Oscillator metrology with software defined radio
OOscillator metrology with software defined radio
Jeff A. Sherman a) and Robert J¨ordens b) National Institute of Standards and Technology, Division of Time and Frequency,Boulder, Colorado, USA (Dated: 6 August 2016)
Analog electrical elements such as mixers, filters, transfer oscillators, isolating buffers,dividers, and even transmission lines contribute technical noise and unwanted envi-ronmental coupling in time and frequency measurements. Software defined radio(SDR) techniques replace many of these analog components with digital signal pro-cessing (DSP) on rapidly sampled signals. We demonstrate that, generically, com-mercially available multi-channel SDRs are capable of time and frequency metrology,outperforming purpose-built devices by as much as an order-of-magnitude. For ex-ample, for signals at 10 MHz and 6 GHz, we observe SDR time deviation noise floorsof about 20 fs and 1 fs, respectively, in under 10 ms of averaging. Examining theother complex signal component, we find a relative amplitude measurement insta-bility of 3 × − at 5 MHz. We discuss the scalability of a SDR-based system forsimultaneous measurement of many clocks. SDR’s frequency agility allows for com-parison of oscillators at widely different frequencies. We demonstrate a novel andextreme example with optical clock frequencies differing by many terahertz: using afemtosecond-laser frequency comb and SDR, we show femtosecond-level time com-parisons of ultra-stable lasers with zero measurement dead-time. a) Electronic mail: jeff[email protected] b) Electronic mail: [email protected] a r X i v : . [ phy s i c s . i n s - d e t ] M a y . OVERVIEW Time is best measured by counting periods of natural or manmade oscillators . Tomaximize temporal resolution we must interpolate between integer periods, a task equivalentto determining an oscillator’s phase. Consider two oscillators with frequency f , the periodsof which can be counted as clocks. Their phase offset ∆ φ ( t k ) (in radians) at a measurementepoch t k can be interpreted as a time offset ,∆ T ( t k ) = ∆ φ ( t k )2 πf . (1)Resolving whether ∆ T is stationary is the most sensitive method for detecting small fre-quency offsets or fluctuations between the oscillators and thus calibrating or characterizingthem as clocks . For continuously running clocks, a linear drift in ∆ T defines a (frac-tional) frequency offset between the oscillators y = [∆ T ( t ) − ∆ T ( t )] / ( t − t ) consistentwith the notion that frequency is the rate of change of phase.In this work, we briefly review existing high-resolution techniques for measuring ∆ T ofradio frequency oscillators. We introduce the software defined radio (SDR) concept in thecontext of time and frequency metrology, and describe basic demonstration experiments validfor many SDR implementations. Finally, we explore SDR’s ability to compare oscillators atdissimilar frequencies and to scale to many-oscillator comparisons. One new SDR applicationis discussed in some detail: phase-coherent measurement of optical clocks via a femtosecondlaser frequency comb. A. Radio techniques in oscillator metrology
Though clock frequencies may be high, measurement bandwidth need not be for com-paring oscillators i and r that are similar in frequency, f i ≈ f r . Since clock oscillators aretypically very stable, a signal at f i − f r is both low in bandwidth and low in absolute fre-quency and therefore amenable to high-precision measurement. Such frequency translationis rooted in radio techniques—transmitters shift a signal of low- to moderate-bandwidthupwards to many megahertz or gigahertz for ease of wide-area propagation while receiversspectrally convert the signal back to its original band with no practical loss in information.The widely applied dual-mixer time-difference (DMTD) technique , illustrated in Fig-ure 1a, is an example of radio frequency translation applied to oscillator metrology. A2 ow-passlow-pass zero-crossdetectormixer zero-crossdetector PLLPLL
Time interval counterFPGA accumulator(frequency tuning word) “start”“stop”transferoscillator a) Generic dual-mixer time-difference (DMTD) scheme b) Software-defined radio (single channel)sampleclockDSP NCONCO DDCDDCDDC clockADC high-passfilter decimation& filtersformatter &network interface outputoutput
PLL
FPGAc)
Software-defined radio (dual channel)sampleclockADC1ADC2 high-passhigh-pass decimation& filtersdecimation& filtersformatter & network interface outputs d) Example SDR output data processing chains complex-divide(single channel vs. reference clock) (dual channel; common frequency)
FIG. 1. Schematic comparison of a) generic dual-mixer time-difference (DMTD), and b) a softwaredefined radio (SDR) described here. Dashed-lines surround digital processing sections. In bothconcepts, f i is the frequency of an oscillator under test. A reference oscillator f r disciplines a digitalclock at fast frequency ν m through a phase-locked-loop (PLL). Both methods gain resolution byspectrally shifting f i to a low frequency f b ; in SDR, the mixer analogue is digital downconversion(DDC) with a synthesized numerically-controlled oscillator (NCO). c) Two oscillators f i and f j are compared in two channels of a single ADC, suppressing noise due to the ν m PLL, its referencetone f r , and the ADC’s aperture jitter. d) While DMTD directly outputs time-offset data; furtherprocessing is performed on the SDR sampled waveform z ( t k ) with a computer to determine timeoffset. We illustrate two simplified processing chains for single- and dual-channel measurements;see text for details. ransfer oscillator is synthesized at f r − f b , slightly offset from input and reference oscilla-tors f i and f r : ( | f i − f r | (cid:28) f b (cid:28) f i,r ). The transfer oscillator is mixed (multiplied) withboth f r and f i tones, creating two signals with frequencies near f b after low-pass filtering. Atime-interval counter (TIC) counts periods of a fast timebase oscillator ν m , also disciplinedby f r , gated by high slew-rate zero-crossing detectors observing the two heterodyne productsnear f b .As a consequence of the spectral conversion, DMTD methods resolve ∆ T ≤ − = 1 THz. DMTDrealizations often employ an offset frequency 1 Hz ≤ f b ≤
