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

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


I. INTRODUCTION AND STATE OF THE ART
In virtually all domains of technology, the RF electronics is going digital via dedicated hardware, FPGA processing, Software Defined Radio techniques, and ADCs and DACs are ubiquitous. This major trend is obviously driven by big telecom companies, for mass consumer products and infrastructure equipment. While basic principles of conversion are rather mature [1,, all the development is confidential. The technical information about converters and digital frequency synthesis is now in magazines [2], [3], [4], [5], [6], [7] and books [8], [9], [10,Chap. 3], [11], [12,, [13].
Converters are available from leading manufacturers (chiefly, Analog Devices, Linear Technology, and Texas Instruments) with several GHz clock speed, 12-16 bits (BUS), and up to 12-13 ENOB (Equivalent Number Of Bits). Highspeed ADCs are generally more complex than DACs, and have inferior tradeoff between ENOB and maximum clock frequency. The reason is that most ADC architectures (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 AM noise, high stability, frequency agility, and programmable amplitude and phase. We have in mind general purpose instruments, Atomic-Molecular-Optics physics and atomic clocks [14], [15], [16], longdistance synchronization via fiber links [17], [18], [19], realtime phase measurements [20], particle accelerators [21], etc.
In this context, we focus on the AM and PM noise of DACs and DDSs. Interestingly, 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 implemented 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 most concerned with SFDR, SINAD, THD and ENOB, and leakage from/to adjacent channels (see for example [1,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, starting from f = 10 Hz. It is often difficult to distinguish the device's phase noise from the contribution of the reference oscillator and of the PN analyzer. The literature says quite little about phase noise in DACs and about how it is measured. Reference [22] provides some useful tips based on general RF/microwave methods for PM noise measurement. The AM noise is neither seen in data sheets nor in technical literature.
We describe the noise as the PSD (Power Spectral Density) of the random phase ϕ(t) and fractional amplitude α(t), denoted with S ϕ (f ) and S α (f ) as a function of the Fourier frequency f . Notice that the more popular quantity L (f ) 1 2 S ϕ (f ) is defined only for phase noise. We use the polynomial law S ϕ (f ) = b 0 +b −1 /f +. . . and S α (f ) = h 0 +h −1 /f +. . . truncated after the flicker term 1/f , as appropriated for twoport components. The reader may refer to [23], [24], [25] for an introduction to PM noise, and to [26] for AM noise.
Measuring some DDSs with various methods, we observed that the white noise coefficient b 0 can be of −165 dBrad 2 /Hz. The flicker coefficient b −1 can be of −135 dBrad 2 at 10 MHz output frequency, and of −110 dBrad 2 at 100-150 MHz output. We published only a part of this at a conference [27]. Anyway, these numerical values define the minimum requirement for the background noise. Figure 1 shows the direct measurement of the DAC noise with commercial PN analyzers. All such instruments achieve reduced background noise by correlating and averaging the output of two channels.
Traditional phase-noise analyzers used for the measurement of DACs. The PM noise spectrum results from the contribution of the two DACs. The FFT analyzer measures the average cross-PSD of the two inputs "ϕ 2 − ϕ 1 ," thus it rejects the noise components present only at one input. Notice that the single-DAC clock fluctuation τ 1 and τ 2 (included in ϕ 1 and ϕ 2 ) are detected, while the common-mode clock fluctuation τc is rejected because it cancels in The classical PN analyzer is based on a saturated mixer close to the quadrature condition, which converts the input phase into a voltage. In our case the quadrature condition can set numerically, provided the symmetry be sufficient to set the two channels at once. A problem is the high saturation power of the mixer (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 be amplified, but amplifiers add complexity and noise. A three-DAC version of Fig. 1-A is also possible, which measures the noise of one DAC and rejects the noise of the other two [22,Fig. 4b].
Conversely, the fully-digital analyzer is based on the direct AD conversion of the input signal [28]. The benefit is obvious, in that the instrument accepts different frequencies at the 'in' and 'ref' inputs, and of course there is no phase adjustment. At the time of writing, such test sets are available from only two brands, Microsemi (formerly Symmetricom) [29] and, Jackson Labs [30, scheduled mid/late 2019]. The Rohde Schwarz FSWP [31], [32] is a digital instrument with down conversion from microwaves, but it is still unclear to us whether the 'ref' input is suitable to our purposes.

