The Statistics of the Cross-Spectrum and the Spectrum Average: Generalization to Multiple Instruments.

This article addresses the measurement of the power spectrum of red noise processes at the lowest frequencies, where the minimum acquisition time is so long that it is impossible to average on a sequence of data record. Therefore, averaging is possible only on simultaneous observation of multiple instruments. This is the case of radio astronomy, which we take as the paradigm, but examples may be found in other fields such as climatology and geodesy. We compare the Bayesian confidence interval of the red noise parameter using two estimators, the spectrum average and the cross-spectrum. While the spectrum average is widely used, the cross-spectrum using multiple instruments is rather uncommon. With two instruments, the cross-spectrum estimator leads to the Variance-Gamma distribution. A generalization to q devices based on the Fourier transform of characteristic functions is provided, with the example of the observation of millisecond pulsars with five radio telescopes (RTs). The simulations show that the spectrum average is by a small amount more efficient than the cross-spectrum, chiefly when the background exceeds the signal. However, some notable differences between their upper limit indicate that it should be wiser to compute both estimators.

Influence of flicker noise and nonlinearity on the frequency spectrum of spin torque nano-oscillators.

The correlation of phase fluctuations in any type of oscillator fundamentally defines its spectral shape. However, in nonlinear oscillators, such as spin torque nano-oscillators, the frequency spectrum can become particularly complex. This is specifically true when not only considering thermal but also colored 1/f flicker noise processes, which are crucial in the context of the oscillator's long term stability. In this study, we address the frequency spectrum of spin torque oscillators in the regime of large-amplitude steady oscillations experimentally and as well theoretically. We particularly take both thermal and flicker noise into account. We perform a series of measurements of the phase noise and the spectrum on spin torque vortex oscillators, notably varying the measurement time duration. Furthermore, we develop the modelling of thermal and flicker noise in Thiele equation based simulations. We also derive the complete phase variance in the framework of the nonlinear auto-oscillator theory and deduce the actual frequency spectrum. We investigate its dependence on the measurement time duration and compare with the experimental results. Long term stability is important in several of the recent applicative developments of spin torque oscillators. This study brings some insights on how to better address this issue.

Cross-Spectrum Measurement Statistics: Uncertainties and Detection Limit.

The cross-spectrum method consists in measuring a signal c(t) simultaneously with two independent instruments. Each of these instruments contributes to the global noise by its intrinsic (white) noise, whereas the signal c(t) that we want to characterize could be a (red) noise. We first define the real part of the cross spectrum as a relevant estimator. Then, we characterize the probability density function (pdf) of this estimator knowing the noise level (direct problem) as a Variance-gamma (VG) distribution. Next, we solve the "inverse problem" due to Bayes' theorem to obtain an upper limit of the noise level knowing the estimate. Checked by massive Monte Carlo simulations, VG proves to be perfectly reliable for any number of degrees of freedom (DOFs). Finally, we compare this method with another method using the Karhunen-Loève transform (KLT). We find an upper limit of the signal level slightly different as the one of VG since KLT better considers the available information.

Phase-Noise and Amplitude-Noise Measurement of DACs and DDSs.

This article proposes a method for the measurement of phase noise (PN, or PM noise) and amplitude noise (AN, or AM noise) of digital-to-analog converters (DACs) and direct digital synthesizers (DDSs) based on the modulation-index amplification. The carrier is first reduced by a controlled amount (30-40 dB) by adding a reference signal of nearly equal amplitude and opposite in phase. Then, residual carrier and noise sidebands are amplified and sent to a conventional PN analyzer. The main virtues of our method are: 1) the noise specs of the PN analyzer are relaxed by a factor equal to the carrier suppression ratio and 2) the capability to measure the AN using a PN analyzer with no need for the analyzer to feature AN measurement. An obvious variant enables AN and PN measurements using an AN analyzer with no PN measurement capability. Such an instrument is extremely simple and easy to implement with a power-detector diode followed by an FFT analyzer. Unlike the classical bridge (interferometric) method, there is no need for external line stretcher and variable attenuators because phase and amplitude controls are implemented in the device under test. In one case (AD9144), we could measure the noise over 10 decades of frequency. The flicker noise matches the exact 1/f law with a maximum discrepancy of ±1 dB over 7.5 decades. Due to the simplicity, reliability, and low background noise, this method has the potential to become the standard method for the AN and PN measurements of DACs and DDSs.

KLTS: A Rigorous Method to Compute the Confidence Intervals for the Three-Cornered Hat and for Groslambert Covariance.

