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This chapter appears in the book "Analog Circuit Design-RF Analog-to- Digital Con-verters; Sensor and Actuator Interfaces; Low-Noise Oscillators, PLLs and Synthesizers", published by Kluwer Academic Publishers, Boston, November 1997, 428 pp. Hard-bound, ISBN 0-7923-9968-4 frequency (IF), then the phase noise around fLO will How Phase Noise Appears in Oscillators downconvert nearby channels on to the same IF. As will shortly emerge, fundamental reasons dictate that the phase Asad A. Abidi noise spectral density is always highest at fLO and falls off at Integrated Circuits & Systems Laboratory frequencies away from it. Thus the largest expected strengths Electrical Engineering Department of the immediately adjacent channels set a limit on the University of California, Los Angeles tolerable phase noise close-in to the receiver LO frequency. In abidi@icsl.ucla.edu http://www.icsl.ucla.edu/aagroup modern wireless communication systems with power control such as GSM and DECT, the allowed strength of nearby ABSTRACT channels must conform to a template roughly inverse to the Wireless transceivers closely specify the phase noise oscillator phase noise specification. in the local oscillator. Yet it is not very well Phase noise in the transmitter LO — very often this is the understood how phase noise is predicted, especially same oscillator as is used in the in oscillators which do not use a passive resonator. It receiver — may overwhelm nearby is also difficult to model flicker noise in the close-in weak channels. This is because the TX ph phase noise spectrum. This is a qualitative phase noise spectral density grows a se noi se discussion of the various physical processes directly with the transmitted signal Nearby Frequency Channels responsible for phase noise production, particularly power, and at a given point in space, in CMOS oscillators, and it offers a common the noise sidebands of a strong Figure 2: Phasenearbyin transmitter LO can overwhelm noise weak channels. treatment of resonator-based oscillators, ring transmitter may be greater than oscillators, and relaxation oscillators. another faded or attenuated signal occupying the same frequency (Figure 2). Why is Oscillator Phase Noise Important? Although the topic of noise in oscillators has engaged The phase noise in the radio receiver’s local oscillator limits classical investigations of a qualitative nature [1-3], Leeson in immunity against nearby interference 1966 was the first to propose a simple intuitive signals. In the transmitter, phase noise phenomenological model [4, 5] relating the level of phase can swamp out nearby channels. In the Unwanted noise in a widely used class of resonator-based oscillators to Channel receiver (Figure 1), the local oscillator LO voltage and current noise sources in the circuit elements. This (LO) frequency, fLO, is tuned to a model has been widely embraced, and serves well to predict Frequency certain frequency offset from the IF oscillator phase noise induced by sources of white noise. desired channel (zero offset in the case Figure phase noise indownconversion 1: Undesired However, while Leeson admits that device flicker (1/f ) noise by LO a receiver. of direct conversion) to downconvert it to the intermediate may determine the phase noise very close to the oscillation frequency, his model cannot explain why. More recently, Then, the waveform of the noisy oscillation is sampled at the phase noise has been discussed in the relaxation oscillator, and grid points, t k , and deviations in the samples from the even in the ring oscillator which is of some interest for CMOS baseline are translated into phase deviations, f( t k ) , by the implementations. However, there is no one satisfactory inflection slope, S, of the 2 waveform . Thus, method to predict phase noise in the latter types of oscillators, f( t k ) = 2pfc ◊ x( tk )/ S , where fc is the nominal oscillation nor one treatment which unifies and differentiates the frequency. This works well for small fluctuations in phase. mechanisms as to how voltage and current noise in the circuit components of an oscillator transform into phase noise. The Phase noise is defined by the variance of f( t k ) , E f( tk )2 . purpose of this paper is to present a progress report on the This is characterized in the frequency domain by the spectral state of qualitative models explaining the production of phase density, Sf ( f ) . After some simplifying assumptions to noise. Ongoing work on this topic seeks to develop these into surmount the problem that the phase noise defined above is a quantitative models validated by measurements of phase noise nonstationary random walk process, it follows from purely in the various types of oscillators. statistical reasoning [6] that Definition of Phase Noise Sf ( f ) µ FG fc / S IJ 2 (1) When asked to visualize phase noise, we first invoke the H f K and the result is the same for every voltage and current in the picture of a noisy sinewave whose oscillator circuit. phase, measured relative to a time The use of an oscillator with phase noise requires the grid set by a noiseless sinewave, is statistics of phase to be converted into the equivalent randomly perturbed at the zero sidebands the phase modulations induce around the crossings (Figure 3). This may be oscillation frequency. We define the phase noise spectral generalized to the non-sinusoidal density, ( f ) , as the ratio of the power density in one phase periodic waveforms x(t) in practical modulation sideband relative to the oscillation power. From Figure 3: A noiseless sinewave compared with a sinewave with phase noise (jittered phase modulation theory for small angles, it follows that: zero crossings). oscillator circuits [6] by first defining i2 f = Sf ( f )/ 2 S d Slope= x(tk) ( f ) = f pk / 2 tk a zero crossing baseline and periodic time grid at every inflection point on, whose waveform inflects at some point. The period and phase noise is say, the rising edge of the noiseless exactly the same for all waveforms within an oscillator, so the most Figure 4: Defining phase jitter. 1 oscillation waveform (Figure 4). convenient waveform may be chosen for measurement. 