Artifacts and Errors in Cross-Spectrum Phase Noise Measurements

This article is intended to give a warning about: (i) the internal processing inside the phase-noise analyzers, (ii) the oscillators whose white phase noise floor seems too low, chiefly the 100-MHz OCXOs, and (iii) the need to introduce in the domain of phase-noise measurements the basic concepts of uncertainty found in the International Vocabulary of Metrology (VIM). The measurement of low-noise quartz oscillators, or of other low-noise oscillators exhibiting a noise floor of the order of $-180$ dBc/Hz or less, relies on the cross-spectrum method. The measurement may take long time, from hours to 1-2 days, for the number of averaged cross spectra to be sufficient to reject the background of the noise analyzer. That said, something anomalous is often seen in a region approximately one decade wide, located where higher-slope noise joins the white floor or the $1/f$ noise. The plot may be quite thick and irregular, and a dip may appear, where the phase noise looks lower than the white noise floor. Such anomalies reveal something worse. The white region of the PM noise spectrum may be affected by gross errors and, in the not-so-rare worst case, is a total nonsense. We address the problem trough a simple experiment, where we insert a dissipative attenuator between the oscillator and the PN analyzer. Surprisingly, in some cases the attenuation results in a lower noise floor. Such erratic behavior is reproducible, having been observed separately in three labs with instruments from the two major brands. We provide the experimental evidence, the full theory, and suggestions to mitigate the problem.


Introduction
Modern analyzers measure the phase noise (PN, or PM noise) by correlation and averaging on the simultaneous measurement of the oscillator under test (DUT) with two separate channels, each consisting on a phase detector and a frequency reference. The DUT noise is extracted after rejecting the background noise of the instrument, assuming that the two channels are statistically independent. After the seminal paper [1], and the early application shown in [2], this choice is adopted by virtually all manufacturers ( Table 1). The dual channel scheme comes in two flavors, with one or two reference oscillators. We focus on the latter because it enables the noise rejection of the reference oscillators, and also of the frequency synthesizers which may be interposed between reference oscillators and phase detectors.
The correlation-and-averaging process rejects the single-channel noise proportionally to 1/ √ m, where m is the number of averaged spectra, that is, 5 dB per factor-of-ten. Nowadays, digital electronics provides a high computing power and memory size for cheap, as compared to the cost and to the complexity of RF and microwave technology. Thus, the theoretical rejection can exceed 30 dB if the experimentalist accepts the long measurement time it takes, ultimately limited by the time-frequency indetermination theorem. However, such rejection cannot be achieved in practice because of fundamental phenomena and artifacts. The thermal energy in the input power splitter [3,4,5,6,7] and impedance matching [8] first caught the attention of the scientific community. These and other problems were addressed in three international workshops [9,10,11].
Most practitioners, naively, believe that a noise analyzer always overestimates the DUT noise because it adds its own background noise. This is not true in the case of the two-channel instruments because the cross spectrum is the frequency-domain equivalent of the covariance. The correlation between channels introduces systematic errors and artifacts, which can be positive or negative. The consequence is that there is no a-priori rule to state whether the instrument over-estimates or under-estimates the DUT noise. A problem is that the noise rejection due to averaging, usually calculated and displayed together with the phase noise, does not account for artifacts and systematic errors. Another problem is that the instruments display the absolute value of the cross spectrum, giving no warning about negative outcomes. The combination of these facts originates erratic and misleading results.
We propose an experiment ( Fig. 1) that reveals the presence systematic errors due to unwanted correlated terms. We focus on the 100 MHz OCXOs because this type of oscillator exhibits the lowest white PM noise floor. However trivial the experiment may seem, nothing even broadly similar has been reported in the literature. We provide all the details related to two specific cases, together with the full theoretical interpretation.

Phase Noise and Thermal Energy
The phase noise is described in terms the power spectral density (PSD) of the random phase ϕ(t), and denoted with S ϕ (f ). A model that is found useful to describe oscillators and components is the polynomial law where the term b 0 is the white PM noise, b −1 /f is the flicker PM noise, b −2 /f 2 is the white FM noise, b −3 /f 3 is the flicker FM noise, b −4 /f 4 is the frequency random walk, and other terms can be added. The quantity L (f ), most often used by the manufacturers, is defined as L (f ) = (1/2)S ϕ (f ) [12]. Because white phase noise is of mostly of additive origin, it holds that where P is the carrier power, and N is the power spectral density of the RF noise. In this context, we prefer the unit W/Hz to J. By analogy with the power spectral density (PSD) N = kT of the thermal noise, we associate to b 0 the equivalent temperature where k = 1.380649×10 −23 J/K (exact) is the Boltzmann constant.

