A sound signal is conveniently represented as a complex number: its modulus (amplitude) reflects loudness, while its argument (phase) indicates the wave's position in time: its instantaneous shift. Traditional neural networks split this unified description into two independent coordinates — roughly speaking, they assign an address on a Cartesian grid: "two steps east, one step north." This approach loses the intrinsic connection: in a real signal, loudness and phase change in a coordinated manner, just as distance and direction are inseparable in a polar coordinate system.

The model, inspired by Ludwig Boltzmann's ideas on entropy as a measure of a system's disorder, operates directly on the pair "distance–direction"—that is, amplitude and phase together, in their natural coupling. To handle sound spectra, where the frequency composition varies over a wide range, a logarithmic scale was introduced into the architecture: the human ear perceives a tenfold increase in power as roughly a doubling of loudness, and logarithmic compression mimics this psychoacoustic feature. This gave rise to the LogPolarBM model, which elegantly includes the classical statistical distributions of Rice and Nakagami as special cases: that is, under certain conditions, its behavior reduces to long-studied patterns of fading signals. In tests on audio data, LogPolarBM surpassed even deep neural networks in accuracy, despite being considerably simpler.

An unexpected extension: the same principle of polar phase representation proved applicable far beyond acoustics — in wireless communications and quantum physics. The phases of wave functions are critically important for quantum computing, where information is encoded precisely in relative phases. The machine's ability to "hear" phase (that is, to extract it without the losses that were inevitable in separate descriptions) allows it to pull out subtle correlations from the signal that eluded previous methods.