstrating the potential of smart-jamming that targets specific mechanisms in a wireless stack. Many proactive mitigation techniques sacrifice performance for guaranteed robustness. In this work, we develop a proactive mitigation approach for multi-carrier wireless links. The approach is formulated within the framework of game theory. We show that the interaction between a multi-carrier multi-rate system (in particular OFDM) and a power-limited jammer can be formulated as a constrained zero-sum or a bimatrix game. We show that the Nash equilibrium strategies can be derived analytically and numerically and we apply them to the special case of IEEE 802.11 OFDM links.

10.1 Introduction

The many desirable characteristics of the Orthogonal Frequency Division Multiplexing systems (OFDM) such as high spectral efficiency, high data rates, robustness in multipath fading channels and ease of implementation have made OFDM the primary physical layer solution for most modern wireless communication systems. Wireless Local Area Networks (WLAN) based on different flavors of the IEEE 802.11 or Wireless Metropolitan Area Networks (WMAN) based on IEEE 802.16 both use OFDM as the main physical layer modulation scheme. Additionally, most leading cellular technologies such as 4G LTE standard rely on OFDM at the physical layer.

Nevertheless, it has been shown in the literature that current implementations of OFDM are vulnerable to a variety of jamming attacks specially due to OFDM sensitivity to channel estimation and synchronization between the transmitter and the receiver [1]. For instance, Clancy and Goergen [2] studies the possibility of jamming the channel estimation procedure as an efficient type of attack. This work suggests that targeting the accuracy of channel state information estimation requires significantly less power than jamming the whole frequency band.

Other work studied jamming attacks that prevents the receiver from ever acquiring proper synchronization [3]. To achieve this goal, the adversary targets the symbol timing estimation algorithm, the first step in the synchronization process. This work suggests that a jammer who exploits the weakness in the timing estimation algorithm can cause massive errors to all synchronization estimates.

Other work focused on targeting the coding and interleaving scheme [4]. The authors discovered that in the case of IEEE 802.11ag, the coded bits are interleaved in a deterministic patterns over the OFDM sub-carriers. They demonstrated that it is possible for an adversary to block Wi-Fi packets at a cost 2–3 orders of magnitude lower than full-band jamming.

Game theory has also been used to study jamming games in OFDM systems, for instance, reference [5] considers jamming in a wireless OFDM network with transmission costs for both jammer and transmitter. This work uses the general-sum framework to model the jamming problem. The numerical example in this work suggests that when the jammer is close to the base station, the jammer should pay less attention to the subchannels with poor quality and spend more energy on the subchannels with good quality which in turn, forces the transmitter to use the resources of the bad quality subchannels.

In this chapter, we study the performance of an adaptive OFDM wireless communication system under power limited jamming using game theoretic approaches. We show that with modest assumptions, this problem can be formulated into either the constrained zero-sum, or the constrained bimatrix games. We presents some of fundamental of these framework here, for a detailed study of these frameworks we refer the reader to [6] and [7].

10.2 Communications and Adversary Models

In this section, we briefly introduce our system model and discuss the motivation behind our work. The details of our model will be discussed in the sections that follows. The transmitter and the receiver are communicating over a wireless noisy channel that is subjected to an adaptive adversary. The communicating nodes are using an adaptive Orthogonal Frequency Division Multiplexing (adaptive OFDM) to communicate. The transmitter adaptively changes the subcarriers’ data rates such that the overall throughout of the wireless link is maximized.

On the other hand, the jammer, also adaptively, jams the OFDM subchannels with different jamming powers, in order to degrade the performance of the wireless link. We assume that the jammer can use arbitrary jamming powers and can jam any subchannel that he wishes but for practical reasons, he must maintain a maximum jamming power and energy. Our goal is to model this jamming problem and study the long term achievable performance of this adaptive OFDM system.

10.2.1 Communications Model Under Jamming

Consider an OFDM wireless communication system with K subchannels where the bandwidth of each subchannel is Δf. The transmitter has an adaptation block which enables him to jointly change/adapt his channel coding rate and modulation scheme for each subcarrier (see Fig. 10.1). For convenience, we assume the transmitter uses time domain channel coding and subchannel bandwidth is sufficiently narrow such that the frequency response characteristics of the subchannels are ideal.

