1 Introduction↩︎

Averting the threat of jamming attacks on wireless communication systems is a problem of vital importance as society becomes ever more reliant on wireless communication infrastructure [2]–[5]. Multi-antenna (or MIMO) processing offers an attractive and effective solution by enabling the mitigation of jammers through spatial filtering [6]. In MIMO-based jammer mitigation, the jammer’s interference is often removed by projecting the receive signal onto the orthogonal complement of the jammer channel’s subspace, which traditionally is estimated in a dedicated training period [7]–[14]. Such an approach has two main disadvantages: First, a dedicated training period for estimating the jammer’s channel reduces the achievable data rate, since no information can be transmitted during the training period. Second, a training-period-based approach is ineffective against smart jammers that transmit only at specific instances to evade estimation [15] or against dynamic jammers that change their subspace through time-varying multi-antenna transmit beamforming [16]. In order to avoid both of these limitations, we propose a novel paradigm for jammer mitigation with MIMO processing that we call joint jammer mitigation and data detection (JMD).

1.1 State of the Art↩︎

The fundamental challenge of jammer mitigation through spatial filtering is that it requires information about the jammer, such as the subspace spanned by the jammer’s channel [7]–[10] or the covariance matrix of the jammer’s interference [8], [17], [18]. Existing methods often assume that the jammer transmits permanently and with static beamforming. Such a behavior would enable one to estimate the required quantities during a dedicated training period in which the legitimate transmitters do not transmit [7], [8], [18] or in which they transmit predetermined symbols (pilots) that carry no information [9]–[13]. Relying on such a static jammer assumption, the receiver can filter the jammer in the subsequent communication period until the wireless channel changes and the jammer training process is started anew. A smart jammer, however, can easily circumvent such mitigation methods by deliberately violating their assumptions: It can pause jamming for the duration of the dedicated training period, so that the receiver learns nothing meaningful [15]. Or, if the jammer has multiple antennas, it can use beamforming to dynamically change its subspace (as well as the interference covariance matrix at the receiver), so that after the training period, the receiver’s filter will no longer match the jammer’s transmit characteristics [16], [19].⁴

In order to mitigate smart jammers, methods have been suggested that attempt to fool the jammer into transmitting during the training period by distributing and randomizing the timing of the training period [7], [9]. However, such methods may not work against jammers that jam only intermittently at random time instants. Similarly, methods have been proposed to mitigate dynamic multi-antenna jammers by perpetually estimating their instantaneous subspace [16]. However, such methods may be effective only against jammers that change their subspace in a sufficiently slow manner. Furthermore, all training-period based mitigation methods are subject to an inherent trade-off between the time that they dedicate to the attempt of estimating the required jammer characteristics and the time that remains for payload data transmission.

In light of these considerations, a more principled approach to MIMO-based jammer mitigation is needed. We have recently proposed MAED [15], which, in hindsight, can be viewed as the first instance of our JMD paradigm. MAED unifies not only jammer mitigation and data detection, but also channel estimation (at the cost of increased computational complexity).

1.2 Contributions↩︎

We propose joint jammer mitigation and data detection (JMD), a novel paradigm for jammer mitigation. The core idea is to estimate and remove the subspace of the jammer interference jointly with detecting the data of an entire transmission frame (or coherence interval). JMD removes the need for a dedicated training period, which enables higher data rates because more time is available for data transmission. Beyond that, considering an entire transmission frame at once enables JMD to deal with smart jammers that try to evade mitigation (i) by jamming only at specific instances or (ii) by dynamically changing their subspace through multi-antenna beamforming.

As in [15], we leverage the fact that a jammer cannot leave its subspace within a coherence interval (which holds also for multi-antenna jammers with time-varying beamforming). Going beyond our work in [15], we show that this fact can be exploited to mitigate the jammer’s contamination of the channel estimate even when the channel is estimated independently of the subsequent jammer mitigation and data detection. This insight opens the door to computationally more efficient signal-processing algorithms for jammer mitigation.⁵ Moreover, our earlier work in [15] only mitigates single-antenna jammers, while this paper shows that JMD is also well suited for the mitigation of multi-antenna jammers—even dynamic ones. We capitalize on these new insights by proposing the JMD-type algorithm SANDMAN (an algorithm that offers all of the jammer mitigation capabilities of MAED [15], but at reduced computational complexity), and by extending MAED to multi-antenna jammers. Furthermore, we present theoretical guarantees for JMD that are stronger than those in [15] and that rely on weaker assumptions. Extensive simulations demonstrate the efficacy of our JMD-type algorithms SANDMAN and MAED for mitigating a wide range of smart and dynamic single- and multi-antenna jammers. This paper therefore provides a full and mature account of the JMD paradigm, while the earlier work in [15] should (in hindsight) be regarded as a precursor that did not yet grasp the full breadth and depth of the JMD paradigm. Compared to the conference version [1], this paper adds most of the theoretical results (and all of the proofs), more comprehensive discussion, and more extensive evaluation.

1.3 Notation↩︎

Matrices and column vectors are represented by boldface uppercase and lowercase letters, respectively. For a matrix \(\mathbf{A}\), the transpose is \(\mathbf{A}^{\text{T}}\), the conjugate transpose is \(\mathbf{A}^{\text{H}}\), the Moore-Penrose pseudoinverse is \(\mathbf{A}^{\dagger}\) (if \(\mathbf{A}\) is full-rank and tall, then \(\mathbf{A}^{\dagger}=(\mathbf{A}^{\text{H}}\mathbf{A})^{-1}\mathbf{A}^{\text{H}}\), and if \(\mathbf{A}\) is full-rank and wide, then \(\mathbf{A}^{\dagger}=\mathbf{A}^{\text{H}}(\mathbf{A}\mathbf{A}^{\text{H}})^{-1}\)), and the Frobenius norm is \(\| \mathbf{A}\|_F\). The columnspace and rowspace of \(\mathbf{A}\) are \(\textit{col}(\mathbf{A})\) and \(\textit{row}(\mathbf{A})\), respectively, and their orthogonal complements are \(\textit{col}(\mathbf{A})^\perp\) and \(\text{row}(\mathbf{A})^\perp\). Horizontal concatenation of two matrices \(\mathbf{A}\) and \(\mathbf{B}\) is denoted by \([\mathbf{A},\mathbf{B}]\); vertical concatenation by \([\mathbf{A};\mathbf{B}]\). The \(N\!\times\!N\) identity matrix is \(\mathbf{I}_N\). The \(\ell_2\)-norm of a vector \(\mathbf{a}\) is \(\|\mathbf{a}\|_2\), and the \(\ell_0\)-“norm” \(\|\mathbf{a}\|_0\) indicates the number of nonzero entries of \(\mathbf{a}\). We use \(\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{C})\) to denote the \(N\)-dimensional circularly-symmetric complex Gaussian distribution with covariance matrix \(\mathbf{C}\in\mathbb{C}^{N\times N}\). Finally, \([1:N]\) denotes the set of integers from \(1\) through \(N\).

2 Transmission Model↩︎

We focus on mitigating jamming attacks in the flat-fading, block-fading multi-user (MU) MIMO uplink.⁶ We consider the following transmission model: \[\begin{align} \mathbf{y}_k = \mathbf{H}\mathbf{s}_k + \mathbf{J}\mathbf{w}_k + \mathbf{n}_k. \label{eq:model} \end{align}\tag{1}\] Here, \(\mathbf{y}_k\in\mathbb{C}^B\) is the BS receive vector at sample instant \(k\), \(\mathbf{H}\in \mathbb{C}^{B\times U}\) is the UE channel matrix, \(\mathbf{s}_k\in\mathcal{S}^U\) contains the sample-\(k\) transmit symbols of \(U\) single-antenna UEs with constellation \(\mathcal{S}\) and is assumed to be uniformly distributed over \(\mathcal{S}^U\), \(\mathbf{J}\in\mathbb{C}^{B\times I}\) is the channel matrix of an \(I\)-antenna jammer,⁷ \(\mathbf{w}_k\in\mathbb{C}^{I}\) is the sample-\(k\) jammer transmit vector, and \(\mathbf{n}_k\sim\mathcal{C}\mathcal{N}(\mathbf{0},N_0\mathbf{I}_B)\) is circularly-symmetric complex Gaussian noise with per-entry variance \(N_0\). We assume that \(B\geq U+I\), i.e., the number of receive antennas is greater than or equal to the number of antennas of the UEs and the jammer combined, and that the matrix \([\mathbf{H},\mathbf{J}]\) has full rank \(U+I\). For simplicity, we take \(\mathcal{S}\) to be QPSK, though other constellations would also be possible [15]. The constellation \(\mathcal{S}\) is scaled to unit symbol power, so that \(\mathop{\mathrm{\mathbb{E}}}_{}\mathopen{}\left[\mathbf{s}_k\mathbf{s}_k^{\text{H}}\right]=\mathbf{I}_U\), and factors related to power control are absorbed into the channel matrix \(\mathbf{H}\). In general, the jammer is a dynamic multi-antenna jammer and is able to dynamically change its jamming activity. Specifically, the jammer transmits \[\begin{align} \mathbf{w}_k = \mathbf{A}_k \tilde{\mathbf{w}}_k, \label{eq:jammer95beamforming} \end{align}\tag{2}\] where, without loss of generality, \(\mathop{\mathrm{\mathbb{E}}}_{}\mathopen{}\left[\tilde{\mathbf{w}}_k\tilde{\mathbf{w}}_k^{\text{H}}\right]=\mathbf{I}_I\) for all \(k\), and \(\mathbf{A}_k\in\mathbb{C}^{I\times I}\) is a beamforming matrix that can change arbitrarily over time (i.e., \(\mathbf{A}_k\) depends on the sample instant \(k\)). In particular, \(\mathbf{A}_k\) can be the all-zero matrix (no jamming at sample instant \(k\)), or some of its rows can be zero (the jammer uses only a subset of its antennas at sample instant \(k\)), or it can be rank-deficient in some other way.

