2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221)
Designing a nFural network (NN) for processing complex signals is a challenging task due to the l... more Designing a nFural network (NN) for processing complex signals is a challenging task due to the lack of bounded and differentiable nonlinear activation functions in the entire complex domain C. To avoid this difficulty, 'splitting', i.e., using uncoupled real sigmoidal functions for the real and imaginary components has been the traditional approach, and a number of fully complex activation functions introduced can only correct for magnitude distortion but can not handle phase distortion. We have recently introduced a fully complex NN that uses a hyperbolic tangent function defined in the entire complex domain and showed that for most practical signal processing problems, it is sufficient to have an activation function that is bounded and differentiable almosf everywhere in the complex domain. In this paper, the fully complex NN design is extended to employ other complex activation functions of the hyperbolic; circular, and their inverse hnction family. They are shown to successfully restore nonlinear amplitude and phase distortions of non-constant modulus modulated signals.
Neural Networks for Signal Processing X. Proceedings of the 2000 IEEE Signal Processing Society Workshop (Cat. No.00TH8501), 2000
One of the challenges in designing a neural network to process complexvalued signals is finding a... more One of the challenges in designing a neural network to process complexvalued signals is finding a suitable nonlinear complex activation function. The main reason for this difficulty is the conflict between the boundedness and the differentiability of complex functions in the entire complex plane, stated by Louiville's theorem. To avoid this difficulty, 'splitting', i.e., using two separate real nonlinear activation functions for the real and imaginary signal components has been the traditional approach. In this paper, we introduce a feed-forward neural network (FNN) architecture employing hyperbolic tangent tunh(z) function defined in the entire complex domain, and compare its performance with the FNN that uses a split complex structure. Since tunh(z) is analytic and bounded almost everywhere in the complex plane, when trained by backpropagation, it can easily outperform the nonanalytic split complex activation function in convergence speed and achievable minimum squared error when the domain is bounded around the unit circle. We demonstrate this property by a n equalization example, equalization of multi-phase shift keying (MPSK) signals corrupted by a multipath channel. The properties of tunh(z) and future directions to combat nonlinear distortions in complex transmission schemes a r e discussed.
2004 IEEE International Conference on Acoustics, Speech, and Signal Processing
A number of complex nonlinear functions are proposed for the independent component analysis (ICA)... more A number of complex nonlinear functions are proposed for the independent component analysis (ICA) of complex-valued data. We discuss the properties of these nonlinearities and show their efficiency in generating the higher order statistics needed for ICA.
An Evaluation of the Effectiveness of Intentional Interference to eLORAN
MILCOM 2006, 2006
Recent studies on the vulnerability of GPS to interference recommend that a backup system be made... more Recent studies on the vulnerability of GPS to interference recommend that a backup system be made available to determine own-aircraft position for both navigation and surveillance services in the National Airspace System (NAS). This backup will provide horizontal-only position ...
A scintillation signal model and a Global Positioning System (GPS)-Wide Area Augmentation System ... more A scintillation signal model and a Global Positioning System (GPS)-Wide Area Augmentation System (WAAS) receiver model are developed. The scintillation signal model is based on a Nakagami-m distribution for intensity and a Gaussian distribution with zero mean for phase. The GPS-WAAS receiver model includes Link 1 (L1) GPS and WAAS carrier-and C/A-code-tracking loops, as well as semicodeless Link 2 (L2) carrier and Y-code tracking capabilities. The results show that noncoherent delay locked loops (DLLs) typically used for code tracking are very robust to both amplitude and phase scintillation. Carrier-phasetracking loops are much more susceptible to scintillation, and the signal-to-noise threshold for reliable carrier tracking is very dependent on the scintillation strength. Fortunately, it appears that the worst case scintillation encountered at midlatitudes, including the United States, does not significantly impact L1 carrier-tracking performance. Semicodeless tracking of the L2 carrier is shown to be very fragile. Even weak scintillation can cause loss of L2 carrier lock for low-elevation satellites. 1996; Aarons and Basu, 1994]. This effect could cause a receiver to "lose lock" on the ranging signals broadcast by Wide Area Augmentation System (WAAS) [Loh et al., 1995] geostationary or GPS satellites, potentially causing a short service outage for one or more aircraft [Pullen et al., 1998]. Scintillation occurs most frequently during the peak of the solar cycle. Scintillation may be severe in equatorial regions (geomagnetic equator + 15 ø) after sunset and, to a somewhat lesser extent, the polar and auroral regions. Scintillation typically has minimum impact in midlatitude regions, e.g., the conterminous United States (CONUS). The aviation community is interested in the answers to the following questions regarding