The three-cornered hat/Groslambert Covariance (GCov) methods are widely used to estimate the stability of each individual clock in a set of three, but no method gives reliable confidence intervals for large integration times. We propose a new KLTS (Karhunen-Loève Tansform using Sufficient statistics) method which uses these estimators to consider the statistics of all the measurements between the pairs of clocks in a Bayesian way. The resulting cumulative density function (CDF) yields confidence intervals for each clock Allan variance (AVAR). This CDF provides also a stability estimator that is always positive. Checked by massive Monte Carlo simulations, KLTS proves to be perfectly reliable even for one degree of freedom. An example of experimental measurement is given.

Equivalence of Open-Loop and Closed-Loop Operation of SAW Resonators and Delay Lines.

Surface acoustic wave (SAW) sensors in the form of two-port resonators or delay lines are widely used in various fields of application. The readout of such sensors is achieved by electronic systems operating either in an open-loop or in a closed-loop configuration. The mode of operation of the sensor system is usually chosen based on requirements like, e.g., bandwidth, dynamic range, linearity, costs, and immunity against environmental influences. Because the limit of detection (LOD) at the output of a sensor system is often one of the most important figures of merit, both readout structures, i.e., open-loop and closed-loop systems, are analyzed in terms of the minimum achievable LOD. Based on a comprehensive phase noise analysis of these structures for both resonant sensors and delay line sensors, expressions for the various limits of detection are derived. Under generally valid conditions, the equivalence of open-loop and closed-loop operation is shown for both types of sensors. These results are not only valid for SAW devices, but are also applicable to all kinds of phase-sensitive sensors.

Phase Noise and Frequency Stability of the Red-Pitaya Internal PLL.

In field-programmable gate array platforms, the main clock is generally a low-cost quartz oscillator whose stability is of the order of 10-9 to 10-10 in the short term and 10-7 to 10-8 in the medium term, with the uncertainty of tens of ppm. Better stability is achieved by feeding an external reference into the internal phase-locked loop (PLL). We report the noise characterization of the internal PLL of Red-Pitaya platform, an open-source embedded system architected around the Zynq 7010 System on Chip, with analog-to-digital and digital-to-analog converters. Our experiments show that, providing an external 10-MHz reference, the PLL exhibits a residual frequency stability of 1.2×10-12 at 1 s and 1.3×10-15 at 4000 s, Allan deviation in 5-Hz bandwidth. These results help to predict the PLL stability as a function of frequency and power of the external reference, and provide guidelines for the design of precision instrumentation, chiefly intended for time and frequency metrology.

Frequency Stability Measurement of Cryogenic Sapphire Oscillators With a Multichannel Tracking DDS and the Two-Sample Covariance.

This paper shows the first measurement of three 100-MHz signals exhibiting fluctuations from 2×10-16 to parts in 10-15 for an integration time τ between 1 s and 1 day. Such stable signals are provided by three cryogenic sapphire oscillators (CSOs) operating at about 10 GHz, also delivering the 100-MHz output via a dedicated synthesizer. The measurement is made possible by a six-channel tracking direct digital synthesizer (TDDS) and the two-sample covariance tool, used to estimate the Allan variance. The use of two TDDS channels per CSO enables high rejection of the instrument background noise. The covariance outperforms the three-cornered hat (TCH) method in that the background converges to zero "out of the box," with no need of the hypothesis that the instrument channels are equally noisy, nor of more sophisticated techniques to estimate the background noise of each channel. Thanks to correlation and averaging, the instrument background (AVAR) rolls off with a slope 1/√m , the number of measurements, down to 10-18 at τ = 104 s. For consistency check, we compare the results to the traditional TCH method beating the 10-GHz outputs down to the megahertz region. Given the flexibility of the TDDS, our methods find an immediate application to the measurement of the 250-MHz output of the femtosecond combs.

Ultracompact reference ultralow expansion glass cavity.

We present the first experimental characterization of our ultracompact, ultrastable laser. The heart of the apparatus is an original Fabry-Perot cavity with 25 mm length and pyramidal geometry, equipped with highly reflective crystalline coatings. The cavity, along with its vacuum chamber and optical setup, fits inside a 30 L volume. We have measured the cavity's thermal and vibration sensitivities and present our first estimation of the cavity fractional frequency instability at σy(1 s)=7.5×10-15.

Cross-Spectrum PM Noise Measurement, Thermal Energy, and Metamaterial Filters.

Virtually all commercial instruments for the measurement of the oscillator PM noise make use of the cross-spectrum method (arXiv:1004.5539 [physics.ins-det], 2010). High sensitivity is achieved by correlation and averaging on two equal channels, which measure the same input, and reject the background of the instrument. We show that a systematic error is always present if the thermal energy of the input power splitter is not accounted for. Such error can result in noise underestimation up to a few decibels in the lowest-noise quartz oscillators, and in an invalid measurement in the case of cryogenic oscillators. As another alarming fact, the presence of metamaterial components in the oscillator results in unpredictable behavior and large errors, even in well controlled experimental conditions. We observed a spread of 40 dB in the phase noise spectra of an oscillator, just replacing the output filter.