In JMD, the receiver will process the receive signal in blocks consisting of \(K\) sample indices (or channel uses). The length \(K\) of these blocks, which we call frames, may be up to the length of a coherence interval, so that the channel matrices \(\mathbf{H}\) and \(\mathbf{J}\) can be assumed to be constant for the duration of a frame. It will be useful to define notation for the receive signal of an entire frame: \[\begin{align} \mathbf{Y}&= \mathbf{H}\mathbf{S}+ \mathbf{J}\mathbf{W}+ \mathbf{N}. \tag{3} \intertext{ Here, \mathbf{Y}=[\mathbf{y}_1,\dots,\mathbf{y}_K]\in\mathbb{C}^{B\times K} is the receive matrix, and the quantities \mathbf{S}\in\mathbb{C}^{U\times K}, \mathbf{W}\in\mathbb{C}^{I\times K}, and \mathbf{N}\in\mathbb{C}^{B\times K} are defined analogously. We divide each frame into a \emph{pilot phase} of length T\geq U and a \emph{data phase} of length D (such that T+D=K). The input-output relations for these phases are denoted by } \mathbf{Y}_T &= \mathbf{H}\mathbf{S}_T + \mathbf{J}\mathbf{W}_T + \mathbf{N}_T \tag{4} \intertext{ and } \mathbf{Y}\!_D &= \mathbf{H}\mathbf{S}_D + \mathbf{J}\mathbf{W}\!_D + \mathbf{N}_D, \tag{5} \end{align}\] respectively. Thus, we have \(\mathbf{Y}=[\mathbf{Y}_T,\mathbf{Y}\!_D]\), \(\mathbf{S}=[\mathbf{S}_T,\mathbf{S}_D]\) with \(\mathbf{S}_T\in\mathbb{C}^{U\times T}\) and \(\mathbf{S}_D\in\mathcal{S}^{U\times D}\), \(\mathbf{W}=[\mathbf{W}_T,\mathbf{W}\!_D]\), and \(\mathbf{N}=[\mathbf{N}_T,\mathbf{N}_D]\). Note that we do not restrict the pilots to the constellation \(\mathcal{S}\). However, we do impose an average power constraint \(\frac{1}{T}\|\mathbf{s}_T\|_2^2=1\) on all rows \(\mathbf{s}_T\) of \(\mathbf{S}_T\). We assume that the columns \(\mathbf{s}_k\) of \(\mathbf{S}_D\) are independent of each other, so that \(\mathbf{S}_D\sim\text{Unif}[\mathcal{S}^{U\times D}]\) (where \(\text{Unif}[\mathcal{A}]\) denotes the uniform distribution over the set \(\mathcal{A}\)). In what follows, we use plain letters for the true transmit signals and channels, letters with a tilde for optimization variables, and letters with a hat for (approximate) solutions to optimization problems. For example, \(\hat{\mathbf{S}}_D\) is an estimate of the data matrix \(\mathbf{S}_D\) obtained by solving an optimization problem with respect to a variable \(\tilde{\mathbf{S}}_D\).

3 Joint Jammer Mitigation and Data Detection↩︎

Existing methods for MIMO jammer mitigation often null a jammer by projecting the receive signal onto the orthogonal complement of the jammer subspace [8]–[10], [16]. That is, they compute \(\mathbf{P}\mathbf{y}_k\), where \(\mathbf{P}= \mathbf{I}_B - \mathbf{J}\mathbf{J}^{\dagger}\) is the orthogonal projection onto \(\textit{col}(\mathbf{J})^\perp\) and has the property \(\mathbf{P}\mathbf{J}=\mathbf{0}\). After this projection, the transmit data can be detected using the effective channel matrix \(\mathbf{H}_\mathbf{P}\triangleq \mathbf{P}\mathbf{H}\), since \[\begin{align} \mathbf{P}\mathbf{y}_k &= \mathbf{P}\mathbf{H}\mathbf{s}_k + \mathbf{P}\mathbf{J}\mathbf{A}_k \tilde{\mathbf{w}}_k + \mathbf{P}\mathbf{n}_k = \mathbf{P}\mathbf{H}\mathbf{s}_k + \mathbf{P}\mathbf{n}_k \\ &\triangleq \mathbf{H}_\mathbf{P}\mathbf{s}_k + \mathbf{n}_{\mathbf{P},k}. \label{eq:projected95model} \end{align}\tag{6}\] This strategy mitigates the jammer regardless of which vectors \(\tilde{\mathbf{w}}_k\) and which beamforming matrices \(\mathbf{A}_k\) the jammer uses (cf. 2 ), since \(\textit{col}(\mathbf{J}\mathbf{A}_k)\subseteq\textit{col}(\mathbf{J})\) for any \(\mathbf{A}_k\). The problem, however, is that of estimating \(\mathbf{J}\)—or \(\textit{col}(\mathbf{J})\)—when the jammer changes \(\mathbf{A}_k\) dynamically such that \(\textit{col}(\mathbf{J}\mathbf{A}_k)\) depends on \(k\). In that case, the receiver will observe different interference subspaces \(\textit{col}(\mathbf{J}\mathbf{A}_k)\) at different instants, but possibly never the “pure” jammer subspace \(\textit{col}(\mathbf{J})\).

To address this problem, we propose JMD. The key idea is to consider jammer subspace estimation and nulling as well as data detection over an entire transmission frame (or coherence interval) jointly. The jammer subspace is identified with the subspace that is not explainable in terms of UE transmit signals; simultaneously, the UE transmit data are estimated while projecting the receive signals onto the orthogonal complement of the jammer subspace estimate. Mathematically, this can be formulated as the following optimization problem:⁸ \[\begin{align} \min_{\substack{\tilde{\mathbf{S}}_D \,\in\, \mathcal{S}^{U\times D},\\ \tilde{\mathbf{P}}\,\in\, \mathscr{G}_{B-I}(\mathbb{C}^B)}} \| \tilde{\mathbf{P}}(\mathbf{Y}\!_D -\mathbf{H}\tilde{\mathbf{S}}_D) \|_F^2. \label{eq:jmd95problem} \end{align}\tag{7}\] Here, \(\tilde{\mathbf{S}}_D\) is the potential estimate of the data matrix \(\mathbf{S}_D\) and \(\tilde{\mathbf{P}}= \mathbf{I}_B - \tilde{\mathbf{J}}\tilde{\mathbf{J}}^{\dagger}\) is the projection onto the orthogonal complement of the potential jammer subspace estimate \(\textit{col}(\tilde{\mathbf{J}})\), with \(\tilde{\mathbf{J}}\in\mathbb{C}^{B\times I}\). The range of \(\tilde{\mathbf{P}}\) is the Grassmannian manifold \(\mathscr{G}_{B-I}(\mathbb{C}^B)=\{\mathbf{I}_B-\tilde{\mathbf{J}}\tilde{\mathbf{J}}^{\dagger}:\tilde{\mathbf{J}}\!\in\!\mathbb{C}^{B\times I}\}\), i.e., the set of orthogonal projections onto \((B\!-\!I)\)-dimensional subspaces of \(\mathbb{C}^B\). The intuition behind this problem formulation is as follows: Using 5 , we can rewrite the objective of 7 as \[\begin{align} \|\tilde{\mathbf{P}}( \mathbf{H}(\mathbf{S}_D-\tilde{\mathbf{S}}_D) + \mathbf{J}\mathbf{W}\!_D + \mathbf{N}_D)\|_F^2. \label{eq:pre95intuition} \end{align}\tag{8}\] If we assume that the thermal noise \(\mathbf{N}_D\) is negligible (\(\mathbf{N}_D=\mathbf{0}\)) compared to the legitimate signals \(\mathbf{H}\mathbf{S}_D\) and to the jammer interference \(\mathbf{J}\mathbf{W}\!_D\), then we may further simplify 8 to \[\begin{align} \textstyle \bigg\|\tilde{\mathbf{P}}\begin{bmatrix}\mathbf{H},\mathbf{J}\end{bmatrix} \begin{bmatrix}\mathbf{S}_D-\tilde{\mathbf{S}}_D \\ \mathbf{W}\!_D \end{bmatrix} \bigg\|_{F \,\displaystyle{.}}^2 \label{eq:intuition} \end{align}\tag{9}\] The justification for this low-noise assumption—which we make repeatedly throughout the paper—is that this paper’s focus is jammer mitigation, not data detection under high thermal noise. In other words, we are concerned with scenarios where communication performance is not primarily limited by thermal noise, but by jamming. The objective in 9 is minimized when the argument of the Frobenius norm is the all-zero matrix, and the projector \(\tilde{\mathbf{P}}\) can null a matrix of rank less than or equal to \(I\). If \(\mathbf{J}\mathbf{W}\!_D\) has rank \(I\), this necessitates that \(\tilde{\mathbf{S}}_D=\mathbf{S}_D\) (by assumption, the matrix \([\mathbf{H},\mathbf{J}]\) has full rank \(U+I\); see ). To make this intuition more rigorous, we introduce the notion of eclipsing, which describes a certain relation between the jammer transmit matrix \(\mathbf{W}\!_D\) and the error matrix \(\mathbf{S}_D-\tilde{\mathbf{S}}_D\) corresponding to a potential symbol estimate \(\tilde{\mathbf{S}}_D\). In this section, we define eclipsing while assuming perfect channel state information (CSI). The more realistic case, in which CSI is estimated using pilots, is considered in .

