Capacity of Gaussian
Arbitrarily-Varying Fading Channels

It is a matter of great importance to study wireless communication channels since they include numerous challenges caused by noise and fading. The wireless environment also allows any uninvited signal to enter the channel. These signals can act as interference or in the worse case as a malicious jammer whose aim is to disrupt the communication between the legitimate users. In this work, we explore how these various signals interact with each other to restrict the overall capacity of the channel.

Goldsmith and Varaiya in [2] studied the point-to-point Gaussian channel with fast fading in which the fading coefficients are drawn i.i.d. from a distribution known to all parties. They determined the capacity of the channel where the fading coefficients are available at the receiver and possibly the transmitter. When the fading coefficients are not available at the transmitter, the capacity is equal to the expected value of the capacity of the corresponding Gaussian channel with the received signal-to-noise ratio affected by the fading gains. If both the transmitter and receiver know the exact channel gains, the transmitter can maximize the capacity by constructing its signal power as a function of the channel gains.

Another line of work studies channels in the presence of a malicious adversary. The adversary can be an active attacker who sends its signal to the channel in order to cease or restrict the communication between the legitimate users. If the adversary’s signal is arbitrarily chosen or is drawn from an unknown distribution, then the channel is called as an arbitrary-varying channel (AVC). We focus on the variant of the problem wherein the adversary’s knowledge is limited to the code of the legitimate user, but it does not have access to the user’s transmitted messages, neither exact nor noisy. Csiszár and Narayan established the capacity of discrete AVC with input and jammer power constraints in [3, 4]. They also derived the capacity for a continuous version of the AVC with input and state (jammer) power constraints in [5].

Csiszár and Narayan in [5] characterized the capacity of the Gaussian arbitrarily-varying channel under the average probability of error and the average power constraints for input and state. It is assumed that the adversary does not have any information about the legitimate signal except the code. It is shown that if the adversary has enough power to forge a message and send it to the channel, then the receiver gets confused and cannot distinguish between the true message and the malicious one. This occurrence is called symmetrizability , causing the capacity to drop to zero. In [5], it is shown that the adversary can symmetrize the channel if and only if it has greater power that the legitimate transmitter. However, if the jammer does not have enough power, then the capacity is equal to the capacity of a standard Gaussian channel with the noise variance increased by the power of the jammer.

The problem of AVC capacity has been also studied for discrete and continuous multi-user channels. The authors in [6] considered the discrete arbitrarily-varying multiple-access memoryless channel. We characterized lower and upper bounds for the capacity of arbitrarily-varying Gaussian interference channel in [7]. List-decoding is also investigated for the discrete and Gaussian AVCs in [8, 9] and in [10], respectively. The list capacity is derived by using list-decoding which decodes a list of messages instead of unique message at the receiver.

The three elements of Gaussian noise, fading, and adversary have been previously combined to study problems with a (passive) eavesdropping adversary, rather than an (active) transmitting adversary. A secrecy capacity problem with slow fading, in which the fading gains are constant over each block length, is considered in [11]. The authors determined the secrecy capacity of the channel where both of transmitter and receiver know the channel state information (CSI) of the main path, but do not have any information about the eavesdropper channel. Furthermore, the capacity is generalized to multiple eavesdroppers in [12]. The problem is also studied in [13] for a specific case of a fast Rayleigh fading eavesdropper channel and a standard Gaussian main channel where the CSI are only known to the eavesdropper.

In this paper, we consider a Gaussian AVC with fast fading on the main path, as illustrated in Fig. 1; we refer to this channel as the Gaussian arbitrarily-varying fading channel (GAVFC). We characterize the capacity of the GAVFC under the average probability of error criterion. Similar to the Gaussian fading channel, we also assume that everyone knows the fading gain distribution including the adversary, but they may or may not know the realization of the gain sequence. Note that the “arbitrarily-varying” aspect of the channel is the adversary’s signal, not the channel gains, which we assume to be random from a known distribution. The receiver always needs the exact fading gains to decode the message, while the adversary and the transmitter may or may not know the exact values of fading gains. Therefore, we derive the capacity of the GAVFC for four cases wherein the channel gains are available at the transmitter and/or adversary as follows:

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  Neither the transmitter nor the adversary knows the channel gains.

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  Only the transmitter knows the channel gains.

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  Only the adversary knows the channel gains.

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  Both the transmitter and the adversary knows the channel gains.

