A Physical Sine-to-Square Converter Noise Model

Transcription

1 A Physical Sine-to-Square Converter Noise Model Attila Kinali Max Planck Institute or Inormatics, Saarland Inormatics Campus, Germany Abstract While sinusoid signal sources are used whenever low phase noise is required, conversion to a square wave-orm is necessary when interacing with digital circuits. Although have been analyzed a ew times in various context, to the best knowledge o the author, there is no complete treatment and explanation o all noise sources within a sine-to-square converter. We attempt to give a quantitative, predictive and physically based noise model o sine-to-square converters without itting parameters other than those imposed by the circuit itsel. I. INTRODUCTION In a lot o settings, a sinusoidal signal is given as an input, be it rom a precision requency source or measurement equipment like a dual mixer time dierence system [], but is required to be used in a digital or quasi digital environment. But or the use in digital electronics, the sinusoidal signal has to be converted into a square wave signal, either explicitly using a sine-to-square converter or implicitly, when the signal enters the digital logic. Although such circuits have been employed or a long time and been analyzed a ew times, a complete noise model is still missing. Especially, there are very ew attempts on a physically based noise model that which can predict the output noise based on circuit parameters and does not need itting parameters. II. RELATED WORK One o the irst analysis o noise in sine-to-square converters was done by Collins []. Collins analyzed the jitter o multistage converters due to white noise with respect to the input slew-rate and noise bandwidth. Although giving insight on how to design multi-stage converters, Collins did not give any insight on the sources o noise and their behavior under dierent conditions. In [3] Sepke et al. analyzed the noise contribution o comparators in analog-to-digital converters. While the circuit model is, on a irst glance, similar, the use o the comparator ater a sample-and-hold circuit changes the analysis considerably and thus its applicability to sine-to-square converters is limited. In [4] Calosso and Rubiola measured the noise in an Field- Programmable Gate Array (FPGA used as a sine-to-square converter. Even though they gave scaling laws or various types o noise, these were not rigorously analyzed and thus could not be related to noise parameters o the circuit directly. III. CIRCUIT MODEL A sine-to-square converter can be modeled by a comparator (or a linear ampliier ollowed by a number o ampliier XXX-X-XXXX-XXXX-X/XX/$XX. c 8 IEEE V in V os t amp V DD,n V out Figure. The circuit model o a sine-to-square converter is simpliied to a noiseless comparator input stage with some hysteresis ±H and all input related noise being lumped together into the oset voltage V os. Noise due to power supply variation V DD,n is modeled using an ampliication stage with a delay o t amp that only depends on the supply voltage. stages. For simplicity, we assume here, that the converter consists o an ideal, noiseless and zero-delay comparator with a hysteresis ±H ollowed by a single noiseless ampliier with a delay t amp (see igure. We use this split also to separate noise contributions due to dierent processes: Any noise that is related directly or indirectly to the input signal is olded into the comparator s input noise which we urther combine with the input-oset voltage or simplicity. All noise related to delays within the circuit are olded into the ampliier. The input signal V i (t = (V V i,am (t sin(ω t ϕ i (t ( with the two noise parts, the amplitude noise V i,am (t and the phase noise ϕ i (t enters the comparator, which has a hysteresis o ±H(t and an input oset voltage o V os (t. The output o the comparator gets urther ampliied and delayed by time t amp (t by the ollowing ampliier. We urther assume that luctuations and noise on the power supply V DD,n do not aect the comparator (e.g. it being an ideally symmetrical dierential pair and model the eect o V DD,n as variations in the ampliier delay t amp. As we are only interested in the phase noise contribution o the ampliier, we will ignore the input phase noise ϕ i (t or the urther analysis. The amplitude noise V i,am (t is included to determine its eect on the output phase noise, due to AM-PM conversion through the hysteresis o the comparator. Multi-stage converters can easily be modeled by series connection o the elementary stage in igure. In longer chains

