We present a formal methodology for identifying a channel in a system consisting of a communication channel in cascade with an asynchronous sampler. The channel is modeled as a multidimensional filter, while models of asynchronous samplers are taken from neuroscience and communications and include integrate-and-fire neurons, asynchronous sigma/delta modulators and general oscillators in cascade with zero-crossing detectors. We devise channel identification algorithms that recover a projection of the filter(s) onto a space of input signals loss-free for both scalar and vector-valued test signals. The test signals are modeled as elements of a reproducing kernel Hilbert space (RKHS) with a Dirichlet kernel. Under appropriate limiting conditions on the bandwidth and the order of the test signal space, the filter projection converges to the impulse response of the filter. We show that our results hold for a wide class of RKHSs, including the space of finite-energy bandlimited signals. We also extend our channel identification results to noisy circuits. 1. Introduction Signal distortions introduced by a communication channel can severely affect the reliability of communication systems. If properly utilized, knowledge of the channel response can lead to a dramatic improvement in the performance of a communication link. In practice, however, information about the channel is rarely available a priori and the channel needs to be identified at the receiver. A number of channel identification methods [1] have been proposed for traditional clock-based systems that rely on the classical sampling theorem [2, 3]. However, there is a growing need to develop channel identification methods for asynchronous nonlinear systems, of which time encoding machines (TEMs) [4] are a prime example. TEMs naturally arise as models of early sensory systems in neuroscience [5, 6] as well as models of nonlinear samplers in signal processing and analog-to-discrete (A/D) converters in communication systems [4, 6]. Unlike traditional clock-based amplitude-domain devices, TEMs encode analog signals as a strictly increasing sequence of irregularly spaced times . As such, they are closely related to irregular (amplitude) samplers [4, 7] and, due to their asynchronous nature, are inherently low-power devices [8]. TEMs are also readily amenable to massive parallelization [9]. Furthermore, under certain conditions, TEMs faithfully represent analog signals in the time domain; given the parameters of the TEM and the time sequence at its output, a time decoding machine (TDM) can recover the encoded
L. Tong, B. M. Sadler, and M. Dong, “Pilot-assisted wireless transmissions: general model, design criteria, and signal processing,” IEEE Signal Processing Magazine, vol. 21, no. 6, pp. 12–25, 2004.
A. A. Lazar and L. T. Tóth, “Perfect recovery and sensitivity analysis of time encoded bandlimited signals,” IEEE Transactions on Circuits and Systems I, vol. 51, no. 10, pp. 2060–2073, 2004.
A. A. Lazar and E. A. Pnevmatikakis, “Faithful representation of stimuli with a population of integrate-and-fire neurons,” Neural Computation, vol. 20, no. 11, pp. 2715–2744, 2008.
H. G. Feichtinger and Gr？chenig K., “Theory and practice of irregular sampling,” in Wavelets: Mathematics and Applications, Studies in Advanced Mathematics, pp. 305–363, CRC Press, 1994.
S. Yan Ng, A continuous-time asynchronous Sigma Delta analog to digital converter for broadband wireless receiver with adaptive digital calibration technique [Ph.D. thesis], Department of Electrical and Computer Engineering, Ohio State University, 2009.
A. A. Lazar, E. A. Pnevmatikakis, and Y. Zhou, “Encoding natural scenes with neural circuits with random thresholds,” Vision Research, vol. 50, no. 22, pp. 2200–2212, 2010, Special Issue on Mathematical Models of Visual Coding.
M. C.-K. Wu, S. V. David, and J. L. Gallant, “Complete functional characterization of sensory neurons by system identification,” Annual Review of Neuroscience, vol. 29, pp. 477–505, 2006.
U. Friederich, D. Coca, S. Billings, and M. Juusola, “Data modelling for analysis of adaptive changes in fly photoreceptors,” Neural Information Processing, vol. 5863, no. 1, pp. 34–48, 2009.
T. W. Berger, D. Song, R. H. M. Chan, and V. Z. Marmarelis, “The neurobiological basis of cognition: identification by multi-input, multioutput nonlinear dynamic modeling,” Proceedings of the IEEE, vol. 98, no. 3, pp. 356–374, 2010.
T. W. Berger, D. Song, R. H. M. Chan et al., “A hippocampal cognitive prosthesis: multi-input, multi-output nonlinear modeling and VLSI implementation,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 20, no. 2, pp. 198–211, 2012.
Z. Song, M. Postma, S. A. Billings, D. Coca, R. C. Hardie, and M. Juusola, “Stochastic, adaptive sampling of information by microvilli in fly photoreceptors,” Current Biology, vol. 22, pp. 1–10, 2012.
F. E. Theunissen, S. V. David, N. C. Singh, A. Hsu, W. E. Vinje, and J. L. Gallant, “Estimating spatio-temporal receptive fields of auditory and visual neurons from their responses to natural stimuli,” Network, vol. 12, no. 3, pp. 289–316, 2001.
O. Schwartz, E. J. Chichilnisky, and E. P. Simoncelli, “Characterizing neural gain control using spike-triggered covariance,” Advances in Neural Information Processing Systems, vol. 14, pp. 269–276, 2002.
W. P. Torres, A. V. Oppenheim, and R. R. Rosales, “Generalized frequency modulation,” IEEE Transactions on Circuits and Systems I, vol. 48, no. 12, pp. 1405–1412, 2001.
E. H. Adelson and J. R. Bergen, “Spatiotemporal energy models for the perception of motion,” Journal of the Optical Society of America A, vol. 2, no. 2, pp. 284–299, 1985.
A. J. Kim, A. A. Lazar, and Y. B. Slutskiy, “System identification of Drosophila olfactory sensory neurons,” Journal of Computational Neuroscience, vol. 30, no. 1, pp. 143–161, 2011.