10 Hz and heterodyne factors10 ≤ f r /f b ≤ , so a TIC must only accurately resolve ∆ T ∼ µ s between the relativelyslow oscillations near f b to discern ( f r /f b ) − (1 µ s) ∼ , ∆ T of 1 ps can be mimicked ormasked by fluctuations of ≈ µ m in electrical length or group delay, so even cables con-tribute to instability (the temperature dependence is of order 0.5 ps m − K − ). Analogcomponents (including ‘digital’ mixers) can contribute flicker-phase noise , amplitude-to-phase-modulation conversion , sensitivity to interference (e.g., channel crosstalk, groundloops, wideband ambient rf), and coupling to the environment (e.g., temperature, humid-ity). The TIC start- and stop-inputs require high-bandwidth, high slew-rate triggers, butthe signals following the mixers are slow sinusoids. Zero-crossing detectors must thereforeboost signal slew rates by ∼ while accurately preserving phase . Without additionalsynthesis steps, DMTD requires f r and f i to be very similar, and among a small set of fre-quencies compatible with the analog processing components. Mixer and filter non-linearityand frequency-dependent group delay complicate maintaining a whole-system ∆ T calibra-tion over arbitrary signal frequencies. Finally, DMTD schemes cannot resolve fluctuationsover time scales shorter than 1 /f b . B. Related work
Some limitations in DMTD can be addressed by replacing certain analog processing stepswith digital implementations. The TIC can be dramatically redesigned with much highereffective ν m . One group replaced the TIC by digitizing the mixed and filtered signals at f b and later eliminated the mixers with high-speed direct sampling of the input signals .4thers have replaced mixer-based spectral down-conversion with aliasing through under-sampling . Early consideration of a direct-sampling system very similar to the presentwork showed plausible limits due to quantization effects alone can be ∆ T < τ / − / for averaging intervals τ . While high-speed samples can be processed entirely in software orwith custom hardware , this work explores oscillator metrology using an inexpensive, com-mercially available, unmodified software defined radio (SDR). We note a similar approach forcharacterizing ADCs . We employ an ADC noise cancelation technique in the time domain,which perhaps is analogous to cross-spectral analysis in the frequency domain. II. SOFTWARE DEFINED RADIOA. “Sample first, ask questions later”
In SDR , signals of interest are sampled by a fast, high-resolution analog-to-digital con-verter (ADC) with little or no analog processing, amplification, or filtering. A numerically-controlled oscillator (NCO), computed synchronously with ADC sampling, takes the place ofthe local oscillator tone in analog radio reception. A digital multiplication of the sampled sig-nal and NCO phasor performs the role of signal mixer. Filtering and sample rate decimationare also performed digitally, reducing noise bandwidth while conserving signal information.Here we focus on SDR receiver functions, but many SDRs are capable of transmission aswell. Since the signal processing chain in SDR is highly-configurable, it has applications inradar, spread-spectrum and multiple-input multiple-output (MIMO) communication, andadvanced protocol demodulation and simulation.SDR seems to suffer a significant disadvantage: noise figures of high-speed ADCs are muchworse than a collection of radio frequency filters, amplifiers, and mixers. On the other hand—especially in the context of precision metrology—analog components are subject to strictimpedance matching requirements and exhibit long-term sensitivity to shock, vibration,supply voltage, temperature, humidity, aging, interference, and signal crosstalk. A lowADC signal-to-noise ratio (SNR) is at least amenable to averaging and process gain, whileenvironmental sensitivities are more pernicious sources of stochastic noise and drift over longdurations. In contrast, digital processing steps are stable, deterministic, and environmentallyinsensitive. 5 . Technical details At the time of writing, the techniques presented here ought to apply to SDRs from atleast ten manufacturers. While we attempt to consider SDR generically, Figure 1b illustratesrelevant components in the SDR receiver studied here (an Ettus USRP N210 except wherenoted ). Field programmable gate array (FPGA) hardware description code and circuitschematics are available for inspection and customization . A receiver daughterboardcouples a ground-referenced input signal (1 to 250 MHz) into a differential ADC via atransformer. The ADC (Texas Instruments ADS62P44) has an analog input bandwidth of450 MHz ( − ± t ap = 150 fs, a significant technical timing uncertainty betweenan idealized sample trigger and actuation of the converter’s sample-and-hold circuitry. Thesample timebase, a voltage-controlled crystal oscillator (VCXO) at ν m = 100 MHz, drivesthe ADC sampling trigger and the FPGA’s digital signal processing (DSP) pipeline. TheSDR includes phase-locked loop (PLL, bandwidth ≈ ν m to a +14 dBm signal at f r = 10 MHz derived from an active hydrogenmaser. In our configuration, this SDR consumes about 10 W of dc power.SDR’s three important DSP tasks are frequency translation, filtering, and data decima-tion. After a high-pass filter suppresses the ADC’s zero-offset, the input signal undergoesdigital down-conversion (DDC), or frequency translation by an NCO tuned to f a = ν m × a , (2)where a is an integer 0 ≤ a < . As in direct digital synthesis (DDS), a phase register accu-mulates the frequency-tuning word a upon every ν m clock cycle, the most significant bits ofwhich are used to derive complex NCO phasor components. However, unlike many DDS im-plementations, SDR often does not use the phase register as an index in a large lookup tableof precomputed trigonometric values. Instead, SDR often implements coordinate rotationdigital computer (CORDIC) to compute NCO phasors in fixed-point arithmetic. Exploit-ing the equivalence between angle rotation and phase accumulation, CORDIC is a succes-sive approximation algorithm built from logical operations well suited to a DSP pipeline:comparisons, bit shifts, and binary addition. After inspection of the two most significant ac-cumulator bits fixes the phase quadrant, this SDR implements K = 20 CORDIC iterationson 24-bit phase words for an approximate angle resolution of tan − (cid:2) − ( K − (cid:3) = 1 . µ rad.6 i gna l ( V ) -101 a) Sample time (s)0 0.2 0.4 0.6 0.8 1 P ha s e (r ad . ) c) R e s i dua l ( m V ) b) FIG. 2. SDR measurement of a signal at f i = f r = 10 MHz, spectrally-shifted by DDC to f b = 10 MHz − (429 , , × − ×