Fig. 10.1

The transmitter model

Without loss of generality, assume the data rates for each subcarrier are chosen from a set of N distinct data rates, denoted by $\mathcal {R}$, i.e.,

$$\displaystyle \begin{aligned} \mathcal{R} = \{R_0 = R_{\max}, \dots, R_i, \dots, R_{N-1} = R_{\min} \}\ \text{(bps)} \qquad || \mathcal{R} || = N \end{aligned}$$

(10.1)

where $R_{\min }$ and $R_{\max }$ denotes the minimum and maximum available data rates for each subcarrier, respectively. Furthermore, assume the channel frequency response is such that (in the absence of jamming) $R_{\max }$ is feasible for all subchannels.¹ Obviously, to maximize the throughput of the wireless link (or equivalently, to maximize the average data rate of OFDM symbols), the maximum achievable rate, $R_{\max }$, must be used for all subcarriers. However, because of jamming, the subchannels’ capacities are not known in advanced and therefore it is not known which rates are feasible prior to transmission.

To overcome this problem, the transmitter randomly assigns the data rates to the subcarriers such that the overall throughput of the wireless link is maximized. Let the column vector ${\mathbf {r}}^{(n)}_{K\times 1}$ denote the transmitter’s strategy for the nth OFDM symbol,

$$\displaystyle \begin{aligned} {{\mathbf{r}}^{(n)}_{K \times 1}}^T = [r_1\ \dots \ r_k\ \dots \ r_K] \qquad \text{where}\ r_k \in \mathcal{R} \end{aligned}$$

(10.2)

that is, for the nth OFDM symbol, the kth element of r ⁽ⁿ⁾ is the data rate at which the kth subcarrier is to be coded/modulated. For each OFDM symbol, the adaptation block selects a vector of K data rates where each rate is selected from the set of available data rates given in (10.1) and passes this vector to the coding/modulation block (see Fig. 10.1). The coding/modulation block transmits the nth OFDM symbol according to r ⁽ⁿ⁾. From this point forward, all strategy vectors are assumed to be per OFDM symbol (i.e., for the nth symbol), and for convenience, we drop the ⁽ⁿ⁾ from the vectors.

Because of jamming, reliable recovery of the transmitted data is not guaranteed for all subchannels therefore, the resulting bit rate per OFDM symbol can be written as

$$\displaystyle \begin{aligned} \text{bit \ rate/symbol} = \sum^{K}_{k=1} \hat r_k \qquad \text{where} \ \hat r_k \triangleq \begin{cases} r_k, & \text{if}\ r_k \leq c_k \\ 0, & \text{if}\ r_k \geq c_k \\ \end{cases} \end{aligned}$$

(10.3)

and c _k, k = 1, …, K denotes the actual channel capacity for the kth subchannel, which in general, is a function of channel frequency response and the jammer power spectral density. For sufficiently narrow Δf, we can assume that the jammer’s power spectral density is flat for all subchannels and therefore, we may express c _k as

$$\displaystyle \begin{aligned} c_k = C(p_k, j_k, N_k) \qquad \text{for} \quad k = 1, \dots , K \end{aligned}$$

(10.4)

where C(p _k, j _k, N _k) denotes the channel capacity function of the wireless link and p _k, j _k and N _k (all in W/Hz) denote the power spectral densities of the transmitter, jammer and channel noise for the kth subchannel (all measured at the receiver front end), respectively.

Let $\hat x_i$ denote the number of subcarriers coded/modulated with $R_i \in \mathcal {R}$, i = 0, …, N − 1, obviously we have

$$\displaystyle \begin{aligned} \sum^{N-1}_{i = 0} \hat x_i = K \qquad \text{and} \qquad 0\leq \hat x_i \leq K \end{aligned}$$

(10.5)

now let $x_i \triangleq \frac {1}{K} \hat x_i$, i.e., x _i denotes the fraction of subcarriers transmitted at rate R _i. Form (10.5) it follows,

$$\displaystyle \begin{aligned} \sum^{N-1}_{i = 0} x_i = 1 \qquad \text{and} \qquad 0\leq x_i \leq 1 \end{aligned}$$

(10.6)