If the jammer does not know the channel gains, we show that the capacity is equal to the capacity of the corresponding fading channel with increased noise variance by the power of the jammer. If the jammer knows the fading gains, then it can choose its signal as a function of the gains, and under some power constraints it can symmetrize the channel and make the capacity zero. Note that if the channel gains are not available at the adversary, it does not have the required channel information to symmetrize the channel. Moreover, all the results still hold if the adversary and the encoder have the channel gains causally or non-causally, except one situation. If the adversary knows the channel gains causally while the encoder knows them non-causally, then the adversary cannot symmetrize the channel since the encoder possesses some extra information that the adversary does not.

The rest of the paper is organized as follows. We describe the GAVFC model and define the various capacities in Sec. II. We state our main theorem including the capacity of the GAVFC in all cases in Sec. III. Before giving the proof of our main theorem, we need some auxiliary lemmas and tools which are presented in Sec. IV. Later, in sections V, VI, VII, and VIII, we provide the converse and achievability proofs of each of the main results in Theorem 1. Finally in the Appendix, we provide a brief proof for the auxiliary results.

Notation: We use bold letters to indicate $n$-length vectors. We employ $\langle\cdot,\cdot\rangle$ and $\circ$ to denote inner product and Hadamard product (element-wise multiplication), respectively. We indicate the positive-part function, 2-norm and the expectation by $|\cdot|^{+}$, $\|\cdot\|$ and $\mathbb{E}[\cdot]$, respectively. Also, for an integer $N$, $[N]$ stands for the set $\{1,2,3,\ldots,N\}$. Notation $\mathbf{I}_{n}$ represents the identity matrix of size $n$. Each of $\log(\cdot)$ and $\exp(\cdot)$ functions has base 2. Moreover, $C(x)=\frac{1}{2}\log(1+x)$, and $X\sim\mathcal{N}(\mu,\sigma^{2})$ denotes Gaussian random variable $X$ with mean $\mu$ and variance $\sigma^{2}$.

The Gaussian arbitrarily-varying fading channel (GAVFC) in Fig. 1 is a point-to-point fading channel with additive Gaussian noise and an intelligent adversary who does not have any information about the transmitted signal except the code. The received signal is given by

where $\mathbf{G}$ is a random sequence of identical and independently distributed (i.i.d.) fast fading channel gains from the legitimate transmitter to the receiver drawn from continuous distribution $f_{G}(g)$ assumed to have positive and finite variance, $\mathbf{x}$ is the $n$-length deterministic vector representing the user’s signal, $\mathbf{s}$ is the adversary signal chosen arbitrarily, and $\mathbf{V}$ is a random $n$-length noise vector distributed as a sequence of i.i.d. zero mean Gaussian random variables with variance $\sigma^{2}$, independent of $\mathbf{x}$, $\mathbf{G}$ and $\mathbf{s}$. Note that the receiver always knows the exact fading coefficients $\mathbf{g}$ while the transmitter and the adversary may not know the gains, know them causally, or know them non-causally.

Define an $\left(N,n\right)$ code for the GAVFC by a message set, an encoding function and a decoding function as follows:

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  Message set $\mathcal{M}=[N]$,

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  Encoding function (one of the following)

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    (No knowledge) $\mathbf{x}(m):\mathcal{M}\to\mathbb{R}^{n}$ where $\mathbf{x}=(x_{1},\ldots,x_{n})$,

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    (Causal) $x_{i}(m,\mathbf{g}^{i})\!\!:\!\mathcal{M}\times\mathbb{R}^{i}\to\mathbb{R}$ where $\mathbf{g}^{i}\!=\!(g_{1},\ldots,g_{i})$ and $\mathbf{x}\!=\!(x_{1},\ldots,x_{n})$ for $i\in[n]$,

  • –

    (Non-causal) $x_{i}(m,\mathbf{g})\!\!:\!\mathcal{M}\times\mathbb{R}^{n}\!\to\!\mathbb{R}$ where $\mathbf{g}\!=\!(g_{1},\ldots,g_{n})$ and $\mathbf{x}\!=\!(x_{1},\ldots,x_{n})$ for $i\!~{}\in~{}[n]$,

