Analog optoelectronic independent component analysis for radio frequency signals
This thesis addresses the problem of blind source separation of signals at radio frequencies. Independent component analysis (ICA), which includes a second-order decorrelation followed by a fourth-order decorrelation, uses signal independence to estimate the original signals from the received mixtures. Until now, ICA has been applied to many applications at or below audio frequencies. The work presented here demonstrates that an optoelectronic implementation using the parallel processing nature of dynamic holography can overcome the computational difficulties associated with algorithmic implementations of ICA.
The holographic nature of a photorefractive crystal combined with the non-linearity of an electro-optic modulator in a feedback loop can be described by a nonlinear dynamical equation. The dynamics can be cast in the form of Lotka-Volterra equations used to study the dynamics of competing populations of species. Although this analogy with the animal world is interesting, the dynamical equation associated with the fourth-order decorrelation system is fascinating. The statistics associated with the original signals, rather than an external potential, determine the dynamics of the system. In particular, the system is multistable, metastable, or monostable depending on whether the probability density functions of the original signals are sub-Gaussian, Gaussian, or super-Gaussian, respectively. The multistable solution, which occurs for sub-Gaussian signals, provides the winner-takes-all behavior required to separate signals. This ability to separate sub-Gaussian signals is advantageous since signals modulated on a sinusoidal carrier are sub-Gaussian. The fourth-order decorrelation system achieves greater than 40 dB signal separation on 200 MHz single-frequency sine waves and greater than 20 dB signal separation for 10 MHz bandwidth signals. The system performance is degraded by 10 to 20 dB when mixed electronically due to imperfections in the mixing circuitry.
The development of a broadband electro-optic modulator capable of modulating to, at least, twice the half-wave voltage was instrumental to achieving radio frequency blind source separation. This compact 532 nm Lithium Niobate modulator has a 300 MHz bandwidth and a half-wave voltage of less than 16 V. To our knowledge, this is the only free-space modulator capable of this modulation depth.
This thesis also advances the theoretical work in the area of optoelectronic signal processing. Three of the main contributors to signal separation degradation are studied to aid in the characterization and improved performance of the fourth-order decorrelation feedback loop. The fourth-order decorrelation system requires a preprocessor, which orthogonalizes the input signal mixtures. The theoretical framework of an optoelectronic system that performs principal component analysis (PCA), one method of orthogonalizing the signal mixtures, is also presented.
The PCA feedback loop looks identical to the fourth-order decorrelation feedback loop, except the electro-optic modulator is used in its linear regime while the photorefractive gain saturates. Because of the physical similarity of the two optoelectronic feedback loops, our hope is that modular designs will aid in the application of this technology to the telecommunications arena.