As a result, for sufficiently large K, the following vector can be well approximated by the following probability vector

$$\displaystyle \begin{aligned} {\mathbf{x}}^{T}_{1\times N} = [x_0\ \dots\ x_i\ \dots\ x_{N-1}] \qquad \text{for} \quad K\gg N \end{aligned}$$

(10.7)

If we assume that the subchannels have nearly same channel characteristics or when the effects of nonideal wireless channel (including different channel gains for subcarriers etc.) have been compensated by appropriate transmission power allocation at the receiver then, the probability vector in (10.7) can be used as an alternative way of representing the transmitter’s strategy. More specifically, the following two vectors may be used interchangeably, to study the optimal transmission strategy and average throughput of the adaptive OFDM system under jamming,²

$$\displaystyle \begin{aligned} {\mathbf{r}}^T_{1\times K} = \left[ r_1\ \dots \ r_k\ \dots r_K \right] \ r_k \in \mathcal{R} \ \overset{\text{or}}{\longleftrightarrow} \ {\mathbf{x}}^{T}_{1\times N} \\ = [x_0\ \dots\ x_i\ \dots\ x_{N-1}] \ \text{s.t.} \ \left\{ \begin{aligned} & 0\leq x_i \leq 1 \\ & \sum^{N-1}_{i = 0} x_i = 1 \\ \end{aligned} \right. \end{aligned}$$

(10.8)

That is, by assuming that the wireless channel impairments have been compensated by the proper transmission power allocation, all subcarriers experience nearly the same channel characteristics across the entire frequency band and as a result, knowing x _N×1 which gives the fractions of subcarriers coded/modulated at available data rates is sufficient to study the performance of the wireless OFDM system under jamming.

Even though the optimal transmission strategy can be computed in terms of x, vector r, which contains the actual transmission rates, must be reconstructed from x in order to code/modulate the input data. Below, we will discuss one possible approach to construct the transmission rate vector from the probability vector. First, the transmitter constructs vector $\widehat {\mathbf {x}}_{N\times 1}$ which contains the number of subcarriers coded/modulated with the available data rates. This can be done by simply multiplying the probability vector x by K and rounding off the results to the closest integer such that the sum remains K, i.e.,

$$\displaystyle \begin{aligned} \widehat{\mathbf{x}}^T_{1\times N} ={\mathrm{round}}\ K {\mathbf{x}}^T_{1\times N} \end{aligned}$$

(10.9)

where round() denotes the rounding to the nearest integer operation (such that the sum remains K). From vector $\widehat {\mathbf {x}}$ the data rate assignment matrix, denoted by P _N×K is constructed. To fill out the elements of P, we simply need to assign ones and zeros to the elements of P such that,

$$\displaystyle \begin{aligned} \begin{aligned} & \sum^{K}_{k = 1} P_{ik} = \hat x_i \qquad \text{for} \quad i = 0, \dots, N -1 \\ & \sum^{N}_{i = 1} P_{ik} = 1 \qquad \text{for} \quad k = 1, \dots, K \\ \end{aligned} \end{aligned}$$

(10.10)

That is, the number of ones in each row of P is equal to the corresponding component of $\widehat {\mathbf {x}}$ and every column contains exactly one 1. For example, one possible data rate assignment matrix is given in (10.11).

$$\displaystyle \begin{aligned} P_{N \times K} = \left[ \begin{array}{llll} 111& 00 & 0000& \dots \\ 000& 11 & 0000& \dots \\ 000& 00 & 1111& \dots \\ \dots & & & \\ \end{array} \right] \begin{array}{l} \leftarrow \text{\# \ of \ 1's = \ } \hat x_0 \\ \leftarrow \text{\# \ of \ 1's = \ } \hat x_1 \\ \leftarrow \text{\# \ of \ 1's = \ } \hat x_2 \\ \\ \end{array} \end{aligned}$$

(10.11)

A 1 at row i and column k of P, indicates that the kth subcarrier is to be coded/modulated with the ith data rate, Finally, the rate vector for the nth OFDM symbol, ${\mathbf {r}}^{(n)}_{K\times 1}$, can be written as

$$\displaystyle \begin{aligned} {{\mathbf{r}}^{(n)}_{1\times K}}^T = {\mathbf{R}}^T_{1\times N}\ {\mathrm{randperm}}^{(n)}\ P_{N\times K} \end{aligned}$$

(10.12)

where R is the vector of available data rates (from (10.1)) and randperm⁽ⁿ⁾ P _N×K denotes randomly permuting columns of P _N×K for n times. As we discuss in Sect. 10.3, for each OFDM symbol, it is necessary to randomly permute columns of P. Figure 10.2 shows the transmitter’s adaptation block.