To say that a jammer is eclipsed therefore means that there exists an incorrect (i.e., distinct from \(\mathbf{S}_D\)) estimate \(\tilde{\mathbf{S}}_D\) of the data matrix \(\mathbf{S}_D\) such that the virtual interference matrix \(\mathbf{\Sigma}(\tilde{\mathbf{S}}_D)\), which appears in 9 , has rank less than or equal to \(I\). Our first theoretical result is that, under the assumption of negligible thermal noise, eclipsing is the only aspect that might hinder successful jammer nulling and data recovery:

(All proofs are in the appendix.) The question is therefore: What are the chances that a jammer is—or is not—eclipsed? This question turns out to be, in general, quite complex. For instance, an immediate consequence of is that any jammer with \(\textit{rank}(\mathbf{W}\!_D)<I\) is eclipsed. The “problem” with such a jammer that only jams with rank smaller than \(I\) is that a projection \(\tilde{\mathbf{P}}\in\mathscr{G}_{B-I}(\mathbb{C}^B)\) can null more dimensions than are occupied by interference, and so can also null any data detection errors \(\mathbf{S}_D-\tilde{\mathbf{S}}_D\) of rank up to \(I-\textit{rank}(\mathbf{W}\!_D)\), which jeopardizes accurate data detection. In that sense, it would therefore be preferable to identify the parameter \(I\) not with the number of physical jammer antennas \(I\), but with the effective rank of the jammer interference, \(\textit{rank}(\mathbf{W}\!_D)\). (Any receive interference \(\mathbf{J}\mathbf{W}\!_D\) with \(\textit{rank}(\mathbf{W}\!_D)<I\) can be rewritten also as a product \(\mathbf{J}'\mathbf{W}_{\!D}'\) for some \(\mathbf{J}'\in\mathbb{C}^{B\times\textit{rank}(\mathbf{W}\!_D)}\).) Nevertheless, we have decided to identify \(I\) with the number of jammer antennas because this is more natural from an operational point of view. We note, however, that our working assumption will therefore be that the jammer transmits with full antenna rank, i.e., \(\textit{rank}(\mathbf{W}\!_D)=I\). As a consequence of this difficulty, we can only give strong theoretical guarantees for the probability of eclipsing for single-antenna jammers.⁹ But we emphasize that our empirical results demonstrate that JMD works well also for multi-antenna jammers; see . For single-antenna jammers, for which we write \(\mathbf{J}\mathbf{W}\!_D\) as \(\mathbf{j}\mathbf{w}_D^{\text{T}}\), the probability of eclipsing turns out to depend on the number of symbols that the jammer jams (i.e., on \(\|\mathbf{w}_D\|_0\)). A jammer that never jams is guaranteed to eclipse; a jammer that jams perpetually eclipses only with minuscule probability:

depicts this probability as a function of the number of jammed symbols \(\|\mathbf{w}_D\|_0\) for different numbers of UEs \(U\).

4 Channel Estimation↩︎

In the previous section, we have assumed that the BS knows the UE channel matrix \(\mathbf{H}\) perfectly. In practice, however, \(\mathbf{H}\) has to be estimated. This is typically done using pilots. In the presence of jamming, the obtained estimate can be contaminated [24]–[27]. We now propose two different approaches for jammer-resilient channel estimation in combination with JMD.

4.1 Linear Channel Estimation↩︎

The first approach consists of simply using a classical linear channel estimator such as least squares (LS) or linear minimum mean square error (LMMSE). It may not be immediately apparent why these estimates should be jammer-resilient. In fact, there is nothing inherently jammer-resilient about them; they become jammer-resilient only when combined with JMD.

The underlying mechanism works as follows: Let the pilot phase be given as in 4 , with a square (\(T=U\)) or wide (\(T>U\)) pilot matrix, and let the corresponding data phase be given as in 5 . If a linear channel estimator is used, then the channel estimation error due to the jammer interference is contained in \(\textit{col}(\mathbf{J})\). For instance, in the case of LS channel estimation, we have \[\begin{align} \hat{\mathbf{H}}&= \mathbf{Y}_T \mathbf{S}_T^{\dagger} = \mathbf{H}+ \underbrace{\mathbf{J}\mathbf{W}_T\mathbf{S}_T^{\dagger}}_{\in\,\textit{col}(\mathbf{J})} + \mathbf{N}_T\mathbf{S}_T^{\dagger}. \label{eq:pilot95contamination} \end{align}\tag{10}\] Therefore, if we simply plug the jammer-contaminated linear channel estimate \(\hat{\mathbf{H}}\) from 10 into 7 as follows, \[\begin{align} \min_{\substack{\tilde{\mathbf{S}}_D \,\in\, \mathcal{S}^{U\times D},\\ \tilde{\mathbf{P}}\,\in\, \mathscr{G}_{B-I}(\mathbb{C}^B)}} \| \tilde{\mathbf{P}}(\mathbf{Y}\!_D - \hat{\mathbf{H}}\tilde{\mathbf{S}}_D) \|_F^2, \label{eq:jmd95problem2} \end{align}\tag{11}\] then the “true” jammer-nulling projection \(\mathbf{P}= \mathbf{I}_B - \mathbf{J}\mathbf{J}^{\dagger}\) also removes the jammer contamination of the channel estimate, since \(\mathbf{P}\hat{\mathbf{H}}=\mathbf{P}\mathbf{H}\). This implies that JMD can naturally incorporate jammer-resilient channel estimation, without requiring extra countermeasures to protect channel estimation from jammer interference.

To characterize the probability of successful data recovery for JMD in combination with linear channel estimation, we provide the following adapted (compared to ) definition of eclipsing:¹⁰

With this modified definition of eclipsing, we obtain the following guarantee for JMD with linear channel estimation:

As in , the question becomes: what is the probability that a jammer eclipses?¹¹ The answer depends fundamentally on whether or not the jammer knows the pilots that are being used by the UEs. Let us first consider the case where the jammer knows the pilots (as well as the transmit constellation \(\mathcal{S}\)).

However, the situation is completely different if the (single-antenna) jammer does not know the pilots. This can be modeled by assuming that a (square) pilot matrix \(\mathbf{S}_T\) is drawn at random according to the power-normalized Haar measure (i.e., drawn uniformly from the set of unitary \(T\times T\) matrices and scaled to \(\|\mathbf{S}_T\|_F^2=U^2\)) [28] and not revealed to the jammer in advance:¹²

Note that the bound in ?? , where the jammer is silent during the pilot phase, coincides with the bound from . The bound in ?? , where the jammer is active during the pilot phase, satisfies \(p_{e,\mathbf{w}_T\neq\mathbf{0}}\leq p_e(\mathbf{w}_D)\) with equality if and only if \(\mathbf{w}_D\) has full support, in which case \(p_{e,\mathbf{w}_T\neq\mathbf{0}}= p_e(\mathbf{w}_D) \approx 4U\cdot2^{-D}\).

It is evident that the guarantees for successful JMD jammer mitigation are fundamentally dependent on whether or not the jammer knows the pilots used for channel estimation. This finding is reminiscent of a more general information-theoretic result which shows that reliable communication in the presence of jamming is possible if the legitimate transmitter and the receiver share a common secret, but not otherwise[29]. Intuitively, the receiver needs a lever that enables it to distinguish between legitimate and illegitimate transmit signals. This lever could be knowledge of the legitimate UEs’ channel matrix \(\mathbf{H}\) (as in ). Or, if ground-truth channel knowledge is not available, it could be a mechanism which guarantees that the jammer cannot replicate a legitimate pilot sequence during the channel estimation phase (as in ).

4.2 Joint Channel Estimation↩︎

Besides linear channel estimation, there is a second possible approach for channel estimation in combination with JMD, namely to estimate the channel jointly with the data and the jammer-mitigating projection. Specifically, the idea is to use a pilot and a data phase as in 4 and 5 , and to then solve \[\begin{align} \min_{\substack{ \tilde{\mathbf{P}}\,\in\, \mathscr{G}_{B-I}(\mathbb{C}^B),\\ \tilde{\mathbf{H}}\,\in\, \mathbb{C}^{B\times U},\\ \tilde{\mathbf{S}}_D \,\in\, \mathcal{S}^{U\times D} }} \big\| \tilde{\mathbf{P}}([\mathbf{Y}_T, \mathbf{Y}\!_D] - \tilde{\mathbf{H}}[\mathbf{S}_T, \tilde{\mathbf{S}}_D]) \big\|_F^2. \label{eq:jmd95problem3} \end{align}\tag{12}\] Analogous to linear channel estimation (cf. ), this approach exploits the fact that the jammer’s interference during the channel estimation and the data phase is contained in the same subspace \(\textit{col}(\mathbf{J})\). Compared to linear channel estimation, joint channel estimation benefits from a joint channel estimation and data detection (JED) performance gain, at the price of having to solve a more complex optimization problem.

To characterize the probability of successful data recovery for JMD in combination with joint channel estimation, we can use the same definition of eclipsing () as in . Furthermore, it turns out that we also obtain the same success guarantee:

Since the definition for eclipsing is the one from , the results of and specify the probability of eclipsing also for the case of joint channel estimation. Therefore, regardless of whether we estimate the channel matrix separately with a linear estimator (as in ) or jointly (as in this section), randomized pilots are necessary for meaningful success guarantees. However, we emphasize that these guarantees do not imply that joint channel estimation has the same performance as linear channel estimation under non-zero noise, i.e., for \(\mathbf{N}\neq\mathbf{0}\).

5 Algorithms↩︎

In the previous two sections, we have introduced the JMD paradigm (see ) and outlined two alternatives for channel estimation (linear and joint; see ). In this section, we provide concrete algorithms, i.e., we propose algorithms for approximately solving the optimization problems in 11 and 12 . The presented algorithms are by no means the only possible algorithms for these optimization problems. The development of other JMD algorithms is left as future work.