The Effect of the Attenuator
Physical insight suggests that the dissipative attenuator can only degrade the signal-to-noise ratio (SNR), which results in increased PM noise. Focusing on the white noise at the attenuator in and out, we use the subscripts "i" and "o" dropping the subscript 0.
Assuming that everything is matched to the characteristic impedance R 0 , the RF white noise at the attenuator output is where kT i is the input noise, A is the voltage gain of the attenuator, 0 < A < 1, and T a is the temperature of the attenuator. The term kA 2 T i means that the input noise kT i is attenuated by the factor A 2 , like any signal. The term k(1 − A 2 )T a is the thermal noise added by the attenuator. This is obvious if one replaces the oscillator with a resistive load R 0 at the temperature T a . In this condition the output is equivalent to a resistor R 0 at the temperature T a , and the total output noise is kT a , independent of A. Equation (4) which is obviously greater than b i = kT i /P i .

The Cross-Spectrum Estimator
Let us start with the instrument internal signals where a(t) and b(t) are the background noise of the channel A and B, they are statistically independent, and have zero mean and equal or similar variance; c(t) is the target signal, that is, the DUT noise. Thus, the statistical properties of c(t) are measured after averaging out a(t) and b(t). Equations (6)- (7)  The reader interested to know more about the method should refer to [15], [16] and [2]. Denoting the discrete Fourier transform with the uppercase letter, as in X(f ) ↔ x(t), the one-sided cross PSD is The denominator T is the acquisition time, the superscript " * " denotes the complex conjugate, and the factor "2" accounts for energy conservation after suppressing the negative frequencies. Dropping the frequency and expanding X = A + C and Y = B + C we get The mathematical expectation E {S yx } is because E{BA * } = 0, E{BC * } = 0, and E{CA * } = 0. All the useful information is in CC * , thus S c > 0. By contrast, all the background noise goes in BA * , BC * and CA * , and under normal circumstances it is equally distributed between real and imaginary part. It is therefore clear that the optimum estimator is where m denotes the average on m realizations. This estimator has two important properties, (i) it is unbiased, and (ii) it is the fastest because it takes in the smallest amount of background noise. A problem with S yx m is that it is not always positive before averaging out the background noise. The negative outcomes cannot be plotted on a logarithmic scale (dB). The FSWP [17,Equation (4)] uses the estimator Albeit the documentation provided by other manufacturers gives little indication about the estimator, we believe that (12) is the most chosen option.
A reason is that it shows no negative values, which could not be represented on a log scale. Another reason is that such estimator is positively biased, and the bias decreases monotonically as m increases. Thus, under normal circumstances | S yx m | converges to E {S c }, after decreasing monotonically during the measurement process. The estimator (12) matches the instrument behavior we observe in the regular use of the instruments, where no attenuator is inserted at the input.   Now we break the hypothesis of statistically independent channels, and we introduce the disturbing signal d(t) ↔ D(f ), the same in the two channels but for the sign, ς x = ±1 and ς y = ±1, as we did in [18] The signal D is either correlated or anticorrelated. Introducing ς = ς x ς y = ±1, and expanding E {S yx } as above, we find Thus, ς is the sign of the correlation coefficient, and the term ςE S d is a systematic bias, positive or negative. The disturbing signal can be (i) the thermal energy in the input power splitter (Sec. 3.2), (ii) the crosstalk between the two channels (Sec. 5.1), and (iii) the AM noise pickup [19] or other effects not considered here.

The Input Power Splitter
Two types of power splitters are mostly used, shown on Fig 2. The loss-free splitter is a 3 dB directional coupler terminated at one input (dark port).
The resistive splitter is a Y network which attenuates the input signal by 6 dB. Denoting with T o equivalent noise temperature at the oscillator output, and with T s the temperature of the power splitter, trite calculation shows that the correlated RF noise is for the 3-dB dissipation-free coupler. Interestingly, (16) is a classical result from Johnson thermometry [20,21], with well known application in microwaves [22,23]. Similarly, we find for the 6-dB resistive coupler. Deriving (16) and (17) from (15), ς does not need to appear explicitly because it always hold that ς = −1.
Because the output power is P o /2 for the 3-dB splitter and P o /4 for the 6-dB splitter, the output SNR is the same, and the white PM noise is Reference [6, Section IV] provides an extension to other less common types of power splitter.