Fig. 10.2

The transmitter’s adaptation block

10.2.2 Jammer Adaptivity Model

The jammer has an adaptation block which allows him to jam individual subchannels with (possibly) different jamming powers (see Fig. 10.3). However, for practical reasons, the jammer’s maximum jamming power per subchannel is limited to $J_{\max }$ (W/Hz). Furthermore, we assume the jammer’s energy budget per OFDM symbols is limited to $E_{\max }$ (Joules). Finally, we assume $E_{\max }$ is such that the jammer cannot use $J_{\max }$ for all subchannels otherwise, the energy constraint would be redundant.

Fig. 10.3

The jammer model

We denote the jammer’s jamming power set by $\mathcal {J}$. Without loss of generality assume, $\mathcal {J}$ is given as follows,

$$\displaystyle \begin{aligned} \mathcal{J} = \{ J_{0} = 0, \dots, J_i, \dots, J_{M-1} = J_{\max} \}\ \text{(W/Hz)} \qquad \text{where} \quad ||\mathcal{J}|| = M \end{aligned}$$

(10.13)

Because of the jammer’s energy constrained, $E_{\max }$, the jammer’s average power, denoted by E[J], is constrained to

$$\displaystyle \begin{aligned} \mathrm{E}[J] \leq \frac{E_{\max}}{T_s} \end{aligned}$$

(10.14)

where T _s is the OFDM symbol duration. Let j _K×1 denote the jammer’s strategy per OFDM symbol (equivalently, jamming vector),

$$\displaystyle \begin{aligned} {\mathbf{j}}^T_{1\times K} = [j_1\ \dots\ j_k\ \dots\ j_K]\ (\text{W/Hz)} \qquad \text{where} \quad j_k \in \mathcal{J} \end{aligned}$$

(10.15)

that is, the kth subchannel is jammed with $j_k \in \mathcal {J}$ (W/Hz). From (10.14) we have,

$$\displaystyle \begin{aligned} T_s \sum^{K}_{k = 1} j_k \varDelta f = T_s \varDelta f \sum^{K}_{k = 1} j_k \leq E_{\max} \end{aligned}$$

(10.16)

$$\displaystyle \begin{aligned} \Rightarrow \qquad \sum^{K}_{k = 1} j_k \leq \frac{E_{\max}}{T_s \varDelta f} {} \end{aligned}$$

(10.17)

Now let $\hat y_i$ denote the number of subchannels jammed with $J_i \in \mathcal {J}$, obviously, we have

$$\displaystyle \begin{aligned} \sum^{K}_{k = 1} j_k = \sum^{M-1}_{i = 0} \hat y_i J_i \end{aligned}$$

(10.18)

It is also clear that,

$$\displaystyle \begin{aligned} \sum^{M-1}_{i = 0} \hat y_i = K \qquad \text{and} \qquad 0\leq \hat y_i \leq K \end{aligned}$$

(10.19)

Define $y_i \triangleq \frac {1}{K} \hat y_i$, from (10.19), we have

$$\displaystyle \begin{aligned} \sum^{M-1}_{i = 0} y_i = 1 \quad \text{and} \qquad 0\leq y_i \leq 1, \qquad \text{for}\ i = 0, \dots, M-1 \end{aligned}$$

(10.20)

Therefore, for sufficiently large K, the following vector can be well approximated by a probability vector

$$\displaystyle \begin{aligned} {\mathbf{y}}^{T}_{1\times M} = [y_0\ \dots\ y_i\ \dots\ y_{M-1}] \qquad \text{for} \quad K \gg M \end{aligned}$$

(10.21)

If we write (10.17) in terms of y _i, i = 1, …, M, we obtain the following constraint on y.

