A Möbius transformation is a complex-valued rational function of the form f(z) = (az + b)/(cz + d), where a, b, c, d are complex constants, that maps circles and lines to circles and lines in the complex plane. In WiFi CSI sensing, this transformation is central to the CSI-ratio model because dividing the complex CSI measurement of one antenna by that of an adjacent antenna corresponds mathematically to a Möbius transformation of the underlying channel response, which cancels shared phase noise and hardware impairments while preserving motion-induced signal variations. This property makes it particularly valuable for device-free sensing, as the resulting ratio signal is more stable and interpretable than raw CSI, and the geometric structure imposed by the Möbius transformation — constraining the signal locus to arcs or circles in the complex plane — enables cleaner feature extraction for tasks such as localization and activity recognition.
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