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  Decoding function $\Theta(\mathbf{y},\mathbf{g}):\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathcal{M}$,

where the rate of the code is $R=\frac{1}{n}\log(N)$. The message $m$ is drawn uniformly from the set $\mathcal{M}$. If the encoder does not know the channel gains, it maps the message to $\mathbf{x}(m)\in\mathbb{R}^{n}$. If the encoder knows the channel gains causally, then it maps the message to $x_{i}(m,\mathbf{g}^{i})\in\mathbb{R}$, and if the encoder knows the channel gains non-causally, then it maps the message to $x_{i}(m,\mathbf{g})\in\mathbb{R}$ where $\mathbf{x}=(x_{1},\ldots,x_{n})$. Given channel gains $\mathbf{g}$ at the receiver, the signal $\mathbf{y}$ is decoded by function $\Theta(\mathbf{y},\mathbf{g})$ to the message $\hat{m}$. Moreover, we assume that if the channel gains are available at the transmitter then the transmitter’s signal satisfies the expected power constraints $\mathbb{E}\left[\|\mathbf{X}(m,\mathbf{G})\|^{2}\right]\leq nP$ for any message $m\in\mathcal{M}$. Otherwise, the power constraint is $\|\mathbf{x}(m)\|^{2}\leq nP$. The same definition applies to the adversary’s signal power constraint, i.e. if the adversary knows the channel gains, the constraint is $\mathbb{E}\left[\|\mathbf{S}(\mathbf{G})\|^{2}\right]\leq n\Lambda$; otherwise, it is $\|\mathbf{s}\|^{2}\leq n\Lambda$. The three parameters $P$, $\Lambda$, and $\sigma^{2}$ as well as the distribution of fading gains $f_{G}(g)$ are known to all parties.

The probability of error $e(\mathbf{s},m)$ for the message $m\in\mathcal{M}$ in the presence of adversary signal $\mathbf{s}\in\mathbb{R}^{n}$ is now given by the probability that $\hat{m}\neq m$. Thus, the average probability of error for a specific $\mathbf{s}\in\mathbb{R}^{n}$ is

If the adversary knows the channel gains non-causally then his signal is denoted by $s_{i}(\mathbf{g})$ for $i\in[n]$. Alternatively, if the adversary knows the gains causally, then the adversary’s action is given by functions $s_{i}(\mathbf{g}^{i})$ for $i\in[n]$ where $\mathbf{s}=(s_{1},\cdots,s_{n})$ and $\mathbf{g}^{i}=(g_{1},\cdots,g_{i})$. Therefore, the average probability of error for this specific $\mathbf{s}(G)\in\mathbb{R}^{n}$ is

Finally, the overall probability of error $P_{e}^{(n)}$ is maximized over all possible choices of jammers’ sequences $\mathbf{s}$ which satisfy either $\mathbb{E}\left[\|\mathbf{S}\|^{2}\right]\leq n\Lambda$ or $\|\mathbf{s}\|^{2}\leq n\Lambda$. Rate $R$ is achievable if there exists a sequence of $\left(2^{nR},n\right)$ codes where $\underset{n\rightarrow\infty}{\lim}P_{e}^{(n)}=0$. The capacity is the supremum of all achievable rates. We denote the capacity of the GAVFC as $C_{\alpha,\beta}$ where $\alpha$ denotes the transmitter’s knowledge, and $\beta$ denotes the adversary’s knowledge; $\alpha$ and $\beta$ can be U, C, or N depending on whether the transmitter or adversary does not know the gains (U $=$ unknown), knows the gains causally (C), or knows the gains non-causally (N). For example, $C_{\text{U,N}}$ is the capacity where the transmitter does not know the gains and the adversary knows the gains non-causally.

We present our results for the capacity of GAVFC whether the fading channel gains $\mathbf{G}$ are available causally or non-causally at the encoder and/or the adversary (the decoder always knows the gains) in the following theorems.

Note that when the encoder knows the gains (5), (7), and (8), the capacity expression includes a maximization of the input power as a function $\varphi(\cdot)$ of the gain, similar to the result in [2]. Similarly, when the jammer knows the gains (6)–(8), the capacity expression includes a minimization that represents the jammer’s choice of noise power as a function $\psi(\cdot)$ of the gain. Moreover, when the jammer knows the gains, with enough power it can symmetrize the channel by mimicking the legitimate signal, thus reducing the capacity to zero. However, in (8) we have assumed that the adversary knows the gains causally and the encoder and the decoder know the gains non-causally. Thus, the encoder and decoder effectively share a secret (the channel gains at the end of the block) unknown to the adversary, so the adversary cannot symmetrize the channel. It is also worth mentioning that for the other cases (except (8)) our proof works exactly the same whether the transmitter and/or the adversary know the gain sequence causally, non-causally, or even memorylessly (i.e., at time $i$, you only know the gain value at time $i$).

