Phase-Noise and Amplitude-Noise Measurementof DACs and DDSs

Claudio E. Calosso∇, A. Carolina Cárdenas Olaya∇,and Enrico Rubiola∃∇

∇ Istituto Nazionale di Ricerca Metrologica INRiM, Italy

∃ FEMTO-ST Institute, UBFC and CNRS, France

http://rubiola.org

June 13, 2019

Abstract

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 (DAC) and Direct Digital Synthesizers (DDS) basedon modulation-index amplification. The carrier is first reduced by a con-trolled amount (30-40 dB) by adding a reference signal of nearly equalamplitude and opposite in phase. Then, residual carrier and noise side-bands are amplified and sent to a conventional PN analyzer. The mainvirtues of our method are: (i) the noise specs of the PN analyzer arerelaxed by a factor equal to the carrier suppression ratio; and, (ii) thecapability to measure the AN using a PN analyzer, with no need for theanalyzer to feature AN measurement. An obvious variant enables ANand PN measurements using an AN analyzer with no PN measurementcapability. Such instrument is extremely simple and easy to implementwith a power-detector diode followed by a FFT analyzer. Unlike the clas-sical bridge (interferometric) method, there is no need for external linestretcher and variable attenuators because phase and amplitude controlis implemented in the device under test. In one case (AD9144), we couldmeasure the noise over 10 decades of frequency. The flicker noise matchesthe exact 1/f law with a maximum discrepancy of ±1 dB over 7.5 decades.Thanks to simplicity, reliability, and low background noise, this methodhas the potential to become the standard method for the AN and PNmeasurement of DACs and DDSs.

1 Introduction and State of the ArtIn virtually all domains of technology, the RF electronics is going digital viadedicated hardware, FPGA processing, Software Defined Radio techniques, andADCs and DACs are ubiquitous. This major trend is obviously driven by bigtelecom Companies, for mass consumer products and infrastructure equipment.While basic principles of conversion are rather mature [Kes04, Chap. 1–3], all

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C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 2

the development is confidential. The technical information about converters anddigital frequency synthesis is now in magazines [Mur15, Neu17, Cor04, Cor04,VB15, Win18] and books [Cla13], [HBM15], [Hoe16, Chap. 3], [Man12], [ML13,Chap. 9–11], [Sym13].

Converters are available from leading manufacturers (chiefly, Analog De-vices, Linear Technology, and Texas Instruments) with several GHz clock speed,12-16 bits (BUS), and up to 12-13 ENOB (Equivalent Number Of Bits). High-speed ADCs are generally more complex than DACs, and have inferior tradeoffbetween ENOB and maximum clock frequency. The reason is that most ADCarchitectures (SAR, pipelined, and subranging flash) employ a DAC.

Our interest is oriented to scientific applications, where the demand for high-purity RF signals is ever growing. The relevant parameters are low PM and AMnoise, high stability, frequency agility, and programmable amplitude and phase.We have in mind general purpose instruments, Atomic-Molecular-Optics physicsand atomic clocks [RJR+17, PPD+18, FCAH+15], long-distance synchroniza-tion via fiber links [PGR+12, CBC+14, LHC+12], real-time phase measurements[Cal13], particle accelerators [Acc], etc.

In this context, we focus on the AM and PM noise of DACs and DDSs. In-terestingly, modern high-speed telecom-oriented DACs have an internal NCO,which makes the DAC very similar to the DDS. If not, the NCO can be imple-mented in FPGA, transferring the data to the DAC via the JESD204B interface.Thus, we refer to the term DAC as a placeholder for both DAC and DDS.

Going through numerous data sheets, we see that manufacturers are mostconcerned with SFDR, SINAD, THD+R and ENOB, and leakage from/to adja-cent channels (see for example [Kes04, Chap. 2] for the definition of these terms).By contrast, phase noise is generally documented only as a typical plot of L (f)in a reference condition. It is often difficult to divide the device’s phase noisefrom the contribution of the reference oscillator and of the PN analyzer. Theliterature says little or nothing about phase noise and about how it is measured,and nothing has been found about measurement methods specific to DACs. TheAM noise is neither seen in datas heets nor in technical literature.

We describe the noise as the PSD (Power Spectral Density) of the randomphase ϕ(t) and fractional amplitude α(t), denoted with Sϕ(f) and Sα(f) as afunction of the Fourier frequency f . Notice that the more popular quantityL (f) , 1

2Sϕ(f) is defined only for phase noise. We use the polynomial lawSϕ(f) = b0 + b−1/f + . . . and Sα(f) = h0 + h−1/f + . . . truncated after theflicker term 1/f , as appropriated for two-port components. The reader mayrefer to [FP09, RW91, Rut78] for an introduction to PM noise, and to [Rub05]for AM noise.

Measuring some DDSs with various methods, we observed that the whitenoise coefficient b0 can be of −165 dBrad2/Hz. The flicker coefficient b−1 canbe of −135 dBrad2 at 10 MHz output frequency, and of −110 dBrad2 at 100-150 MHz output. We published only a part of this at conference [CGR12].Anyway, these numerical values define the minimum requirement for the back-ground noise.

Figure 1 shows the direct measurement of the DAC noise with commer-cial PN analyzers. All such instruments achieve reduced background noise bycorrelating and averaging the output of two channels.