5.1 The SANDMAN Algorithm↩︎

We start by providing SANDMAN, an algorithm for JMD in conjunction with linear channel estimation as expressed through the optimization problem in 11 . Since solving 11 exactly is difficult, we solve it approximately. A key difficulty is that, due to the discreteness of \(\mathcal{S}\), the problem in 11 is NP-hard even when fixing \(\tilde{\mathbf{P}}\) and solving only for \(\tilde{\mathbf{S}}_D\) [30]. We thus relax the constraint set \(\mathcal{S}\) to its convex hull \(\mathcal{C}\triangleq \textit{conv}(\mathcal{S})\).¹⁴

To promote symbol estimates at, or near, the corners of \(\mathcal{C}\), we add a concave regularizer \(-\|\tilde{\mathbf{S}}_D\|_F^2\) weighted by \(\alpha>0\) to the objective in 11. We colloquially refer to the resulting constraint and regularizer as a box prior. The use of this box prior is motivated by our assumption (cf. ) that the transmit constellation is QPSK, so that the corners of \(\mathcal{C}\) correspond to the constellation symbols \(\mathcal{S}\). If larger constellations such as 16-QAM or 64-QAM are used, other signal priors are more effective [15]. The relaxed problem is \[\begin{align} \min_{\substack{\tilde{\mathbf{S}}_D \,\in\, \mathcal{C}^{U\times D},\\ \tilde{\mathbf{P}}\,\in\, \mathscr{G}_{B-I}(\mathbb{C}^B)}} \big\| \tilde{\mathbf{P}}(\mathbf{Y}\!_D - \hat{\mathbf{H}}\tilde{\mathbf{S}}_D) \big\|_F^2 - \alpha\|\tilde{\mathbf{S}}_D\|_F^2. \label{eq:sandman95problem} \end{align}\tag{13}\] Due to the nonconvex constraint set \(\mathscr{G}_{B-I}(\mathbb{C}^B)\), the relaxed problem is still non-convex. But the following result holds:

This theorem suggests using an alternating minimization strategy, since solving 13 for either \(\tilde{\mathbf{S}}_D\) or \(\tilde{\mathbf{P}}\) is straightforward as long as the other quantity is fixed:

5.1.1 Solving for \(\tilde{\mathbf{S}}_D\)↩︎

To solve 13 with respect to \(\tilde{\mathbf{S}}_D\) (for fixed \(\tilde{\mathbf{P}}\)), we use forward-backward splitting (FBS) [31]. FBS is a method for solving optimization problems of the form \[\begin{align} \min_{\tilde{\mathbf{s}}}~ f(\tilde{\mathbf{s}}) + g(\tilde{\mathbf{s}}), \label{eq:fbs95problem} \end{align}\tag{14}\] where \(f\) is convex and differentiable, and \(g\) is convex but not necessarily differentiable. FBS solves such problems iteratively by computing \[\begin{align} \tilde{\mathbf{s}}^{(t+1)} = \text{prox}_g\big(\tilde{\mathbf{s}}^{(t)} - \tau^{(t)}\nabla f(\tilde{\mathbf{s}}^{(t)}); \tau^{(t)}\big), \label{eq:fbs95iteration} \end{align}\tag{15}\] where \(\tau^{(t)}\) is the stepsize at iteration \(t\), \(\nabla f(\tilde{\mathbf{s}})\) is the gradient of \(f\) in \(\tilde{\mathbf{s}}\), and \(\text{prox}_g\) is the proximal operator of \(g\), defined as [32] \[\begin{align} \text{prox}_g(\tilde{\mathbf{s}}; \tau) = \arg\!\min_{\tilde{\mathbf{x}}~~~~} \tau g(\tilde{\mathbf{x}}) + \frac{1}{2} \|\tilde{\mathbf{s}}- \tilde{\mathbf{x}}\|_2^2. \end{align}\] FBS solves convex problems exactly (for a sufficient number of iterations with suitable stepsizes \(\tau^{(t)}\)), but it is also effective for approximately solving non-convex problems [31]. To solve the problem in 13 , we define the functions \(f\) and \(g\) as \[\begin{align} f(\tilde{\mathbf{S}}_D) &= \big\| \tilde{\mathbf{P}}(\mathbf{Y}\!_D - \hat{\mathbf{H}}\tilde{\mathbf{S}}_D) \big\|_F^2, \\ g(\tilde{\mathbf{S}}_D) &= - \alpha\big\|\tilde{\mathbf{S}}_D\big\|_F^2 + \chi_\mathcal{C}(\tilde{\mathbf{S}}_D), \end{align}\] where \(\chi_\mathcal{C}\) acts entrywise on \(\tilde{\mathbf{S}}_D\) as the indicator function of \(\mathcal{C}\), \[\begin{align} \chi_\mathcal{C}(\tilde{s}) = \begin{cases} 0 &: \tilde{s} \in \mathcal{C}\\ \infty &: \tilde{s} \notin \mathcal{C}. \end{cases} \end{align}\] The gradient of \(f\) in \(\tilde{\mathbf{S}}_D\) is given as \[\begin{align} \nabla f(\tilde{\mathbf{S}}_D) = -2\,\hat{\mathbf{H}}^{\text{H}}\tilde{\mathbf{P}}(\mathbf{Y}\!_D - \hat{\mathbf{H}}\tilde{\mathbf{S}}_D). \end{align}\] The proximal operator of \(g\) acts entrywise on \(\tilde{\mathbf{S}}_D\) and is given as \(\text{prox}_g(\tilde{s}; \tau) = \text{clip}(\tilde{s}/(1-\tau\alpha); \sqrt{\frac{1}{2}})\) when \(\alpha\tau<1\) (where \(\text{clip}(z;a)\) clips the real and imaginary part of \(z\in\mathbb{C}\) to the interval \([-a,a]\)), and otherwise as \(\arg\!\min_{\tilde{x} \in \{\pm\sqrt{\frac{1}{2}} \pm i\sqrt{\frac{1}{2}} \} } |\tilde{s} - \tilde{x}|^2\).

5.1.2 Solving for \(\tilde{\mathbf{P}}\)↩︎

According to , we can solve 13 with respect to \(\tilde{\mathbf{P}}\) (for fixed \(\tilde{\mathbf{S}}_D\)) by calculating the \(I\) principal left-singular vectors \([\mathbf{u}_1,\dots,\mathbf{u}_I]=\mathbf{U}_I\) of \(\mathbf{Y}\!_D - \hat{\mathbf{H}}\tilde{\mathbf{S}}_D\). Instead of performing an exact but computationally expensive singular value decomposition (SVD), we approximate \(\mathbf{U}_I\) with the power method from [33]. We perform a single power iteration per dimension. The resulting procedure is outlined in .

The SANDMAN algorithm alternates between descent steps in \(\tilde{\mathbf{S}}_D\) and approximate computations of \(\tilde{\mathbf{P}}\) for a total of \(t_\text{max}\) iterations. We choose \(\alpha=2.5\), and the stepsizes \(\tau^{(t)}\) are selected using the Barzilai-Borwein criterion [34] detailed in [35]. SANDMAN is summarized in and has a computational complexity of \(O(t_\text{max}UDB)\), i.e., its complexity is linear in the number of UEs \(U\), the number of data samples \(D\), and the number of BS antennas \(B\).

One key feature of SANDMAN is that, for single antenna jammers for which \((\tilde{\mathbf{J}}^{(t)})^{\dagger}=(\tilde{\mathbf{j}}^{(t)})^{\dagger} = (\mathbf{j}^{(t)})^{\text{H}}/\|\mathbf{j}^{(t)}\|^2\), it requires no matrix inversion. This makes SANDMAN particularly attractive for hardware implementation. In fact, the first (and so far the only) jammer-mitigating MU-MIMO receiver ASIC [22] is based on SANDMAN, mainly thanks to the fact that no matrix inversion is required.

5.2 The MAED Algorithm↩︎

MAED is an algorithm for JMD in conjunction with joint channel estimation as expressed through the optimization problem in 12 . A version of MAED for single-antenna jammers has originally been proposed in [15].¹⁵ Estimating the channel jointly enables MAED to achieve the detection gains associated with joint channel estimation and data detection (JED) [36]–[40]. Other than estimating the channel jointly as well, however, MAED works similarly to SANDMAN.

To derive the MAED algorithm, we start by noting that the objective in 12 is quadratic in \(\tilde{\mathbf{H}}\). The optimal \(\tilde{\mathbf{H}}\) as a function of \(\tilde{\mathbf{P}}\) and \(\tilde{\mathbf{S}}\triangleq[\mathbf{S}_T,\tilde{\mathbf{S}}_D]\) is therefore equal to \[\begin{align} \tilde{\mathbf{H}}= \tilde{\mathbf{P}}\mathbf{Y}\tilde{\mathbf{S}}^{\dagger}. \end{align}\] By plugging this expression back into 12 , we obtain an optimization problem which only depends on \(\tilde{\mathbf{P}}\) and \(\tilde{\mathbf{S}}\). Furthermore, as in 13 , we also relax the constraint set \(\mathcal{S}\) to its convex hull \(\mathcal{C}\) and add a concave regularizer \(-\alpha\|\tilde{\mathbf{S}}_D\|_F^2\) to promote symbol estimates near the constellation points. The resulting optimization problem is therefore \[\begin{align} \min_{\substack{ \tilde{\mathbf{P}}\,\in\, \mathscr{G}_{B-I}(\mathbb{C}^B),\\ \tilde{\mathbf{S}}=[\mathbf{S}_T,\tilde{\mathbf{S}}_D]:\, \tilde{\mathbf{S}}_D \,\in\, \mathcal{C}^{U\times D} }} \big\| \tilde{\mathbf{P}}\mathbf{Y}(\mathbf{I}_K - \tilde{\mathbf{S}}^{\dagger}\tilde{\mathbf{S}}) \big\|_F^2 - \alpha\|\tilde{\mathbf{S}}_D\|_F^2. \end{align}\] As in SANDMAN, we now use an alternating minimization strategy where we alternate between FBS-based descent steps in \(\tilde{\mathbf{S}}\) and approximate optimization steps in \(\tilde{\mathbf{P}}\). The gradient required for FBS is \[\begin{align} \nabla f(\tilde{\mathbf{S}}) = - (\mathbf{Y}\tilde{\mathbf{S}}^{\dagger})^{\text{H}}\tilde{\mathbf{P}}\mathbf{Y}(\mathbf{I}_K-\tilde{\mathbf{S}}^{\dagger}\tilde{\mathbf{S}}). \end{align}\] The proximal operator \(\text{prox}_g\) maps the first \(T\) columns of \(\tilde{\mathbf{S}}\) to \(\mathbf{S}_T\) and acts entrywise on the remaining columns as \(\text{prox}_g(\tilde{s}; \tau) = \text{clip}(\tilde{s}/(1-\tau\alpha); \sqrt{\frac{1}{2}})\) when \(\alpha\tau<1\) (where \(\text{clip}(z;a)\) clips the real and imaginary part of \(z\in\mathbb{C}\) to the interval \([-a,a]\)), and otherwise as \(\arg\!\min_{\tilde{x} \in \{\pm\sqrt{\frac{1}{2}} \pm i\sqrt{\frac{1}{2}} \} } |\tilde{s} - \tilde{x}|^2\).