Characterization of zero-bias microwave diode power detectors at cryogenic temperature.

We present the characterization of commercial tunnel diode low-level microwave power detectors at room and cryogenic temperatures. The sensitivity as well as the output voltage noise of the tunnel diodes is measured as functions of the applied microwave power. We highlight strong variations of the diode characteristics when the applied microwave power is higher than a few microwatts. For a diode operating at 4 K, the differential gain increases from 1000 V/W to about 4500 V/W when the power passes from -30 dBm to -20 dBm. The diode white noise floor is equivalent to a Noise Equivalent Power of 0.8 pW/Hz and 8 pW/Hz at 4 K and 300 K, respectively. Its flicker noise is equivalent to a relative amplitude noise power spectral density Sα(1 Hz) = - 120 dB/Hz at 4 K. Flicker noise is 10 dB higher at room temperature.

The Ω Counter, a Frequency Counter Based on the Linear Regression.

This paper introduces the Ω counter, a frequency counter-i.e., a frequency-to-digital converter-based on the linear regression (LR) algorithm on time stamps. We discuss the noise of the electronics. We derive the statistical properties of the Ω counter on rigorous mathematical basis, including the weighted measure and the frequency response. We describe an implementation based on a system on chip, under test in our laboratory, and we compare the Ω counter to the traditional Π and Λ counters. The LR exhibits the optimum rejection of white phase noise, superior to that of the Π and Λ counters. White noise is the major practical problem of wideband digital electronics, both in the instrument internal circuits and in the fast processes, which we may want to measure. With a measurement time τ , the variance is proportional to 1/τ(2) for the Π counter, and to 1/τ(3) for both the Λ and Ω counters. However, the Ω counter has the smallest possible variance, 1.25 dB smaller than that of the Λ counter. The Ω counter finds a natural application in the measurement of the parabolic variance, described in the companion article in this Journal [vol. 63 no. 4 pp. 611-623, April 2016 (Special Issue on the 50th Anniversary of the Allan Variance), DOI 10.1109/TUFFC.2015.2499325].

Characterization of the Individual Short-Term Frequency Stability of Cryogenic Sapphire Oscillators at the 10(-16) Level.

We present the characterization of three cryogenic sapphire oscillators (CSOs) using the three-cornered-hat method. Easily implemented with commercial components and instruments, this method reveals itself very useful to analyze the fractional frequency stability limitations of these state-of-the-art ultrastable oscillators. The best unit presents a fractional frequency stability better than 5 ×10(-16) at 1 s and below 2 ×10(-16) for [Formula: see text].

The Parabolic Variance (PVAR): A Wavelet Variance Based on the Least-Square Fit.

This paper introduces the parabolic variance (PVAR), a wavelet variance similar to the Allan variance (AVAR), based on the linear regression (LR) of phase data. The companion article arXiv:1506.05009 [physics.ins-det] details the Ω frequency counter, which implements the LR estimate. The PVAR combines the advantages of AVAR and modified AVAR (MVAR). PVAR is good for long-term analysis because the wavelet spans over 2τ, the same as the AVAR wavelet, and good for short-term analysis because the response to white and flicker PM is 1/τ(3) and 1/τ(2), the same as the MVAR. After setting the theoretical framework, we study the degrees of freedom and the confidence interval for the most common noise types. Then, we focus on the detection of a weak noise process at the transition-or corner-where a faster process rolls off. This new perspective raises the question of which variance detects the weak process with the shortest data record. Our simulations show that PVAR is a fortunate tradeoff. PVAR is superior to MVAR in all cases, exhibits the best ability to divide between fast noise phenomena (up to flicker FM), and is almost as good as AVAR for the detection of random walk and drift.

Phase noise in RF and microwave amplifiers.