100 MHz) ≈ buffered samples)of z ( t k ) = I ( t k ) + iQ ( t k ) data. b) The residual amplitude of I ( t k ) after removing a best-fit single-tone. The noise is predominately white, but modulation related to f b and proportional to | I ( t k ) | is clearly observed. c) The instantaneous phase evolves as 2 πf b t ; here we plot arg z ( t k ) wrappedinto − π < arg z ( t k ) ≤ π . CORDIC approximates resampling the real input signal into a complex frame rotating at f a ,adding negligible quantization noise (approximately equivalent to σ x ( τ ) = 0 . τ / − / at 10 MHz). The SDR ultimately truncates the resulting signal to 16 bits of resolution foreach of the in-phase ( I ) and quadrature-phase ( Q ) components.Transmission and manipulation of output samples z ( t k ) = I ( t k ) + iQ ( t k ) at the physicalsample rate ν m would require ≈ divides the samplerate by an integer 1 ≤ n cic ≤ ,each accomplishing a rate division of 2 and antialias filtering. Within their passbands thesefilters have a linear phase/frequency dependence and thus are shape-preserving in the time-domain. Figure 2 shows data acquired with typical settings, n cic = 25 and both half-band7lters enabled, which yields a decimation of n dec = 4 n cic = 100 and ν m /n dec = 10 samplesper second, requiring 32 Mbps of network and buffering resources. ADC quantization noisepower, which is nearly uniform in density (‘white’) over a Nyquist bandwidth of ± ν m / n − (see Appendix B). The final DSP section queues and formats z ( t k ) along with metadata such as hardware time-stamping and drives their transmissionto a general-purpose data acquisition computer. Application programming interfaces areavailable for several languages, free tools like gnuradio , and commercial data processingpackages. III. DEMONSTRATION EXPERIMENTS
We now outline our study of SDR’s suitability for oscillator metrology. We first discussphase measurements over intervals of a few seconds, the analysis of which includes informa-tion about fast fluctuations up to ν m / (2 n dec ) in frequency. Then, we consider measurementnoise over several hours to days using methods which average over fast fluctuations. We findthat over intervals greater than about 10 ms, ADC aperture jitter is likely a limiting tech-nical noise source. We demonstrate a promising solution available in many SDRs: a second,independent ADC channel is synchronously sampled such that aperture jitter and manyother noises subtract in common-mode. We consider application of SDR in a many-clockinter-comparison, and to the problem of optical frequency and phase metrology. Finally,we briefly describe measurement performance of a 6 GHz microwave tone beyond the ADCbandwidth, and the instability of amplitude measurements in two SDR models. A. Phase of ADC input vs. the sampling timebase
Consider the arrangement in Figure 1b where f i is approximately known and stationary,and f r is treated as a frequency reference ( f i need not be similar to f r ). We choose theinteger a so f a (see Eq. 2) is close to f i . Absent technical noise, ν m = 10 f r due to the mastertimebase’s PLL, making f a exactly computable. The SDR output samples, z ( t k ), representthe input signal spectrally shifted to a low frequency f b = f i − f a ; the sample epochs are t k = k × ( n dec /ν m ). The signal phase, arg z ( t k ) ≡ tan − [Im z ( t k ) / Re z ( t k )] (see Figure 2c)is a time-integral of angular frequency 2 πf b and so evolves in time as 2 πf b t k + φ , where8 )b) Fourier frequency (Hz) P ha s e PS D ( d B c / H z ) -160-140-120-100 Free-running VCXO ADCMaser c) T i m e de v i a t i on ( s ) -5 -4 -3 -2 -1 -14 -13 -12 Single ADC channelDual ADC channels
Averaging interval (s) P ha s e ( m r ad . ) -0.200.2 Sample time (s)
FIG. 3. a) Unwrapped phase signal arg z ( t k ) when f i and f r derive from the same 10 MHz oscillator(0.1 mrad corresponds to ∆ T ≈ . f a isremoved. Here, n dec = 100. b) A Fourier transform of arg z ( t k ) (black data) yields a single-sidedphase noise power spectral density (PSD). At high Fourier frequencies we observe a white noise of ≈ −
140 dBc/Hz (green dashed line), consistent with the ADC’s SNR, signal power, and decimationfiltering. At low Fourier frequencies we observe technical noise roughly tracking the rising noisedensity of the ν m VCXO (red dashed line), which the PLL cannot fully suppress. A hydrogen masernoise specification (black dot-dash) provides context. The relative PSD between two ADC channels(blue data) has much improved flicker noise. c) Time deviation σ x ( τ ) in one-channel (black) andtwo-channel (blue) modes; n dec = 500. Solid points derive from short streams of arg z ( t k ) sampleswithout averaging. Open circles result from pre-averaging streams in 1 s chunks. White phase noiseof 1 . τ / − / (black dashed line) is equivalent to ≈
86 dB SNR (see Appendix A). The bluedashed line represents a further 6 dB improvement. A red dashed line marks the ADC’s aperturejitter t ap . For τ (cid:29)
10 s, we expect environmental coupling to dominate both measurement modes.See text for further detail. includes technical offsets such as cable delays. The tan − function is evaluated withindependent numerator and denominator arguments, removing a phase-quadrant ambiguity.Generally, all SDRs are capable of this mode of measurement, though those without theability to reference ν m will suffer in accuracy.To analyze the phase noise floor of this configuration, we split a single 10 MHz oscillatorinto the f i and f r inputs. The amplitude at the ADC is kept near half-scale to avoiddistortion (typical input power was ≈ z ( t k ) = 2 π (1 − a/ )( kn dec /
