Measuring the Power Law Phase Noise of an RF Oscillator with a Novel Indirect Quantitative Scheme

electronics

Article Measuring the Power Law Phase Noise of an RF Oscillator with a Novel Indirect Quantitative Scheme

Xiaolong Chen 1,* , Cuiling Peng 1 , Huiting Huan 1, Fushun Nian 1,2 and Baoguo Yang 2 1 School of Mechano-Electronic Engineering, Xidian University, Xi’an 710071, China 2 Science and Technology on Electronic Test & Measurment Laboratory, Qingdao 266555, China * Correspondence: [email protected]

Received: 30 April 2019; Accepted: 8 July 2019; Published: 9 July 2019

Abstract: In conventional phase noise metrology, the phase noise of an oscillator is measured by instruments equipped with specialized and sophisticated devices. Such hardware-based testing usually requires high-performance and costly apparatuses. In this paper, we carried out a novel phase noise measurement method based on a mathematical model. The relationship between the phase noise of a radio frequency oscillator and its power spectral density (PSD) was established, different components of the power law phase noise were analyzed in the frequency domain with their characteristic parameters. Based on the complete physical model of an oscillator, we fitted and extracted the parameters for the near-carrier Gaussian and the power law PSD with Levenberg-Marquardt optimization algorithm. The fitted parameters were used to restore the power law phase noise with considerable precision. Experimental validation showed an excellent agreement between the estimation from the proposed method and the data measured by a high-performance commercial instrument. This methodology can be potentially used to realize fast and simple phase noise measurement and reduce the overall cost of hardware.

Keywords: Leeson’s power law model; phase noise; power spectral density; parameter ﬁtting

1. Introduction Phase noise occurs in almost all sorts of oscillators [1], ranging from radio frequency (RF) [2] to optics [3], and presents a direct impact on the performance of modern communication, radar and navigation systems in terms of reliability, detectability, and accuracy [4]. The critical step of phase noise abatement should be particularly considered in fiber-optical communication [5], and the determination of phase noise draws attention from engineers and researchers in the RF electronic community [6,7]. Modern instrumental measurement of the noise performance of an oscillator can be sorted into three methodologies, namely the direct phase detection, delay-line discrimination, and digital phase demodulation [8]. The frequency and phase demodulation are particularly important and used widely for commercial instruments [9]. However, these approaches usually adopt either a delay-line homodyne [10] or a phase-lock-loop to discriminate the phase fluctuation and measure the noise [11]. A reference source must be used, and its frequency should be maintained the same (synchronized) as the signal-under-test (SUT). Practically, the phase noise of the reference source must be at least one order lower than the SUT in order to realize reliable measurement. The accuracy of the phase noise measurement depends heavily on the performance of phase-locked loop (PLL) and other functional circuits. These requirements have technical difficulty and can inevitably increase the total cost of the measurement. In the community of signal noise measurement, elaborate designs of testing apparatuses have been materialized in both the time-domain and frequency-domain to precisely measure the phase noise performance of an RF free oscillator. However, such straightforward measurement relies on