Hardware Architectures
The FSWP [24,17] is based on the SDR (Software Defined Radio) technology after down-converting the input to an appropriate IF (see [25] for a modern treatise of SDR). The mixers are used in the linear region because linearity prevents the AM noise from polluting the phase noise measurement. The use of I-Q mixers enables to unwrap the phase, and to measure beyond the IF. Two operating modes are used, depending on the Fourier frequency. Up to 1 MHz, the input RF signal is down converted to 1.3 MHz. Beyond 1 MHz, the reference synthesizers are set close to the input frequency, keeping the beat note below 10 Hz. In both cases, I and Q of the down-converted signal are digitized, and the phase information is extracted in FPGA. The FSWP uses a 3-dB coupler as the input power splitter (actually, three different couplers are switched, for < 1 GHz, 1−8 GHz, and 8−50 GHz). The E5052B [26] is based on direct phase detection with double-balanced mixers as the phase-to-voltage converters. The mixers are saturated at both inputs, and driven with synchronous signals kept in quadrature. The mixer output is digitized and processed. The power splitter is a Y resistive network.

Experiments and Results
The experiment consists of the measurement of the white noise floor after inserting various dissipative attenuators in the path from the oscillator under test to the phase noise analyzer, as shown on Figure 1.
Two oscillators are tested, (A) a Wenzel 501-04623E, and (B) a Wenzel 501-25900B "Golden Citrine," both 100-MHz OCXOs intended for the lowest-noise applications. The former dates more than 20 years ago. The latter is the top low-PM-noise oscillator by Wenzel. According to the spectra published on the web pages of several manufactures, (B) is the OCXO that exhibits the lowest white noise we have found, below −190 dBc/Hz [27]. The oscillator is clamped on a vibration-damping breadboard, of the same type commonly seen in optical experiments. A 150-MHz low-pass filter (MiniCircuits SLP-150) is inserted at the oscillator output. The attenuation is obtained by stacking small-size SMA attenuators at the filter output, close to the oscillator. The attenuators (Radiall brand) are intended for DC to 18 GHz. In most of the tests, the phase-noise analyzer is a Rohde Schwarz FSWP 26 with high-stability OCXO and cross-spectrum options. The phase-noise analyzers are referenced to a T4Science Hydrogen maser, in turn monitored vs other masers of the same type. The power is measured with a Rohde & Schwarz power meter, which replaces temporarily the phasenoise analyzer before each measurement. The attenuation is evaluated as the power ratio. All the experiments are done in a Faraday cage with usual isolation transformer and EMI filters. Temperature and humidity are stabilized to 22 ± 0.5 • C and 50% ± 10% by a PID control, which also guarantee a drift smaller than 0.2 K/hour. The environment control is probably overdone for PM noise measurements, yet it helps to get conservative results. Figure 3 shows the phase noise spectra of the oscillator (A), observed with different values of the attenuation between 0 dB and 27 dB. The experimental data (dots) on Fig. 4 are the white PM noise from Fig. 3, averaged on a suitable region 2-3 decades wide. Surprisingly, the observed floor does not match the "attenuator only" plot. The latter is calculated from (5). Instead, the floor decreases monotonically from 0 dB to 15 dB attenuation, and it increases monotonically beyond.
Measuring the oscillator (A) with a Keysight E5052B, we see that the white PM noise decreases monotonically with the attenuation, attends a minimum at 9 dB, and increases at higher attenuation (Fig. 5). We could not push the attenuation beyond 15 dB because the carrier power falls below the minimum for the E5052B.
The anomalously low white PM noise when an attenuator is introduced was first observed by one of us (AR) in his radio amateur lab at home, measuring a Wenzel 501-04538F 10 MHz OCXO with a FSWP 8.
Comparing      Fig. 5. Figure 3-B shows the same plots of Fig. 3-A, just separated for better readability. The most interesting fact is the appearance of dips at 1-1.5 kHz for attenuation of ≥ 18 dB. Figures 6 and 7 refer to the same experiment of Fig. 3 and 4, but for the oscillator (B). In this case the white noise floor increases monotonically with the attenuation, but there is a significant discrepancy between the experi-        mental data and the "attenuator only" floor predicted by (5). Additionally, dips are seen on Fig. 6-B at 2-20 kHz, more noticeable than on Fig. 3

-B.
Inspired by the theory (Sec. 3.1), we hacked a FSWP at the Rohde Schwarz R&D facility in München, extracting { S yx (f ) } and { S yx (f ) }. This instrument is of the same type of that we have in Besancon. In München we measured a third oscillator (C), a 100-MHz Wenzel 501-25900B "Golden Citrine" OCXO, same brand and type of (B). The DUT is connected via a 3-dB attenuator, and the FSWP had internal 5-dB attenuation mechanically switched for better impedance matching. Additionally, there is a 2.4 dB (typical) loss inside the FSWP, before the power splitter. All losses accounted for, the signal level at the power splitter input is 8.