10) (neglecting a fixed φ ). In software, we subtract this deterministic trend,the magnitude of which is made small by an appropriate choice of a , and interpret residualfluctuations as measurement noise. Figure 3a shows a typical residual phase signal, a Fouriertransform of which yields the phase noise power spectral density (Figure 3b).The black curve in Figure 3c depicts a complementary statistical measure: the oscillatortime deviation σ x ( τ ) = τ √ mod σ y ( τ ), where mod σ y ( τ ) is the modified Allan deviation .Briefly, σ x ( τ ) characterizes the predictability of phase (in time units, see Eq. 1) as a functionof averaging interval τ . Over roughly 20 µ s < τ < µ s we observe behavior consistentwith white-phase noise, σ x ( τ ) ≈ . τ / − / . Regrettably, σ x ( τ ) stops decreasing withfurther averaging, and besides an oscillation peak (related to modulation at f b ) appears lim-ited to a flicker-floor roughly consistent with the ADC’s t ap = 150 fs. Reducing input powerincreases the white-phase instability, but otherwise these performance limits persist overmany instrument configurations: rf-coupling method (dc-coupled op-amp vs. transformer),choice of heterodyne f b , decimation factor n dec , stock vs. quiet linear power supply, etc. B. Phase of one ADC channel vs. another
To do better we must reduce the influence of phase noise in ν m and the ADC aperturejitter t ap . Fortunately, many SDRs can process two independent ADC channels which aresampled synchronously (specifically, the two ADC channels exist on the same chip). Toexamine residual noise in this differential configuration, we split the same oscillator intothe three inputs ( f i , f j , and f r ) as shown in Figure 1c, though it is not crucial that the f r input be identical to either of the others. Since f i = f j , the same NCO frequency f a is usedto DDC both channels, giving the same deterministic trend to both output phase signals.The phase signals should be unwrapped before subtraction because, due to noise and small10hase offsets, 2 π -discontinuities can appear at different sample epochs. Example single- anddual-channel signal processing chains are illustrated for comparison in Figure 1d.Blue curves in Figures 3b and 3c show the significant improvement in phase noise and timedeviation from dual-channel operation. A flicker-floor of σ x ≈
20 fs now appears roughlyan order-of-magnitude below t ap and persists over 1 ms < τ <
500 s. It also improves byan order-of-magnitude upon the typical noise floor of the DMTD instrument ( σ x ≈
300 fs).While we lack detailed knowledge of the ADC, we posit that each channel’s sample-and-hold circuitry shares a trigger-input threshold-detector. After this element, circuit paths,component/process variation, and environmental non-uniformity on the ADC chip are likelyminute. ADC voltage-reference fluctuations and phase noise in the ν m PLL (and its reference, f r ) are also highly common to both sampled channels. Remaining non-common elementsinclude off-chip transmission lines, coupling transformers, and on-chip ADC sampling cir-cuitry. We show later that similar common-mode suppression is present in a different ADCwith much larger t ap = 1 ps.In this mode of operation, it is less important that ν m be locked to a high-quality oscillatorbecause phase noise in ν m will be highly-common between the two sampled inputs. Noise isnot completely suppressed, however. We found slightly better performance, at the level of20 % in σ x , when ν m was referenced to a hydrogen maser versus the SDR’s quartz oscillator.We hypothesize that parasitic coupling of the digital sample clock at ν m is slightly imbalancedbetween the two ADC inputs. This feature is likely specific to the SDR model and circuitlayout. C. Instability over long averaging intervals
Maximum decimation in an SDR still results in several megabits per second of data perchannel. As a practical matter for long-duration measurements, we reduce this data streamas it is acquired to one recorded ∆ T value per second. This step reduces measurementbandwidth to ≈ z ( t k ) over groups of N = ν m /n dec samples per second, and the phase estimation routine discussed in Appendix A. Issues relatedto windowed averaging here are analogous to those in frequency meters .The SDR measurement stability does not degrade much over intervals of several hours, an11 i m e o ff s e t ( p s ) -1000100200300400 SDR 1ch. + 50 psSDR 2ch. + 25 psCommercial DMTD
MJD - 57247 (days) T i m e o ff s e t ( p s ) -10-50510 DMTD - SDR 1ch.DMTD - SDR 2ch. a)b)
SDR “warm-up”transient Human activitynear apparatusapprox. 31 hours
FIG. 4. a) A comparison of two hydrogen masers over five days (MJD is the modified Julian date),using a commercial instrument based on DMTD (red) and the SDR described here (black/blue forsingle-/dual-channel mode, respectively). From each time series we subtract a linear phase trendcorresponding to the masers’ frequency difference of y = 8 . × − . We introduce 25 ps and50 ps offsets for visual clarity. b) We show the differences of each SDR measurement with thatof the DMTD. Some technical noise features are understood and annotated; it is not yet knownwhether the DMTD, SDR, or both systems contribute to the ∼
31 h periodic modulation. important requirement for an atomic-clock measurement system . We undertook no specialenvironmental stabilization beyond standard laboratory conditions (ambient temperaturecontrol of ≈ . σ y ( τ ) = 7 × − ( τ / − through τ = 10 s. 12 veraging duration (s) T o t a l A ll an de v i a t i on -17 -16 -15 -14 -13 Commercial DMTD (ST0014 vs. ST0010)Commercial DMTD residualsSDR 2-ch. (ST0014 vs. ST0010)SDR 2-ch. residualsSDR 1-ch. residuals
FIG. 5. Fractional frequency instability σ y ( τ ) of hydrogen masers (NIST masers ST0014 vs.ST0010) as measured by a DMTD commercial instrument (red, solid) and the SDR two-channeltechnique (blue, solid) described here. From τ ≥
200 s both techniques become identically limitedby maser frequency fluctuations. Open points show typical residual instabilities of the DMTDinstrument (red), the single-channel SDR method (black), and the two-channel SDR method (blue).The blue dashed line is an eye guide placed at σ y ( τ ) = 7 × − ( τ / − . Both DMTD and SDRmethods yield one datum per second, but the effective measurement bandwidth of the DMTDinstrument is known to be (cid:29)
D. Clock comparison with software radio