Electronics 2019, 8, 767; doi:10.3390/electronics8070767 www.mdpi.com/journal/electronics Electronics 2019, 8, 767 2 of 9 high-performance hardware and does not consider the nature of the phase noise of a source. Recently, studies that focused on the generation mechanism of the phase noise were being carried out by many groups. The mathematical models of phase noise allowed phase noise of an oscillator to be characterized by mathematical manipulations [12]. Given substantial advantages of cost-effectiveness and simplified setup, phase noise evaluated by software-based mathematical algorithms has great potential in practical measurement. Theoretical [13] and experimental [14] works have been contributed to the fundamentals of phase noise models both in the time and frequency domain during the past decades [6]. In the frequency domain, phase noise was usually characterized by its power spectral density (PSD), which can be derived as the Fourier transform of the autocorrelation function [15]. Suggested by Leeson et al. [16,17], the spectrum of the phase noise of a free oscillator can be intuitively depicted as a power-law relation. The independent power indices indicate five sorts of phase noise which were studied by parts [17] or separately [18] in terms of their PSD where the factor of each power law term indicates the magnitude of its corresponding noise component. However, such “heuristic derived” [16] factors in the Leeson’s model, have not been determined directly, only qualitative conclusions were drawn from simulation from many of the presented works. Among these works, Chorti et al. [19] developed a comprehensive power law model with respect to the PSD of a free oscillator by analyzing zero-mean Gaussian stochastic phase noise in the frequency domain, which essentially connects the power law phase noise of the oscillator to its measurable power spectrum density. Unfortunately, the work mainly focused on theoretical simulation and did not move forward to the phase of experimental validation, or as proposed in this work, determine the phase noise quantitatively. In this work, we proposed a novel phase noise measurement method with the application of an extended form of the Leeson’s power law theory proposed by Chorti et al. [19]. The testing modality is based on a mathematical model which correlates the phase noise components of the SUT with its power spectral density. The PSD data were measured and fitted to the model obtaining parameters with respect to the phase noise model, and consequently, the phase noise can be determined. The mathematical manipulation algorithm was predesigned with software programs which can reduce the use of complicated circuits or hardware and simplify the testing devices. To validate this methodology, a group of contrast experiments were carried out with error analysis. The results indicate that the proposed method exhibited similar accuracy with respect to a high-performance commercial phase noise analyzer.

2. System Model

2.1. Testing Modality In order to resolve the main confinement of high-performance functional hardware, this research developed a mathematical phase noise measurement modality. The testing schematic is depicted in Figure1. The SUT was output from an RF signal source and its PSD was measured by a spectrum analyzer. As discussed above, the phase noise of the oscillator under test can be well described with power-law relation. With this knowledge of SUT, its phase noise was obtained further by fitting the PSD data to a composite power-law expression, and the coefficients were extracted. To validate the performance of measurement, a high-performance phase noise analyzer was used to test the same SUT. The commercial phase noise analyzer was equipped with specially designed hardware and sophisticated modules for phase noise measurement, therefore, its measurement can be regarded as a reliable reference. The measurement results were compared with the ones acquired by the proposed method. Electronics 2019, 8, 767 3 of 9 Electronics 2019, 8, x FOR PEER REVIEW 3 of 10

RF Oscillating signal

Spectrum analyzer

PSD data

Optimization / Fitting Composite theoretical model

Restoring Coeffiicients of Phase noise PSD power law model

Figure 1. Schematic of phase noise measurement modality. Figure 1. Schematic of phase noise measurement modality. 2.2. Phase Noise Model 2.2. Phase Noise Model The mathematical derivation of phase noise PSD should be sought and correlated with the PSD of theThe oscillator. mathematical Basically, derivation a non-ideal of phase oscillator noise PSD can should be expressed be sought by and a combination correlated with of noise the PSD and a ofsinusoidal the oscillator. signal Basically, [20], written a non in-ideal a complex oscillator form can as be expressed by a combination of noise and a sinusoidal signal [20], written in a complex form as V(t) = [V0 + a(t)] exp i[2π f0t + ϕ(t)] (1) V( t )= [ V00 + a ( t )]exp i [2 f t + ( t )] (1) wherewhere a(at)(t )and and ϕ(t()t )denote denote the the amplitude amplitude and and phase phase noise noise,, respectively. respectively. The The phase phase noise noise becomes becomes dominantdominant when when the the offset oﬀset frequency frequency range range is is small small and and close close to to the the central central frequency frequency f0f 0[2[121], ],within within whichwhich our our measurements measurements were taken.taken. InIn thethe frequency frequency domain, domain, the the single single sideband sideband PSD PSD of phase of phase noise noiseL(f ) isL(f) of is main of main interest interest in practical in practical measurement measurement and and expressed expressed as as