2 Input signal t V os H H with: t os (t = t delay V os os (t = ω V os (t (5 t H (t = t delay H (t = ω V (t (6 t AM (t = t delay V i,am i,am (t H ω V i,am (t (7 The delay through the ampliier depends on the supply voltage V DD, thus any noise on the supply voltage will modulate the delay. As the delay depends on many actors like architecture, process, temperature etc., we simpliy the relationship to t os t H t amp Output signal t t amp (t = t amp V DD V DD,n (t O(V DD,n = cv DD,n (t O(V DD,n cv DD,n (t, with the circuit dependent parameter c. This, o course, removes time dependent delay variations due to e.g., aging or temperature, which can be signiicant at long time scales. But these contributions are easy to add later in the analysis and are let out, at the moment, or simplicity. (8 Figure. The zero crossing o the input signal gets delayed by the oset voltage V os and the hysteresis ±H and the delay t amp through the ampliier stage. The oset voltage and hysteresis related delays t os and t H depend not only on the values o V os and H respectively, but also on the slew-rate o the input signal. The ampliier delay t amp only depends on the supply voltage V DD. or with suiciently large input amplitude, there will be a slewrate saturation. This can be used to simpliy all ollowing stages to single ampliier stages, without the comparator, and old the input noise into the variation o the ampliier delay t amp. I ilters are used between stage, like in the case o Collins style sine-to-square converters [] care has to be taken to account or the change in noise properties in each stage. IV. NOISE SOURCES Assuming the hysteresis H o the comparator is symmetric around the zero point with oset voltage V os, i.e., V os ± H, and both are small enough such that the small angle sine approximation can be used around the zero-crossing point (see igure, then the propagation delay through the comparator is t delay (t = t os (t t H (t t amp (t ( = V os(t H(t ω (V V i,am (t t amp(t. (3 As we are interested in the variation o the delay, we will be looking at t delay. Assuming noise is small one can split the contributions: t delay (t = t os (t t H (t t AM (t t amp (t (4 V. NOISE TRANSLATION AND SCALING The phase noise is deined by S ϕ ( = ϕ rms( (9 with the phase luctuation ϕ measured over a bandwidth o Hz [5]. As the phase relates to delay with ϕ = ω t we can write S ϕ ( = ω t ( with x denoting, inormally speaking, the average over all x(t with a measurement interval /. Or more ormally, the absolute value o the Fourier coeicient at requency o the signal x(t: x = x(te πjt dt ( For reasons o being concise, we ignore here the mathematical details o integrating over time series o random signals, which might potentially be non-continuous and assume all random signals are o inite bandwidth and thus integrable. We also assume all integrals go over inite time (measurement intervals in order or them to be deined in case o / α noises, which otherwise would lead to ininite signal power. For a discussion o integration over random signals see e.g., [6] and [7]. As all discussed noise sources are assumed to be independent, we can write S ϕ ( = ω t delay = V H V 4 os V i,am ω c V DD,n (

3 Dierent requency scaling or dierent noise sources becomes already evident. The input related noise processes do not scale with ω while the delay related noise does scale with ω. These are the ϕ-type and x-type noises, respectively, as discussed in [4]. A. Impulse Sensitivity Function (ISF In [8] Egan noted that white phase noise gets aliased due to sampling. Formally, this can be described by using the ISF as introduced by Hajimiri and Lee in [9]. We slightly modiy it to adapt it or the more general setting o sine-to-square converters: ϕ(t = t Γ(τn(τ dτ (3 with Γ(t being the ISF and n(t being the eecting noise. Please note that Γ(t is implicitly also a unction o the circuit and its parameters, which also include the input signal. In other words, i the shape or amplitude o the input signal changes, this will potentially result in a change o the shape o Γ(t. The ISF or a sine-to-square converter can be approximated by a comb o alternating positive and negative rectangular pulses: Γ(t = n= Π ( t τ w nt n= Π ( t τ w nt τ d (4 with a period T = π/ω and a pulse width o τ w. τ d denotes the phase shit between the positive and the negative pulses and is related to the duty cycle o the output signal and depends, in our circuit model, on the input signal amplitude V and the oset voltage V os. In a irst order approximation, For a 5 % duty cycle τ d = T /. For simplicity, we assume that the positive and negative pulse, which relate to the positive and negative zero crossing respectively, are o the same magnitude and width, which is not necessarily the case. It should be noted, that τ w depends on the output slew-rate o the converter, thus is proportional to T. Hence, or slewrate limited converters, τ w becomes independent o T (in irst order. The Fourier series o the unction Γ(t can be expressed as Γ(t = τ w T τ w T n= n= sinc sinc ( nω τ w e jnωτw e jnωt ( nω τ w e jnωτw e jnωt e jnωτd (5 Looking at the Fourier series directly explains two phenomena reported in [4]: The /ω scaling o white noise and the /ω and /ω scaling o licker noise: B. Scaling o White Noise Under the assumption that τ d = T / the Fourier transorm o Γ(t in equation (5 becomes a Dirac comb like structure with Dirac pulses at odd multiples o ω due to its periodic nature and because the even harmonics cancel out. These Dirac pulses δ( (n ω have approximately constant amplitude a n up to the requency /(πτ d rom which on they decay with / or db/dec. The multiplication with noise in equation (3 results in a mixing process (c.. [7] that converts all noise in distance ω to one o the Dirac pulses a n δ( (n ω down into the signal passband around ω with an amplitude that is proportional to the amplitude o the Dirac pulse. Because the noise in each down-converted requency region is independent, the total down converted noise becomes a geometric sum S ϕ,white,total a n (6 n = a n a n (7 n τ d ω n> τ d ω a a (8 τ d ω k k= ( = a (9 τ d ω k k= There harmonic series in equation (9 grows slowly and can be approximated by H n = n k= k = ln n γ O(/n, with γ being the Euler-Mascheroni constant (c.. [, section..7]. Even though H = and thus S ϕ,white,total =, the sum is in reality limited. One reason is that the ISF edges have a inite steepness, which adds a second sinc term to Γ(t and thus a second corner requency ater which it decays with 4 db/dec or /. Another is the limited bandwidth o the circuit, which acts similarly by adding a cut o requency, ater which the noise (and signal decay with an additional db/dec. The sum H (r = k= k, r > is bounded by r a small constant (e.g., H ( = π /6 [], thus we can express the total white phase noise as: ( S ϕ,white,total a c BW ( τ d ω ω a c BW ( ω with c BW and c BW being (noise bandwidth depending constants o the circuit. We conclude that the total white noise o the sine-to-square converter gets an additional scaling with a actor o /ω due to aliasing induced by the periodicity o the ISF. Thus we end up with: S ϕ,white ( ω V os ω V H ω V 4 i,am ω c VDD,n ( The proportionality actor o equation ( depends on the equivalent noise bandwidth, respectively how many harmonics contribute to aliasing, and on the ratio τ w /T rom Γ(t. In case τ d T /, then Γ(t will also have even harmonics. For white noise, the even harmonics will act the same way as the odd harmonics and add to the proportionality actor o equation (. For most systems, one can saely assume that the duty cycle will be close to 5 % and thus the even

