Over-the-Air Computation Systems: Optimization, Analysis and Scaling Laws

TL;DR

This paper considers an optimization problem to minimize the computation mean-squared error of the K sensors’ signals at the receiver by optimizing the transmitting-receiving (Tx-Rx) policy, under the peak power constraint of each sensor.

Abstract

For future Internet of Things (IoT)-based Big Data applications (e.g., smart cities/transportation), wireless data collection from ubiquitous massive smart sensors with limited spectrum bandwidth is very challenging. On the other hand, to interpret the meaning behind the collected data, it is also challenging for edge fusion centers running computing tasks over large data sets with limited computation capacity. To tackle these challenges, by exploiting the superposition property of a multiple-access channel and the functional decomposition properties, the recently proposed technique, over-the-air computation (AirComp), enables an effective joint data collection and computation from concurrent sensor transmissions. In this paper, we focus on a single-antenna AirComp system consisting of $K$ sensors and one receiver (i.e., the fusion center). We consider an optimization problem to minimize the computation mean-squared error (MSE) of the $K$ sensors' signals at the receiver by optimizing the transmitting-receiving (Tx-Rx) policy, under the peak power constraint of each sensor. Although the problem is not convex, we derive the computation-optimal policy in closed form. Also, we comprehensively investigate the ergodic performance of AirComp systems in terms of the average computation MSE and the average power consumption under Rayleigh fading channels with different Tx-Rx policies. For the computation-optimal policy, we prove that its average computation MSE has a decay rate of $O(1/\sqrt{K})$, and our numerical results illustrate that the policy also has a vanishing average power consumption with the increasing $K$, which jointly show the computation effectiveness and the energy efficiency of the policy with a large number of sensors.

Figures (9)

• Figure 1: Illustration of the AirComp system.
• Figure 2: Achievable MSE region of a two-sensor MAC system, where the red broken line is the Pareto front of the sum-of-MSE minimization problem, and the dashed diagonal line indicates that the sum of MSE is larger than $(K-1)=1$, which can be obtained from \ref{['region1']} and \ref{['region2']}, and the horizontal and vertical dashed lines indicate the outer boundaries of the achievable MSE region, i.e., $\mathsf{MSE}_k \leq 1$.
• Figure 3: The average critical number of the computation-optimal policy versus the number of sensors.
• Figure 4: The average computation MSE versus $K$.
• Figure 5: The standard deviation of $\mathsf{MSE}/K$ versus $K$.

Theorems & Definitions (39)

• Lemma 1a
• Remark 1
• Lemma 1b
• Lemma 1c
• proof
• Corollary 1
• Lemma 2a: Switching structure
• proof
• Lemma 2b: Consistency
• Lemma 2c: Monotonicity