Interpretation
The dips found at 1-1.5 kHz in Fig. 3, and also at 2-20 kHz in Fig. 6, suggest that S ϕ (f ) changes sign at these points. The presence of such dips were predicted by a simulation in [5, Fig. 3] and [5, Fig. 1(b)]. Common sense suggests that S ϕ (f ) > 0 for f < f dip , rather vice versa, because the oscillator's S ϕ (f ) is quite large at low f , and cannot be corrupted by artifacts. This is experimentally confirmed on the oscillator (C), as shown in Fig. 8.
From the theoretical standpoint, the combined effect of the attenuator (5) and of the power splitter (18) results in at the attenuator output. This contains two systematic effects: the attenuator noise (positive), and the thermal energy of the power splitter (negative). At high attenuation (A → 0), the RF spectrum associated to the noise sidebands tends to kT a . In this condition, (19) predicts b o < 0 because at the equilibrium temperature T s inside the instrument is obviously higher than the attenuator (and room) temperature T a . Let us start from the old Wenzel 501-04623E (Fig.4). Using the absolutevalue estimator, the expected b o is Fitting the experimental points with (20) fails because there results a too high T s . Because isolation between channel cannot be perfect, we replace T s with T s − ςT c , where ςT c expresses the crosstalk given in terms of a temperature, and ς has the same meaning as in (15). Accordingly, (20) rewrites as Notice that there are two unknowns in (21), T i and ςT c − T s . The former is dominant at no attenuation (A = 1), where the observed PM noise is rather high. The latter is dominant at high attenuation (A → 0). Because ςT c − T s appears as a single quantity in (21), separating ςT c from T s is somewhat artificial, but it is useful in that it provides physical insight. We assume T a = 295 K (23 • C) and T s = 320 K (47 • C) a convenient round number quite plausible for the instrument inside. Fitting the data of Fig. 4 with (21) results in T i = 4528 K and T c = 122 K. This is the curve labeled "full model." Using b i = kT i /P i , with P i = 9.6 mW (+9.8 dBm at A = 1), we get b i = 6.5×10 −10 rad 2 /Hz (−171.9 dBrad 2 /Hz). Comparing this value to the readout (−172.4 dBrad 2 /Hz, at A = 1), the instrument introduces a bias of −0.5 dB due to the combined effect of power splitter and crosstalk.
Removing the absolute value in (21) yields which results in b o > 0 up to 15 dB attenuation, and in b o < 0 beyond. Rewriting the polynomial model (1) for the absolute-value estimator  Fig. 3-A, we find the solid lines overlapped to the experimental spectra of Fig. 3-B. The model matches the experiment, and predicts precisely the dips. These dips occurr at ≥ 18 dB attenuation, where b o < 0. Now we turn our attention to the Wenzel 501-25900B "Golden Citrine," the oscillator (B). Looking at Fig. 6-A and Fig. 7, we notice that the white noise floor increases monotonically increasing the attenuation, and the dips are present for all the values of the attenuation -albeit these dips are not clear at 0 dB and 6 dB because of insufficient averaging. This indicates that b o < 0 in all cases. Evaluating (23) with the same T a , T s and T c as above, we find T eq = T i = 50 K. The model fits well the experimental data, as shown on Fig. 7. The temperature of 50 K is equivalent to a white noise floor of −200 dBrad 2 /Hz at +18.5 dBm (70.5 mW) output power, with no attenuation. Finally, (23) predicts precisely the dips seen on at Fig. (6)-B.

The Origin of the Crosstalk
Trying to understand the crosstalk, we look at the part of the FSWP where the strongest and the weakest signals come close to one another, which is the input mixer. Let us put numbers together with this idea. For linear conversion, the LO signal should not be lower than +20 dBm. The phase noise of a state-of the art synthesizer at 100 MHz carrier is of the order of −160 dBrad 2 /Hz. For reference, the R&S SMA100A synthesizer with the lowphase-noise option SMA-B22 has a white floor of this order [28, data sheet, p. 12]. At +20 dBm power, the white-noise sidebands are of −140 dBm/Hz, that is, 10 −17 W/Hz. The crosstalk kT c we search for is of 1.7×10 −21 W/Hz with T c = 122 K, This is 38 dB smaller than the LO sidebands. A coupling of the order of −38 dB due to leakage is quite plausible for a good mixer circuit. Besides, the absence of discontinuity in the spectrum (Figs. 3 and 6) at 1 MHz indicates that the crosstalk does not depend on the operating mode, which excludes some other parts of the instrument. Anyway, this interpretation is just a guess, not based on the internal design nor on specific measurements.

