Analog Computing with Hybrid Couplers and Phase Shifters

Analog computing with microwave signals can enable exceptionally fast computations, potentially surpassing the limits of conventional digital computing. For example, by letting some input signals propagate through a linear microwave network and reading the corresponding output signals, we can instantly compute a matrix-vector product without any digital operations. In this paper, we investigate the computational capabilities of linear microwave networks made exclusively of two low-cost and fundamental components: hybrid couplers and phase shifters, which are both implementable in microstrip. We derive a sufficient and necessary condition characterizing the class of linear transformations that can be computed in the analog domain using these two components. Within this class, we identify three transformations of particular relevance to signal processing, namely the discrete Fourier transform (DFT), the Hadamard transform, and the Haar transform. For each of these, we provide a systematic design method to construct networks of hybrid couplers and phase shifters capable of computing the transformation for any size power of two. To validate our theoretical results, a hardware prototype was designed and fabricated, integrating hybrid couplers and phase shifters to implement the $4\times4$ DFT. A systematic calibration procedure was subsequently developed to characterize the prototype and compensate for fabrication errors. Measured results from the prototype demonstrate successful DFT computation in the analog domain, showing high correlation with theoretical expectations. By realizing an analog computer through standard microwave components, this work demonstrates a practical pathway toward low-latency, real-time analog signal processing.