II. PRINCIPLES AND METHOD
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 that correlation is not necessary. This solves two problems at once. The first is that the correlation instruments rely on the hypothesis that the two channels are statistical independent. This is sometimes untrue, and anyway hard to check. Gross errors are possible if the experimentalist has not a deep understanding (see, for example [33], [34]). The second problem is the measurement time, chiefly with the digital instruments because the noise of the DACs under test is often lower than that of the input ADCs. In fact, the noise rejection is proportional to 1/ √ m, where m is the number of the FFTs averaged. This means 5 dB per factor of 10. Accordingly, if a FFT starting from 1 mHz takes 2000 s acquisition time, averaging over 100 spectra for 10 dB noise rejection takes a measurement time of 2 days and 7.5 hours. A large m is often necessary with the correlation instruments, to reject the single-channel noise and to smooth the spectrum. By contrast, our method does not rely on noise rejection by averaging. A comparatively smaller m is needed, only for smoothing. An extensive treatise abut the noise rejection and the spectrum smoothing is available in [35]. The scheme of Fig. 2 illustrates our method to measure the noise of two equal devices under test (DUT). The rectangles labeled A 1 , A 2 , B, C and D show the spectrum in the critical points, in log scale. The two signals V 1 and V 2 are combined in the directional coupler so that almost all the carrier power goes at the Σ output, and a small power goes to the ∆ output. However, the noise sidebands are equally split between the Σ and ∆ output because these signals are not coherent (spectra B and C). Residual carrier and noise sidebands are amplified (spectrum D) and detected by the PN analyzer or by the AN analyzer.
Our method derives from an early idea for the PN measurement of microwave two-port devices [36]. In the original version, the reference carrier crosses the DUT, which contributes its own noise. The carrier is completely suppressed by subtracting a copy of the reference through a magic Tee (a waveguide directional coupler), and the DUT noise sidebands are down-converted to DC by synchronous detection. Microwave amplification of the noise sidebands was added later to boost the gain and to reduce the background noise [37]. The whole system is equivalent to a Wheatstone bridge powered by an AC signal (the microwave carrier), followed by a lockin amplifier. The lowest background noise is achieved after adjusting the system for the maximum carrier rejection [38], [39]. By contrast, we leave a carefully controlled amount of carrier, as proposed in [40] for different purposes. There results modulation-index amplification, with optional AM-to-PM and PM-to-AM conversion depending on the phase relationships.
The presence of the residual carrier is essential in that it makes the signal suitable to the measurement with all-digital PN analyzers, or with a simple AN analyzer implemented with a power-detector diode ($100-300 for a packaged and connectorized detector) and a FFT spectrum analyzer [26]. Thanks to the AM-to-PM and PM-to-AM conversion, a PN analyzer with no AN measurement capability enables the measurement of both AN and PN. Likewise, an AN analyzer with no PN measurement capability. The modulation-index amplification relaxes the noise specs of the PN analyzer, or of the AN analyzer, by the same factor. The RF gain (40-60 dB) is needed for the power to match the input range of the PN or AN analyzer.
The background white noise is due the noise F kT of the amplifier, where F is 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, the background is b 0 = 2F kT /P , where P is the carrier power at the directional-coupler input, and the factor '2' accounts for the inherent 3-dB loss in the coupler. The background 1/f PM noise is that of the amplifier, divided by the carrier suppression ratio. This happens because the 1/f phase noise in components results from parametric up-conversion of the near-dc noise [41].
For our purposes, it is convenient to set V 1 and V 2 close to the nominal amplitude V 0 , but for a small difference in amplitude (β 1 and β 2 ) and phase (γ 1 and γ 2 ), so that (3) Figure 3 shows these signals and how they combine to form V ∆ . The latter can be written as where η and θ describe the static amplitude and phase of the carrier seen at the coupler output, and (t) and ψ(t) describe the amplitude and phase modulation. We will show that 1/η is the modulation-index amplification. The quantity η has to satisfy | + jψ| η 1. First, η 1 is necessary for the amplification to be useful. Second, η | + jψ| is needed to prevent V ∆ from sweeping 0. If this happens, AM and 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 β 1 , β 2 , γ 1 and γ 2 to appropriate values. Notice that β 1 and Interplay between signals at the output of the directional coupler. In the actual RF circuit, all signals are scaled up by a factor V 0 / √ 2, omitted for graphical clarity. β 2 , and likewise γ 1 and γ 2 , are not equal in the general case, but they can be assumed equal for η 1. Accordingly, it is useful to set β 1 = β 2 = β/2 and γ 1 = γ 2 = γ/2, which defines β and γ. By inspection on Fig. 3, it holds that β 1 and γ θ for η 1.