The classical PN analyzer is based on a saturated mixer close to the quadra-ture condition, which converts the input phase into a voltage. In our case the

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 3

(A) Saturated mixer

(B) Fully digital

NCO

NCO

τ1

τ2

τc

FFT

anal

yzer

DAC

DAC

in

ref 2

ref 1PM-noise analyzer

∆

∆

ADCNCO

NCO

τ1

τ2

τc

DAC

DAC ADC

ADC

ADC

CORDIC

FFT

anal

yzer

in

ref

PM-noise analyzer

CORDIC

CORDIC

CORDIC φ2

φ1

φ2

φ1

Figure 1: Traditional phase-noise analyzers used for the measurement of DACs.The PM noise spectrum results from the contribution of the two DACs. No-tice that the single-DAC fluctuation τ1 and τ2 are detected, while the clockfluctuation τc is rejected.

quadrature condition can set numerically, provided the symmetry be sufficientto set the two channels at once. A problem is the high saturation power of themixer (7-15 dBm), compared to the low output power of the DACs (≈0 dBm).The power splitters introduce additional 3 dB intrinsic loss. The signal can beamplified, but amplifiers add complexity and noise.

Conversely, the fully-digital analyzer is based on the direct AD conversionof the input signal [GHR+04]. The benefit is obvious, in that the instrumentaccepts different frequencies at the ‘in’ and ‘ref’ inputs, and of course thereis no phase adjustment. At the time of writing there are only two options,Microsemi (formerly Symmetricom) [Mic] and, Jackson Labs [Pha, scheduledmid/late 2019]. The Rohde Schwarz FSWP [FSW, FR16] is a digital instrumentwith down conversion from microwaves, but it has not a suitable ‘ref’ input.

Our method relies on modulation-index amplification by a factor 1/η 1,introduced later. After amplification, the AM and PM noise is so high thatcorrelation is not necessary. This solves two problems at once. The first isthat the correlation instruments rely on the hypothesis that the two channelsare statistical independent. This is sometimes untrue, and anyway hard tocheck. Gross errors are around the corner if the experimentalist has not a deepunderstanding (see, for example [NHH14, GGR+17]). The second problem isthe measurement time, chiefly with the digital instruments because the noise ofthe DACs under test is often lower than that of the input ADCs. In fact, the

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 4

B

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N

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yzer

in

ref

FFT

anal

yzerAM detect

coupler

V1

V2

VΣ

V∆NCO DAC

A1

A2

DACNCO

τ1

τ2

τc

AN analyzerD

C

Figure 2: Measurement method. The rounded rectangles show the relevantspectra in log scale.

noise rejection is proportional to 1/√m, where m is the number of the FFTs

averaged [RV10]. This means 5 dB per factor of 10. Accordingly, if a FFTstarting from 1 mHz takes 2000 s acquisition time, averaging over 100 spectrafor 10 dB noise rejection takes a measurement time of 2 days and 7.5 hours. Bycontrast, our method needs a small m, only to smooth the measured spectra.

2 Principles and MethodWe compare two equal devices under test (DUT) using the scheme of Figure 2.The rectangles labeled A1, A2, B, C and D show the spectrum in the criticalpoints, in log scale. The two signals V1 and V2 are combined in the directionalcoupler so that almost all the carrier power goes at the Σ output, and a smallpower goes to the ∆ output. However, the noise sidebands are equally splitbetween the Σ and ∆ output because these signals are not coherent (spectra Band C). Residual carrier and noise sidebands are amplified (spectrum D) anddetected by the PN analyzer or by the AN analyzer.

Our method derives from an early idea for the PN measurement of microwavetwo-port devices [San68]. In the original version, the reference carrier crossesthe DUT, which contributes its own noise. The carrier is completely suppressedby subtracting a copy of the reference through a magic Tee (a waveguide di-rectional coupler), and the DUT noise sidebands are down-converted to DCby synchronous detection. Microwave amplification of the noise sidebands wasadded later to boost the gain and to reduce the background noise [Lab82]. Thewhole system is equivalent to a Wheatstone bridge powered by an AC signal(the microwave carrier), followed by a lock-in amplifier. The lowest backgroundnoise is achieved after adjusting the system for the maximum carrier rejection[ITW98, RGG99]. By contrast, we leave a carefully controlled amount of carrier,as proposed in [Wal97] for different purposes. There results modulation-indexamplification, with optional AM-to-PM and PM-to-AM conversion dependingon phase relationships. The presence of the residual carrier is essential in thatthe signal is suitable to the measurement with all-digital PN analyzers, or witha simple AN analyzer implemented with a power-detector diode ($100-300 fora packaged and connectorized detector) and a FFT spectrum analyzer [Rub05].

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 5

Thanks to the AM-to-PM and PM-to-AM conversion, a PN analyzer with no ANmeasurement capability enables the measurement of both AN and PN. Likewise,an AN analyzer with no PN measurement capability. The modulation-index am-plification relaxes the noise specs of the PN analyzer, or of the AN analyzer, bythe same factor. The RF gain (40-60 dB) is needed for the power to match theinput range of the PN or AN analyzer.

The background white noise is due the noise FkT of the amplifier, where Fis the amplifier noise figure (1-2 dB, that is, 1.25 to 1.6), and kT ≈ 4×10−21

W/Hz is the thermal energy at room temperature. Converted into PM noise, thebackground is b0 = 2FkT/P , where P is the carrier power at the directional-coupler input, and the factor ‘2’ accounts for the inherent 3-dB loss in thecoupler. The background 1/f PM noise is that of the amplifier, divided bythe carrier suppression ratio. This happens because the 1/f phase noise incomponents results from parametric up-conversion of the near-dc noise [BR12].