1 Z+ + = = −  Lf()1 Sf () Rt ∞ ()exp(2 iftdt ) (2) L( f ) = S2ϕ( f ) = 0 Rϕϕ(t) exp( 2πi f t)dt (2) 2 0 − In Equation (2), f indicates the frequency offset with respect to the nominal central frequency f0 and RIn( Equationt) is the autocorrelation (2), f indicates the of frequency phase fluctuation. offset with respect Equation to the (2) nominal can also central be expressed frequency inf 0 termsand R ϕϕ of( t) frequencyis the autocorrelation noise by ofrewriting phase fluctuation. the relative Equation frequency (2) can deviation also be expressed in the in time terms domain of frequency as 1 noise by rewriting−1 the relative frequency deviation in the time domain as y(t) = (2π f0)− dϕ(t)/dt, y( t )= (2 f ) d ( t ) dt , and the PSD for the frequency noise can be expressed as and the PSD0 for the frequency noise can be expressed as f 2 = 2 Sy ()() f2 f S f (3) S ( f ) =f0 S ( f ) (3) y 2 ϕ f0 To analyze the SUT from an oscillator, the PSD of V(t) should be derived. Taking Fourier transform withTo analyzerespect to the V(t) SUT yields from an oscillator, the PSD of V(t) should be derived. Taking Fourier transform with respect to V(t) yields ith()iϕ(t)i VV(()f )= =δ()δ f( f )  e e (4)(4) ∗ F wherewhere  denotesδ denotes the the Dirac Dirac delta delta function function and andthe nominal the nominal central central frequency frequency f0 is normalizedf 0 is normalized by using by theusing coordinate the coordinate of offset of o frequencyffset frequency f. Forf . the For case the case of a of relatively a relatively precise precise and and high high-performance-performance oscillator,oscillator, (ϕt)( tis) issmal smalll and and we we consider consider the the Taylor Taylor expansion expansion of of the the second second term term in inEquation Equation (4), (4), viz. viz. 2 exp[iϕ(t)] 1 + iϕ(t)2 1/2ϕ (t) + , the PSD of V(t) would be written as exp[i ()] t 1 +≈ i  () t − 12  ()− t + , the PSD··· of V(t) would be written as

 SV( f ) = δ( f ) + R( f ) + ϕ( f0)ϕ∗( f ) = δ( f ) + R( f ) + Sϕ( f ) (5) SfV ()=δ()()()() fRf + + f0 f =δ())() fRfSf + ( +  (5) wherewhere RR(f)( f comes) comes from from the the nonlinear nonlinear terms terms of of (ϕt)( t and) and can can only only be be neglected neglected for for the the small small-angle-angle approximation. In other cases, SV( f ) should be discussed independently [18]. approximation. In other cases, SfV () should be discussed independently [18].

Electronics 2019, 8, 767 4 of 9

The power law relation that interprets phase noise in the frequency domain has been discussed by A. Demir [13], S. Yousefi [18], C. Greenhall [22], and N. Ashbby [23], etc. Qualitatively, the PSD of the phase noise term Sϕ(f ) exhibits a Gaussian shape when close to the carrier, or f ~ 0, which is followed by the so-called “power law region” [24]. The latter can be concisely expressed as the summation of hα fα over α = 0, 1, 2, 3 and 4, representing white phase noise, flicker phase noise, white frequency − − − − modulated (FM) noise, flicker FM phase noise, and random walk FM phase noise respectively. Each hα stands for the magnitude of the corresponding noise component. Considering the practical bandwidth of commercial instruments, such five sorts of noise can be expressed in the PSD form of both SV( f ) and Sϕ( f ) as summarized in Table1[ 17–19,22]. It can be seen that for a large offset frequency f, Sϕ( f ) gradually approaches SV( f ), which conforms well with the small-angle approximation. For the cases of flicker FM and random walk FM phase noise, they will generate Gaussian components in the PSD of the oscillator which becomes significant for a small f. It should be mentioned that, although the factors hα are all constants upon the foregoing analysis, they would vary with respect to different carrier f 0, which results in a more strict estimation model; our experiments nevertheless showed phase noise approximated even with even constant factors agreed perfectly with data measured by a commercial phase noise analyzer in practice.

Table 1. The power law phase noise occurs in SV(f ) and Sϕ(f )[19].