Combining (1)-(2)-(3) gives The AM and PM associated with V ∆ result from where is the voltage swing of (5), and is the carrier. Recalling that β 1 and γ θ for η 1, Equations (6) can be elegantly rewritten as The matrix represents a rotation by −θ, which comes from the fact that I and Q (in-phase and quadrature voltage swing) is projected on V ∆ . The term 1/η means that I and Q are now referred to a carrier of amplitude η 1, which results in proportionally larger fractional-amplitude or phase swing. Equation (9) emphasizes two key concepts 1) The modulation index is amplified by a factor 1/η 2) The rotation enables to preserve the character of AM and PM (θ = 0), to interchange AM and PM (θ = π/2), or to take any combination of AM and PM.
Replacing and ψ with their spectra, (9) becomes which enables the measurement options listed in Table I. 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 fluctuation τ c is rejected, which includes the clock (Fig. 2). By contrast, correlation in the AM noise is more subtle because it originates from the power supply and from the voltage reference. Insufficient PSRR (Power-Supply Rejection Ratio) may result in correlated noise, and multiple DACs in a chip may share the reference.
We introduce the approximation that S α1 S α2 and S ϕ1 S ϕ2 , based on the fact that the two DACs are nominally equal. Thus, it makes sense to approximate (15) as averaging the two matrices From the geometrical properties shown on Figure 4, the Pythagoras theorem gives Putting (20)- (23) into (15) and expanding (18), we get a general expression for A . The derivation of A and the subsequent manipulations are done with a symbolic math app (Wolfram Mathematica). Since the full expression of A is too complex and hard to read, we omit it and we discuss three relevant cases. The first case is which is equivalent to (10), and validates on rigorous basis the simplified approach of Section II-A. 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 small angle error ζ, and we expand (15) in series truncated to the second order of η and ζ. For θ = 0, replacing θ → ζ gives The third case is the accuracy of (18) for θ = π/2, analyzed in the same way as before but for θ = π/2 + ζ. Hence, The finite gain results in small errors, negligible in most practical cases. For example, 1/η 2 = 100 (20 dB) results in 1.1×10 −2 dB error in (25), and in −26 dB coupling between AM and PM in (26). Of course, the gain must be well calibrated because of the 1/η 2 factor in (15). The angle error deserves more attention. For example, ζ = 0.1 (5.7 • ) results in −4.4×10 −2 dB error and −20 dB coupling between AM and PM in both (25) and (26). The PM noise is generally dominant because of the jitter in the clock distribution, and may pollute the AM-noise measurement.

III. EXPERIMENTS
We measured the two channels of an AD9144 with the scheme of Fig. 5. The four channels of the AD9144 go in pairs, with two NCOs. The two channels we measure can be considered as separated devices, but for a small crosstalk and for the small jitter τ c in the common-mode clock path. The AD9144 is driven by a Z-Board ZC706 via the JESD204B interface. The ZC706 is a computer based on a Zynq-7000, with flash memory, DDR3 memory, Ethernet, USB, SD card slot, HDMI video, and SPI and FMC connectors for daughterboards. It runs Linaro linux. In turn, the Zynq-7000 is a System on Chip (SoC), which is basically a microprocessor and a FPGA on the same chip.
The AD9144 is actually an AD9144-FMC-EBZ card [42], which contains the AD9144 chip, the AD9516-0 clock generator, the output baluns, and auxiliary 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 amplitude, but in our case such speed is needed only for the amplitude because phase and frequency are static parameters, set only once.