We investigate the approximation ability of a multi layer perceptron (MLP) network when it is ext... more We investigate the approximation ability of a multi layer perceptron (MLP) network when it is extended to the complex domain. The main challenge for processing complex data with neural networks has been the lack of bounded and analytic complex nonlinear activation functions in the complex domain, as stated by Liouville's theorem. To avoid the conflict between the boundedness and the analyticity of a nonlinear complex function in the complex domain, a number of ad hoc MLPs that include using two real-valued MLPs, one processing the real part and the other processing the imaginary part, have been traditionally employed. However, since nonanalytic functions do not meet the Cauchy-Riemann conditions, they render themselves into degenerative backpropagation algorithms that compromise the efficiency of nonlinear approximation and learning in the complex vector field. A number of elementary transcendental functions (ETFs) derivable from the entire exponential functionez that are analyt...
The contents of this document reflect the views of the author and The MITRE Corporation and do no... more The contents of this document reflect the views of the author and The MITRE Corporation and do not necessarily reflect the views of the Federal Aviation Administration or the Department of Transportation. Neither the FAA nor the DOT makes any warranty or guarantee, expressed or implied, concerning the content or accuracy of these views.
Journal of Vlsi Signal Processing Systems for Signal Image and Video Technology, 2002
Designing a neural network (NN) to process complex-valued signals is a challenging task since a c... more Designing a neural network (NN) to process complex-valued signals is a challenging task since a complex nonlinear activation function (AF) cannot be both analytic and bounded everywhere in the complex plane . To avoid this difficulty, 'splitting', i.e., using a pair of real sigmoidal functions for the real and imaginary components has been the traditional approach. However, this 'ad hoc'
2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221)
Designing a nFural network (NN) for processing complex signals is a challenging task due to the l... more Designing a nFural network (NN) for processing complex signals is a challenging task due to the lack of bounded and differentiable nonlinear activation functions in the entire complex domain C. To avoid this difficulty, 'splitting', i.e., using uncoupled real sigmoidal functions for the real and imaginary components has been the traditional approach, and a number of fully complex activation functions introduced can only correct for magnitude distortion but can not handle phase distortion. We have recently introduced a fully complex NN that uses a hyperbolic tangent function defined in the entire complex domain and showed that for most practical signal processing problems, it is sufficient to have an activation function that is bounded and differentiable almosf everywhere in the complex domain. In this paper, the fully complex NN design is extended to employ other complex activation functions of the hyperbolic; circular, and their inverse hnction family. They are shown to successfully restore nonlinear amplitude and phase distortions of non-constant modulus modulated signals.
Neural Networks for Signal Processing X. Proceedings of the 2000 IEEE Signal Processing Society Workshop (Cat. No.00TH8501), 2000
One of the challenges in designing a neural network to process complexvalued signals is finding a... more One of the challenges in designing a neural network to process complexvalued signals is finding a suitable nonlinear complex activation function. The main reason for this difficulty is the conflict between the boundedness and the differentiability of complex functions in the entire complex plane, stated by Louiville's theorem. To avoid this difficulty, 'splitting', i.e., using two separate real nonlinear activation functions for the real and imaginary signal components has been the traditional approach. In this paper, we introduce a feed-forward neural network (FNN) architecture employing hyperbolic tangent tunh(z) function defined in the entire complex domain, and compare its performance with the FNN that uses a split complex structure. Since tunh(z) is analytic and bounded almost everywhere in the complex plane, when trained by backpropagation, it can easily outperform the nonanalytic split complex activation function in convergence speed and achievable minimum squared error when the domain is bounded around the unit circle. We demonstrate this property by a n equalization example, equalization of multi-phase shift keying (MPSK) signals corrupted by a multipath channel. The properties of tunh(z) and future directions to combat nonlinear distortions in complex transmission schemes a r e discussed.