The tests described above demonstrate the low instability of the SDR technique; herewe discuss time accuracy. Two 5 MHz signals, sourced by hydrogen masers (NIST masersST0010 and ST0014), are input into the two SDR ADC channels. A non-linear frequencydoubler converted one of these to create the f r = 10 MHz PLL reference. The maser signalswere measured simultaneously by a commercial system based on DMTD. Figure 4a showsexcellent agreement between the methods. The time-series of the difference between the13ata sets (Figure 4b) reveals technical noise in one or both measurement systems, somedetails of which are not yet understood. An initial transient of about 15 ps in magnitude isa repeatable ‘warm-up’ SDR characteristic lasting several minutes. Key component temper-atures, measured with platinum resistive thermometers attached with thermally conductiveepoxy, increase by 5 K to 10 K in these first several minutes of operation. We also observea periodic variation (of roughly 31 h) with an amplitude of order 5 ps. Such a variationwould contribute < − to fractional frequency instability, which is of marginal signifi-cance in the inter-comparison of maser clocks. Figure 5 shows the frequency instability ofthe maser comparisons and typical SDR and DMTD residual instabilities. At averagingintervals of τ ≈ s, the single-channel SDR technique is comparable with the commer-cial DMTD instrument; the two-channel SDR technique outperforms both by almost anorder-of-magnitude. E. Multi-channel operation
Some commercial DMTD instruments accept 16 or more input oscillators, where onechannel is permanently assigned a special role as reference for the TIC timebase ν m . Thetwo-channel SDR scheme presented here is scalable to an unlimited number of channels,and it is possible but not necessary that one oscillator be assigned a special role. Figure 6asketches a scheme whereby multiple SDR instruments are arranged in a ‘ring,’ immune to anysingle oscillator or SDR fault. A ‘hub’ model (Figure 6b), where one oscillator is distributedto all measurement nodes is also possible. Simultaneous implementation of the one-channelSDR technique using a distinct f r oscillator provides a ‘backup hub’ mode of operationwith degraded performance. In a scaled deployment, it may be desirable to increase thedecimation performed in hardware, perform the phase computation and averaging itselfin the FPGA, and/or distribute the software data processing among multiple connectedcomputers. We estimate that, per measurement channel, the material cost of a SDR solutionis a factor of two or more below competitive multi-channel DMTD instruments.14 DR 00 1 2 3 m a) Multiple SDRs: ring configuration b) Multiple SDRs: hub configuration with backupSDR 0 0SDR 1 SDR 2 ...... SDR m-1 12 3 m FIG. 6. Multiple SDRs can scale for coherent many-oscillator comparisons in flexible arrangements,the choice of which will depend on which failure modes are judged most likely. a) For example, ina ‘ring’ configuration, each SDR node produces the two-channel differential signal f j − f i , a one-channel signal f i − f r and unique one-channel residual f j − f r . Phase data collection for all oscillatorsis uninterrupted with any single node failure. b) In a ‘hub’ configuration, the oscillator indexed ‘0’is distributed to an ADC channel in each node as part of a two-channel differential measurement.To protect the network against failure of oscillator ‘0’, oscillator ‘1’ provides a shared PLL referenceto all nodes, enabling one-channel measurements of all oscillators as a ‘degraded backup’. Here,junctions imply distinct distribution amplifier channels; differential amplifier and cable delays mustbe accounted for when comparing oscillator phase differences. F. Optical oscillator measurement
Optical atomic frequency references now exceed the performance of official primary stan-dards based on microwave frequencies by factors of 1000 in stability and potentially 100in accuracy . Generally, optical frequency references operate by disciplining a pre-stabilized laser oscillator to an atomic resonance in neutral atoms or single trapped ions .Direct phase and frequency comparisons between two lasers at f α and f β are only possi-ble if they are sufficiently close to create a heterodyne beatnote on a photodiode or othertransducer. Otherwise, a now standard technique employs a broadband femtosecond laserfrequency comb (FLFC, or comb) as a common heterodyne oscillator spanning hundreds of15 m p li t ude ( m V ) -50050 Sample time (ms)
Sample time (s) T i m e o ff s e t ( f s ) -505 a)b)c) FIG. 7. Tracking optical phase with SDR (see text for details). a) The SDR down-converts aheterodyne between a femtosecond laser frequency comb (FLFC) stabilized to reference laser f α ,and laser f β to an audio tone of ≈
140 Hz. We plot the complex components of the SDR output z ( t k ). b) Laser β is transmitted to the FLFC heterodyne via an uncompensated fiber optic link.By shaking the fiber, we observe and can coherently track resulting phase fluctuations. c) Dividingarg z ( t k ) by 2 πf β , we cast phase fluctuations as time instability of the optical oscillator β . Aconstant phase and frequency offset are suppressed in the plot. terahertz . A FLFC spectrum consists of many optical modes, whose absolute frequen-cies can be expressed as f n = nf rep + f ceo for many thousands of consecutive integers n .The comb’s pulse repetition rate, f rep , scales inversely with the laser resonator length, and | f ceo | < f rep depends on the details of the intra-cavity dispersion. For our purposes, it issufficient to note that both degrees of freedom correspond to radio frequencies controllableby phase-lock techniques.We measured and tracked phase fluctuations between two laser oscillators using a FLFCand the SDR. A Ti:sapphire FLFC with f rep ≈ f α ≈
259 THz. We used self-referencing interferometry to stabilize f ceo . A second ultra-stable laser , f β ≈
282 THz,interfered with another comb mode to make a heterodyne tone f o near 160 MHz on anamplified photodiode. Independent characterizations have determined frequency instability16oors of ≤ × − for laser α and 1 × − for laser β . Due to the comb’s phaselocks, fluctuations of f o are directly related to the fluctuations between the α and β laseroscillators; the required comb mode integers for absolute determinations can be obtained bylow-resolution wavemeter measurements of f α and f β .Traditionally, only gated frequency measurements are made of f o , discarding informationabout phase fluctuations. A DMTD scheme to track phase is impractical: generally, f o canappear at any frequency up to f rep / f o fluctuationsand drift are typically too large. In contrast, the SDR has a high input bandwidth, a tunableNCO for down-converting arbitrary f o , and tracks phase information over very short intervals ν m /n dec ≤ µ s with no dead time.Since the ADC sample clock ν m = 100 MHz, f o ≈
160 MHz appears in the third ± ν m / − .
005 860 MHz. We set the NCO f a = − .