Most of the tests were done in Torino with the exact scheme of Fig. 5, and with 1/η from 20 dB to 49 dB. Table 2 shows the operating parameters. However, preliminary tests were done in Besançon using a slightly different configuration, and no power detector. The output V Σ was used as the carrier-rejection monitor, and later in the experiment as the reference of the PN analyzer. This saves one coupler on the RF path, and results in 3-dB lower background noise.
The AD9144 data sheet does not indicate the ENOB. Instead, it gives the typical white noise −162 dBm/Hz (N = 6.3×10 −20 W/Hz) with 150 MHz single-tone output, full amplitude, and 983.04 MHz sampling frequency. These conditions are quite similar to ours. The white AN follows from S α = N/P , where P is the output power. The PN follows from S ϕ = N/P + J, where J is the additional contribution of the clock jitter, unknown here and not impacting on the AN. From the specs, we expect S α = −164 dB/Hz with +2 dBm output power, and S α = −161 dB/Hz for the noise of two channels.
Accounting for 7 dB loss from the DAC output to the input of the RF amplifier (low-pass filter and two 3-dB couplers), and adding the independent contribution of two DACs, the expected noise is of −166 dBm/Hz at the input of the RF amplifier. This value is 8 dB higher than the thermal noise at room temperature (−174 dBm/Hz). Allowing a noise figure of 2 dB, the amplifier contributes +1 dB to the noise limit.
The AN measurement is performed with a single-channel system (power detector, amplifier and FFT analyzer). The Schottky-diode detector has two modes of operation. The true power detection (V out ∝ P in ) relies on the linearization of the exponential V (I) characteristics of the diode, which is valid for the peak voltage V in V T = kT /q. This corresponds to approximately −20 dBm input power on 50 Ω. At higher power, peak detection occurs (V out ∝ V in ). We operate the detector at −3 dBm, which is in the peak-detection region. This operating mode is safe for regular AN measurement, but it would not be suitable to other applications requiring true quadratic detection, like beating two RF tones.
The system needs adjustment and calibration. We first set η and θ to the desired value by adjusting phase and amplitude of V 1 and V 2 . With reference to Figure 6, we start with V 1 and V 2 just below the full-scale range. In this condition, V 1 − V 2 is determined by the arbitrary phase difference, plus the small amplitude asymmetry. Then, we adjust phase and amplitude alternatively for maximum carrier suppression by monitoring the residual carrier. The monitor is an oscilloscope triggered by the 125-MHz reference at one output of the AD9144, or a spectrum analyzer. The spectrum analyzer is easier to use, but the oscilloscope gives access to the sign of the error signal. Manually adjusting phase and amplitude for maximum   carrier rejection, we found that the binary search suffers from convergence instability due uncertainty and fluctuations. This problem is solved by using a sub-binary search, which is a binary-like search with smaller steps. Of course, it takes a larger number of iterations. After the maximum carrier rejection, we set the desired amount of residual carrier by changing V 2 . We recommend to check on AM-PM gain and crosstalk by adding a digital modulation to the DAC under test.
IV. RESULTS Figure 7 shows the PM noise and AM noise of the AD9144 measured with different gain levels. All the spectra in this Section refer to the total noise of the two output channels, hence the noise of one channel is 3 dB lower.
In the first experiment ( Fig. 7 A and 7 B) we use the PN analyzer to measure both the PM and AM noise. Figure 7 A shows the raw phase noise spectra as displayed by the PN analyzer, for different values of the gain 1/η from 20 dB to 49 dB. Figure 7 B reports the same spectra, corrected for the gain. The plots overlap perfectly on almost all the frequency span, indicating that the value of 1/η is not critical. In the upper half-decade, the noise is some 1 dB higher at lower gain. We did not investigate further on this small discrepancy. Likewise, Figure 7 C and 7 D show the AM noise measured with the PN analyzer. The residual carrier is orthogonal to the input carrier (θ = π/2), so that the system performs AMto-PM conversion. The raw spectra (Fig. 7 C) are correctly given in rad 2 /Hz, as displayed by the PN analyzer. The same spectra, corrected for the gain, are shown Figure 7 D. This is the AM noise of the two channels of the AD9144. The unit is dB/Hz, as appropriate. As before, the results overlap on 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 probably due to the power supply. We exclude the AD9144 internal reference because it is a common-mode signal, and has at most a second-order effect on the amplitude noise.