2.1 Modulation-Index AmplificationThe directional coupler in Figure 2 delivers the output

V∆ =1√2

(V2 − V1) , (1)

where V1 and V2 are the output signals of the two converters, and the factor 1/√

2is due to energy conservation in the absence of loss. These signals have randomfractional amplitude α1(t) and α2(t), and random phase ϕ1(t) and ϕ2(t). Weassume that |α1| 1, |α2| 1, |ϕ1| 1 and |ϕ2| 1, hence eα+jϕ ' 1+α+jϕ.

For our purposes, it is convenient to set V1 and V2 close to the nominalamplitude V0, but for a small difference in amplitude (β1 and β2) and phase (γ1

and γ2), so that

V1 = V0 (1− β1) e−iγ1eα1+jϕ1 (2)

V2 = V0 (1 + β2) e+iγ2eα2+jϕ2 . (3)

Figure 3 shows these signals and how they combine to form V∆. The latter canbe written as

V∆ = ηV0√

2ejθeε+jψ , (4)

where η and θ describe the static amplitude and phase of the carrier seen at thecoupler output, and ε(t) and ψ(t) describe the amplitude and phase modulation.We will show that 1/η is the modulation-index amplification. The quantity ηhas to satisfy

∣∣eε+jψ∣∣ η 1. First, η 1 is necessary for the amplificationto be useful. Second, η

∣∣eε+jψ∣∣ is needed to prevent V∆ from sweeping 0. Ifthis happens, PM are no longer defined. Because AM and PM noise are small,these conditions allow a large range for η.

The desired amount of residual carrier is obtained by setting β1, β2, γ1 andγ2 to appropriate values. Notice that β1 and β2, and likewise γ1 and γ2, are notequal in the general case, but they can be assumed equal for η 1. Accordingly,it is useful to set β1 = β2 = β/2 and γ1 = γ2 = γ/2, which defines β and γ. Byinspection on Fig. 3, it holds that β 1 and γ θ for η 1.

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 6

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/2<latexit sha1_base64="2GAQO9F/CW+elRmxwUcDd/4UNFg=">AAAB6XicZZDNSgMxEMdn/az1q+rRS7AVPJS6ux70ZsGLxwr2A9qlZNNsG5pklyQrlKUPoSdRb76OL+Bb+Ahm2x60HQj58Z8ZZv4TJpxp47rfztr6xubWdmGnuLu3f3BYOjpu6ThVhDZJzGPVCbGmnEnaNMxw2kkUxSLktB2O7/J8+4kqzWL5aCYJDQQeShYxgo2VOpUeNfjSr/RLZbfmzgKtgreA8u0PzKLRL331BjFJBZWGcKx113MTE2RYGUY4nRZ7qaYJJmM8pF2LEguqg2y27xSdW2WAoljZJw2aqX87Miy0wGZUtX9eoXPQExFWQ1HNE0pHemmGiW6CjMkkNVSS+Ygo5cjEKLeNBkxRYvjEAiaK2S0RGWGFibHHKVr73rLZVWj5Ne+q5j/45bo7vwMU4BTO4AI8uIY63EMDmkCAwwu8w4czdp6dV+dtXrrmLHpO4F84n79qeo4b</latexit>

V = ej<latexit sha1_base64="Xnir8WFRJal/fExzbBHoGf/Z5ls=">AAAB/nicZZDJSgNBEIZr3I1b1KOXxih4CGEmHvQiCnrwqGCi4MTQ06kxbXoWumuEMAT0ZfQk6s2H8AV8Cx/BnsSDyw9Nf9RC1V9BqqQh1/1wxsYnJqemZ2ZLc/MLi0vl5ZWmSTItsCESleiLgBtUMsYGSVJ4kWrkUaDwPOgdFvnzW9RGJvEZ9VNsRfw6lqEUnGyoXV7baLb9I1TE93wkzvAqv/Gpa3Gw0S5X3Jo7FPsP3jdU9j9hqJN2+d3vJCKLMCahuDGXnptSK+eapFA4KPmZwZSLHr/GS4sxj9C08qGJAdu0kQ4LE21fTGwY/dmR88hEnLpV+xcVpgDTj4JqEFWLhDah+TODwt1WLuM0I4zFaESYKUYJK27BOlKjINW3wIWWdksmulxzQfZiJWvf+2v2PzTrNW+7Vj+tVw7qozvADKzBOmyBBztwAMdwAg0QcA+P8AKvzp3z4Dw5z6PSMee7ZxV+yXn7AltRlpo=</latexit>

PM1

AM1

<latexit sha1_base64="yO1WSrJ1TBwulQ/1aSPJB6jp4TU=">AAAB6XicZZBLSgNBEIZr4ivGV9Slm8ZEcBHCzLjQnQE3LiOYByRD6On0JE16HnTXCCHkELoSded1vIC38Aj2TLLQpKDpj7+qqPrLT6TQaNvfVmFjc2t7p7hb2ts/ODwqH5+0dZwqxlsslrHq+lRzKSLeQoGSdxPFaehL3vEnd1m+88SVFnH0iNOEeyEdRSIQjKKRutU+jjnS6qBcset2HmQdnCVUbn8gj+ag/NUfxiwNeYRMUq17jp2gN6MKBZN8XuqnmieUTeiI9wxGNOTam+X7zsmFUYYkiJV5EZJc/dsxo6EOKY5r5s8qdAZ6Gvo1P6xlCaUDvTIDgxtvJqIkRR6xxYgglQRjktkmQ6E4Qzk1QJkSZkvCxlRRhuY4JWPfWTW7Dm237lzV3Qe30nAXd4AinME5XIID19CAe2hCCxhIeIF3+LAm1rP1ar0tSgvWsucU/oX1+QsjjY6Y</latexit>

PM2

AM2

Figure 3: Interplay between signals at the output of the directional coupler. Inthe actual RF circuit, all signals are scaled up by a factor V0/

√2, omitted for

graphical clarity.