006 000 MHz in orderto obtain an audio beat note | f b | ≈
140 Hz. Figure 7a shows the output sample data undernormal conditions; Figure 7b shows directly observable phase noise created by vigorouslyshaking the uncompensated fiber optics coupling laser β to the comb. It is importantto appreciate that a radian of optical phase remains unscaled by mixing with the comb tomake f o , nor is it scaled by the DDC process f o → f b in the SDR. So, treating laser α as areference, we can derive the time fluctuations of laser β by unwrapping and dividing the f b phase arg z ( t k ) by a factor 2 π ×
282 THz, following Eq. 1. Figure 7c shows the result: well-resolved femtosecond-level temporal instability between two would-be optical clocks, lasers α and β . In measurement of optical heterodyne tones, the SDR noise floor is negligible.A multi-channel SDR arrangement monitoring several FLFC heterodyne beat notes couldform the measurement basis for an optical time scale , meaning an ensemble of optical os-cillators statistically weighted to produce a robust and reliable ‘average clock’ . Relatedtechnology is approaching a high level of readiness, including robust fiber-FLFC designs ,stabilized ‘flywheel’ lasers with frequency instabilities σ y ≤ × − , and optical atomicstandards characterized at the 10 − uncertainty level . G. Microwave frequencies
Microwave frequencies far beyond the ADC input bandwidth are measurable by SDRmodels that incorporate an analog mixer and microwave local-oscillator (LO) synthesizer ref-17 ) b)
FIG. 8. a) Time deviation of differential phase measurements of 5 MHz and 6 GHz signals. For5 MHz, the SDR featured a 12-bit ADC with 1 ps aperture jitter. The instability floor is more thantwo orders of magnitude lower than t ap , indicating excellent common-mode suppression of technicalnoise. For 6 GHz, the SDR featured a LO synthesizer and analog mixer front-end to translate thesignal into the ADC bandwidth. Though phase-noise performance is made worse by these elements,the high signal frequency leads to a time stability floor of 1 fs, roughly an order-of-magnitude betterthan the results at 10 MHz (Figure 3c). b) We also investigated amplitude measurement instability(normalized to input amplitude) of two-channel signals in these SDR models. In both plots, dataat longer τ are obtained by additional software decimation by a factor of 2500 prior to storage.These data were acquired in an unstabilized office environment and with the sample clock ν m un-referenced. Shaded bands indicate standard statistical uncertainties. erenced to the same source as ν m . In a separate investigation, we tested a SDR (Ettus USRPB210) featuring such a front-end (Analog Devices AD9361) capable of down-converting two ≤ −
22 dBm) signal and set the SDR’s programmable ampli-fiers to 49 dB to use the full ADC range. ν m was set to 30.72 MHz, and n dec to 32. Dueto the mixer front-end, we observed significantly higher phase noise than the results in sec-tion III B: a white noise floor at −
123 dBc/Hz and flicker noise of −
90 dBc/Hz ( f / − .However, given the much higher carrier frequency, the equivalent time deviation limits were σ x ( τ ) = 20 as ( τ / − / over short intervals and a flicker floor of 1 fs, as shown in Figure 8a.18 . Amplitude metrology Though we have so far ignored it, the amplitude of a complex sampled SDR signal isalso available as (cid:112) I ( t k ) + Q ( t k ). In a separate investigation, we studied the relativeamplitude instability limit of signals input into two ADC channels. We tested a SDR (EttusUSRP B100) with a 12-bit ADC (Analog Devices AD9862), ν m = 64 MHz, n dec = 64, and f i = f j = 5 MHz. As shown in Figure 8b, we observed a relative amplitude instability floorof 3 × − over the averaging interval 0 . ≤ τ ≤
100 s. The 6 GHz configuration, describedin section III G, achieved an amplitude instability floor of 5 × − . IV. CONCLUSIONS
Generally, SDR receivers are little more than high-speed signal samplers followed by aseries of digital filters designed to reduce data rate and noise bandwidth. However, these fewingredients are sufficient for several recipes in high-precision time and frequency metrology.Phase/time-offset measurements using unmodified SDR hardware can exceed the stabilityperformance of a commercially-available instrument based on the classic DMTD design whileoffering increased flexibility. SDR measurement of phase using two input channels differen-tially reduces the influence of technical timing noise and has demonstrated a maser clockfrequency resolution σ y ≤ − within 10 s of averaging. Over several days of continuoushydrogen maser measurement, the SDR technique appears highly accurate, with relativelylow environmental noise coupling in a typical laboratory environment. SDR hardware isscalable to coherently measure any number of oscillators at almost any radio or microwavefrequency. We have shown the SDR can resolve relative oscillator amplitude fluctuationsbelow the part-per-million level. Finally, we have demonstrated that SDR can be usefullyemployed in the comparison of ultra-stable optical clocks and oscillators by measuring het-erodyne products of clocks with a femtosecond laser frequency comb.Useful extensions of this work could include a long-term frequency comparison of atomic-clocks’ output signals at multiple frequencies (e.g., 5 MHz and 100 MHz), and integrationof a many-channel fast ADC into an SDR architecture for better multi-channel scalability.Alternatively, the transmission functions of the SDR could be employed in active phase-noisecompensation in optical or FLFC interferometry applications.19IST’s Time and Frequency Division funded this investigation. The work is a contribu-tion of NIST and not subject to U.S. copyright. The authors thank Judah Levine for helpfuldiscussions, Joshua Savory (maser comparisons), Franklyn Quinlan (FLFC measurements),and Roger Brown for careful reading of the manuscript. Appendix A: Spectral estimation of frequency and phase
In the single-channel setup of Figure 1b, a f i known only to within a Nyquist bandwidth ν m / f a overits full range. Without loss of generality, we suppose f i < ν m / f a such that | f i − f a | (cid:28) ν m /n dec ; in other words, the DDC frequency must bewithin the decimated Nyquist zone. The sign of the sampled ‘beatnote’ f b = f i − f a is fixedby the sense of temporal phase rotation in z ( t k ), or equivalently, the phase relationship of itsreal and imaginary components. The problem of high resolution determination of f i reducesto spectral estimation on groups of N samples of z ( t k ) to estimate f b . Though no closed-form solution exists generally for spectral estimation , our circumstances are unusuallyfavorable: z ( t k ) consists of a single, low-frequency tone f b , with high SNR and little harmonicdistortion. Though computationally intensive, an optimal un-biased frequency estimatorgiven these assumptions is the argument ˆ f b maximizing the basic periodogram function | P ( f ) | , where P ( f ) = 1 N N − (cid:88) k =0 z ( t k ) e − i πft k . (A1)For signals like ours, | P ( f ) | is well-approximated by a quadratic polynomial near its