While we have stated the theorem by writing the capacities in terms of optimization over the $\varphi$ and/or $\psi$ functions, these expressions can be computed by solving for the optimizing functions. In particular, the optimum value of $\varphi^{*}(g)$ in (5) is $\left|\lambda-\frac{\Lambda+\sigma^{2}}{g^{2}}\right|^{+}$ where $\lambda$ is obtained by $\mathbb{E}[\varphi^{*}(G)]=P$, and the optimum value of $\psi^{*}(\cdot)$ in (6) is a function of gain $g$ as follows

and $\lambda$ is obtained by solving $\mathbb{E}\psi^{*}(G)=\Lambda$. Moreover, the optimum values of $\varphi^{*}(g)$ and $\psi^{*}(g)$ in (7) are

where $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ are found by solving $\mathbb{E}\varphi(G)=P$, $\mathbb{E}\psi(G)=\Lambda$ and $\mathbb{E}G^{2}\varphi(G)=\Lambda$, respectively. Finally, the optimum values of $\varphi^{*}(g)$ and $\psi^{*}(g)$ in (8) are

where $\lambda_{1}$ and $\lambda_{2}$ can be obtained by solving $\mathbb{E}\varphi(G)=P$ and $\mathbb{E}\psi(G)=\Lambda$, respectively.

In the Fig. 2, the capacity of GAVFC with Rayleigh fading is shown for $P=1,\sigma^{2}=0.25,0<\Lambda<5$ and the Rayleigh distribution scale parameter $\sigma_{R}=1$ whether the channel gains are available at the encoder and/or the adversary. However, $C$ is the capacity of Gaussian arbitrary-varying channel with no fading. Note that when the transmitter’s knowledge increases, the capacity increases, whereas when the adversary’s knowledge increases, the capacity decreases. On the other hand, the knowledge of adversary about the channel gains may decrease the capacity, and in this case if the adversary’s power exceeds $2$, the capacity will be zero by the symmetrizability.

The proofs of the different capacity variants all follow a similar pattern, so we have attempted to reduce the redundancy in our presentation. We provide essentially one converse proof for each capacity expression in the theorem:

1. 1.

   Sec. V-A: Converse for $C_{U,U}$.

2. 2.

   Sec. VI-A: Converse for $C_{N,U}$. This also bounds $C_{C,U}$ which is trivially upper bounded by $C_{N,U}$.

3. 3.

   Sec. VII-A: Converse for $C_{U,C}$. This also bounds $C_{U,N}$ which is trivially upper bounded by $C_{U,C}$.

4. 4.

   Sec. VIII-A: Converse for $C_{C,C}$. This also bounds $C_{C,N}$ which is trivially upper bounded by $C_{C,C}$. Essentially the same proof also works for $C_{N,N}$. The case $C_{N,C}$ requires a different proof also covered in this section.

We provide essentially three achievability proofs:

1. 1.

   Sec. VI-B: Achievability for $C_{C,U}$. This also bounds the $C_{N,U}$, which is trivially lower bounded by $C_{C,N}$. It also provides a bound for $C_{U,U}$, because the same proof works assuming $\varphi(g)=P$ (i.e., the encoder’s power is independent of the channel gain).

2. 2.

   Sec. VIII-B: Achievability for $C_{C,N}$. This also bounds $C_{N,N}$ and $C_{C,C}$, which are trivially lower bounded by $C_{C,N}$. It also provides a bound for $C_{U,N}$ and $C_{U,C}$, because the same proof works again assuming $\varphi(g)=P$.

3. 3.

   Sec. VIII-C: Achievability for $C_{N,C}$. This case is different from all others in that there is effectively a shared secret between encoder and decoder.

Before proceeding to the proofs, we first define the typical set for continuous random variables $X_{1},\ldots,X_{k}$ with probability density function $f_{X_{1},\ldots,X_{k}}(x_{1},\ldots,x_{k})$ as follows:

where $h(X_{A})$ is the differential entropy of $(X_{i}:i\in A)$. Next, we define the typical set for continuous random variables $X_{1},\ldots,X_{k}$ with probability density function $f_{X_{1},\ldots,X_{k}}(x_{1},\ldots,x_{k})$ and a discrete random variable $\tilde{G}$ with probability mass function $P_{\tilde{G}(\tilde{g})}$ as follows:

where $H(\tilde{G})$ and $h(X_{A}|\tilde{G})$ denote the entropy of $G$ and the conditional differential entropy of $X_{A}$ given $\tilde{G}$.

Throughout the achievability proofs, we will utilize several lemmas including the joint typicality lemma and conditional typicality lemma for Gaussian random variables given in [10]. In addition, we will need the following two lemmas; they show that with high probability a Gaussian codebook satisfies several desirable properties. The proof is given in the Appendix.