2004 IEEE International Conference on Acoustics, Speech, and Signal Processing
A number of complex nonlinear functions are proposed for the independent component analysis (ICA)... more A number of complex nonlinear functions are proposed for the independent component analysis (ICA) of complex-valued data. We discuss the properties of these nonlinearities and show their efficiency in generating the higher order statistics needed for ICA.
An Evaluation of the Effectiveness of Intentional Interference to eLORAN
MILCOM 2006, 2006
Recent studies on the vulnerability of GPS to interference recommend that a backup system be made... more Recent studies on the vulnerability of GPS to interference recommend that a backup system be made available to determine own-aircraft position for both navigation and surveillance services in the National Airspace System (NAS). This backup will provide horizontal-only position ...
A scintillation signal model and a Global Positioning System (GPS)-Wide Area Augmentation System ... more A scintillation signal model and a Global Positioning System (GPS)-Wide Area Augmentation System (WAAS) receiver model are developed. The scintillation signal model is based on a Nakagami-m distribution for intensity and a Gaussian distribution with zero mean for phase. The GPS-WAAS receiver model includes Link 1 (L1) GPS and WAAS carrier-and C/A-code-tracking loops, as well as semicodeless Link 2 (L2) carrier and Y-code tracking capabilities. The results show that noncoherent delay locked loops (DLLs) typically used for code tracking are very robust to both amplitude and phase scintillation. Carrier-phasetracking loops are much more susceptible to scintillation, and the signal-to-noise threshold for reliable carrier tracking is very dependent on the scintillation strength. Fortunately, it appears that the worst case scintillation encountered at midlatitudes, including the United States, does not significantly impact L1 carrier-tracking performance. Semicodeless tracking of the L2 carrier is shown to be very fragile. Even weak scintillation can cause loss of L2 carrier lock for low-elevation satellites. 1996; Aarons and Basu, 1994]. This effect could cause a receiver to "lose lock" on the ranging signals broadcast by Wide Area Augmentation System (WAAS) [Loh et al., 1995] geostationary or GPS satellites, potentially causing a short service outage for one or more aircraft [Pullen et al., 1998]. Scintillation occurs most frequently during the peak of the solar cycle. Scintillation may be severe in equatorial regions (geomagnetic equator + 15 ø) after sunset and, to a somewhat lesser extent, the polar and auroral regions. Scintillation typically has minimum impact in midlatitude regions, e.g., the conterminous United States (CONUS). The aviation community is interested in the answers to the following questions regarding
We investigate the approximation ability of a multi layer perceptron (MLP) network when it is ext... more We investigate the approximation ability of a multi layer perceptron (MLP) network when it is extended to the complex domain. The main challenge for processing complex data with neural networks has been the lack of bounded and analytic complex nonlinear activation functions in the complex domain, as stated by Liouville's theorem. To avoid the conflict between the boundedness and the analyticity of a nonlinear complex function in the complex domain, a number of ad hoc MLPs that include using two real-valued MLPs, one processing the real part and the other processing the imaginary part, have been traditionally employed. However, since nonanalytic functions do not meet the Cauchy-Riemann conditions, they render themselves into degenerative backpropagation algorithms that compromise the efficiency of nonlinear approximation and learning in the complex vector field. A number of elementary transcendental functions (ETFs) derivable from the entire exponential functionez that are analyt...
The contents of this document reflect the views of the author and The MITRE Corporation and do no... more The contents of this document reflect the views of the author and The MITRE Corporation and do not necessarily reflect the views of the Federal Aviation Administration or the Department of Transportation. Neither the FAA nor the DOT makes any warranty or guarantee, expressed or implied, concerning the content or accuracy of these views.
Journal of Vlsi Signal Processing Systems for Signal Image and Video Technology, 2002
Designing a neural network (NN) to process complex-valued signals is a challenging task since a c... more Designing a neural network (NN) to process complex-valued signals is a challenging task since a complex nonlinear activation function (AF) cannot be both analytic and bounded everywhere in the complex plane . To avoid this difficulty, 'splitting', i.e., using a pair of real sigmoidal functions for the real and imaginary components has been the traditional approach. However, this 'ad hoc'
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