$$\displaystyle \begin{aligned} \sum^{K}_{k = 1} j_k = K \sum^{M}_{i = 1} y_i J_i \leq \frac{E_{\max}}{T_s \varDelta f} \end{aligned}$$

(10.22)

Therefore, (10.21) can be used as an alternative way of representing the jammer’s strategy. More specifically, the following two vectors can be used interchangeably to study the optimal jamming strategy and average performance degradation of the wireless link³

$$\displaystyle \begin{aligned} \begin{aligned} {\mathbf{j}}^T_{1\times K} = \left[ j_1\ \dots \ j_k\ \dots j_K \right], \ j_k \in \mathcal{J} \ \overset{\text{or}}{\longleftrightarrow} \ & {\mathbf{y}}^{T}_{1\times M} = [y_0\ \dots\ y_i\ \dots\ y_{M-1}] \\ & \text{s.t.} \quad \left\{ \begin{aligned} & 0\leq y_i \leq 1 \\ & \sum^{M-1}_{i = 0} y_i = 1 \\ & \sum^{M-1}_{i = 0} y_i J_i = {\mathbf{y}}^T \mathbf{J} \leq \frac{E_{\max}}{K T_s \varDelta f}\\ \end{aligned} \right. \end{aligned} \end{aligned}$$

(10.23)

Even though the optimal jamming strategy can be computed in terms of y, as it is shown in Fig. 10.3, vector j ⁽ⁿ⁾, which contains the actual jamming powers for the nth OFDM symbol, must be passed to the jamming block in order to allocate the available jamming power accordingly. The jamming vector can be constructed from the jamming probability vector in the exact same manner that the rate vector was constructed. Here we only provide the results and refer the reader to Sect. 10.2.1 for the details.

$$\displaystyle \begin{aligned} {{\mathbf{j}}^{(n)}_{1\times K}}^T = {\mathbf{J}}^T_{1\times M}\ {\mathrm{randperm}}^{(n)}\ P_{M \times K} \end{aligned}$$

(10.24)

where P _M×K denotes the jamming assignment matrix, and is constructed in the exact same way as (10.11) was constructed, and randperm⁽ⁿ⁾ denotes randomly permuting columns of P for n times. Figure 10.4 shows the jammer’s adaptation block in detail.

Fig. 10.4

The jammer’s adaptation block

The random permutation block is necessary in the jammer’s adaptation block to randomize the jamming vector for every OFDM symbol before passing it to the jamming block. Otherwise, the subchannels will be jammed with a fixed power and the jamming problem simplifies to a simple water filling problem [8].

10.3 Proactively Optimizing the Average Throughput of Adaptive OFDM in the Presence of Adaptive Jammers

We consider the problem of maximizing the average throughput of the wireless link under jamming by randomly adapting the data rates of the subcarriers.⁴ Obviously, in absence of the jammer, the optimal strategy to maximize the average throughput is to use the maximum data rate for all subcarriers. However, in the presence of the jammer, the capacities of the subcarriers are not known in advance.

By assigning different data rates to the subcarriers, the transmitter can increase the possibility that some of the subcarriers overcome the jamming (see Fig. 10.5). Furthermore, data rates assignments must be done randomly, that is, for every OFDM symbol the data rate assignment pattern must be randomized. This randomization is necessary since static data rate assignment pattern would make higher data rates more vulnerable to jamming as these data rates are easier to jam.

Fig. 10.5

Adaptive OFDM under jamming

Consider a typical wireless OFDM system such as IEEE 802.11, without loss of generality, assume the set of available data rates in the OFDM system, $\mathcal {R}$ is sorted in a decreasing order, i.e.,

$$\displaystyle \begin{aligned} \mathcal{R} = \{ R_0 = R_{\max}, \dots, R_i, \dots, R_{N - 1} = R_{\min} \}\ \text{bps} \end{aligned}$$

(10.25)

where R _i, i = 0, …, N are the available data rates of the OFDM system for the subcarriers. It can be shown that to jam a rate adaptive wireless system with N rates, it sufficient to use no more than N + 1 jamming powers [6]. Therefore, WLOG, we assume the jammer is using the following jamming set

$$\displaystyle \begin{aligned} \mathcal{J} = \{ J_0 = 0, \dots, J_j, \dots, J_N = J_{\max} \}\ \text{W/Hz} \end{aligned}$$