As in SANDMAN, we choose \(\alpha=2.5\) and compute the stepsizes using the Barzilai-Borwein criterion of [35]. The resulting algorithm is summarized in and has computational complexity \(O(t_\text{max}UKB)\). While this is the same asymptotic complexity order as for SANDMAN, MAED requires the computation of a pseudoinverse (of \(\tilde{\mathbf{S}}\in\mathbb{C}^{U\times K}\)) in every algorithm iteration, which makes it less attractive for hardware implementation.

6 Evaluation↩︎

6.1 Simulation Setup↩︎

We evaluate our JMD-type methods SANDMAN and MAED using simulations. The channel vectors are generated with QuaDRiGa [41]. We simulate a MU-MIMO system with \(B=32\) BS antennas and \(U=16\) single-antenna UEs at a carrier frequency of 2 GHz using the 3GPP 38.901 urban macrocellular (UMa) channel model [42]. All antennas are omnidirectional. The BS antennas are arranged as a uniform linear array (ULA) and spaced at half wavelength. The UEs are uniformly distributed at distances between \(10\) m and \(250\) m in a \(120^\circ\) angular sector in front of the BS, and with a minimum angular separation of \(1^\circ\) between any two UEs. We assume \(\pm3\) dB per-UE power control. The specific jammer model varies between the different experiments, see below. In general, we consider \(J\geq1\) jammers placed randomly in the same area as the UEs, with a minimum angular separation of \(1^\circ\) between any two jammers as well as between any jammer and any UE. Every jammer is equipped with \(\frac{I}{J}\geq1\) antennas arranged as a ULA with half-wavelength spacing that faces towards the BS.

To be able to compare our methods with training-period-based methods, we now consider communication frames of length \(L=K+R\) instead of \(K\). That is, we replace 3 with \[\begin{align} \mathbf{Y}= \mathbf{H}\mathbf{X}+ \mathbf{J}\mathbf{W}+ \mathbf{N}\in \mathbb{C}^{B\times L}, \end{align}\] where the columns of \(\mathbf{X}\) consist of \(R\) (for “redundancy”) evenly distributed all-zero columns used as a jammer training period as well as the columns of \(\mathbf{S}=[\mathbf{S}_T,\mathbf{S}_D]\).¹⁶ The submatrices of \(\mathbf{Y}\) consisting of the columns of the jammer training period, the pilot phase, and the data phase are denoted \(\mathbf{Y}_J\in\mathbb{C}^{B\times R}\), \(\mathbf{Y}_T\in\mathbb{C}^{B\times T}\), and \(\mathbf{Y}_D\in\mathbb{C}^{B\times D}\), respectively. Our JMD methods will use \(R=0\) since they do not require a jammer training period. We assume a frame duration of \(L=100\) channel uses.

We consider QPSK transmission, and we use a square pilot matrix (\(T=U=16\)). The pilots are selected as rows of a \(U\times U\) Haar matrix (normalized to unit symbol energy) and not revaled to the jammer. The signal-to-noise ratio (SNR) is \[\begin{align} \textit{SNR} \triangleq\frac{\mathop{\mathrm{\mathbb{E}}}_{\mathbf{s}_k}\mathopen{}\left[\|\mathbf{H}\mathbf{s}_k\|_s^2\right]}{\mathop{\mathrm{\mathbb{E}}}_{\mathbf{n}_k}\mathopen{}\left[\|\mathbf{n}_k\|_2^2\right]} = \frac{\|\mathbf{H}\|_F^2}{BN_0}. \end{align}\] Furthermore, we characterize the receive power of the jammer interference relative to the receive power of the average UE as \[\begin{align} \rho \triangleq\frac{\frac{1}{L}\|\mathbf{J}\mathbf{W}\|_F^2}{\frac{1}{U}\mathop{\mathrm{\mathbb{E}}}_{\mathbf{s}_k}\mathopen{}\left[\|\mathbf{H}\mathbf{s}_k\|_2^2\right]} = \frac{\frac{1}{L}\|\mathbf{J}\mathbf{W}\|_F^2}{\frac{1}{U}\|\mathbf{H}\|_F^2}, \end{align}\] where we deterministically scale the interference \(\mathbf{J}\mathbf{W}\) to some specified \(\rho\). As performance metrics, we consider the uncoded bit error rate (BER) and a metric called modulation error ratio (MER) between the data symbols \(\mathbf{S}_D\) and their estimate \(\hat{\mathbf{S}}_D\), \[\begin{align} \textit{MER}\triangleq \sqrt{\frac{\mathbb{E}\big[\|\hat{\mathbf{S}}_D - \mathbf{S}_D\|_F^2\big]}{\mathbb{E}\big[\|\mathbf{S}_D\|_F^2\big]}}. \end{align}\] We use the MER as a surrogate for error vector magnitude (EVM), which the 3GPP 5G NR technical specification requires to be below 17.5% [43] for QPSK transmission.

6.2 Higher Data Rates↩︎

The first advantage of JMD compared to training-period based mitigation schemes is increased data rates since no symbol time slots are reserved for estimating the jammer’s channel.

Jammer Model↩︎

The rate advantage due to the absence of a training period is shown for the following jammer model:

Single-antenna barrage jammer↩︎

One single-antenna jammer transmits i.i.d. circularly-symmetric complex Gaussian noise during the entire frame. The jammer power is \(\rho=30\) dB.

Baselines↩︎

The following receivers are used as baselines:

POS-JED↩︎

This receiver uses \(0\leq R \leq L-T\) channel uses in each frame for estimating the jammer subspace as the \(I\) principal left singular vectors of \(\mathbf{Y}_J\). This reduces the number of channel uses for data transmission from \(D=K-T\) to \(D=K-T-R\). The jammer is mitigated by first projecting \(\mathbf{Y}_T\) and \(\mathbf{Y}_D\) onto the orthogonal complement of the estimated subspace, followed by performing FBS-based joint channel estimation and data detection analogous to MAED.

G-POS-JED↩︎

This receiver serves as an upper bound to the achievable performance of MAED. It works analogous to MAED but is furnished with ground-truth knowledge of the jammer channel matrix \(\mathbf{J}\). It therefore uses \(R=0\) and sets \(\tilde{\mathbf{P}}^{(t)}\) on line 7 of to the optimal projector \(\mathbf{P}= \mathbf{I}_B - \mathbf{J}\mathbf{J}^{\dagger}\).

POS-BOX↩︎

Like POS-JED, this receiver uses \(0\leq R \leq L-T\) channel uses of each frame for estimating the jammer subspace as the \(I\) principal left singular vectors of \(\mathbf{Y}_J\). The jammer is mitigated by first projecting \(\mathbf{Y}_T\) and \(\mathbf{Y}_D\) onto the orthogonal complement of the estimated subspace, followed by performing LS channel estimation and FBS-based data detection with a box prior analogous to SANDMAN.

G-POS-BOX↩︎

This receiver serves as an upper bound to the achievable performance of SANDMAN. It works analogous to SANDMAN but is furnished with ground-truth knowledge of the jammer’s channel matrix \(\mathbf{J}\). It therefore uses \(R=0\) and fixes \(\tilde{\mathbf{P}}^{(t)}\) on line 7 of to the optimal projector \(\mathbf{P}= \mathbf{I}_B - \mathbf{J}\mathbf{J}^{\dagger}\).

All of the above algorithms run \(t_\text{max}=30\) iterations.

Results↩︎

In order to demonstrate the rate increase due to the absence of a jammer training period, we consider the tradeoff between the lowest SNR for which a receiver satisfies \(\text{MER}\leq 17.5\%\) and the ratio \[\begin{align} r = \frac{L-T-R}{L-T}, \end{align}\] which is the number of channel uses \(L-T-R\) per communication frame that are available for data transmission, normalized with respect to the maximum number of samples \(L-T\) available without a jammer estimation phase. The ratio \(r\) translates directly to the achieved data rate (measured in terms of bits/s). shows the results:

Since SANDMAN and MAED as well as G-POS-BOX and G-POS-JED have no jammer training period (\(R=0\)), they achieve \(r=1\). In contrast, the detection accuracy of POS-BOX and POS-JED increases with the redundancy \(R\) (a longer training period yields a better estimate of the jammer channel, which enables more precise nulling), at the expense of \(r\). Our results demonstrate that SANDMAN and MAED, which can utilize the receive data of the entire frame to estimate the jammer subspace, achieve virtually the same performance as their genie-assisted counterparts G-POS-BOX and G-POS-JED. In contrast, POS-BOX and POS-JED can only use a subset of the receive samples to estimate the jammer interference, and so perform significantly worse even when dedicating a significant fraction of the frame to jammer training. In fact, POS-BOX does not reliably outperform SANDMAN even when using almost the entire frame for jammer training (so that no almost no channel uses remain for data transmission, \(r\to0\)). The performance of POS-JED even deteriorates when \(r\to0\) (since joint channel estimation and data detection degenerates when there are very few data symbols), and it is always at least \(0.18\) dB away from the performance of MAED.

6.3 Mitigating Smart, Distributed, and Multi-Antenna Jammers↩︎

In this experiment, we analyze the ability of JMD to mitigate smart jammers that might not jam during any training phase but only at specific instants, or spatially distributed single-antenna jammers, or multi-antenna jammers that use dynamic beamforming to change their interference subspace.

Jammer Models↩︎

Besides the single-antenna barrage jammer (), we consider the following types of jammers:

Single-antenna data jammer↩︎

A single-antenna jammer that does not jam the training period and pilot phase, and that transmits i.i.d. complex Gaussian noise with \(\rho=30\) dB in the data phase. Note that this jammer is smart (i.e., protocol-aware).

Single-antenna pilot jammer↩︎

A single-antenna jammer that does not jam the training period and data phase, and that transmits i.i.d. complex Gaussian noise with \(\rho=30\) dB during the pilot phase. Note that this jammer is smart, too.