Combining (1)-(2)-(3) gives

V∆ =V0√

2

[(1 +

β

2

)ejγ/2eα1+jϕ1 −

(1− β

2

)e−jγ/2eα1+jϕ1

]. (5)

The AM and PM associated to V∆ result from

ε = <SC

and ψ = =

SC

, (6)

where

S =V0√

2

[(1 +

β

2+ j

γ

2

)(α2 + jϕ2)−

(1− β

2− j γ

2

)(α1 + jϕ1)

](7)

is the voltage swing of (5), and

C = ηV0√

2ejθ (8)

is the carrier. Recalling that β 1 and γ θ for η 1, Equations (6) can beelegantly rewritten as[

εψ

]=

1

η

[cos θ sin θ− sin θ cos θ

] [α2 − α1

ϕ2 − ϕ1

](9)

The matrix represents a rotation by θ, which comes from the fact that I andQ (in-phase and quadrature voltage swing) is projected on V∆. The term 1/ηmeans that I and Q are now referred to a carrier of amplitude η 1, whichresults in proportionally larger amplitude or phase swing. Equation (9) empha-sizes two key concepts

1. The modulation index is amplified by a factor 1/η

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 7

Table 1: Measurement Options.

Operation PN-analyzer only AN-analyzer only Signal vectors

θ = 0AN and PNamplification

Sψ 'Sϕ2 + Sϕ1

η2Sε '

Sα2 + Sα1

η2

V1

V2

V∆

θ = π/2AN-PN crossamplification

Sψ 'Sα2 + Sα1

η2Sε '

Sϕ2 + Sϕ1

η2 V1

V2V∆

2. The rotation enables to preserve the character of AM and PM (θ = 0), tointerchange AM and PM (θ = π/2), or to take any combination of AMand PM.

Replacing ε and ψ with their spectra, (9) becomes[SεSψ

]=

1

η2

[cos2 θ sin2 θsin2 θ cos2 θ

] [Sα2 + Sα1

Sϕ2 + Sϕ1

], (10)

which enables the measurement options listed in Table 1.It is important to remember (10) relies on the assumption that the noise of

the two converters is uncorrelated because the system is insensitive to common-mode noise. This is clear with the PM noise: the common-mode time fluctuationτc is rejected, which includes the clock (Fig. 2). By contrast, correlation in theAM noise is more subtle because it originates from the power supply and fromthe voltage reference. Insufficient PSRR (Power-Supply Rejection Ratio) mayresult in correlated noise, and multiple DACs in a chip may share the reference.

2.2 Exact AnalysisWe evaluate the errors due to the hypothesis that η 1. With reference toFigure 3, β1 and β2, and likewise γ1 and γ2 are not equal in the general case.Combining (1)-(3) gives

V∆ =V0√

2

[(1 + β2) ejγ2eα1+jϕ1 − (1− β1) e−jγ1eα1+jϕ1

]. (11)

In low noise conditions, the associated voltage swing is

S =V0√

2

[(1 + β2) ejγ2 (α2 + jϕ2)− (1− β1) e−jγ1 (α1 + jϕ1)

]. (12)

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 8

Putting the above expression of S in (6) gives

ε =1 + β2

η

[α2 cos(θ − γ2) + ϕ2 sin(θ − γ2)

]+

− 1− β1

η

[α1 cos(θ + γ1) + ϕ1 sin(θ + γ1)

](13)

ψ =1 + β2

η

[−α2 cos(θ − γ2) + ϕ2 sin(θ − γ2)

]+

− 1− β1

η

[−α1 cos(θ + γ1) + ϕ1 sin(θ + γ1)

]. (14)

Using the matrix expression of the above for the spectra of uncorrelated DACs,similar to (10), yields[

SεSψ

]=

(1 + β2)2

η2

[A2

] [Sα2

Sϕ2

]+

(1− β1)2

η2

[A1

] [Sα1

Sϕ1

](15)

with [A2

]=

[cos2(θ − γ2) sin2(θ − γ2)sin2(θ − γ2) cos2(θ − γ2)

](16)

[A1

]=

[cos2(θ + γ1) sin2(θ + γ1)sin2(θ + γ1) cos2(θ + γ1)

]. (17)

We introduce the approximation that Sα1 ' Sα2 and Sϕ1 ' Sϕ2, basedon the fact that the two DACs are nominally equal. Thus, it makes sense toapproximate (15) as [

SεSψ

]=

1

η2

[A] [Sα2 + Sα1

Sϕ2 + Sϕ1

], (18)

averaging the two matrices[A]

=1

2

(1 + β2)

2 [A2

]+ (1− β1)

2 [A1

]. (19)

From the geometrical properties shown on Figure 4, the Pythagoras theoremgives

(1 + β2)2

=(

1 +η

2cos θ

)2

+(η

2sin θ

)2

(20)

(1− β1)2

=(

1− η

2cos θ

)2

+(η

2sin θ

)2

(21)

γ2 = arctan

( η2 sin θ

1 + η2 cos θ

)(22)

γ1 = arctan

( η2 sin θ

1− η2 cos θ

)(23)