(10.26)

When the wireless channel is nearly flat or, when the jamming is the dominant cause of the noise at the receiver front end (which is the typical case for most jamming scenarios), it can be assumed that each rate in $\mathcal {R}$ can tolerate up to a certain level of jamming power which is the same for all subchannels and it is completely lost otherwise (since every subchannel experiences nearly the same channel or the jamming is the dominant factor of the noise). As we will see shortly, this assumption greatly simplifies the analytical results and allows us to express the results in closed form expressions.

Assume $R_i \in \mathcal {R}$, 0 ≤ i ≤ N − 1, can be recovered for any jamming power less than J _i+1, 0 ≤ i ≤ N − 1. That is, R ₀ can only tolerate J ₀ and no rate can tolerate J _N. Furthermore, let the transmitter’s and the jammer’s strategies be,⁵

$$\displaystyle \begin{aligned} \widehat{\mathbf{x}}^T = \left[ \hat x_0\ \dots\ \hat x_i\ \dots\ \hat x_{N-1} \right]_{1\times N} \qquad \hat x_i: \text{\# \ of \ subchannels \ to \ be \ sent \ with}\ R_i \end{aligned}$$

(10.27)

and

$$\displaystyle \begin{aligned} \widehat{\mathbf{y}}^T = \left[ \hat y_0\ \dots\ \hat y_i\ \dots\ \hat y_N \right]_{1\times (N + 1)} \qquad \hat y_i: \text{\# \ of \ subchannels \ jammed \ with}\ J_i \end{aligned}$$

(10.28)

where $\widehat {\mathbf {x}}$ and $\widehat {\mathbf {y}}$ are defined and constructed as discussed in Sects. 10.2.1 and 10.2.2, respectively. Since the transmitter and the jammer randomize their respective strategies independently, the partial average throughput from rate R ₀, and denoted by T ₀, becomes

$$\displaystyle \begin{aligned} T_0 = \hat x_0 \left(\frac{\hat y_0}{K} \right) R_0 \end{aligned}$$

(10.29)

that is because $\hat x_0$ is the number of subcarriers coded/modulated with R ₀ and ${\hat y_0}/{K}$ is the probability that a subchannel is jammed with J ₀. Similarly, the partial average throughput from data rate R _i is

$$\displaystyle \begin{aligned} T_i = \hat x_i \left( \frac{1}{K} \sum^{i}_{j = 0} \hat y_j \right) R_i \end{aligned}$$

(10.30)

Therefore, the average throughput per OFDM symbols becomes,

$$\displaystyle \begin{aligned} \begin{aligned} T = \sum^{N-1}_{i=0} T_i & = \sum^{N-1}_{i = 0} \hat x_i \left( \frac{1}{K} \sum^{i}_{j = 0} \hat y_j \right) R_i\\ & = K \sum^{N-1}_{i = 0} \sum^{i}_{j = 0} \frac{\hat x_i}{K} \frac{\hat y_j}{K} R_i\\ & = K \sum^{N-1}_{i = 0} \sum^{i}_{j = 0} x_i y_j R_i \\ & = {\mathbf{x}}^T K R_{N\times (N+1)} \mathbf{y} \end{aligned} \end{aligned}$$

(10.31)

where, x and y are defined in (10.7) and (10.21), respectively and the matrix R _N×(N+1) is a lower triangular matrix where the nonzero elements of the rows of R are equal to the data rates, i.e.,

$$\displaystyle \begin{aligned} R_{N \times ( N+1 )} = \begin{bmatrix} R_0 & 0 & & \dots & & 0\\ \vdots & \ddots& \ddots& & & 0\\ R_i & & R_i & 0 & \dots & \vdots\\ \vdots & & & \ddots& \ddots&\\ R_{N-1} & & \dots & & R_{N-1} & 0\\ \end{bmatrix} \end{aligned}$$