Inside the Oscillator
We address the question of the origin of T eq , and why it can be smaller than the room temperature. From our purposes, the oscillator consists of a core (the auto-oscillator in strict sense), a buffer, and an output filter (Fig. 9). The attenuation in the filter stopband is generally achieved by reflecting the power back to the generator's internal impedance.
The conventional oscillators may have a lowpass or bandpass RLC filter at the output to suppress the harmonic distortion and to solve other practical problems ( Fig. 9-A). Such filter cannot have a bandwidth smaller than a few MHz at 100 MHz carrier because the quality factor Q of these resonators is of the order of 10-20 in practical conditions. As a consequence, the output impedance is reasonably matched in the whole Fourier-frequency span, and the white noise is chiefly the noise of the sustaining amplifier, where the carrier is the weakest.
In the thermally limited quartz oscillator, a quartz resonator is present between the core and the buffer. Such filter can be the main resonator if the carrier is extracted from the resonator's ground pin [29], [30,, or a second quartz resonator ( Fig. 9-B). Out of the resonator bandwidth ν 0 (1 ± 1/2Q), the quartz is a high impedance circuit, thus the noise of the sustaining amplifier is not transmitted to the buffer. The noise associated to the resonator's motional resistance is also rejected, for the same reason. The buffer (a common-base amplifier) has low input impedance and low noise figure, thus the white noise is chiefly limited by the physical temperature of the collector resistor R C at the output. Such oscillators may have an additional RLC output filter of the same type discussed before.
In the sub-thermally limited quartz oscillator, a quartz resonator or a quartz filter is introduced in series to the output [31, Fig. 7], with no further amplification. The output filter has a small cutoff frequency even with the low Q imposed by the heavy load condition. For example, taking Q = 5000 at 100 MHz, the cutoff frequency is f c = 10 kHz. For comparison, a good resonator at this frequency has Q > 10 5 , unloaded. Out of the bandwidth ν 0 (1 ± 1/2Q), the output impedance is quite high (|Z o | 50 Ω), which gives the appearance of a cold source. There no violation of the second principle because the filter is obviously in thermal equilibrium with the environment. However, the electrical access to the thermal energy is open. In this condition, the input power splitter of the noise analyzer is reasonably well matched only in the pass band, and nearly open circuit in the stopband. In the stopband, the expected cross spectrum relates to the thermal energy of the power splitter (and to the crosstalk, if any), wich has negative sign in the correlation.
Simple attempts to measure the output impedance failed because the impedance analyzers do not work in the presence of the strong carrier at the input. We disassembled two 100 MHz oscillators, a Wenzel 501-04623E and a Wenzel Citrine, the same type as the oscillator (A) and (B), respectively. The oscillator (A) is of the conventional type, with a RLC filter at the output. The white PM noise limited by the signal-to-noise ratio in the sustaining amplifier. The oscillator (B) is of the sub-thermally limited type, with a quartz resonator in series to the output. Albeit we did not reverse-engineer the oscillator, the two values of T eq , 4528 K and 50 K, are consistent with the oscillator architecture.

