Precision frequency sources such as quartz oscillators, masers, and passive atomic frequency standards are affected by phase and frequency instabilities including both random and deterministic components. It is of prime importance to have a comprehensive characterization of these instabilities in order to be able to assess the potential utility of each source. For that purpose, many parameters have been proposed especially for dealing with random fluctuations. Some of them have been recommended by the IEEE Subcommittee on Frequency Stability and later by Study Group 7 on "Standard Frequencies and Time Signals" of the International Radio Consultative Committee (CCIR). Others are not so widely used but show interesting capabilities. This paper aims at giving a broad review of parameters proposed for phase and frequency instability characterization, including both classical widely used concepts and more recent less familiar approaches. Transfer functions that link frequency-domain and time-domain parameters are emphasized because they provide improved understanding of the properties of a given time-domain parameter or facilitate introducing of new parameters. As far as new approaches are concerned, an attempt has been made to demonstrate clearly their respective advantages. To this end, some developments that did not appear in the original references ate presented here, e.g, the modified three sample variance Σ y 2(τ), the expressions of 〈δy- T 2〉, the intetpretation of structure functions of phase and its relations with Σ y 2(τ) and the Hadamard variance. The effects of polynomial phase and frequency drifts on various parameters have also been pointed out in parallel with those of random processes modeled by power-law spectral densities.

Characterization of phase and frequency instabilities in precision frequency sources: Fifteen years of progress

In this paper the general correlation spectrum of an oscillator with white and f -α noise sources is derived from the Langevin equations describing the deterministic and stochastic behaviour of an oscillator by perturbation methods. The treatment of f -α noise and the influence of the finite measuring time on the noise spectra are included in a time domain calculation. The theory is applied to a Van der Pol oscillator exhibiting parametric sideband amplification and to a realized oscillator demonstrating the applicability of the theory to technically relevant circuits

Analysis of white and f−α noise in oscillators

An integrated voltage-controlled oscillator (VCO) at a frequency of 2 GHz is implemented in a f/sub T/= 25 GHz standard bipolar process. The phase noise of the VCO is -136 dBc/Hz at 4.7 MHz frequency offset. The LC-resonator uses vertically coupled on-chip inductors and integrated tuning diodes. Due to the poor performance of integrated resonators on silicon ICs, oscillators with phase noise meeting requirements of wireless applications are difficult to integrate. With fully integrated designs only the standards for cordless phones, for instance DECT, can be achieved. The critical point in the DECT-specification is the emission of the transmitter due to intermodulation in the third adjacent channel, that must be <-47 dBm. This value is measured with an integration bandwidth of 1 MHz centered at the nominal center frequency. With a channel-spacing of 1.728 MHz the third adjacent channel is located 5.184 MHz from the actual transmit channel frequency. The beginning of the integration bandwidth is at an offset frequency of 4.684 MHz related to the nominal frequency of the transmit channel. This is the offset frequency, at which the specification must be met. The resulting noise requirement is -132 dBc/Hz at a offset frequency of 4.684 MHz, when the integration bandwidth and the transmit output power of 25 dBm are taken into account.

A fully integrated VCO at 2 GHz

This paper is devoted to one aspect of the study of phase noise in oscillators: how an oscillator reacts to internal noise that occurs in the active element. The following theoretical analysis will lead us to express the Leeson's model in a more general form.

Phase Noise in Oscillators: A Mathematical Analysis of Leeson's Model

Measures of frequency and phase instability in the time and frequency domains are reviewed. The particular choice as to which domain is used depends on the application. However, mathematical conversions from one domain to the other can present problems. Most of the major manufacturers now specify instability characteristics of their standard in terms of these recommended measures. This work thus defines and formalizes the general practice of more than a decade. >

Standard terminology for fundamental frequency and time metrology

A recently developed theoretical analysis has shown that it is possible to measure the Allan variance (a time-domain measure of frequency instability) without any statistical treatment of data from an electronic counter. The measurement is made via high-pass filtering of phase noise with a test set similar to the one used for frequency-domain measurements. The unique test set described in this paper relies on this principle and is capable of measuring the short-term frequency instability of the best quartzcrystal oscillators in both time and frequency domains.

Measurement of Frequency Stability in Time and Frequency Domains via Filtering of Phase Noise

La variance de Hadamard permet d’effectuer, a l’aide d’un frequencemetre, l’analyse spectrale du bruit de frequence des oscillateurs dans une gamme de frequences accessible a un nombre restreint d’analyseurs de spectre; en outre la bande equivalente d’analyse peut etre rendue tres faible. Une etude mathematique detaillee de cette technique de mesure est presentee dans cet article. Les proprietes caracteristiques et les limitations inherentes au principe meme de cette technique sont mises en evidence a partir de l’etude de sa fonction de transfert; l’influence de la nature du spectre analyse sur la validite des resultats obtenus est etudiee. Enfin, les ameliorations de cette technique qui peuvent resulter d’une plus grande complexite du processus de mesure sont presentees.

Gerard Sauvage

Papers

Phase Noise in Oscillators: A Mathematical Analysis of Leeson's Model

Measurement of Frequency Stability in Time and Frequency Domains via Filtering of Phase Noise

Analyse spectrale du bruit de fréquence des oscillateurs par la variance de Hadamard