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 9

2cos

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-1

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2sin

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2cos

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1 1<latexit sha1_base64="VYWXhZFMktrqlzti5v8SOSuX6WY=">AAAB7HicZZDNSgMxEMdn/az1q+rRS7AVPKxltx70ZsGLxwr2A9qlZNNsG5pklyQrlNK30JOoN5/GF/AtfASz2x60HQj58Z8ZZv4TJpxp43nfztr6xubWdmGnuLu3f3BYOjpu6ThVhDZJzGPVCbGmnEnaNMxw2kkUxSLktB2O77J8+4kqzWL5aCYJDQQeShYxgo2VuhX/shdSg/t+pV8qe1UvD7QK/gLKtz+QR6Nf+uoNYpIKKg3hWOuu7yUmmGJlGOF0VuylmiaYjPGQdi1KLKgOpvnKM3RulQGKYmWfNChX/3ZMsdACm5Fr/6xCZ6AnInRD4WYJpSO9NMNEN8GUySQ1VJL5iCjlyMQoc44GTFFi+MQCJorZLREZYYWJsfcpWvv+stlVaNWq/lW19lAr1935HaAAp3AGF+DDNdThHhrQBAIxvMA7fDjSeXZenbd56Zqz6DmBf+F8/gJDZI8k</latexit>

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2sin

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<latexit sha1_base64="yO1WSrJ1TBwulQ/1aSPJB6jp4TU=">AAAB6XicZZBLSgNBEIZr4ivGV9Slm8ZEcBHCzLjQnQE3LiOYByRD6On0JE16HnTXCCHkELoSded1vIC38Aj2TLLQpKDpj7+qqPrLT6TQaNvfVmFjc2t7p7hb2ts/ODwqH5+0dZwqxlsslrHq+lRzKSLeQoGSdxPFaehL3vEnd1m+88SVFnH0iNOEeyEdRSIQjKKRutU+jjnS6qBcset2HmQdnCVUbn8gj+ag/NUfxiwNeYRMUq17jp2gN6MKBZN8XuqnmieUTeiI9wxGNOTam+X7zsmFUYYkiJV5EZJc/dsxo6EOKY5r5s8qdAZ6Gvo1P6xlCaUDvTIDgxtvJqIkRR6xxYgglQRjktkmQ6E4Qzk1QJkSZkvCxlRRhuY4JWPfWTW7Dm237lzV3Qe30nAXd4AinME5XIID19CAe2hCCxhIeIF3+LAm1rP1ar0tSgvWsucU/oX1+QsjjY6Y</latexit>

Figure 4: Geometrical properties from Figure 3.

Putting (20)-(23) into (15) and expanding (18), we get a general expression for[A]. This expression is omitted here because it too complex and hard to read.

We discuss two relevant cases, derived the results with a symbolic math app.The first case is

limη→0

[A]

=

[cos2(θ) sin2(θ)sin2(θ) cos2(θ)

], (24)

which is equivalent to (10) and validates our hypotheses.The second case is the accuracy of (18) for θ = 0, still under the assumption

that the noise of the two DACs is equal. For this purpose, we allow a smallangle error ζ, and we expand (15) in series truncated to the second order of ηand ζ. For θ = 0, replacing θ → ζ gives

[A]

=

[1 + η2/4− ζ2 ζ2

ζ2 1 + η2/4− ζ2

]. (25)

The third case is the accuracy of (18) for θ = π/2, analyzed in the same way asbefore but for θ = π/2 + ζ. Hence,

[A]

=

[ζ2 + η2/4 1− ζ2

1− ζ2 ζ2 + η2/4

]. (26)

The finite gain results in small errors, negligible in most practical cases. Forexample, 1/η2 = 100 (20 dB) results in 1.1×10−2 dB error in (25), and in −26dB coupling between AM and PM in (26). Of course, the gain must be wellcalibrated because of the 1/η2 factor in (15). The angle error deserves moreattention. For example, ζ = 0.1 (5.7) results in −0.44 dB error and −20 dBcoupling between AM and PM in both (25) and (26). The PM noise is generallydominant because of the jitter in the clock distribution, and may pollute theAM-noise measurement.

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 10

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Figure 5: Experimental setup.

3 ExperimentsWe measured the two channels of an AD9144 with the scheme of Fig. 5. Thefour channels of the AD9144 go in pairs, with two NCOs. The two channels wemeasure can be considered as separated devices, but for a small crosstalk andfor the small jitter τc in the common-mode clock path. The AD9144 is drivenby a Z-Board ZC706 via the JESD204B interface. The ZC706 is a computerbased on a Zynq-7000, with flash memory, DDR3 memory, Ethernet, USB, SDcard slot, HDMI video, and SPI and FMC connectors for daughterboards. It

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 11

Table 2: Operating Parameters of Fig. 5.

η V∆ VM Gain Vin VD VDC

−20 −22 −26 40 +10 −3 86

−30 −32 −36 40 0 −3 86

−40 −42 −46 60 +10 −3 86

−49 −51 −55 60 +1 −2 86

dB dBm dBm dB dBm dBm mV‘V ’ given in dBm means ‘the power of the signal V .’

runs Linaro linux. In turn, the Zync-7000 is a System on Chip (SoC), which isbasically a microprocessor and a FPGA on the same chip.