Noise Type α Sϕ(f) SV(f) Comments B is the bandwidth of the 2 h0B testing instruments. SV(f ) and White phase noise 0 2 2+ 2 h0 4π f B Sϕ(f ) become constants only if B f 2ν1 (2π)− h 1 h 1 Flicker phase noise 1 + − 1+−ν 0 < ν1 1 − f 1 ν1 f 1 | | | | h 2 h 2 SV(f ) and Sϕ(f ) become − White FM noise 2 f−2 π2h2 + f 2 2 2 2 identical if f >> π h − − 2 − q is a function of ν3 as (0 < ν3 1): 2 π f 3ν3 Flicker FM h 3 − − 1 q2 h 3 2 ν /4 3 3 ν3 e− + − = 3 f − q 3 ν3 q 2 √π π− phase noise − f − r · | | 1 | | Γ( ν3)h 3 2 − (2 ν )Γ( 3 1 ν ) − 3 2 − 2 3 p f 2 f is the starting oﬀset q p 2πh 4 4 Random walk FM h 4 2 e− − + frequency for f noise, p is a − 2π h 4 − 4 p2 f 2+ f 4 − phase noise − h 4 constant and H(*) denotes the −4 H( f f ) f − Heaviside step function.

2.3. Factor Approximation The analytical forms of the oscillator’s PSD and concomitant phase noise consist of ten unknown parameters, five of which being the power-law factors, viz. hα. To conduct efficient parameter approximation and optimization, we simplify our data fitting process and assume that: (i) For commercial testing instruments, their bandwidth is usually much larger than the SUT and thus the white phase noise component in SV( f ) is the same as its counterpart in Sϕ( f ), (ii) with ν1 being a very small positive value for the flicker noise of an oscillator, the identity also stands with respect to flicker phase noise and results in a simple 1/f spectrum, (iii) parameters p and f are actually connected and functions of h 4 [19]. For a real oscillator, h 4 and p are usually small and the effect of the power law − − starting frequency f is limited, which will be verified by the experimental results in the following section. With the above assumptions, only six parameters need to be determined, namely the Gaussian Electronics 2019, 8, 767 5 of 9

component q and the power law factors hα.. The objective function F(f ) for optimization can be written, based on the least square criterion [25], as

n 2o F( f ) = min [10lg[SV( f )] Pe( f )] (6) hα,q,p −

2 q 2 1 π f p p f SV( f ; hα, q, p) = exp 2 + 2 exp + q q 2π h 4 2πh 4 − − − (7) + 1 + ( 2 2− + 2) 1 + 3 + 4 h0 h 1 f − h 2 π h 2 f − − h 3 f − h 4 f − − − − − − where Pe(f) is the experimental tested PSD in the logarithmic coordinate. It should be noted that the objective function Equations (6) and (7) had been modiﬁed by omitting the Heaviside step function but adding parameter p in the variable list. Such setting violates the equations with respect to power unity and PSD continuity [19], hence, causes a perceptible discrepancy for frequencies close to the carrier, nonetheless, it has limited impact upon the determination of phase noise within the power law region, but would eﬀectively decrease the computational complexity. The constraints for the variables are ( 1 > hα, p, q > 0 2 (8) p 2π h 4 ≤ − in which the upper bound for the factors of all terms in Equation (7) is set to 1 to comply with the energy conservation of an oscillator’s PSD. Equations (6)–(8) can be optimized by a gradient-based Levenberg-Marquardt algorithm [23]. The Levenberg-Marquardt algorithm is the most widely used non-linear least squares algorithm. By giving the initial iteration value, the optimal solution can be obtained by gradient descent faster.