Discussion and Conclusions
The main outcome of our experiments is the evidence that pushing the noise rejection too far by averaging on a large number of data may result in misleading or grossly wrong results. The reason is in residual correlated effect, not under control. In general terms, under-estimating the DUT noise is obviously worse than over-estimating it.
The second outcome is that impedance matching in the whole analysis bandwidth is a critical issue. Sub-thermally limited oscillators exploit impedance mismatch in the filter stopband to deliver the lowest noise. Erratic results occur in the stopband because the power splitter delivers anti correlated output signals. From a different standpoint, the benefit of a subthermally limited oscillator is unclear to us if the oscillator is intended to be a part of a system at room temperature.
One idea is new, that the (anti-)correlated noise inside the instrument can be modeled with a temperature (T s +T c ). Let us look at T s and T c separately. Because (18) is based on simple and well-established physics, a software correction inside the instrument could compensate for T s in a reliable way. Likewise, (5) culd be used to correct for the effect of a switchable dissipative attenuator, if present. Unlike T s , there is no general way to compensate for T c . We have no a priori reason to trust it as as a constant in the carrierfrequency range (4 decades), nor as reproducible parameter across different specimens or architectures. The brute force approach of putting the power splitter in a liquid-He cryostat [7] is not effective because of the crosstalk. In our experiment 70% of the bias error is due to the power splitter, 30% to the crosstalk. However, compensating for T s alone is a general solution, and mitigates the problem.
The real-part estimator S yx m is superior to the traditional estimator S yx m in that (i) it converges faster because the background noise in S yx m is not taken in, and (ii) it reveals the negative, nonsensical outcomes.
Measuring the oscillator (A), one may be satisfied of the spectra taken with no attenuation (A = 1) because • Two instruments from the major brands, with similar correlation algorithm but radically different in the RF architecture and in the detection principle, are in a perfect agreement.
• The systematic error in the white noise, revealed by our rather complex experiment, is of a mere −0.5 dB, not alarming.
By contrast, the white noise floor measured on the oscillator (B) is a complete nonsense because S yx (f ) m < 0 inside the instrument. Unlike most domains of metrology (mass, length, etc.), a PM noise spectrum consists of hundreds or thousands of points on the S ϕ (f ) plot. The common ditto too much information is no information rises the question of the nature of the measurand. General experience indicates that the polynomial law (1) describes well the PM noise spectrum of quartz ad dielectric oscillators, thus a small number (4-5) of parameters b n tell the whole story. In optics, some additional terms appear, like bumps and blue noise, which call for a small number of additional coefficients [32]. Regardless of the model we choose, a small number of missing points, like the negative spurs we have seen, is not a real nuisance and can be ignored. The experienced scientist does this after visual inspection. By contrast, an irregular behavior over a wide frequency range has to be taken seriously.
Ultimately, some concepts found in the International Vocabulary of Metrology (VIM) [33] and in the Guide to the Expression of Uncertainty in Measurement (GUM) [34] (see also [35,36]) should be introduced in phase noise measurements. From the VIM: Type-A and Type-B evaluation of uncertainty (2.28 and 2.29), the influence quantities (2.52), the definitional uncertainty (2.27), and the null measurement uncertainty (4.29).
Because none of us is a true expert of uncertainty in metrology, subtleties may escape from our attention. However, this article shows that the assessment of uncertainty in PM noise is still at a too rudimentary stage. The following digression is intended to stimulate a discussion, with no intention of stating rules.
The single-channel background of the instrument is chiefly zero-average Gaussian noise with white or flicker spectral distribution. Thus, it falls in the A-type uncertainty, which can be reduced with statistical processing on the time series. By contrast, the correlated or anti-correlated effects contribute to the B-type uncertainty. They can be quantified, and in some cases partially corrected, only after understanding the system in depth. This is the case of T s and T c .
We have seen that the output impedance Z o (f ) produces erratic results if it changes significantly in the analysis bandwidth. This opens the question of whether Z o (f ) goes in the definitional uncertainty (it is inside the DUT), it goes in the B-type uncertainty, or if it is an influence quantity. The role of impedance mismatch is well known in microwave noise measurements [37,38], but these concepts have not been transposed to PM noise.
The combined uncertainty is u = u 2 A + u 2 B , where, u A can be reduced by increasing the number of averages, not u B . This is where we see the importance of the null measurement uncertainty (Fig. 10). Suppose that the numerical outcome of the experiment is v ± u. In (A) and (B), we see v ± u > 0, likely with a smaller u in (B) because of averaging on a larger number of spectra (smaller u A ). By contrast, in (C) the uncertainty bar hits negative values because v − u < 0. This is not a valid result because S ϕ (f ) cannot be negative. We would rather replace the result with S ϕ (f ) = 0, with a detection threshold u. In this case, u takes the meaning of the minimum PM noise that can be detected. The case (D) should be discarded at sight.

Disclaimer
Our strong statements require an equally strong disclaimer about the commercial products we refer to. We experimented on them because they were on hand at the right time, as opposite to gathering parts with this research in mind. By no means we criticize these products, nor we endorse them. The problems and the inconsistencies we describe relate to unintended, strange, or weird use of these products. Driven by the genuine scientific curiosity, we share our knowledge with the ultimate intent to contribute to better understanding the physics and the technology of phase noise metrology. We hope that no misunderstanding will arise, and we apologize if this will happen.