4 harmonics will be small. Hence it is possible to ignore the eects o even harmonics in a irst order approximation. C. Scaling o Flicker Noise Flicker noise is, initially, only present around DC, thus the irst harmonic up converts the licker noise into the signal band. Hence, the licker component derives rom equation ( directly as: S ϕ,licker ( V os V H V 4 i,am ω c VDD,n D. Scaling in a Multi-Stage Sine-to-Square Converter (3 I multiple gain stages are used in a sine-to-square converter, then each stage acts upon the noise and thus the harmonics o the ISF o each stage alias noise noise into the signal band. Even i all stages are the same, each stage will have a dierent Γ(t as τ w will change with the slew rate o the input signal o each stage. Thus a simple multiplication o S ϕ with the number o stages will, in general, not lead to an accurate result. Nevertheless one can derive scaling rules quite easily: For white noise, the harmonics o the additional stages each up convert noise, which is then down converted into the signal band by the ollowing stage. As the equivalent noise bandwidth o each stage individually stays constant, the scaling rules in section V-B remain unchanged and thus equation ( is still valid with only a larger proportionality actor. For licker noise, the up conversion and the ollowing down conversion o consecutive stages change the behavior slightly. I the duty cycle is exactly 5 %, then only odd harmonics will exist, and hence none o the present harmonics will see any licker noise in a distance o ω. Thus only the irst harmonic o each stage will up convert licker noise into the signal band and, as with white noise, equation (3 is still valid with a slightly larger proportionality actor. But, due to the presence o V os, the duty cycle will deviate rom 5 % and give rise to even harmonics. Please note, that not only the DC component o V os will lead to even harmonics, but also its higher requency noise component. Hence the scaling o licker noise will change its properties depending on the requency spectrum o V os. Because now there are harmonics at a spacing o ω, the previously up converted licker noise is seen by a harmonic at a distance o ω and is thus down converted again into the signal band. Unlike the aliasing o white noise, the aliased licker noise ultimately has the same origin, thus all down converted licker noise components are correlated. Thus or S ϕ, this leads to a scaling proportional to ω From equation (4 we see that the power o the even harmonics relates to the power o the odd harmonics with exp (jω τ d = sin (ω o τ d. Assuming τ d T / we can replace τ d by it s deviation (noise value rom T /: τ d,n = τ d T / = ω V V os (t (4 leading to sin (ω o τ d = sin (ω o τ d,n ( = sin V os (t V V os (t V (5 To evaluate the eects o equation (5 on the noise spectrum, we have to take into account, that τ d,n represents a jitter value due to V os. As such, it is subject to same aliasing and thus scaling laws as S ϕ. I the ampliication o the irst converter stage is large, we can saely assume that τ d,n is dominated by the irst stage. I we also assume that the noise equivalent bandwidth is large and thus (the jitter τ d,n is dominated by white noise. We then can ignore the contribution and scaling due to licker noise. Due to aliasing o white noise, we get an additional scaling term o /ω, as we have already seen with equation (. Thus the power o the even harmonics will scale approximately with /(ω V V os. The scaling due to aliasing will act dierently on dierent types o noise. While all input related noise sources will see the ull eect o aliasing, the V DD related noise component will not. As the V DD related noise acts as a delay in each stage o the multi stage converter, it will only see part o the licker noise aliasing, depending on which stage was the source o the noise. Thus V DD related noise will see an additional scaling actor between and ω depending on the exact structure o the sine-to-square converter and which stages contribute how much to the output noise. Putting the arguments above together, we can conclude that S ϕ,licker o a multistage sine-to-square converter gets an additional ω term due to aliasing o correlated noise and an /ω term due to the power scaling o the even harmonics: S ϕ,licker,multi ( ω S ϕ,licker ( ω V os ω ω H V 4 V i,am ωα c V DD,n (6 with α, the scaling actor o the V DD related noise, being between and 3. VI. EXAMPLE ANALYSIS OF MEASUREMENTS Calosso and Rubiola presented measurements o a Cyclone III FPGA in [4]. Using the analysis in the previous sections, we will review the noise analysis o Calosso and Rubiola. For convenience, we have reproduced the plot o the noise measurement in igure 4. For high oset requencies, the /ω scaling o white ϕ-type noise is nicely visible. Similarly, or low oset requencies and high signal requencies, the ω scaling o licker x-type noise shows up. But or small oset requencies and small signal requencies the behavior changes. To ully understand the noise in this