The AD9144 is actually an AD9144-FMC-EBZ card [AD9], which containsthe AD9144 chip, the AD9516-0 clock generator, the output baluns, and auxil-iary functions. This card is plugged onto the FMC connector of the Z-Board,and delivers +2 dBm power on 50 Ω load on each output (SMA connector).The speed of the JESDB204B sufficient to set instantaneous phase and am-plitude, but in our case such speed is needed only for the amplitude becausephase and frequency are static parameters, set only once. The sampling fre-quency is of 1 GHz, and the DAC output frequency is of 125 MHz. This choice(fck/fout = 8, a small round number) eliminates the pseudo-random spurs, andmakes it easy to measure the ‘true’ random noise inherent in the component (see[NS87, TW05] for the origin of such spurs). The 1-GHz clock is generated bya Rohde Schwarz SMG 801 synthesizer driven by its internal low-noise OCXO.The jitter of this clock is rejected because the clock is common mode. There isno need to feed the PN analyzer directly with the OCXO signal because the PNanalyzer has a reference input, derived from the DAC output. The noise of suchreference has negligible effect because of the modulation-index amplification.

Most of the tests were done in Torino with the exact scheme of Fig. 5, andwith 1/η is from 20 dB to 49 dB. Table 2 shows the operating parameters.However, preliminary tests were done in Besançon using a slightly differentconfiguration. We had less flexibility in the choice of the gain, and no powerdetector. The output VΣ was used as the carrier-rejection monitor, and later inthe experiment as the reference of the PN analyzer. This saves one coupler onthe RF path, and results in 3-dB lower background noise.

The AD9144 data sheet does not indicate the ENOB. Instead, it givesthe typical white noise −162 dBm/Hz (N = 6.3×10−20 W/Hz) with 150 MHzsingle-tone output, full amplitude, and 983.04 MHz sampling frequency. Theseconditions are quite similar to ours. The white AN follows from Sα = N/P ,where P is the output power. The PN follows from Sϕ = N/P + J , where J isthe additional contribution of the clock jitter, unknown here and not impactingon the AN. From the specs, we expect Sα = −164 dB/Hz with +2 dBm outputpower, and Sα = −161 dB/Hz for the noise of two channels.

Accounting for 7 dB loss from the DAC output to the input of the RFamplifier (low-pass filter and two 3-dB couplers), and adding the independentcontribution of two DACs, the expected noise is of −166 dBm/Hz at the inputof the RF amplifier. This value is 8 dB higher than the thermal noise at room

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 12

DAC 1DAC 2

reference

≈FSR – 0.5 dBinitial state(random)

DAC 2

null the carrier

PNAN

AN

PN

initial state(null)

AM PM amplification

AM PM cross-amplification

(A) Full carrier suppression (B) Add a small carrier

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Figure 6: Adjustment procedure.

temperature (−174 dBm/Hz). Allowing a noise figure of 2 dB, the amplifiercontributes +1 dB to the noise limit.

The AN measurement is performed with a single-channel system (powerdetector, amplifier and FFT analyzer). At 86 mV output, the power detectoris no longer in the quadratic region, and detects the fluctuations of the RFpeak voltage. This operating mode is safe for regular AN measurement, but itwould not be suitable to other applications, like beating two tones in the Poundfrequency control. For reference, a flicker of −100 dB/Hz is equivalent to anamplitude stability of 11.8 ppm (use σ2 = 2 ln(2) h−1 with h1 = 10−100/10, thusσ = 1.18×10−5).

Before using, the system needs a simple adjustment and calibration. Wefirst need to set η and θ to the desired value by adjusting phase and amplitudeof V1 and V2. With reference to Figure 6, we start with V1 and V2 just belowthe full-scale range. In this condition, V1 − V2 is determined by the arbitraryphase difference, plus the small amplitude asymmetry. Then, we adjust phaseand amplitude alternatively for maximum carrier suppression by monitoringthe residual carrier. The monitor is an oscilloscope triggered by the 125-MHzreference at one output of the AD9144, or a spectrum analyzer. The spectrumanalyzer is more comfortable to use, but the oscilloscope gives access to thesign of the error signal. Setting phase and amplitude manually, we found thata sub-binary search is advantageous. Albeit it takes more iterations than the‘true’ binary search, it is conservative vs uncertainty and noise in the monitor,and it simplifies the decision about the next iteration.

After the maximum carrier rejection, we set the desired amount of residualcarrier by changing V2. We recommend to check on AM-PM gain and crosstalkby adding a digital modulation to the DAC under test.

4 ResultsFigure 7 shows the PM noise and AM noise of the AD9144 measured withdifferent gain levels. All the spectra in this Section refer to the total noise ofthe two output channels, hence the noise of one channel is 3 dB lower.

In the first experiment (Fig. 7A and 7B) we use the PN analyzer to measureboth the PM and AM noise. Figure 7A shows the raw phase noise spectra as

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 13

displayed by the PN analyzer, for different values of the gain 1/η from 20 dB to49 dB. Figure 7B reports the same spectra, corrected for the gain. The plotsoverlap perfectly on almost all the frequency span, indicating that the value of1/η is not critical. In the upper half-decade, the noise is some 1 dB higher atlower gain. We did not investigate further on this small discrepancy. Likewise,Figure 7C and 7D show the AM noise measured with the PN analyzer. Theresidual carrier is orthogonal to the input carrier (θ = π/2), so that the systemperforms AM-to-PM conversion. The raw spectra (Fig. 7C) are correctly givenin rad2/Hz, as displayed by the PN analyzer. The same spectra, corrected forthe gain, are shown Figure 7D. This is the AM noise of the two channels ofthe AD9144. The unit is 1/Hz, as appropriate. As before, the results overlapon almost the full span, but for a small discrepancy in the upper half decade.Here, the noise measured with the lower gain (20 dB) is some 1.5 dB higher.The bump at 100 kHz, 5 dB above the asymptotic approximation, is probablydue to the power supply. We exclude the AD9144 internal reference because itis a common-mode signal, and has at most second-order effect on the amplitudenoise.