Single-antenna sparse jammer↩︎

A non-smart single-antenna jammer that only jams for a (randomly selected) single symbol time per communication frame, with \(\rho=30\) dB.

Distributed single-antenna barrage jammers↩︎

Four distributed and statistically independent single-antenna jammers transmit i.i.d. complex Gaussian noise with \(\rho=30\) dB.

Slowly varying multi-antenna jammer↩︎

A non-smart four-antenna jammer where the \(\tilde{\mathbf{w}}_k\) (see 2 ) are white Gaussian random vectors with \(\rho=30\) dB. The matrices \(\mathbf{A}_k\) are constructed as follows: Only the leftmost column \(\mathbf{a}_{k,1}\) of \(\mathbf{A}_k\) is nonzero. For randomly selected instants \(k_1<\dots<k_M, M=5\), the vector \(\mathbf{a}_{k_m,1}\) is fixed to randomly drawn vectors \(\{\mathbf{a}^{(m)}\}_{m=1}^M\). For \(k_m\!<\!k\!<\!k_{m+1}\), \(\mathbf{a}_{k,1}\) interpolates between \(\mathbf{a}^{(m)}\) and \(\mathbf{a}^{(m+1)}\).

Abruptly varying multi-antenna jammer↩︎

A non-smart four-antenna jammer that transmits only with a random subset of one or two of its antennas at each instant \(k\). To this end, a randomly selected subset of the rows of the beamforming matrix \(\mathbf{A}_k\) (see 2 ) is populated with i.i.d. \(\mathcal{C}\mathcal{N}(0,1)\) entries, and the other rows are set to zero. The beamforming matrix \(\mathbf{A}_{k+1}\) is equal to \(\mathbf{A}_k\) with probability \(0.95\) and otherwise is redrawn (including the support set of the rows of \(\mathbf{A}_{k+1}\)). The vectors \(\tilde{\mathbf{w}}_k\) are white Gaussian random vectors with \(\rho=30\,\)dB.

Baselines↩︎

We compare SANDMAN and MAED with the baselines from . The training period-based methods POS-BOX and POS-JED use a redundancy of \(R=4\) and \(D=80\) data transmission symbols per frame (yielding a relative rate of \(r=80/84\approx0.95\)) while the other methods use \(R=0\) and \(D=84\) data transmission symbols (yielding \(r=1\)). All iterative algorithms run for \(t_\text{max}=30\) iterations when facing a single-antenna jammer, and for \(t_\text{max}=50\) iterations when facing distributed or multi-antenna jammers. The algorithms are provided with knowledge of the number \(I\) of jammer antennas. Other than that, the configurations of the algorithms do not depend on the type of jammer being faced. For context, we also include the following unmitigated receiver as an additional baseline:

Unmitigated↩︎

This receiver does not mitigate the jammer. It performs LS channel estimation and (jammer-oblivious) LMMSE data detection.

Results↩︎

depicts the results. The performance of the G-POS-BOX and G-POS-JED baselines depends on the jammer type only via the number of jammer antennas.¹⁷ In contrast, the performance of the unmitigated receiver suffers to varying degrees—but always significantly—under the different types of jammers: the least harmful jammers for an unmitigated receiver are the pilot jammer and the sparse jammer (with a BER of around \(1\%\) at high SNR), and the most harmful are the data jammer as well as distributed and multi-antenna jammers , , (with a BER of \(20\%\) - \(50\%\) even at high SNR). As expected, the training period-based baselines POS-BOX and POS-JED mitigate the barrage jammers and successfully, but fail against all smart or dynamic jammers.¹⁸

In contrast, the JMD-type methods SANDMAN and MAED are able to mitigate all jammers more or less successfully:

• They achieve virtually the same performance as their genie-assisted counterparts G-POS-BOX and G-POS-JED against the barrage jammers and as well as against the data jammer and the pilot jammer . This indicates that SANDMAN and MAED null these jammers essentially perfectly. Note also that SANDMAN and MAED outperform their training period-based counterparts POS-BOX and POS-JED even against the barrage jammers and (as expected based on ).

• Also against the smart jammers and , SANDMAN and MAED achieve virtually the same performance as their genie-assisted counterparts. In contrast, their training period-based counterparts fail completely against such smart jammers that do not jam during the training period.

• Even against the sparse jammer , which might be expected to be their Achilles heel (based on the discussion in and ), SANDMAN and MAED perform outperform the unmitigated baseline as well as their training-period based counterparts. They manage to achieve a BER of below \(1\%\) at high SNR.

• Against the dynamic multi-antenna jammers and , SANDMAN and MAED achieve BERs significantly below \(1\%\) at high SNR. In contrast, the BERs of their training period-based counterparts POS-BOX and POS-JED remain significantly above \(1\%\) even against such non-smart dynamic jammers. Note, however, that also SANDMAN and MAED eventualy hit an error floor due to difficulty of disentangling a high-dimensional and time-varying interference subspace from the signal subspace. The error floor of SANDMAN is lower than that of MAED because the optimization problem approximated by SANDMAN is less complex, and thus less prone to misconvergence, than the one approximated by MAED.

In summary, SANDMAN and MAED outperform their training period-based counterparts POS-BOX and POS-JED for every considered jammer type (sometimes decivisely). They also consistently outperform the unmitigated baseline. This demonstrates the suitability of JMD-based methods for mitigating smart and/or dynamic single-antenna jammers, distributed jammers, or multi-antenna jammers.

7 Discussion and Conclusion↩︎

We have proposed joint jammer mitigation and data detection (JMD), a novel paradigm for mitigating jammers in MIMO systems that does not use a dedicated jammer training period. As a result, JMD is able to mitigate smart jammers regardless of (i) when they are active and (ii) how they vary their multi-antenna transmit beamforming. We have provided theoretical success guarantees for the case of smart single-antenna jammers, and we have proposed two JMD-type algorithms (SANDMAN and MAED) whose efficacy against single- and multi-antenna jammers has been demonstrated through extensive simulations.

At the moment, JMD still exhibits certain drawbacks: (i) we only provide success guarantees for single-antenna jammers, (ii) JMD requires the number of jammer antennas (or the rank of jamming interference) to be known at the receiver in advance, and (iii) the JMD-type algorithms SANDMAN and MAED exhibit an error floor against sparse jammers (which are prone to eclipsing, see and ) and against multi-antenna jammers (for which solving the respective optimization problems is challenging). All of these issues can be remedied by combining JMD with a recently developed technique in which the transmit signals are transformed using a linear time-domain transform. Preliminary results are shown in [45] and will be detailed more fully in future work.

Proofs↩︎

7.1 Proof of↩︎

We start by noting that the Frobenius norm in 7 is nonnegative and evaluates to zero if and only if its argument is the all-zero matrix. That \(\{\hat{\mathbf{P}},\hat{\mathbf{S}}_D\} = \{\mathbf{I}_B - \mathbf{J}\mathbf{J}^{\dagger}, \mathbf{S}_D\}\) is a minimizer follows then directly from \[\begin{align} \!\!\hat{\mathbf{P}}(\mathbf{Y}\!_D - \mathbf{H}\hat{\mathbf{S}}_D) &= (\mathbf{I}_B - \mathbf{J}\mathbf{J}^{\dagger}) (\mathbf{H}\mathbf{S}_D + \mathbf{J}\mathbf{W}\!_D - \mathbf{H}\mathbf{S}_D)\!\! \\ &= (\mathbf{I}_B - \mathbf{J}\mathbf{J}^{\dagger})\mathbf{J}\mathbf{W}\!_D \\ &= \mathbf{0}, \end{align}\] since \((\mathbf{I}_B - \mathbf{J}\mathbf{J}^{\dagger})\mathbf{J}= \mathbf{0}\). It remains to show that \(\{\hat{\mathbf{P}},\hat{\mathbf{S}}_D\}\) is the only minimizer of 7 . For this, we rewrite the argument of the Frobenius norm in 7 as \[\begin{align} \tilde{\mathbf{P}}(\mathbf{H}\mathbf{S}_D + \mathbf{J}\mathbf{W}\!_D - \mathbf{H}\tilde{\mathbf{S}}_D) &= \tilde{\mathbf{P}}\begin{bmatrix}\mathbf{H},\mathbf{J}\end{bmatrix} \begin{bmatrix} \mathbf{S}_D - \tilde{\mathbf{S}}_D\\ \mathbf{W}\!_D \end{bmatrix}. \label{eq:data95objective95rewritten} \end{align}\tag{16}\] Since \(\tilde{\mathbf{P}}\in\mathscr{G}_{B-I}(\mathbb{C}^B)\), it can null a matrix only if that matrix has rank less than or equal to \(I\). But by assumption, the matrix \([\mathbf{H},\mathbf{J}]\) has full column rank \(U+I\). So its product with \([\mathbf{S}_D - \tilde{\mathbf{S}}_D; \mathbf{W}\!_D]\) can give a matrix whose rank does not exceed \(I\) only if the rank of \([\mathbf{S}_D - \tilde{\mathbf{S}}_D; \mathbf{W}\!_D]\) itself does not exceed \(I\). By our assumption that the jammer is not eclipsed, this can be the case only if \(\tilde{\mathbf{S}}_D = \mathbf{S}_D\), in which case 16 simplifies to \(\tilde{\mathbf{P}}\mathbf{J}\mathbf{W}\!_D\). Since the jammer is not eclipsed, the rank of \(\mathbf{W}\!_D\) itself is equal to \(I\). Hence \(\textit{col}(\mathbf{J}\mathbf{W}\!_D)=\textit{col}(\mathbf{J})\), so that \(\tilde{\mathbf{P}}\mathbf{J}\mathbf{W}\!_D=\mathbf{0}\) if and only if \(\tilde{\mathbf{P}}=\mathbf{I}_B - \mathbf{J}\mathbf{J}^{\dagger}\). \(\blacksquare\)

7.2 Proof of↩︎

Since signals from the pilot phase play no role for this result, it will be convenient to simply call \(\mathbf{w}_D\) the jammer transmit signal (instead of “the transmit signal during the data phase”).

