László T. Tóth

Perfect recovery and sensitivity analysis of time encoded bandlimited signals

A time encoding machine is a real-time asynchronous mechanism for encoding amplitude information into a time sequence. We investigate the operating characteristics of a machine consisting of a feedback loop containing an adder, a linear filter, and a noninverting Schmitt trigger. We show that the amplitude information of a bandlimited signal can be perfectly recovered if the difference between any two consecutive values of the time sequence is bounded by the inverse of the Nyquist rate. We also show how to build a nonlinear inverse time decoding machine (TDM) that perfectly recovers the amplitude information from the time sequence. We demonstrate the close relationship between the recovery algorithms for time encoding and irregular sampling. We also show the close relationship between time encoding and a number of nonlinear modulation schemes including FM and asynchronous sigma-delta modulation. We analyze the sensitivity of the time encoding recovery algorithm and demonstrate how to construct a TDM that perfectly recovers the amplitude information from the time sequence and is trigger parameter insensitive. We derive bounds on the error in signal recovery introduced by the quantization of the time sequence. We compare these with the recovery error introduced by the quantization of the amplitude of the bandlimited signal when irregular sampling is employed. Under Nyquist-type rate conditions, quantization of a bandlimited signal in the time and amplitude domains are shown to be largely equivalent methods of information representation.

Analysis of timing jitter in bandpass sigma-delta modulators

The effect of clock jitter on the dynamic range of discrete-time (DT) and continuous-time (CT) sigma-delta modulators is addressed. It is shown that clock jitter in DT modulators mixes with the input signal, while for CT modulators, the jitter mixes with the out-of-band quantization error and elevates the passband noise. The signal-to-noise ratio of CT modulators is shown to be more susceptible to clock jitter than their DT counterparts. Analytical and simulation results are provided.

Time encoding and perfect recovery of bandlimited signals

A time encoding machine is a real-time asynchronous mechanism for encoding amplitude information into a time sequence. We investigate the operating characteristics of a machine consisting of a feedback loop containing an adder, a linear filter and a Schmitt trigger. We show how to recover, loss-free, the amplitude information of a bandlimited signal from the time sequence.

An Overcomplete Stitching Algorithm for Time Decoding Machines

We investigate a class of finite-dimensional time decoding algorithms that: (1) is insensitive with respect to the time-encoding parameters; (2) is highly efficient and stable; and (3) can be implemented in real time. These algorithms are based on the observation that the recovery of time encoded signals given a finite number of observations has the property that the quality of signal recovery is very high in a reduced time range. We show how to obtain a local representation of the time encoded signal in an efficient and stable manner using a Vandermonde formulation of the recovery algorithm. Once the signal values are obtained from a finite number of possibly overlapping observations, the reduced-range segments are stitched together. The signal obtained by segment stitching is subsequently filtered for improved performance in recovery. Finally, we evaluate the complexity of the algorithms and their computational requirements for real-time implementation.

Noise analysis of a class of oscillators

This paper presents noise analysis of a class of oscillators, which can be modeled by a positive feedback system with a frequency-selective Mth-order filter, an ideal comparator, and a white noise source. An explicit analytical expression for the output power spectral density is derived. A simplified expression is obtained for a special case when the Mth-order filter is replaced by a second-order bandpass filter. The general expression is shown to reduce to a well-known result if a high and-factor filter is further assumed. Theoretical results presented here are verified by experiment.

On the robustness of an analog VLSI implementation of a time encoding machine

Time encoding is a mechanism for representing the information contained in a continuous time, bandlimited, analog signal as the zero-crossings of a binary signal. Time decoding algorithms have been developed that make a perfect recovery of time encoded bandlimited signals possible. We consider a simple one-opamp active RC implementation of the time encoder and investigate the robustness in performance of the time decoder when the former is subject to non-idealities of the analog VLSI realization. We show that up to a constant scaling factor, delay and offset the input signal can be accurately reconstructed even if the opamp has a finite DC gain and finite bandwidth and the circuit exhibits parameter offsets. We develop an experimental upper bound for the reconstruction error that can be used in the design of the encoder. The analytical results are verified with numerical simulations.

Fast recovery algorithms for time encoded bandlimited signals

Time encoding is a real-time asynchronous mechanism of mapping amplitude information into a time sequence. We investigate fast algorithms for the recovery of time encoded bandlimited signals and construct an algorithm that has provably low computational complexity. We also devise a fast algorithm that is parameter-insensitive.

Further results for noise in active RC and MOSFET-C filters

Noise due to resistors and op amps in active RC and MOSFET-C filters is considered. Bounds for the signal-to-resistive noise ratio and signal-to-op amp noise ratio in state-space filters are given.

A time decoding realization with a CNN

Time encoding is a novel real-time asynchronous mechanism for encoding amplitude information into a time sequence. The analog bandlimited input is fed into a simple nonlinear neuron-like circuit that generates a strictly increasing time sequence based on which the signal can be reconstructed. The heart of the reconstruction is solving a system of ill-conditioned linear equations. The paper shows that the equations can be manipulated so that the reconstruction becomes feasible using a cellular neural network (CNN) with a banded system matrix. In particular, the system is first transformed into a smaller well-conditioned system, and, then, the Lanczos process is used to lay it out into a set of even smaller systems characterized by a set of tridiagonal matrices. Each of these systems can directly be solved by CNNs, whereas the preprocessing (transformation and Lanczos algorithm) and simple postprocessing phases can be partly or fully implemented by using the digital capabilities of the CNN universal machine (CNN-UM). Each step of the proposed formulation is confirmed by numerical (digital) simulations.

Sensitivity analysis of time encoded bandlimited signals

A time encoding machine, consisting of a feedback loop containing an adder, an integrator and a Schmitt trigger, encodes amplitude information into a time sequence. We demonstrate how to construct a time decoding machine that perfectly recovers the amplitude information from the time sequence and is trigger parameter insensitive. We derive bounds on the error in signal recovery introduced by the quantization of the time sequence. We compare these with the recovery error introduced by the quantization of the amplitude of the bandlimited signal when irregular sampling is employed. Under Nyquist-type rate conditions, quantization of a bandlimited signal in the time and amplitude domains are shown to be largely equivalent methods of information representation.