In the second experiment, we measure amplitude and phase noise using theAN detector (Figure 7E and 7F). The maximum frequency is limited by thefull span of the FFT analyzer, which is of 100 kHz. The AM noise spectraoverlap well on most of the span (Fig. 7E). A small spread, ±1 dB, shows upin the upper decade. The almost-flat region beyond 10 kHz is actually thebump already seen in the AM noise spectra (Figure 7D). The PM noise spectra(Fig. 7 F) overlap well, with a maximum spread of ±1.5 dB in the upper decade.In both cases, the higher noise is observed with the lowest gain, 20 dB. This isascribed to the background noise of the AM detector, which cannot be rejected.

Because a white floor is not visible in any plot of Fig. 7, we can only evaluatethe upper bound and check on the consistency with the design parameters.From the value Sα = −161 dB/Hz based on the data sheet, and accounting for1 dB contribution of the RF amplifier, we expect Sα = −160 dB/Hz for thetwo converters. This is exactly equal to the lowest value seen on Fig. 7D (40dB/rad and 49 dB/rad gain) at f = 850 kHz. Inspecting on Sϕ, the lowestvalue seen on Fig. 7D (40 dB/rad and 49 dB/rad) is of −159 dB/Hz at f = 850kHz, that is, 1 dB higher than the AN. This indicates that the measured valuesare consistent with the design.

Figure 8 compares the above results with 30 dB gain. The two AM noisespectra overlap, and likewise the two PM noise spectra. This confirms thatthe two variants of the method, AN analyzer and PN analyzer, give equivalentresults.

The flicker of amplitude is h−1 = 8×10−12 (−110 dB/Hz at 1 Hz) up to a fewkHz, corrupted by a bump at 100 kHz. Such flicker is equivalent to a fractionalamplitude stability σα = 3.3×10−6 (Allan deviation). This is found using theclassical formula σ2 = 2 ln(2) h−1, which holds for flicker [FP09, CR16].

The flicker of phase takes two levels, b1 = 4×10−11 rad2 (−104 dBrad2) atlower f and b1 = 2.8×10−11 (−105.5 dBrad2) at higher f , with a discrepancyof 1.5 dB. The average of these two values is b1 = 3.4×10−11 rad2. The latter,converted into time-fluctuation PSD is Sx = k−1/f with k−1 = 5.5×10−29 s2.Using σ2 = 2 ln(2) h−1, we find a time fluctuation σx = 8.7 fs. Having testedonly one frequency, we are still unable to divide the time type noise (the timefluctuation σx is independent of the carrier frequency ν) and the phase type

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 14

(A) P

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Figure 7: Phase noise and amplitude noise of the AD9144, two channels. Thefour options are shown, with modulation-index amplification and AM-PM crossamplification, and the two types of detection, AM and PM. The spectra in (B)are the same of (A), but in (B) we account for the gain. Likewise, (D) and (C).

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 15

h–1 = 8x10–12

b–1 = 4x10–11exact 1/ƒ

AN

PN

exact 1/ƒ

b–1 = 2.8x10–11exact 1/ƒ

1.5 dB

AD9144, two channels

125 MHz out1 GHz sampling30 dB (x-)gain

b–1 = 3.4x10–11average

File: Results-synthesisC.E.Calosso, A.C.Cárdenas Olaya, E.Rubiola, June 2019

Figure 8: Comparison of the results with 30 dB gain. The full span of the FFTanalyzer limits the AM → AM and PM → AM spectra to 100 kHz.

AD9144, two channels

7.5 decades

125 MHz out1 GHz sampling50 dB gain

exact 1/ƒ slope

Frac

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File: Results-wide-spanC.E.Calosso, A.C.Cárdenas Olaya, E.Rubiola, June 2019

Figure 9: Flicker noise observed on the widest span. Because θ is not calibrated,the measured quantity (right-hand scale) is a combination of AM and PM.

noise (the phase fluctuation in rad is independent of ν, and σx scales as 1/ν)that combine into this result (see [CR16] for definitions and properties of phase-type and time-type noise).

Figure 9 shows the 1/f noise measured on the widest span. The experimentwas done in Besançon with the alternate configuration mentioned. The calibra-tion is unfortunately less reliable than in the other cases. The gain is of 50 dB,but the phase of the residual carrier was not correctly set. Thus, we assessed θ

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 16

a posteriori using the flicker coefficients of

Sψ =1

η2

[sin2(θ)Sα + cos2(θ)Sϕ

]. (27)

The observed quantity is η2Sψ = c−1/f with c−1 = 1.5×10−11 (from Fig. 9),and the reference quantities are h−1 = 8×10−12 and b−1 = 3.4×10−11 (averageof the two levels shown), found on Figure 8. Solving

c1 = h−1 (1− cos 2θ) + b−1 cos2 θ , (28)

we find cos2 θ = 0.26, and finally θ = 1.03 rad (59). Accordingly, the observedresult is

η2Sψ = [0.74Sα + 0.26Sϕ] . (29)

Regardless of the accuracy of θ, the relevance of this result is the observationof the flicker noise with exact 1/f slope over 7.5 decades, with a maximumdiscrepancy of 1 dB.

5 New PerspectivesThe lesson learned, combined with our past experience [Rub05, RG02], suggeststhe schemes shown on Figure 10. These schemes are neither peer-reviewednor tested, but they rely on solid experience. Two ideas are used in differentconfigurations, yet for the same purpose.

The first idea (Fig. 10A-C-D) is that V∆ can be split into two branches,amplified, and sent to the two inputs of a dual-channel noise analyzer. The noiseof the two branches is statistically independent, and rejected by correlation andaveraging.