(10.32)

therefore, the optimal transmission/jamming and the average throughput of the wireless link at the equilibrium is the solution of the following maxmin problem,

$$\displaystyle \begin{aligned} T({\mathbf{x}}^*, {\mathbf{y}}^*) = \max_{\mathbf{x}}\min_{\mathbf{y}} {\mathbf{x}}^T K R_{N\times (N+1)} \mathbf{y} \quad \text{s.t.} \quad \left\{ \begin{aligned} & {\mathbf{x}}^T \mathbf{1} = 1 \\ & {\mathbf{y}}^T \mathbf{1} = 1 \\ & {\mathbf{y}}^T \mathbf{J} \leq \frac{E_{\max}}{KT_s \varDelta f} \\ \end{aligned} \right. \end{aligned}$$

(10.33)

where, x ^∗ and y ^∗ denote the optimal transmission and jamming strategies, respectively. The maxmin problem in (10.33) can be solve analytically and numerically (Sect. 10.4).

10.4 Generalized Interactions: Constrained Bimatrix Games

In zero-sum game framework, it is usually assumed that the players have perfect knowledge of the game and the actions that are available to the other players, and they use this knowledge to compute their respective optimal strategies. In such a case, the zero-sum framework fully captures the conflicting goals of the players. Moreover, the equilibrium solution of the zero-sum game guarantees a minimum payoff regardless of the other player’s strategy [9]. For a more comprehensive study of constrained zero-sum games in wireless jamming see [6]. 

However, in some jamming scenarios, having perfect knowledge of the system parameters (or available actions) may not be a feasible option or too costly for a player. In addition, players may have objectives that are not exactly the opposite of each other, for example, the transmitter may wish to minimize the average error probability while the jammer wishes to minimize the average throughput of the system (as opposed to maximizing the average error probability).

In such scenarios, a more appropriate framework to model the communication system under jamming would be a bimatrix game instead of a zero-sum game.⁶ In bimatrix games it is no longer required that the sum of the players’ payoffs to be zero (or a constant value) [9]. As a result, players can have different objectives and the respective payoffs can be defined based on the players’ goals and their knowledge of the game (which in general may be imperfect). Such a formulation, encompasses a variety of situations from full competition to full cooperation.

Additionally, in standard zero-sum and bimatrix games there are no additional restrictions on players’ mixed-strategies, i.e., players may choose any probability distribution over their respective action sets (pure-strategies). However, there exist scenarios where, due to practical reasons, not all mixed-strategies are permitted and/or feasible.

Such scenarios demand for a more general framework to study them. In this chapter, we study a linearly constrained bimatrix game to overcome these limitations. In constrained games, the players’ mixed-strategies not only have to be a probability distributions but they must satisfy some additional linear constraints too⁷ (Fig. 10.6 shows the classification of standard and constrained games). Detailed study of necessary and sufficient conditions under which the existence of the Nash equilibrium is guaranteed as well as a systematic approach to find the NE is beyond the scope of this chapter, but can be found in [7].

Fig. 10.6

Classification of standard and constrained games

10.5 Performance Analysis: The Case of IEEE 802.11 OFDM

In IEEE 802.11, the 20 MHz channel bandwidth is subdivided into 52 subchannel with a separation of Δf = 20∕64 = 0.3125 MHz. Of the total 52 subcarriers, 48 carry data and 4 are pilot subcarriers, however, for convenience, in our analysis we assume the number of subcarriers is K = 64 and all the subcarriers carry data. The OFDM symbol interval is T _s = 4 μs (which includes a 0.8 μs cyclic prefix), which results in the symbol rate R _s = 0.25 MSymbols/s. Table 10.1 shows some of the IEEE 802.11 physical layer parameters including the modulation schemes and code rates used for the subcarriers. The last column in Table 10.1 shows the resulting data rates for the subcarriers. Hence, the set of available data rates for the subcarriers is

$$\displaystyle \begin{aligned} \mathcal{R} = \left\{R_0 = 1.125,\ R_1 = 1.0,\ 0.75,\ 0.5,\ 0.375,\ 0.25,\ 0.1875,\ R_7 = 0.125\right\}_{\text{Mbps}} \end{aligned}$$

(10.34)

Table 10.1 IEEE 801.11 modulation schemes and code rates

Suppose that we have a base station that is communicating with a mobile user and the total transmission power for the 64 subcarriers is 200 mW. Then transmission power per subcarrier becomes,

$$\displaystyle \begin{aligned} P_T = \frac{0.2}{64} = 0.0031\ \text{W} = -25\ \text{dBW} \end{aligned}$$