5 FPGA dbm sine n 47 µ 33 k k n output 68k µ µ k k µ Figure 3. The circuit used in [4]. The FPGA acts as as a sine-to-square converter and buer. The operation ampliier orms an integrator to stabilize the duty cycle o the output and steer the input oset voltage to that eect. Courtesy o Claudio Calosso and Enrico Rubiola. area we irst have to look at the circuit that was used in [4], which is shown in igure 3. The FPGA acts as a sine-to-square converter with multiple stages. There are two output buers used, one or the output, which is ed to the noise measurement equipment and one that is ed, through a low pass ilter, into an integrator. The integrator adjusts the oset voltage o the input signal such that the duty cycle is kept at 5 %. Analysis o the transer characteristics o this stabilization circuit reveals a 3 db requency at approximately 8 mhz. Armed with this inormation understanding igure 4 becomes easy. For oset requencies below 8 mhz the integrator in igure 3 compensates any deviation rom 5 % duty cycle, thus eliminating the second order harmonics and aliasing o licker noise. Hence only (unaliased x-type noise is visible. In the range between approximately 8 mhz and khz when white noise becomes dominant, we see aliased ϕ-type noise, due to the multi-stage nature o the FPGA or lower signal requencies. For higher signal requencies, licker noise scales with ω, thus α o equation (6 must be close to, which in turn would suggest, that most o the V DD related noise originates rom the output stage or the last ew stages. VII. CONCLUSION Starting rom a simple circuit model, we have analyzed how input and power supply noise aect the sine-to-square conversion. The scaling o the additive noise in dependence o the input requency is has been shown or dierent noise contributors and under dierent settings. These scaling laws then have been used to explain the noise measurements o [4]. REFERENCES [] D. W. Allan and H. Daams, Picosecond time dierence measurement system, in 9th Annual Symposium on Frequency Control, May 975, pp [] O. Collins, The design o low jitter hard limiters, IEEE Transactions on Communications, vol. 44, no. 5, pp. 6 68, May 996. [3] T. Sepke, P. Holloway, C. G. Sodini, and H. S. Lee, Noise analysis or comparator-based circuits, IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 56, no. 3, pp , March 9. [4] C. E. Calosso and E. Rubiola, Phase noise and jitter in digital electronics, in 4 European Frequency and Time Forum (EFTF, June 4, pp [5] IEEE standard deinitions o physical quantities or undamental requency and time metrology random instabilities, IEEE Std 39-8, Feb 9. [6] B. Øksendal, Stochastic Dierential Equations, 6th ed. Springer, 3. [7] A. Lapidoth, A Foundation in Digital Communication. New York: Cambridge University Press, 9. [8] W. F. Egan, Modeling phase noise in requency dividers, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 37, no. 4, pp , July 99. [9] A. Hajimiri and T. H. Lee, A general theory o phase noise in electrical oscillators, IEEE Journal o Solid-State Circuits, vol. 33, no., pp , Feb 998. [] D. E. Knuth, The Art o Computer Programming, Volume : Fundamental Algorithms, 3rd ed. Redwood City, CA, USA: Addison Wesley Longman Publishing Co., Inc., 997.

6 x-type Multistage Figure 4. The noise measurement o [4], slightly adapted. The white noise region shows the /ω scaling o aliased ϕ-type noise. The licker noise region shows both the unaliased x-type noise at high ω and aliased licker noise at low ω. At low oset requencies, the aliasing o licker noise goes away due to duty cycle stabilization in igure 3. Courtesy o Claudio Calosso and Enrico Rubiola.