The second idea (Fig. 10B-E-F) is to use two reference DACs (‘1’ and ‘2’)to generate two separate ‘V∆.’ The correlation-and-averaging process extractsthe statistical characteristics of VDUT, and rejects the noise of V1 and V2.

The configuration of Fig. 10A is the straightforward application of the firstidea. After rejecting the noise of the RF amplifier and of the AM detector, thecross-spectrum converges to the total noise PSD of the two DACs.

Figure 10B enables the measurement of a single DAC by correlating twoindependent branches, each of which amplifies the modulation index. The noiseof the two reference DACs (V1 and V2) is independent, and only VDUT is commonto the two branches.

Some PN analyzers provide access to all the four inputs. This is exploitedin Fig. 10C-F. In the case of the analog instruments, two amplifiers are neededto increase V3 and V4 to the saturation level of the mixer. The Symmetricominstruments [Mic] have only two accessible inputs, thus they are not suitableto these schemes. By contrast, the forthcoming PhaseStation [Pha] has all thefour ADC inputs accessible.

Figure 10C is the obvious remake of Fig 10A, after replacing the AM de-tector with a PM detector. The two additional DACs, ‘3’ and ‘4,’ provide thephase reference to the PN analyzer. One more adjustment is needed, to set V3

and V4 in quadrature to the amplified version of V∆.Figure 10D is equivalent to Fig. 10C, but for the digital analyzer instead of

the classical saturated-mixer instrument. In this case there is no need to set the

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 17

AM detect

V2

VΣ1

V∆1

NCO DAC

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V∆2

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anal

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(A) Correlation AM detector (B) Single-DAC, correlation AM detector

(C) Correlation, analog PM detector (D) Correlation, digital PM detector

(E) Single-DAC, correlation, analog PM detector (F) Single-DAC, correlation, digital PM detector

V1

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VΣ2

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V∆

NCO DAC

NCO DAC

NCO DAC

NCO DACFF

T an

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V4

V3

a

b

c

d

Figure 10: New schemes, under consideration for the next experiments.

phase of V3 and V4. Trusting the modulation-index amplification, it is possibleto save one converter (‘4’) feeding the phase references ‘c’ and ‘d’ with the samesignal (V3). The reason is that the digital instrument accept arbitrary phase(and frequency) at the reference inputs.

Figure 10E is the obvious remake of Fig 10B, replacing the AM detectorwith a PM detector. Because only VDUT is common to the two branches, thisconfiguration enables the measurement of a single DAC by correlating the twoindependent branches, each of which amplifies the modulation index. One maybe tempted to save two DACs by feeding the phase references ‘c’ and ‘d’ withVΣ1 and VΣ2. This is probably not a good idea. A problem is that the phaseof such signals cannot be controlled independently of V∆, thus the quadraturecondition has to be set by other means. Another problem is that even a smallchange in the impedance seen by VΣ impacts dramatically on V∆, thus the gain1/η and on the phase θ.

Figure 10F is equivalent to Fig. 10E, but for the classical analog instrumentreplaced with a fully digital PN analyzer. As before, it is possible to save one

C. E. Calosso, A. C. Cárdenas Olaya and E. Rubiola, June 13, 2019 18

converter (‘4’) feeding the phase references ‘c’ and ‘d’ with the same signal (V3),but the signals ‘VΣ’ should not be re-used.

6 ConclusionsWe have proposed a method for the measurement of PM noise AM noise ofDACs and DDSs, and we have proved the concept by measuring an AD9144.

The value of the method is in its reliability and simplicity. Implementationand use require modest skill in analog RF electronics, and only standard skillin programming and using the target DACs and DDSs. Our experiments relyon commercial parts only, like the Z-Board and DAC daughterboards, and onready-to-use connectorized RF modules. Under no circumstance we had todesign and implement ad hoc electronics. The low background noise is inherentin the principle, and easy to achieve. The cross spectrum comes, optionally,only after modulation-index amplification. Consequently, none of the knownflaws of the cross spectrum can threat the reliability of the result. Tuningand calibration can be automated because it is done entirely by setting integernumbers in the target converter. Thanks to all these characteristics, our methodhas the potential to become the standard method for the measurement of AMnoise and PM noise of DACs and DDSs.

The experimentalist has two options, using a dedicated PN analyzer or asimple power detector and a general-purpose FFT analyzer. Complexity andbackground noise are equivalent, and the results overlap.

The power-detector option does not require a PN analyzer, which is a spe-cialized and expensive instrument. The FFT analyzer can be implemented inGNU radio on the Z-Board, after adding an ADC daughterboard. Overall, thisversion probably fits in the budget of a radio amateur or hobbyist.

The other option requires a commercial PN analyzer. In fact, the design andimplementation of such instrument is definitely not simple. The virtue of a fully-digital PN is the wide dynamic range, inherent in the CORDIC algorithm usedto extract the random phase modulation from the sampled input [MVTB+09].

As a fringe benefit, we observed the noise of the AD9144 on a frequency spanof 10 decades. The flicker noise matches the exact 1/f law, with a maximumdiscrepancy of ±1 dB over 7.5 decades. The flicker noise of an electronic deviceover such a wide frequency range has probably never been reported before.

AcknowledgmentsThis work is funded by the ANR Programme d’Investissement d’Avenir underthe Oscillator IMP project (ANR-11-EQPX-0033-OSC-IMP) and the First-TFnetwork (ANR-10-LABX-48-01), and by the Région Bourgogne Franche Comté.

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