maxi-mum. We therefore implemented Brent’s method of one-dimension parabolic interpolation to efficiently search for ˆ f b . A lower resolution FFT-based spectral estimator seeds this non-linear search with an initial guess. Unlike such FFT-based methods, no windowing functionor zero-padding must be applied to the sampled data prior to | P ( f ) | maximization, andthere is no need to make N a power-of-2. The search also yields an optimal estimator for thesingle-tone amplitude, ˆ A b = | P ( ˆ f b ) | . In the limit of high SNR and spectrally-uniform uncor-related (‘white’) noise, periodogram maximizing spectral estimates converge with maximumlikelihood and non-linear least-squares fit results.Figure 5 (black open circles) shows measured frequency instability of a f i = f r = 10 MHz20ignal, which surpasses that of a commercial frequency meter of comparable cost. Impor-tantly, note that the SDR measurement instability decreases as τ − , compared to manyfrequency meters’ instability ∝ τ − / . This difference is attributable to non-zero dead-timeand frequency quantization in commercial meter readings. The interval N ( ν m /n dec ) − isanalogous to a ‘gate interval’ of a traditional frequency meter. With SDR, this parametermay be chosen during or after data acquisition since z ( t k ) data can be stored. Barringinterruption in data transmission, this method of frequency analysis has zero ‘dead-time’intervals during which the input oscillations are unmeasured.We continue the spectral estimation method to determine phase offset measurements fromsets of N complex waveform samples z ( t k ). If unknown, we first find the ˆ f b maximizing theperiodogram function | P ( f ) | from Eq. A1. Then, the optimal estimate of the signal’s phaseis ˆ φ b = tan − (cid:34) Im P ( ˆ f b )Re P ( ˆ f b ) (cid:35) . (A2)Successive estimates of phase on continuously sampled data will evolve asˆ φ b ( t k ) = φ + 2 πf b t k (A3)= φ + 2 π ( f i − f r ) t k + 2 πf r (cid:18) − ν m f r a (cid:19) t k , (A4)where φ is the initial phase offset and, for the SDR described in section II B, ν m /f r = 10.The final term, the result of our choosing a heterodyne offset frequency, is exactly computablein terms of f r and is removable in post-processing. Subtracting it using complex phaserotation neatly avoids 2 π discontinuities, leaving us only with a phase growing linearly withthe frequency difference of interest f i − f r . Phase discontinuities must still be expected andhandled over time intervals τ ≥ π/ ( f i − f r ). The variance of a single ˆ φ b estimate using N (cid:29) var (cid:16) ˆ φ b (cid:17) ≥ N . (A5)As SNR ∝ /N itself (due to process gain), the bound for variance in the phase estimatoris independent of the sample density N under optimal noise conditions, remaining inverselyproportional to the SNR and total observation duration. Combining this result with Eq. 1,the resulting theoretical bound on time deviation is σ x ( τ ) = 12 πf i (cid:113) var( ˆ φ b ) ≥ . × − s ( τ / − / , (A6)21 ABLE I. Decimating low-pass filters in the SDR ideally improve SNR proportionally to thedecimation factor n dec . We a slightly worse empirical scaling ∝ n . . Here we compare themeasured SNR for a constant, half-scale, f i = f r = 10 MHz maser-referenced tone under differentdecimation settings. f b ≈ n dec Expected SNR (dB) Observed SNR (dB) Excess noise (dB)20 81.5 74.5 7.040 84.5 76.7 7.8100 88.5 79.8 8.7200 91.5 82.1 9.4500 95.5 85.2 10.3 where f i = 10 MHz, N = 10 samples per second, and the effective SNR ≈
86 dB (seeAppendix B). Observations in Figure 3 (black solid points) are consistent with this noiselimit over short averaging intervals.
Appendix B: Decimation fidelity in practice
Ideally, in the presence of uniform Gaussian noise, the SNR of ADC samples should beimproved by a factor of the decimation ratio n dec since the CIC and half-band decimatingfilters approximate an ideal low-pass filter. Alternatively, with SNR is expressed in dB,SNR (ideally observed) = SNR ADC + 10 log n dec . (B1)However, finite precision in the numerical filters, and the presence of non-Gaussian noise,such as spurs and input noise near the sample clock ν m , result in slightly worse performance.We observe an approximate n . improvement with 20 ≤ n dec ≤
500 as shown in Table I.
REFERENCES Certain commercial equipment, instruments, or materials are identified in this paper forinformational purposes only. Such identification does not imply recommendation or en-dorsement by the National Institute of Standards and Technology, nor does it imply thatthe materials or equipment identified are necessarily the best available for the purpose.22
David W Allan and James Barnes. A modified ‘Allan Variance’ with increased oscillatorcharacterization ability. In
Thirty Fifth Annual Frequency Control Symposium. 1981 , pages470–475. IEEE, 1981. David W Allan and Howard Daams. Picosecond time difference measurement system. In , volume 1, pages 404–411, 1975. DW Allan and H Hellwig. Time deviation and time prediction error for clock specifica-tion, characterization, and application. In
IEEE 1978 Position Location and NavigationSymposium , volume 1, pages 29–36, 1978. Maurice G Bellanger, Jacques L Daguet, and Guy P Lepagnol. Interpolation, extrapola-tion, and reduction of computation speed in digital filters.
Acoustics, Speech and SignalProcessing, IEEE Transactions on , 22(4):231–235, 1974. BJ Bloom, TL Nicholson, JR Williams, SL Campbell, M Bishof, X Zhang, W Zhang,SL Bromley, and J Ye. An optical lattice clock with accuracy and stability at the 10 − level. Nature , 506(7486):71–75, 2014. G Brida. High resolution frequency stability measurement system.
Review of ScientificInstruments , 73(5):2171–2174, 2002. Andrea C Cardenas-Olaya, Enrico Rubiola, Jean-M Friedt, Massimo Ortolano, SalvatoreMicalizio, and Claudio E Calosso. Simple method for ADC characterization under theframe of digital PM and AM noise measurement. In
Frequency Control Symposium &the European Frequency and Time Forum (FCS), 2015 Joint Conference of the IEEEInternational , pages 676–680. IEEE, 2015. Gilles Cibiel, Myrianne R´egis, Eric Tournier, and Oliver Llopis. AM noise impact on lowlevel phase noise measurements.
Ultrasonics, Ferroelectrics, and Frequency Control, IEEETransactions on , 49(6):784–788, 2002. Steven T Cundiff and Jun Ye. Colloquium: Femtosecond optical frequency combs.
Reviewsof Modern Physics , 75(1):325, 2003. Ettus Research circuit schematics and technical information. http://files.ettus.com/ . Ettus Research code repository. https://github.com/EttusResearch/uhd . Tara M Fortier, Albrecht Bartels, and Scott A Diddams. Octave-spanning Ti:Sapphirelaser with a repetition rate > Optics Letters , 31(7):1011–1013, 2006. 23 TM Fortier, A Rolland, F Quinlan, FN Baynes, AJ Metcalf, A Hati, A Ludlow, N Hinkley,M Shimizu, T Ishibashi, et al. Digital-photonic synthesis of ultra-low noise tunable signalsfrom RF to 100 GHz. arXiv preprint arXiv:1506.03095 , 2015. GNURadio team. GNURadio: the free and open software radio ecosystem. http://gnuradio.org/ . Charles Greenhall, Dave Howe, and Donald B Percival. Total variance, an estimator oflong-term frequency stability.
