Phase noise is a topic of theoretical and practical interest in electronic circuits, as well as in other fields, such as optics. Although progress has been made in understanding the phenomenon, there still remain significant gaps, both in its fundamental theory and in numerical techniques for its characterization. In this paper, we develop a solid foundation for phase noise that is valid for any oscillator, regardless of operating mechanism. We establish novel results about the dynamics of stable nonlinear oscillators in the presence of perturbations, both deterministic and random. We obtain an exact nonlinear equation for phase error, which we solve without approximations for random perturbations. This leads us to a precise characterization of timing jitter and spectral dispersion, for computing of which we have developed efficient numerical methods. We demonstrate our techniques on a variety of practical electrical oscillators and obtain good matches with measurements, even at frequencies close to the carrier, where previous techniques break down. Our methods are more than three orders of magnitude faster than the brute-force Monte Carlo approach, which is the only previously available technique that can predict phase noise correctly.

Phase noise in oscillators: a unifying theory and numerical methods for characterization

This paper presents a complete set of blocks implemented in the popular MATLAB SIMULINK environment, which allows designers to perform time-domain behavioral simulations of switched-capacitor (SC) sigma-delta (/spl Sigma//spl Delta/) modulators. The proposed set of blocks takes into account most of the SC /spl Sigma//spl Delta/ modulator nonidealities, such as sampling jitter, kT/C noise, and operational amplifier parameters (white noise, finite DC gain, finite bandwidth, slew rate and saturation voltages). For each block, a description of the considered effect as well as all of the implementative details are provided. The proposed simulation environment is validated by comparing the simulated behavior with the experimental results obtained from two actual circuits, namely a second-order low-pass and a sixth-order bandpass SC /spl Sigma//spl Delta/ modulator.

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Behavioral modeling of switched-capacitor sigma-delta modulators

An updated version of a 1985 tutorial paper on active filters using operational transconductance amplifiers (OTAs) is presented. The integrated circuit issues involved in active filters (using CMOS transconductance amplifiers) and the progress in this field in the last 15 years is addressed. CMOS transconductance amplifiers, nonlinearised and linearised, as well as frequency limitations and dynamic range considerations are reviewed. OTA-C filter architectures, current-mode filters, and other potential applications of transconductance amplifiers are discussed.

CMOS transconductance amplifiers, architectures and active filters: a tutorial

A new companding current-mode (CI) integrator principle is proposed. The integrator function is linear despite being composed of nonlinear parts. Its application promises monolithic filters with improved noise performance suitable for low supply voltages. A class AB circuit implementation is proposed which features an extended dynamic range.

Companding current-mode integrator: A new circuit principle for continuous-time monolithic filters

Thermal noise represents a major limitation on the performance of most electronic circuits. It is particularly important in switched circuits, such as the switched-capacitor (SC) filters widely used in mixed-mode CMOS integrated circuits. In these circuits, switching introduces a boost in the power spectral density of the thermal noise due to aliasing. Unfortunately, even though the theory of noise in SC circuits is discussed in the literature, it is very intricate. The numerical calculation of noise in switched circuits is very tedious, and requires highly sophisticated and not widely available software. The purpose of this paper is twofold. It provides a tutorial description of the physical phenomena taking place in an SC circuit while it processes noise (Sections II-III). It also proposes some specialized but highly efficient algorithms for estimating the resulting sampled noise in SC circuits, which need only simple calculations (Sections IV-VI ). A practical design procedure, which follows directly from the estimate, is also described. The accuracy of the proposed estimation algorithms is verified by simulation using SpectreRF. As an example, it is applied to the estimation of the total thermal noise in a second-order low-distortion delta-sigma converter.

Design-oriented estimation of thermal noise in switched-capacitor circuits

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.

Analysis of timing jitter in bandpass sigma-delta modulators

The use of companding in signal processing is considered. The applications discussed include noise reduction as well as increasing selectivity in single sinusoid detection.

Companding in signal processing

Noise in Gm-C filters is discussed. Topology-independent bounds for the signal-to-noise ratio (SNR) are derived, assuming the total power dissipation or the total capacitance are specified. Expressions for certain products of noise with frequently used quantities are derived, and are shown to be invariant under certain actions, which are sometimes taken in order to improve the SNR. Finally, specification-dependent bounds for SNR are derived.

Noise in Gm-C filters

Analog signals can be fully encoded as an asynchronous time sequence generated by a time encoding machine (TEM). With knowledge of the parameters of the time encoding machine, the asynchronous time sequence can be decoded using a non-linear time decoding machine. In one embodiment, the non-linear time decoding machine generates a set of weighted Dirac delta functions centered in the intervals of the asynchronous time sequence of the time encoding machine output. The weighting of each of the delta functions is determined by the parameters of the TEM as well as the values of the time sequence. The generation of the series of delta functions is a nonlinear operation (500, 510, 520). The input signal can be recovered from the series of weighted delta functions by (530) applying this series through an impulse response filter.

Time encoding and decoding of a signal

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.

L. Toth

Papers

Analysis of timing jitter in bandpass sigma-delta modulators

Companding in signal processing

Noise in Gm-C filters

Time encoding and decoding of a signal

Noise analysis of a class of oscillators