Finite Blocklength Covert Communication over Quasi-Static Multiple-Antenna Fading Channels

Abstract

The white book released by the International Telecommunications Union (ITU) calls for extremely high-security and low-latency communication over fading channels. Under the low-latency requirement, the corresponding fading model is quasi-static fading while high-security can be achieved via covert communication. In response to the call of ITU, we study the finite blocklength performance of optimal codes for covert communication over quasi-static multi-antenna fading channels, under the covertness metric of Kullback-Leibler (KL) divergence. In particular, we study all four cases regarding the availability of channel state information (CSI) for legitimate transmitter and receiver, and assume that the warden knows perfect CSI for the channel from the legitimate transmitter to itself. Specifically, we show that, when the blocklength is $n$, the first-order covert rate satisfies the square root law, scaling as $Θ(n^{-\frac{1}{2}})$ with the coefficient determined by the traces of the channel matrices of the legitimate users and the warden, and the second-order rate vanishes. In contrast to the non-covert result of Yang et al. (TIT, 2014), we show that CSI availability at the legitimate users does not affect the finite blocklength performance for covert communication. Furthermore, we reveal the significant spatial diversity gain provided by multiple-antenna systems for covert communication. For the covertness analysis, we extend the quasi-$η$-neighborhood framework to fading channels and address challenges arising from the random channel matrices. For the reliability analysis, due to the vanishing power imposed by the covertness constraint, we refine the non-covert analysis by Yang et al. (TIT, 2014), by carefully controlling higher-order terms and exploiting the properties of covert outage probability.

Figures (7)

• Figure 1: System model for covert communication over quasi-static MIMO fading channels.
• Figure 2: Plot of the empirical CDF of the maximal singular value of a $2\times 2$ MIMO Rayleigh fading channel and a Rician fading channel with $K=10$, obtained via $10^6$ independent Monte–Carlo trials. The vertical markers indicate the spectral norm levels beyond which the tail probabilities fall below $10^{-6}$. This confirms that a reasonable upper bound on the channel gain exists for typical quasi-static fading channels.
• Figure 3: Plot of first-order asymptotics $R_1(n,\varepsilon,\delta)$ as a function of blocklength $n$ for a $2\times 2$ MIMO Rician fading channel with different $\bm{\lambda}_0$, where $\varepsilon=0.01$, $\delta=0.1$ and $\mathbf{H}_\mathrm{b}$ is as in Example \ref{['example:rician fading']}, based on $10^6$ independent Monte–Carlo trials. As observed, $R_1(n,\varepsilon,\delta)$ decreases as $\sqrt{\lambda_0}$ increases, reflecting the inverse relationship between the covert transmission rate and Willie’s admissible SNR. This behavior is also consistent with the square root law.
• Figure 4: Plot of first-order covert rate $\sqrt{n}R_1(n,\varepsilon,\delta)$ as a function of number of antennas $N_\mathrm{a}$ for MIMO Rician fading channels, where $n=1000$, $\varepsilon=0.01$, $\delta=0.1$, $N_\mathrm{b}=N_\mathrm{w}=N_\mathrm{a}$, $\bm{\lambda}_0=\mathbf{I}_{N_\mathrm{a}}$ and $\mathbf{H}_\mathrm{b}$ is as in Example \ref{['example:rician fading']}, based on $10^6$ independent Monte–Carlo trials. Increasing the number of transmit antennas significantly improves the covert transmission rate. Specifically, when $N_\mathrm{a}=16$, compared with the single antenna case of $N_\mathrm{a}=1$, $\sqrt{n}R_1(n,\varepsilon,\delta)$ is improved by a factor of $\frac{3.23}{0.21}\approx 15.4$; when $N_\mathrm{a}=64$, the improvement factor increases to $\frac{10.43}{0.21}\approx 49.7$.
• Figure 5: Plot of non-asymptotic achievability and converse bounds for $\sqrt{n}R^*(n,\varepsilon,\delta)$ as a function of blocklength $n$ for a $2\times 2$ MIMO Rician fading channel, where $\varepsilon=0.01$, $\delta=0.1$, $\bm{\lambda}_0=\mathbf{I}_2$ and $\mathbf{H}_\mathrm{b}$ is as in Example \ref{['example:rician fading']}, based on $10^7$ independent Monte–Carlo trials. This setting yields the first-order covert rate $\sqrt{n}R_1(n,\varepsilon,\delta) = 0.53$. For comparison, the approximation for $\sqrt{n}R^*(n,\varepsilon,\delta)$ corresponding to a MIMO AWGN channel is also shown, which is reported in liu2026covertMIMO with $a=0.53$ in \ref{['eq:theoretical benchmark awgn']}. As the blocklength grows to $5000$, the achievability bound on $\sqrt{n}R^*(n,\varepsilon,\delta)$ for the quasi-static MIMO fading channel can reach $\frac{0.38}{0.53}\approx 71.7\%$ of $\sqrt{n}R_1(n,\varepsilon,\delta)$, whereas that for the MIMO AWGN channel only achieves $\frac{0.24}{0.53}\approx 45.3\%$. This indicates that $R^*(n,\varepsilon,\delta)$ converges rapidly to $R_1(n,\varepsilon,\delta)$ under quasi-static fading.

Theorems & Definitions (19)

• Example 1: Rayleigh Fading
• Example 2: Rician Fading
• Example 3: Nakagami Fading
• Definition 1
• Theorem 1
• Definition 2
• Example 4: Angle Threshold Decoding
• Lemma 2
• Lemma 3
• Lemma 4