We start by defining the coset \[\begin{align} \mathfrak{E}(\mathbf{S}_D) \triangleq \Big\{ \mathbf{E}(\tilde{\mathbf{S}}_D; \mathbf{S}_D)=\mathbf{S}_D - \tilde{\mathbf{S}}_D : \tilde{\mathbf{S}}_D \in \mathcal{S}^{U\times D}\!\setminus\!\{\mathbf{S}_D\} \Big\}. \end{align}\] Note that, by definition, the coset \(\mathfrak{E}(\mathbf{S}_D)\) does not include the all-zero matrix. We can now rewrite as follows:

Thus, the jammer eclipses if (i) \(\mathfrak{E}(\mathbf{S}_D)\) includes a matrix \(\mathbf{E}(\tilde{\mathbf{S}}_D; \mathbf{S}_D)\) whose rows are all collinear and (ii) \(\mathbf{w}_D^{\text{T}}\) is collinear with these rows. Without loss of generality, for all \(\mathbf{E}(\tilde{\mathbf{S}}_D; \mathbf{S}_D)\in\mathfrak{E}(\mathbf{S}_D)\), denote the \(u\)th row of \(\mathbf{E}(\tilde{\mathbf{S}}_D; \mathbf{S}_D)\) by \(\mathbf{e}_{(u)}(\tilde{\mathbf{s}}_{(u)};\mathbf{s}_{(u)})^{\text{T}}\), where \(\tilde{\mathbf{s}}_{(u)}^{\text{T}}\) and \(\mathbf{s}_{(u)}^{\text{T}}\) denote the corresponding rows of \(\tilde{\mathbf{S}}_D\) and \(\mathbf{S}_D\), respectively. We then define the cosets \[\begin{align} \mathfrak{e}_{(u)}(\mathbf{s}_{(u)}) \!\triangleq \! \big\{\mathbf{e}_{(u)}(\tilde{\mathbf{s}}_{(u)};\mathbf{s}_{(u)})\!=\tilde{\mathbf{s}}_{(u)}\!-\mathbf{s}_{(u)} : \tilde{\mathbf{s}}_{(u)}\!\!\in\!\mathcal{S}^D\!\setminus\!\{\mathbf{s}_{(u)}\} \!\big\} \end{align}\] for \(u=1,\dots,U\). Note that, by definition, \(\mathfrak{e}_{(u)}(\mathbf{s}_{(u)})\) does not include the all-zero vector. We have the following lemmas:

and give rise to the following corollary:

To complete the proof of , it now only remains to show that the probability \(q(\mathbf{w}_D)\) as defined in ?? is at most \[\begin{align} \frac{2^{\|\mathbf{w}_D\|_0}-1}{4^{\|\mathbf{w}_D\|_0-1}}. \end{align}\] For this, we define the concept of an eclipsing-optimal jammer.

now follows from and , where the approximation for large \(\|\mathbf{w}\|_0\) follows by approximating \(q(\mathbf{w}_D)\) as \(4\cdot2^{-\|\mathbf{w}\|_0}\) and then using a first-order Taylor approximation around \(q(\mathbf{w}_D)=0\). \(\blacksquare\)

7.3 Proof of↩︎

Using the theorem’s assumptions and 10 , we rewrite the argument of the optimization objective in 11 as \[\begin{align} & \tilde{\mathbf{P}}(\mathbf{Y}\!_D - \hat{\mathbf{H}}\tilde{\mathbf{S}}_D) \\ &= \tilde{\mathbf{P}}(\mathbf{H}\mathbf{S}_D + \mathbf{J}\mathbf{W}\!_D - (\mathbf{H}+ \mathbf{J}\mathbf{W}_T\mathbf{S}_T^{\dagger}) \tilde{\mathbf{S}}_D) \\ &= \tilde{\mathbf{P}}\begin{bmatrix}\mathbf{H},\mathbf{J}\end{bmatrix} \begin{bmatrix} \mathbf{S}_D - \tilde{\mathbf{S}}_D\\ \mathbf{W}\!_D - \mathbf{W}_T\mathbf{S}_T^{\dagger}\tilde{\mathbf{S}}_D \end{bmatrix} \end{align}\] From here on, the proof is very similar to the one of (considering the modified notion of eclipsing as defined in ), and so is omitted. \(\blacksquare\)

7.4 Proof of↩︎

Assume without loss of generality that the jammer knows the pilot sequence of the first UE, i.e., the first row of \(\mathbf{S}_T\), which we denote \(\mathbf{s}_{T,1}^{\text{T}}\). Then the jammer can transmit \(\mathbf{w}_T=\mathbf{s}_{T,1}\) and \(\mathbf{w}_D\in\mathcal{S}^D\). We therefore have \(\mathbf{w}_T^{\text{T}}\mathbf{S}_T^{\dagger}=[1,0,\dots,0]\) and so \[\begin{align} \mathbf{\Sigma}&= [\mathbf{S}_D-\tilde{\mathbf{S}}_D;\mathbf{w}_D^{\text{T}} - \mathbf{w}_T^{\text{T}}\mathbf{S}_T^{\dagger}\tilde{\mathbf{S}}_D] \\ &= [\mathbf{S}_D-\tilde{\mathbf{S}}_D;\mathbf{w}_D^{\text{T}} - \tilde{\mathbf{s}}_{D,1}^{\text{T}}], \end{align}\] where \(\tilde{\mathbf{s}}_{D,1}^{\text{T}}\) is the first row of \(\tilde{\mathbf{S}}_D\). In that case, consider for \(\tilde{\mathbf{S}}_D\) the matrix which is equal to \(\mathbf{S}_D\) on all rows except the first row, where it is equal to \(\mathbf{w}_D^{\text{T}}\). If \(\mathbf{w}_D^{\text{T}}\neq\mathbf{s}_{D,1}^{\text{T}}\), then \(\tilde{\mathbf{S}}_D\neq\mathbf{S}_D\) and \(\mathbf{\Sigma}\) is a matrix of rank one, meaning that the jammer eclipses.

Since \(\mathbf{s}_{D,1}^{\text{T}}\) is drawn uniform at random from \(\mathcal{S}\), the probability that \(\mathbf{w}_D^{\text{T}}\neq\mathbf{s}_{D,1}^{\text{T}}\) is equal to \(1-4^{-D}\). \(\blacksquare\)

7.5 Proof of↩︎

If the jammer does not jam during the pilot phase, \(\mathbf{w}_T=\mathbf{0}\), then eclipsing with channel estimation coincides with eclipsing with perfect CSI (cf. ), and the result follows from .

If the jammer does jam during the pilot phase, \(\mathbf{w}_T\neq\mathbf{0}\), then \(\mathbf{w}_T\) is independent of \(\mathbf{S}_T\) (since the jammer does not know \(\mathbf{S}_T\)). We now focus on the last row of \(\mathbf{\Sigma}\), which is \(\mathbf{w}_D^{\text{T}}-\mathbf{w}_T^{\text{T}}\mathbf{S}_T^{\dagger}\tilde{\mathbf{S}}_D\). Since \(\mathbf{S}_T\) is Haar distributed (and therefore unitary) up to a scale-factor, \(\mathbf{S}_T^{\dagger}=\frac{1}{T}\mathbf{S}_T^{\text{H}}\) is Haar distributed up to a scale-factor, too. Hence, \(\mathbf{x}^{\text{T}}\triangleq\mathbf{w}_T^{\text{T}}\mathbf{S}_T^{\dagger}\) is uniformly distributed over the complex \(U\)-dimensional sphere of radius \(\|\mathbf{w}_T\|_2/T\) [28]. We can therefore also write \(\mathbf{x}= \frac{\|\mathbf{w}_T\|_2}{T \|\mathbf{z}\|_2}\mathbf{z}\), where \(\mathbf{z}\sim\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{I}_U)\), and rewrite the last row of \(\mathbf{\Sigma}\) as \(\mathbf{w}_D^{\text{T}} - \frac{\|\mathbf{w}_T\|_2}{T \|\mathbf{z}\|_2}\mathbf{z}^{\text{T}}\tilde{\mathbf{S}}_D\). Here, \(\mathbf{z}^{\text{T}}\tilde{\mathbf{S}}_D\) is a (transposed) complex Gaussian vector with covariance matrix \(\tilde{\mathbf{S}}_D^{\text{H}}\tilde{\mathbf{S}}_D\), so that its entries are complex circularly-symmetric scalar Gaussians with variance \(U\) (corresponding to the energy of the columns of \(\tilde{\mathbf{S}}_D\)). Hence, the last row \(\mathbf{w}_D^{\text{T}}-\mathbf{w}_T^{\text{T}}\mathbf{S}_T^{\dagger}\tilde{\mathbf{S}}_D\) has full support \(D\) with probability one. Since this row does not depend on \(\mathbf{S}_D\), we can simply refer to , with the full-support (with probability one) vector \(\mathbf{w}_D^{\text{T}}-\mathbf{w}_T^{\text{T}}\mathbf{S}_T^{\dagger}\tilde{\mathbf{S}}_D\) in lieu of the potentially sparse vector \(\mathbf{w}_D^{\text{T}}\), and the result follows. \(\blacksquare\)

7.6 Proof of↩︎

This theorem was already proved for the special case of a single-antenna jammer in [15]. The extension to multi-antenna jammers is straightforward using arguments as in the proofs of and and so is omitted. \(\blacksquare\)