3. Results and Discussion To validate the phase noise measurement capability by the mathematical relationship between the power spectrum and power law phase noise model, a testing system was established. In the experimental arrangement, the SUT was a carrier signal directly output from an RF signal source (R&S®SMW200A), and its PSD was measured by a spectrum analyzer (R&S®FSW). R&S®SMW200A has good phase noise performance (single-sideband phase noise 90 dBc/Hz for 1 GHz carrier ≤ and frequency offset >10 Hz, the overall noise pattern is close to the theoretical power-law model according to the instrumental specification [26]) which is conducive to experimental verification. The mathematical model gives predictions for the phase noise of the RF source at different carrier frequencies. The reference phase noise data was offered by a high-performance phase noise analyzer (R&S®FSWP). A group of error and accuracy analysis was carried out for further analysis. The capability of phase noise measurement by the proposed method was validated at a series of carrier frequencies. The carrier frequency of 10 MHz, 100 MHz, and 600 MHz was used as examples for phase noise analysis. The normalized single sideband PSD of the SUT at 600 MHz carrier obtained by the spectrum analyzer is shown in Figure2 with the abscissa denotes the frequency o ffset (in Hz). In order to enhance the optimization, it would be very helpful to qualitatively analyze the power spectrum prior, to generate a reliable guess and narrow the variable bounds. The parameters after fitting are used to restore phase noise. Intuitively, one can observe a baseline for frequency larger than ~10 kHz in spite of some fluctuations. The actual values of these fluctuations are very small in linear coordinates. Since the least square method is used in the calculation process, the error between experimental and numerical values is the smallest. The baseline can be mainly characterized by white 13 15 phase noise [17] and the constraint for h 4 can be further narrowed as 10− > h 4 > 10− . Similarly, − − 14 8 the lower and upper bound of the rest power-law coefficients can be modified as from 10− to 10− consecutively. Levenberg-Marquardt iteration is able to find the optimized fitting with fast convergence with proper guesses and narrowed boundary limit for hα, α = 0, 1, 2, 3 and 4. The optimized − − − − fitting results are shown in Figure2 (red-dashed-line). Electronics 2019, 8, x FOR PEER REVIEW 6 of 10

Marquardt iteration is able to find the optimized fitting with fast convergence with proper guesses and narrowed boundary limit for hα, α = 0, −1, −2, −3 and −4. The optimized fitting results are shown in Figure 2 (red-dashed-line). The extracted parameters for the PSD of the three groups of SUT are presented in Table 2. The value of ν3 is derived from the expression of q and conforms well to the desired feature of ν3→0 described in [17] and [18]. Due to the absence of f in the model, p is fitted out to be a very small number, and the spectrum of near-carrier range is resulted from the superposition of Gaussian and −4 f components without a knee point. This feature obviously violates the equations of power unity and spectral continuity, and that is the reason for the larger discrepancy between the experimental and fitted spectrum close to the carrier (< ~4 kHz). For larger offset frequency range, the fitting curve correctly predicts the trend and change pattern of the oscillator’s PSD.

Table 2. The controlled variables extracted by the best fitting of power spectral density (PSD).

Controlled Variable Value (10 MHz) Value (100 MHz) Value (600 MHz) q 0.240 0.500 0.500 ν3 9.00 × 10−4 2.00 × 10−4 2.0 × 10−4 p 3.95 × 10−15 7.54 × 10−19 4.55 × 10−16 h0 1.69 × 10−14 1.55 × 10−14 1.28 × 10−14 h−1 1.36 × 10−10 1.51 × 10−10 1.58 × 10−10 h−2 1.36 × 10−8 2.33 × 10−8 3.79 × 10−8 −7 −7 −7 Electronics 2019, 8, 767 h−3 4.91 × 10 4.31 × 10 3.22 × 10 6 of 9 h−4 8.81 × 10−10 1.81 × 10−13 8.13 × 10−11

-30 Measured PSD of 600MHz oscillator -40 Fitting from Eq.(6) -50

-60

-70

-80

-90

-100

Power spectral density(dB) Power spectral -110

-120

-130

-140 3 4 5 10 10 10 Frequency(Hz)

FigureFigure 2. 2.The The PSD PSD data data ﬁtting fitting by by the the composite composite theoretical theoretical model model (f (0f0= = 600600 MHz,MHz, RMSERMSE= = 7.53).