Understanding amplifier phase noise is a critical issue in many fields of engineering and physics, such as oscillators, frequency synthesis, telecommunication, radar, and spectroscopy; in the emerging domain of microwave photonics; and in exotic fields, such as radio astronomy, particle accelerators, etc. Focusing on the two main types of base noise in amplifiers, white and flicker, the power spectral density of the random phase φ(t) is Sφ(f) = b(0) + b(-1)/f. White phase noise results from adding white noise to the RF spectrum in the carrier region. For a given RF noise level, b(0) is proportional to the reciprocal of the carrier power P(0). By contrast, flicker results from a near-dc 1/f noise-present in all electronic devices-which modulates the carrier through some parametric effect in the semiconductor. Thus, b(-1) is a parameter of the amplifier, constant in a wide range of P(0). The consequences are the following: Connecting m equal amplifiers in parallel, b(-1) is 1/m times that of one device. Cascading m equal amplifiers, b(-1) is m times that of one amplifier. Recirculating the signal in an amplifier so that the gain increases by a power of m (a factor of m in decibels) as a result of positive feedback (regeneration), we find that b(-1) is m(2) times that of the amplifier alone. The feedforward amplifier exhibits extremely low b(-1) because the carrier is ideally nulled at the input of its internal error amplifier. Starting with an extensive review of the literature, this article introduces a system-oriented model which describes the phase flickering. Several amplifier architectures (cascaded, parallel, etc.) are analyzed systematically, deriving the phase noise from the general model. There follow numerous measurements of amplifiers using different technologies, including some old samples, and in a wide frequency range (HF to microwaves), which validate the theory. In turn, theory and results provide design guidelines and give suggestions for CAD and simulation. To conclude, this article is intended as a tutorial, a review, and a systematic treatise on the subject, supported by extensive experiments.

New-generation of cryogenic sapphire microwave oscillators for space, metrology, and scientific applications.

This article reports on the characterization of cryogenic sapphire oscillators (CSOs), and on the first test of a CSO in a real field installation, where ultimate frequency stability and continuous operation are critical issues, with no survey. Thanks to low-vibration liquid-He cryocooler design, Internet monitoring, and a significant effort of engineering, these oscillators could bridge the gap from an experiment to a fully reliable machine. The cryocooler needs scheduled maintenance every 2 years, which is usual for these devices. The direct comparison of two CSOs demonstrates a frequency stability of 5 × 10(-16) for 30 s ≤ τ ≤ 300 s integration time, and 4.5 × 10(-15) at 1 day (1 × 10(-14) typical). Two prototypes are fully operational, codenamed ELISA and ULISS. ELISA has been permanently installed the new deep space antenna station of the European Space Agency in Malargüe, Argentina, in May 2012. ULISS is a transportable version of ELISA, modified to fit in a small van (8.5 m(2) footprint). Installation requires a few hours manpower and 1 day of operation to attain full stability. ULISS, intended for off-site experiments and as a technology demonstrator, and has successfully completed two long-distance travels.

Unprecedented long-term frequency stability with a microwave resonator oscillator.

This article reports on the long-term frequency stability characterization of a new type of cryogenic sapphire oscillator using an autonomous pulse-tube cryocooler as its cold source. This new design enables a relative frequency stability of better than 4.5 x 10(-15) over one day of integration. To the best of our knowledge, this represents the best long-term frequency stability ever obtained with a signal source based on a macroscopic resonator.

Phase noise of whispering gallery photonic hyper-parametric microwave oscillators.

We report on the experimental study of phase noise properties of a high frequency photonic microwave oscillator based on four wave mixing in calcium fluoride whispering gallery mode resonators. Specifically, the oscillator generates approximately 8.5 GHz signals with -120 dBc/Hz at 100 kHz from the carrier. The floor of the phase noise is limited by the shot noise of the signal received at the photodetector. We argue that the performance of the oscillator can be significantly improved if one uses extremely high finesse resonators, increases the input optical power, supersaturates the oscillator, and suppresses the residual stimulated Raman scattering in the resonator. We also disclose a method of extremely sensitive measurement of the integral dispersion of millimeter scale dielectric resonators.

Dynamic instabilities of microwaves generated with optoelectronic oscillators.

We introduce a time-domain model to study the dynamics of optoelectronic oscillators. We show that, due to the interaction between nonlinearity and time delay, the envelope amplitude of ultrapure microwaves generated by optoelectronic oscillators can turn unstable when the gain is increased beyond a given critical value. Our analytical predictions are confirmed by numerical simulations and experiments.

The effect of AM noise on correlation phase-noise measurements.

We analyze the phase-noise measurement methods in which correlation and averaging is used to reject the background noise of the instrument. All the known methods make use of a mixer, used either as a saturated-phase detector or as a linear-synchronous detector. Unifortunately, AM noise is taken in through the power-to-dc-offset conversion mechanism that results from the mixer asymmetry. The measurement of some mixers indicates that the unwanted amplitude-to-voltage gain is of the order of 5-50 mV, which is 12-35 dB lower than the phase-to-voltage gain of the mixer. In addition, the trick of setting the mixer at a sweet point--off the quadrature condition--where the sensitivity to AM nulls, works only with microwave mixers. The HF-VHF mixers do not have this sweet point. Moreover, we prove that if the AM noise comes from the oscillator under test, it cannot be rejected by correlation. At least not with the schemes currently used. An example shows that at some critical frequencies the unwanted effect of AM noise is of the same order-if not greater--than the phase noise. Thus, experimental mistakes are around the corner.