In the second experiment, we measure amplitude and phase noise using the AN detector ( Figure 7 E and 7 F). The maximum frequency is limited by the full span of the FFT analyzer, which is of 100 kHz. The AM noise spectra overlap well on most of the span (Fig. 7 E). A small spread, ±1 dB, shows up in the upper decade. The almost-flat region beyond 10 kHz is actually the bump 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 is ascribed 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 evaluate the upper bound and check on the consistency with the design parameters. From the value S α = −161 dB/Hz based on the data sheet, and accounting for 1 dB contribution of the RF amplifier, we expect S α = −160 dB/Hz for the two converters. This is exactly equal to the lowest value seen on dB/Hz at f = 850 kHz, that is, 1 dB higher than the AN. This indicates that the measured values are consistent with the design. Figure 8 compares the above results with 30 dB gain. The two AM noise spectra overlap, and likewise the two PM noise spectra. This confirms that the two variants of the method, AN analyzer and PN analyzer, give equivalent results.
The flicker of phase takes two levels, b −1 = 4×10 −11 rad 2 h -1 = 8x10 -12  (−104 dBrad 2 ) at lower f and b −1 = 2.8×10 −11 (−105.5 dBrad 2 ) at higher f , with a discrepancy of 1.5 dB. The average of these two values is b 1 = 3.4×10 −11 rad 2 . The latter, converted into time-fluctuation PSD is S x = k −1 /f with k −1 = 5.5×10 −29 s 2 . Using σ 2 = 2 ln(2) k −1 , we find a time fluctuation σ x = 8.7 fs. This value is a combination of time type noise (σ x is independent of the carrier frequency ν) and the phase type noise (S ϕ (f ) is independent of ν, and σ x scales as 1/ν). The definitions and the properties of these types of noise are detailed in [43]. Determining the amount of phase type noise and time type noise requires testing at multiple frequencies, which we have still not done. Figure 9 shows the 1/f noise measured on the widest span. The experiment was done in Besançon with the alternate configuration mentioned. The calibration 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 a posteriori using the flicker coefficients of The observed quantity is η 2 S ψ = 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 (average of the two levels shown), found on Figure 8. Solving we find cos 2 (θ) = 0.26, and finally θ = 1.03 rad (59 • ). Accordingly, the observed result is Regardless of the accuracy of θ, the relevance of this result is the observation of the flicker noise with exact 1/f slope over 7.5 decades, with a maximum discrepancy of 1 dB.

V. CONCLUSIONS
We have proposed a method for the measurement of PM noise and AM noise of DACs and DDSs, and we have proved the concept by measuring an AD9144.
The value of the method is in its reliability and simplicity. Implementation and use require modest skill in analog RF electronics, and only standard skill in programming and using the target DACs and DDSs. Our experiments rely on commercial parts only, like the Z-Board and DAC daughterboards, and on ready-to-use connectorized RF modules. Under no circumstance we had to design and implement ad hoc electronics. The low background noise is inherent in the principle, and easy to achieve. The cross spectrum comes, optionally, only after modulation-index amplification. Consequently, none of the known flaws of the cross spectrum can threat the reliability of the result. Tuning and calibration can be automated because it is done entirely by setting integer numbers in the target converter. Thanks to all these characteristics, our method has the potential to become the standard method for the measurement of AM noise and PM noise of DACs and DDSs.
The experimentalist has two options, using a dedicated PN analyzer or a simple power detector and a general-purpose FFT analyzer. Complexity and background noise are equivalent, and the results overlap.
The power-detector option does not require a PN analyzer, which is a specialized and expensive instrument. The FFT analyzer can be implemented in GNU radio on the Z-Board, after adding an ADC daughterboard. Overall, this version probably fits in the budget of a radio amateur or hobbyist.
The other option requires a commercial PN analyzer. In fact, the design and implementation of such instrument is definitely not simple. The virtue of a fully-digital PN is the wide dynamic range, inherent in the CORDIC algorithm used to extract the random phase modulation from the sampled input [44].
As a fringe benefit, we observed the noise of the AD9144 on a frequency span of 10 decades. The flicker noise matches the exact 1/f law, with a maximum discrepancy of ±1 dB over 7.5 decades. The flicker noise of an electronic device over such a wide frequency range has probably never been reported before.