TheTo validate extracted the parameters correctness for theof the PSD proposed of the three meth groupsod, the of SUTphase are noise presented of the in same Table oscillator2. The value was of ν is derived from the expression of q and conforms well to the desired feature of ν 0 described measured3 at 10 MHz, 100 MHz, and 600 MHz, respectively, by a commercial high-performance3→ phase innoise [17, 18analyzer]. Due (R&S to the® absenceFSWP), again, of f in the the result model, of 600p is MHz fitted is out given to beas aan very example small shown number, in Figure and the 3. 4 spectrumThe PSD data of near-carrier were obtained range multiple is resulted times, from and the the superposition extracted phase of Gaussiannoise was andaveraged.f − components The results withoutshow that a kneethe phase point. noise This retrieved feature obviously from the calc violatesulation the is equations in good agreement of power unitywith the and measured spectral continuity,phase noise and and that satisfies is the the reason power for law the model. larger Addi discrepancytionally, Figure between 4 plots the the experimental average error and between fitted spectrumthe phase close noise to restored the carrier and (< measured~4 kHz). with For larger respect off toset different frequency frequency range, the offsets. fitting The curve error correctly is well predictsbelow 5% the for trend all frequency and change offsets. pattern Also, of thethe oscillator’spiecewise maximum PSD. error is shown in Figure 5, with the frequency offset range being divided into four subsections consecutively. It can be observed that at Table 2. several differentThe frequencies, controlled variables the errors extracted are larger by the in best near-carrier fitting of power range spectral (< 100 density Hz), because (PSD). of the superpositionControlled of VariableGaussian and Value untruncated (10 MHz) f-4 component. Value (100 For MHz) the frequency Value offset (600 above MHz) hundreds of hertz, the discrepancy reduces gradually and is smallest for 1–10 kHz range. The SUT PSD deduced q 0.240 0.500 0.500 4 4 4 ν3 9.00 10− 2.00 10− 2.0 10− × 15 × 19 × 16 p 3.95 10− 7.54 10− 4.55 10− × 14 × 14 × 14 h0 1.69 10− 1.55 10− 1.28 10− × 10 × 10 × 10 h 1 1.36 10− 1.51 10− 1.58 10− − × 8 × 8 × 8 h 2 1.36 10− 2.33 10− 3.79 10− − × 7 × 7 × 7 h 3 4.91 10− 4.31 10− 3.22 10− − × 10 × 13 × 11 h 4 8.81 10− 1.81 10− 8.13 10− − × × ×

To validate the correctness of the proposed method, the phase noise of the same oscillator was measured at 10 MHz, 100 MHz, and 600 MHz, respectively, by a commercial high-performance phase noise analyzer (R&S® FSWP), again, the result of 600 MHz is given as an example shown in Figure3. The PSD data were obtained multiple times, and the extracted phase noise was averaged. The results show that the phase noise retrieved from the calculation is in good agreement with the measured phase noise and satisfies the power law model. Additionally, Figure4 plots the average error between the phase noise restored and measured with respect to different frequency offsets. The error is well below 5% for all frequency offsets. Also, the piecewise maximum error is shown in Figure5, with the frequency offset range being divided into four subsections consecutively. It can be observed that at several different frequencies, the errors are larger in near-carrier range (< 100 Hz), because of the superposition of Gaussian and untruncated f -4 component. For the frequency offset above hundreds of hertz, the discrepancy reduces gradually and is smallest for 1–10 kHz range. The SUT PSD deduced phase noise curve shows excellent agreement with the commercial apparatus measured data. The error generally agrees well with the fitting results of the power spectrum. By referring to the reliable reference, this proposed method is proved to be stable and accurate. Electronics 2019, 8, x FOR PEER REVIEW 7 of 10 phase noise curve shows excellent agreement with the commercial apparatus measured data. The Electronics 2019, 8, x FOR PEER REVIEW 7 of 10 error generally agrees well with the fitting results of the power spectrum. By referring to the reliable reference,phase noise this curve proposed shows method excellent is proved agreement to be withstable the and commercial accurate. apparatus measured data. The errorTo generally summarize agrees the well testing with procedure the fitting withresults this of thescheme, power we spectrum. need a spectrumBy referring analyzer to the reliableand a computerreference, embedded this proposed with method an optimization is proved algorithm. to be stable The and procedure accurate. is as follows: (i) testing the SUT with spectrumTo summarize analyzer the and testing obtain procedure its PSD with with the this be stscheme, resolution we thatneed can a bespectrum achieved, analyzer (ii) PSD and data a wascomputer sent to embedded the computer with to an implement optimization data algorithm. fitting with The the procedure composite is power as follows: law model (i) testing and theextract SUT coefficients,with spectrum (iii) analyzer the phase and nois obtaine is its measured PSD with by the calculating best resolution the thatpower can law be achieved, expression (ii) with PSD datathe obtainedwas sent coefficients to the computer and plotting to implement the curves. data fitting with the composite power law model and extract coefficients,Despite the(iii) compact the phase and nois effectivee is measured testing scheme by calculating, the measurement the power method law expression proposed iswith based the onobtained sheer power coefficients law model. and plotting In fact, the there curves. is also Gaussian component of the phase noise which has been neglectedDespite the in compactthis scenario. and effectiveFor a real testing oscillator, scheme the precision, the measurement of this method method would proposed be limited is based for smallon sheer frequency power offset law model. range. InFurthermore, fact, there is the also perf Gaussianormance component of the measurement of the phase is closely noise relatedwhich hasto thebeen SUT neglected noise feature, in this which scenario. should For abe real known oscillator, first, forthe a precision better signal of this source, method which would is less be noisylimited and for hassmall a more frequency regular offset shape. range. The errorFurthermore, of the proposed the perf methodormance is of small the measurement compared with is closelystraightforward related to measurement.the SUT noise However,feature, which if the should signal besource known is un first,der-performed, for a better signal the physical source, whichmodel is is less incapable noisy and of interpretinghas a more regularthe phase shape. noise The constituent, error of the the proposed calculated method value is by small the comparedproposed withmethod straightforward will have a largemeasurement. deviation fromHowever, the actual if the value. signal source is under-performed, the physical model is incapable of interpreting the phase noise constituent, the calculated value by the proposed method will have a

