Over-the-Air Computation over Balanced Numerals

Abstract

In this study, we propose a new approach to compute the majority vote (MV) function based on modulation on conjugate-reciprocal zeros (MOCZ) and introduce three different methods. The proposed methods rely on the fact that when a linear combination of polynomials is evaluated at one of the roots of a polynomial in the combination, that polynomial does contribute to the evaluation. To utilize this property, each transmitter maps the votes to the zeros of a Huffman polynomial, and the corresponding polynomial coefficients are transmitted. The receiver evaluates the polynomial constructed by the elements of the superposed sequence at conjugate-reciprocal zero pairs and detects the MV with a direct zero-testing (DiZeT) decoder. With differential and index-based encoders, we eliminate the need for power-delay information at the receiver while improving the computation error rate (CER) performance. The proposed methods do not use instantaneous channel state information at the transmitters and receiver. Thus, they provide robustness against phase and time synchronization errors. We theoretically analyze the CERs of the proposed methods. Finally, we demonstrate their efficacy in a distributed median computation scenario in a fading channel.

Federated Learning Over-the-Air by Retransmissions

“…For instance, one could consider the possibility of a distributed channel code. With analog communication, this appears to be inapplicable, but with recent ideas of one-bit digital Over-the-Air Computation, there might be possibilities to explore in this direction [44], [45]. Additionally, one could consider combining the retransmission scheme with device selection for further improvements, as suggested in [46].…”

Computing Functions Over-the-Air Using Digital Modulations

“…This is because we only map the input and output using the modulation vectors obtained from Problem P 4 . The main complexity comes from solving the optimization in (9). This optimization must be done before setting up the communication system.…”

“…We evaluate the performance of Problem P 4 for different functions and different numbers of nodes. In the next subsection, we compare the performance between ChannelComp, a standard digital transmission scheme, OFDMA, in which the modulation vector is obtained from (9), and each node uses different frequency channels, and with AirComp [26], which uses analog modulation. We compare these schemes in cases where the input values of function f are continuous and discrete.…”

“…To evaluate the performance of ChannelComp, we solve the relaxed optimization problem in (9) where the channel is considered ideal and noiseless. This experiment is repeated for four functions, i.e., summation, product, maximum, and quadratic, which are symmetric functions, with a quantization level q " 8 for each node.…”

“…However, AirComp entirely relies on analog communication, which is difficult for reliable communications due to channel implications [9]. In addition, AirComp requires analog hardware systems for deploying analog modulation, which is a drawback due to the limited number of current wireless devices that support analog modulations.…”