(10.35)

Furthermore, assume that the combined transmitter/receiver antenna gains and the losses from other sources result in a L _T = 90 dB attenuation in the received signal power. Then, the power of the received signal per subcarrier is,

$$\displaystyle \begin{aligned} (P_R)_{\mathrm{dB}} = (P_T)_{\mathrm{dB}} - (L_T)_{\mathrm{dB}} = -25 - 90 = -115\ \text{dBW} \end{aligned}$$

(10.36)

The power spectral density of additive noise at the receiver front end is N ₀ = 4.1 × 10⁻²¹ W/Hz. Therefore the signal to noise ratio at the receiver front end becomes,

$$\displaystyle \begin{aligned} \text{SNR}_{\text{dB}} = (P_R)_{\mathrm{dB}} - (\varDelta f N_0)_{\text{dB}} \cong 34 \end{aligned}$$

(10.37)

and from (10.37) the capacity of the subcarriers is

$$\displaystyle \begin{aligned} C_{\mathrm{AWGN}} = \varDelta f \log \left( 1 + \text{SNR} \right) = 3.52\ \text{Mbps} \end{aligned}$$

(10.38)

Since the capacity of the link is greater than max_i R, all the available data rates are feasible (when the jammer is not active). Now, suppose that the jammer’s combined antenna gain and losses from other sources results in a L _J = 60 dB attenuation in the jamming signal measured at the receivers front end. To make the channel capacity drop below the data rate (i.e., make the data rates infeasible), the jammer needs to increase the channel noise at the receiver front end by

$$\displaystyle \begin{aligned} P_{JR,i} = \varDelta f J_i = \frac{P_R}{ 2^{R_i/\varDelta f} - 1 } - \varDelta f N_0 \quad \text{(W)} \end{aligned}$$

(10.39)

where R _i’s are given in (10.34). Hence, the jammer’s transmission power becomes

$$\displaystyle \begin{aligned} (P_{JT,i})_{\mathrm{dB}} = (P_{JR,i})_{\mathrm{dB}} + (L_J)_{\mathrm{dB}} \end{aligned}$$

(10.40)

If we substitute the numerical values, the jammer’s strategy set becomes

$$\displaystyle \begin{aligned} \begin{aligned} \mathcal{J} & = \{0, 0.0003, 0.0004, 0.0007, 0.0015, 0.0024, 0.0042, 0.0061, 0.0098\} \\ &\text{W/subchannel} \\ & = \{-\infty, -35.2, -34.2, -31.4, -28.1, -26.2, -23.8, -22.2, -20.1 \} \\ &\text{dBW/subchannel} \\ \end{aligned} \end{aligned}$$

(10.41)

The optimal transmission and jamming strategies and the expected value of the game at the Nash equilibrium can be derived analytically (and numerically) in the same manner as was presented in Sect. 10.4. For Instance, it can be shown that the minimum average jamming power needed to force the lowest rate for the jammer that is using optimal strategy is

$$\displaystyle \begin{aligned} J_{\mathrm{TH}} = K\varDelta f R_7 \sum^{7}_{i = 1} \left( R^{-1}_{i} - R^{-1}_{i-1} \right) J_j = 0.2133\ \text{W} = -6.7 \ \text{dBW} \end{aligned}$$

(10.42)

whereas the average jamming power that the Barrage noise jammer require to achieve the same average throughput is

$$\displaystyle \begin{aligned} J_{\mathrm{barrage}} = \frac{KP_R}{ 2^{R_{\min}/K\varDelta f} - 1 } - K\varDelta f N_0 = 0.39 \ \text{W} = -4.1 \ \text{dBW} \end{aligned}$$

(10.43)

This shows a gain of 2.6 dB for the strategic jammer.

10.6 Concluding Remarks

In this chapter, we studied the performance of an adaptive OFDM wireless communication system under power limited jamming. We showed that with modest assumptions, this problem can be formulated into the constrained zero-sum or constrained bimatrix game. We in particular applied this framework to the IEEE 802.11 with OFDM physical layer.