Electronicslarge deviation2019, 8, 767 from the actual value. 7 of 9

-90 Measured by the commencial phase noise analysizer(FSWP)

-100 Measured by the proposed method -90 Measured by the commencial phase noise analysizer(FSWP) -110 -100 Measured by the proposed method

-120 -110

-130

Phase noise (dBc/Hz) noise Phase -120

-140 -130 Phase noise (dBc/Hz) noise Phase 101 102 103 104 -140 Frequency(Hz)

101 102 103 104 Frequency(Hz) Figure 3. Comparison between fitting and model restoring (600 MHz).

Figure 3. Comparison between ﬁtting and model restoring (600 MHz). Figure 3. Comparison between fitting and model restoring (600 MHz). 8% 10M 7% 100M 8% 600M 6% 10M 7% 100M 5% 600M 6% 4%

Error 5% 3% 4% 2% Error 3% 1% 2%

0 1 2 3 4 10 10 10 10 1% Frequency(Hz)

Electronics 2019, 8, x FOR PEER REVIEW0 1 2 3 4 8 of 10 10 10 10 10 FigureFigure 4. 4.The The error error between between the the phase phase noise noise restored restored and and measured. measured. Frequency(Hz)

Figure 4. The error between the phase noise restored and measured.

FigureFigure 5. 5.The The piecewise piecewise representation representation of of the the maximum maximum error, error, subsections subsections 1 to1 to 4 represent4 represent frequency frequency oﬀoffsetset range range 10–100 10–100 Hz, Hz, 100 100 Hz–1 Hz–1 kHz, kHz, 1–10 1–10 kHz, kHz, and and 10–100 10–100 kHz kHz respectively. respectively.

4. ConclusionTo summarize the testing procedure with this scheme, we need a spectrum analyzer and a computer embedded with an optimization algorithm. The procedure is as follows: (i) testing the SUT In this work, we developed a novel phase noise measurement methodology based on a modified with spectrum analyzer and obtain its PSD with the best resolution that can be achieved, (ii) PSD composite law model. It proved that the phase noise of an oscillator can be determined quantitatively data was sent to the computer to implement data ﬁtting with the composite power law model and by the direct computation proposed in this work, which significantly simplifies phase noise measurement. Regarding the disadvantages of some conventional direct measurement by complicated hardware-based apparatuses, this method only requires a spectrum analyzer and determines phase noise components with the mathematical fitting of the PSD data. Several assumptions were made to simplify the model and improve the optimization and parameter calculation process. Experiments showed that the SUT PSD deduced phase noise agreed well with that of measured by a commercial phase noise analyzer. In most applications, the use of phase noise analyzer may increase the complexity of the system. The proposed method uses the physical model of the SUT phase noise to evaluate its actual phase noise performance, relatively accurate phase noise can be obtained, avoiding the use of hardware-based phase noise detection adopted in phase/frequency demodulation and reducing the overall experimental cost. In addition, this method can be extended to determine phase noise of modulated signals such as pulse modulated signals which is usually hard to determine by a phase noise analyzer. The accuracy and robustness of this methodology can be further improved by establishing a more precise model to characterize the near-carrier feature in the future. As mentioned before, the starting offset frequency of the power law region should be involved in the composite model and distinguished from near-carrier Gaussian component. The inherent Gaussian pattern of phase noise PSD should be considered and tested for more oscillators, which would facilitate a more compact and improved modality.