Ultrasonics, Ferroelectrics, and Frequency Control, IEEETransactions on , 46(5):1183–1191, 1999. J Grove, J Hein, J Retta, P Schweiger, W Solbrig, and SR Stein. Direct-digital phase-noisemeasurement. In
Frequency Control Symposium and Exposition, 2004. Proceedings of the2004 IEEE International , pages 287–291. IEEE, 2004. Sebastian H¨afner, Stephan Falke, Christian Grebing, Stefan Vogt, Thomas Legero, MikkoMerimaa, Christian Lisdat, and Uwe Sterr. 8 × − fractional laser frequency instabilitywith a long room-temperature cavity. Optics Letters , 40(9):2112–2115, 2015. Thomas P Heavner, Elizabeth A Donley, Filippo Levi, Giovanni Costanzo, Thomas EParker, Jon H Shirley, Neil Ashby, Stephan Barlow, and SR Jefferts. First accuracyevaluation of NIST-F2.
Metrologia , 51(3):174, 2014. N Hinkley, JA Sherman, NB Phillips, M Schioppo, ND Lemke, K Beloy, M Pizzo-caro, CW Oates, and AD Ludlow. An atomic clock with 10 − instability. Science ,341(6151):1215–1218, 2013. Eugene Hogenauer. An economical class of digital filters for decimation and interpolation.
Acoustics, Speech and Signal Processing, IEEE Transactions on , 29(2):155–162, 1981. James Jespersen and Jane Fitz-Randolph.
From sundials to atomic clocks: understandingtime and frequency . Courier Corporation, 1999. SM Kay.
Modern spectral estimation: theory and application. 1988 . Prentice Hall, 1999. T Kessler, C Hagemann, C Grebing, T Legero, U Sterr, F Riehle, MJ Martin, L Chen,and J Ye. A sub-40-mhz-linewidth laser based on a silicon single-crystal optical cavity.
Nature Photonics , 6(10):687–692, 2012. G Paul Landis, Ivan Galysh, and Thomas Petsopoulos. A new digital phase measurementsystem. Technical report, DTIC Document, 2001. Judah Levine. Invited review article: The statistical modeling of atomic clocks and thedesign of time scales.
Review of Scientific Instruments , 83(2):021101, 2012.24 Long-Sheng Ma, Peter Jungner, Jun Ye, and John L Hall. Delivering the same opticalfrequency at two places: accurate cancellation of phase noise introduced by an optical fiberor other time-varying path.
Optics Letters , 19(21):1777–1779, 1994. Joe Mitola. The software radio architecture.
Communications Magazine, IEEE , 33(5):26–38, 1995. Ken Mochizuki, Masaharu Uchino, and Takao Morikawa. Frequency-stability measure-ment system using high-speed ADCs and digital signal processing.
Instrumentation andMeasurement, IEEE Transactions on , 56(5):1887–1893, 2007. Nathan R Newbury. Searching for applications with a fine-tooth comb.
Nature Photonics ,5(4):186–188, 2011. Joel Phillips and Ken Kundert. Noise in mixers, oscillators, samplers, and logic an intro-duction to cyclostationary noise. In
Custom Integrated Circuits Conference, 2000. CICC.Proceedings of the IEEE 2000 , pages 431–438. IEEE, 2000. N Poli, CW Oates, P Gill, and GM Tino. Optical atomic clocks.
Rivista del NuovoCimento , 36(12), 2013. William H Press, Saul A Teukolsky, William T Vetterling, and Brian P Flannery.
Numer-ical recipes in C , volume 2. Cambridge university press, 1996. Ivan Prochazka, Petr Panek, and Jan Kodet. Note: Precise phase and frequency com-parator based on direct phase-time measurements.
Review of Scientific Instruments ,85(12):126110, 2014. WJ Riley. Handbook of frequency stability analysis. Technical report, 2008. Stefania R¨omisch, Steven R Jefferts, and Thomas E Parker. A digital time scale at theNational Institute for Standards and Technology. In
General Assembly and ScientificSymposium, 2011 XXXth URSI , pages 1–4. IEEE, 2011. Enrico Rubiola. On the measurement of frequency and of its sample variance with high-resolution counters.
Review of Scientific Instruments , 76(5):4703, 2005. Enrico Rubiola and Fran¸cois Vernotte. The cross-spectrum experimental method. arXivpreprint arXiv:1003.0113 , 2010. Laura B Ruppalt, David R McKinstry, Keir C Lauritzen, Albert K Wu, Shawn Phillips,Salvador H Talisa, et al. Simultaneous digital measurement of phase and amplitude noise.In
Frequency Control Symposium (FCS), 2010 IEEE International , pages 97–102. IEEE,2010. 25 Laura C Sinclair, Ian Coddington, William C Swann, Greg B Rieker, Archita Hati, KanaIwakuni, and Nathan R Newbury. Operation of an optically coherent frequency comboutside the metrology lab.
Optics Express , 22(6):6996–7006, 2014. Petre Stoica and Randolph L Moses.
Spectral analysis of signals . Pearson/Prentice HallUpper Saddle River, NJ, 2005. Donald Barrett Sullivan, David W Allan, David A Howe, and Fred L Walls.
Charac-terization of clocks and oscillators . US Department of Commerce, National Institute ofStandards and Technology, 1990. Patrizia Tavella and Claudine Thomas. Comparative study of time scale algorithms.
Metrologia , 28(2):57, 1991. Masaharu Uchino and Ken Mochizuki. Frequency stability measuring technique usingdigital signal processing.
Electronics and Communications in Japan (Part I: Communica-tions) , 87(1):21–33, 2004. Th Udem, Ronald Holzwarth, and Theodor W H¨ansch. Optical frequency metrology.
Nature , 416(6877):233–237, 2002. Ichiro Ushijima, Masao Takamoto, Manoj Das, Takuya Ohkubo, and Hidetoshi Katori.Cryogenic optical lattice clocks.
Nature Photonics , 9(3):185–189, 2015. Jack E Volder. The CORDIC trigonometric computing technique.
Electronic Computers,IRE Transactions on , EC-8(3):330–334, 1959. L ˇSojdr, J ˇCerm´ak, and R Barillet. Optimization of dual-mixer time-difference multiplier.In
Frequency and Time Forum, 2004. EFTF 2004. 18th European , pages 588–594. IET,2004. Fred L Walls and David W Allan. Measurements of frequency stability.