7.7 Proof of↩︎

We first prove convexity in \(\tilde{\mathbf{S}}_D\) for fixed \(\tilde{\mathbf{P}}\): For fixed \(\tilde{\mathbf{P}}\), the objective in 13 can be rewritten as \[\begin{align} \!\!\! \hat{f}(\tilde{\mathbf{S}}_D) \triangleq& \mathop{\mathrm{\mathbb{T} r}}\big(\mathbf{Y}\!_D^{\text{H}}\tilde{\mathbf{P}}\mathbf{Y}\!_D \nonumber \\ & + \tilde{\mathbf{S}}_D^{\text{H}}[\hat{\mathbf{H}}^{\text{H}}\tilde{\mathbf{P}}\hat{\mathbf{H}}-\alpha \mathbf{I}_U]\tilde{\mathbf{S}}_D - 2\Re\{\mathbf{Y}\!_D^{\text{H}}\tilde{\mathbf{P}}\hat{\mathbf{H}}\tilde{\mathbf{S}}_D\}\big).\!\! \label{eq:traceform} \end{align}\tag{19}\] \(\hat{f}\) is a real-valued function of the complex matrix \(\tilde{\mathbf{S}}_D\). Following [46], we represent the dependence on this complex input matrix by using both \(\tilde{\mathbf{S}}_D\) and \(\tilde{\mathbf{S}}_D^\ast\), giving rise to the four different Hessian matrices \(\mathcal{H}_{\tilde{\mathbf{S}}_D, \tilde{\mathbf{S}}_D^\ast} \hat{f}, \mathcal{H}_{\tilde{\mathbf{S}}_D^\ast, \tilde{\mathbf{S}}_D^\ast} \hat{f}, \mathcal{H}_{\tilde{\mathbf{S}}_D, \tilde{\mathbf{S}}_D} \hat{f}\), and \(\mathcal{H}_{\tilde{\mathbf{S}}_D^\ast, \tilde{\mathbf{S}}_D} \hat{f}\) of \(\hat{f}\). The objective is convex if the matrix \[\begin{align} \mathcal{H}\hat{f} = \begin{bmatrix} \mathcal{H}_{\tilde{\mathbf{S}}_D, \tilde{\mathbf{S}}_D^\ast} \hat{f} & \mathcal{H}_{\tilde{\mathbf{S}}_D^\ast, \tilde{\mathbf{S}}_D^\ast} \hat{f} \\ \mathcal{H}_{\tilde{\mathbf{S}}_D, \tilde{\mathbf{S}}_D} \hat{f}& \mathcal{H}_{\tilde{\mathbf{S}}_D^\ast, \tilde{\mathbf{S}}_D} \hat{f} \end{bmatrix} \end{align}\] is positive semidefinite. Recognizing that the second-order derivatives of the constant and affine terms in 19 are zero, and following [46], we obtain \[\begin{align} \mathcal{H}\hat{f} = \begin{bmatrix} [\hat{\mathbf{H}}^{\text{H}}\tilde{\mathbf{P}}\hat{\mathbf{H}}\!-\!\alpha \mathbf{I}_U]^{\text{T}}\mathop{\mathrm{\otimes}}\mathbf{I}_U & \mathbf{0} \\ \mathbf{0} & [\hat{\mathbf{H}}^{\text{H}}\tilde{\mathbf{P}}\hat{\mathbf{H}}\!-\!\alpha \mathbf{I}_U] \mathop{\mathrm{\otimes}}\mathbf{I}_U \end{bmatrix}. \!\! \end{align}\] \(\mathcal{H}\hat{f}\) is positive semidefinite if all its eigenvalues are non-negative. The eigenvalues of \(\mathcal{H}\hat{f}\) are simply the eigenvalues of \(\hat{\mathbf{H}}^{\text{H}}\tilde{\mathbf{P}}\hat{\mathbf{H}}\!-\!\alpha \mathbf{I}_U\), and they are all non-negative as long as \(\alpha\leq \lambda_\text{min}\), where \(\lambda_\text{min}\) is the smallest eigenvalue of \(\hat{\mathbf{H}}^{\text{H}}\tilde{\mathbf{P}}\hat{\mathbf{H}}\).

We now turn to the minimization with respect to \(\tilde{\mathbf{P}}\), and define \(\mathbf{E}\triangleq \mathbf{Y}\!_D - \hat{\mathbf{H}}\tilde{\mathbf{S}}_D\). Note that any \(\tilde{\mathbf{P}}\in \mathscr{G}_{B-I}(\mathbb{C}^B)\) can be written as \(\tilde{\mathbf{P}}= \mathbf{I}_B - \mathbf{Q}\mathbf{Q}^{\text{H}}\), where \(\mathbf{Q}\in\mathbb{C}^{B\times I}\) consists of \(I\) orthonormal columns. The second term of 13 does not depend on \(\tilde{\mathbf{P}}\), and since the squaring of the norm does not affect the minimizing argument (which is what we are interested in), it suffices to consider \[\begin{align} \min_{\tilde{\mathbf{P}}\,\in\, \mathscr{G}_{B-I}(\mathbb{C}^B)} \| \tilde{\mathbf{P}}\mathbf{E}\|_F &= \min_{\mathbf{Q}} \| \mathbf{E}- \mathbf{Q}\mathbf{Q}^{\text{H}}\mathbf{E}\|_F \tag{20} \\ &\geq \min_{\tilde{\mathbf{E}}\,:\,\mathop{\mathrm{rank}}{\tilde{\mathbf{E}}}\leq I} \|\mathbf{E}- \tilde{\mathbf{E}}\|_F. \tag{21} \end{align}\] Here, 21 follows since the matrix \(\mathbf{Q}\mathbf{Q}^{\text{H}}\mathbf{E}\) has at most rank \(I\) (since \(\mathbf{Q}\) has dimensions \(B\times I\)), so the optimization range in 20 is a subset of the optimization range in 21 . By the Eckart-Young-Mirksy theorem, 21 is minimized for \(\tilde{\mathbf{E}}= \mathbf{U}_I \boldsymbol{\Sigma}_I \mathbf{V}_I^{\text{H}}\), where \(\boldsymbol{\Sigma}_I = \text{diag}(\sigma_1, ..., \sigma_I)\) is the diagonal matrix whose diagonal entries are the \(I\) largest singular values of \(\mathbf{E}\), and \(\mathbf{U}_I\) and \(\mathbf{V}_I\) consist of the corresponding left- and right-singular vectors, respectively [47]. But if we let \(\mathbf{E}=\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{\text{H}}\) be the singular-value decomposition of \(\mathbf{E}\) and choose \(\mathbf{Q}=\mathbf{U}_I\), then \(\mathbf{Q}\mathbf{Q}^{\text{H}}\mathbf{E}= \mathbf{U}_I \boldsymbol{\Sigma}_I \mathbf{V}_I^{\text{H}}\), meaning that 21 holds with equality and that 20 is minimized for \(\mathbf{P}=\mathbf{I}_B - \mathbf{U}_I \mathbf{U}_I^{\text{H}}\). \(\blacksquare\)

References↩︎

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1. A short version of this paper has been presented at IEEE ICC 2023 [1].↩︎

2. The work of GM and CS was supported in part by an ETH Research Grant.↩︎

3. GM and CS are with the Department of Information Technology and Electrical Engineering, ETH Zurich, Switzerland; e-mail: gimarti@ethz.ch and studer@ethz.ch↩︎

4. Another hard-to-mitigate threat is posed by multiple single-antenna distributed jammers, which also cause high-rank interference [20], [21].↩︎

5. In particular, this insight has enabled the first application-specific integrated circuit (ASIC) implementation of an algorithm that mitigates smart jammers in multi-user MIMO [22].↩︎

6. Our methods are also translatable to other MIMO contexts. For example, an extension to MIMO-OFDM in frequency-selective channels is possible but not straightforward, see [23].↩︎

7. The model in 1 can also represent distributed single- or multi-antenna jammers with a total of \(I\) antennas. We consider this case in .↩︎

8. For the moment, we neglect estimation of the UE channel matrix \(\mathbf{H}\) and assume perfect channel knowledge. Channel estimation is considered in . In this section, signals from the pilot phase 4 are therefore ignored.↩︎

9. Also, it is evident from our proofs that the combinatorics involved in the analysis are quite involved already for single-antenna jammers, and we expect them to significantly increase in difficulty for multi-antenna jammers.↩︎

10. This definition of eclipsing coincides with [15] for \(I\!=\!1\) and so can be seen as a generalization of the concept of eclipsing to multi-antenna jammers.↩︎

11. As in , we can characterize this probability analytically for the single-antenna jammer case, and we refer to the simulations of for showing the efficacy against multi-antenna jammers empirically.↩︎

12. Note that the following result is much stronger than [15], which assumes that both \(\mathbf{w}_D\) and \(\mathbf{w}_T\) are distinct from zero, and which gives a nontrivial bound only if the number of UEs satisfies \(U>3\).↩︎

13. Because of the projection \(\tilde{\mathbf{P}}\) in 12 , the optimal value \(\hat{\mathbf{H}}\) of \(\tilde{\mathbf{H}}\) itself is not uniquely determined—only its composition with the projection \(\hat{\mathbf{P}}\) is.↩︎

14. This relaxation, which allows us to use gradient-based methods for optimization, is only temporary—ultimately, entrywise rounding has to be used for converting a penultimate estimate \(\tilde{\mathbf{S}}_D\in\mathcal{C}^{U\times D}\) to one that is contained in \(\mathcal{S}^{U\times D}\). For simplicity, our algorithms omit this detail.↩︎

15. Apart from the number of jammer antennas, the MAED version provided in [15] differs slightly from the one provided in this paper. The conference version of this paper [1] has therefore referred to the newer version of MAED as “MAED 2.0.” For simplicity, however, we have decided to return to the original name “MAED,” since these minor differences between the algorithms do not seem to justify a new name.↩︎

16. Spreading the training period over the frame is the most sensible strategy against dynamic multi-antenna jammers as considered in , so this gives the strongest possible baseline.↩︎

17. Since the jammer is perfectly nulled using ground-truth knowledge, there is no residual jamming interference. However, each jammer antenna entails the loss of one degree of freedom after nulling, see [44].↩︎

18. POS-BOX and POS-JED often perform even worse than the unmitigated method (see, e.g., the pilot jammer ). This is due to the fact that the nonlinear data detectors of POS-BOX and POS-JED try to fit the receive signal with a signal model that is accurate only if the jammer subspace is correctly estimated, and which otherwise can fail catastrophically. In contrast, the linear detection of the unmitigated baseline is less susceptible to model mismatches.↩︎

19. This is also the reason why 18 does not hold for \(\mathbf{e}=\boldsymbol{0}\): the right-hand-side is equal to \(1\), but the left-hand-side is equal to \(0\), cf. ?? .↩︎