Funding: This work was supported by the National Natural Science Foundation of China, Grant No. 61471282, 61727804, 61801358 and 61805187, and the Fundamental Research Funds for the Central Universities under Grants No. 8002/20101196703. The authors acknowledge an open fund (grant No. LSIT201812D) granted by Key Laboratory of Spectral Imaging Technology, Chinese Academy of Sciences.

Conflicts of Interest: The authors declare no conflicts of interest.

References

1. Jurgo, M.; Navickas, R. Structure of All-Digital Frequency Synthesiser for IoT and IoV Applications. Electronics 2019, 8, 29.

Electronics 2019, 8, 767 8 of 9 extract coefficients, (iii) the phase noise is measured by calculating the power law expression with the obtained coefficients and plotting the curves. Despite the compact and effective testing scheme, the measurement method proposed is based on sheer power law model. In fact, there is also Gaussian component of the phase noise which has been neglected in this scenario. For a real oscillator, the precision of this method would be limited for small frequency offset range. Furthermore, the performance of the measurement is closely related to the SUT noise feature, which should be known first, for a better signal source, which is less noisy and has a more regular shape. The error of the proposed method is small compared with straightforward measurement. However, if the signal source is under-performed, the physical model is incapable of interpreting the phase noise constituent, the calculated value by the proposed method will have a large deviation from the actual value.

4. Conclusions In this work, we developed a novel phase noise measurement methodology based on a modified composite law model. It proved that the phase noise of an oscillator can be determined quantitatively by the direct computation proposed in this work, which significantly simplifies phase noise measurement. Regarding the disadvantages of some conventional direct measurement by complicated hardware-based apparatuses, this method only requires a spectrum analyzer and determines phase noise components with the mathematical fitting of the PSD data. Several assumptions were made to simplify the model and improve the optimization and parameter calculation process. Experiments showed that the SUT PSD deduced phase noise agreed well with that of measured by a commercial phase noise analyzer. In most applications, the use of phase noise analyzer may increase the complexity of the system. The proposed method uses the physical model of the SUT phase noise to evaluate its actual phase noise performance, relatively accurate phase noise can be obtained, avoiding the use of hardware-based phase noise detection adopted in phase/frequency demodulation and reducing the overall experimental cost. In addition, this method can be extended to determine phase noise of modulated signals such as pulse modulated signals which is usually hard to determine by a phase noise analyzer. The accuracy and robustness of this methodology can be further improved by establishing a more precise model to characterize the near-carrier feature in the future. As mentioned before, the starting offset frequency of the power law region should be involved in the composite model and distinguished from near-carrier Gaussian component. The inherent Gaussian pattern of phase noise PSD should be considered and tested for more oscillators, which would facilitate a more compact and improved modality.

Author Contributions: Conceptualization, X.C., C.P., H.H., F.N. and B.Y.; methodology, X.C; formal analysis, X.C., F.N. and B.Y.; modeling, C.P. and H.H.; validation, X.C., C.P. and H.H.; data curation, C.P. and H.H.; writing—original draft preparation, X.C., C.P. and H.H.; writing—review and editing, X.C., C.P. and H.H. Funding: This work was supported by the National Natural Science Foundation of China, Grant No. 61471282, 61727804, 61801358 and 61805187, and the Fundamental Research Funds for the Central Universities under Grants No. 8002/20101196703. The authors acknowledge an open fund (grant No. LSIT201812D) granted by Key Laboratory of Spectral Imaging Technology, Chinese Academy of Sciences. Conﬂicts of Interest: The authors declare no conﬂicts of interest.

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