2 By accounting only for deviations in the zero crossing times, this 1 Even in oscillation waveforms such as a perfect triangle wave with no definition captures the phase noise by after passing the noisy oscillation inflection, there is bound to be another voltage or current in the circuit through a limiter. Voltage and Current Noise into Phase Noise zero dependence on voltage, current, and temperature. Then It is convenient to think of any oscillator as an unstable closed we can definitively say that noise cannot modulate the loop circuit, comprising one or more active devices to provide resonant frequency. At most, noise sources within the gain or negative resistance and one or more passive reactances oscillator loop will experience some gain and add to the to set timing. In any practical electronic system, the oscillator oscillation, thereby inducing phase noise. Leeson has given a will connect to other circuits, such as an amplifier or a mixer. simple and satisfactory model [4] for these conditions. Voltage and current noise sources exist everywhere in the Let us model a harmonic oscillator as a frequency- system, and after adding to the oscillation they will all create dependent block with transfer function vn phase noise in the sense of the above definition. However, the F( jw ) in negative feedback with an 1 Oscillator noise sources within the oscillator loop exert a much more amplifier whose gain is absorbed into this output profound influence on phase noise than do the noise sources transfer function (Figure 5). In keeping with the Barkhausen criterion for F(jw ) outside the loop. This is for the following distinct reasons: oscillation, the loop has infinite gain at Figure 5: Block Diagram of feed- • Noise in the oscillator loop may be substantially enhanced back harmonic oscillator. the oscillation frequency, fc , and finite by the sharp frequency selectivity of the loop, and thus become the dominant source of phase noise gain at all other frequencies, i.e. F( j 2pfc ) = 1 . Consider a white noise source, vn , within the loop. Then the noise • Noise in the oscillator loop may directly modulate the appearing at the output in series with the oscillation is: oscillation frequency 1 1 • Noise in the oscillator loop may modulate a reactance, and vout = vn = - vn (2) 1 - F( jw ) F ¢( jw ) ◊( w - w c ) thereby the oscillation frequency This is more conveniently expressed in These three processes are now illustrated at work in three terms of the frequency offset from Noise T.F. different types of oscillators: a resonator-based oscillator, such oscillation, fm ∫ f - fc . as with an LC tuned circuit; a relaxation oscillator, such as a The frequency selection of the multivibrator; and a ring oscillator, comprising a cascade of oscillating loop leads to a noise active delay stages operating in the large signal mode. fc Frequency transfer function which is singular at Phase Noise in Resonator-Based Oscillators Figure 6: Noise transfer function fc , and for small offsets declines around the oscillation frequency. inversely with fm (Figure 6). Thus, Additive Noise noise frequencies around fc are preferentially amplified, and Harmonic oscillators use passive reactances, such as an LC add to the oscillation waveform to create phase noise (Figure tuned circuit or an RC network, to define the oscillation frequency. Let us suppose these reactances are constant, with 7). The phase noise in an LC resonator, where Although this phenomenon had been observed all along at 2Q very small offset frequencies in oscillators, it was attributed to F ¢( jw c ) ª d –F( jw c ) = , is given by dw wc parametric flicker fluctuations in the circuit component LM F I 2 1 OP vn2 values, such as fluctuations in resistance. However, when this 1 f ( fm ) = 1+ c 2 MN GH JK fm2 PQ A2 2Q (3) was found to be the typical character of phase noise in microwave GaAs FET oscillators at offset frequencies beyond This expression gives the two principal factors regulating 1 MHz [7], a search began in earnest to explain how the phase noise. First is the net baseband flicker noise known to exist in MESFETs could Additive noise with strong fc spectrum noise-to-signal ratio within the appear at the oscillation frequency. It was recognized that oscillator found by dividing the Leeson’s linear model does not account for the effects of output due to all the noise nonlinearity on noise in an oscillator which self-limits the sources by the oscillation oscillation amplitude. The oscillator can no longer be treated Figure 7: Noise accentuated around fc causes amplitude, A. Second is the as operating at a fixed bias point, with small noise signal phase jitter. half-power bandwidth of the superimposed. In reality, the bias point varies considerably loaded resonator, fc / 2Q . This over the oscillation amplitude, suggests that aside from lowering the noise levels in the and owing to the voltage and v n 2 Composite noise adding to oscillator components, the oscillation amplitude must be designed current dependence of the pco large, and a high Q resonator must be used. U ) lo transconductance, output ss nve rsi o n ( wit h co n v. Frequency Upconverted Noise conductance, FET gate capacitance, and of noise itself, Additive noise explains the observed phase noise spectrum 9: Upconversion of baseband (1/f) the baseband flicker noise is Figureto oscillation frequency. with a 20 dB/decade slope close to the noise upconverted to fc (Figure 9) [8, 9]. The upconverted noise (fm) oscillation frequency. However, in practice at ~1/f3 then enters into the oscillator loop according to Leeson’s small fm a 30 dB/decade slope is observed ~1/f2 model. As in any mixing process where neither signal drives (Figure 8). This is associated with flicker the circuit into clipping, the upconverted noise depends on (1/f ) noise in the oscillator’s active devices. fm(log) the oscillator signal, and grows with oscillation amplitude, A. However, the additive noise model cannot Figure 8: Typical close-in The resultant phase noise spectral density is therefore explain how flicker noise at low frequencies phase noise spectrum. independent of A. appears around the oscillation frequency, fc . In contrast to MESFETs, the insulating gate of the independent of the oscillation amplitude A and the final phase MOSFET supports much larger oscillation amplitudes. Even noise spectral density depends inversely on A2 . at frequencies as high as 1 GHz, CMOS resonator-based oscillators will support rail-to-rail voltage swings. This makes Phase Noise in Relaxation Oscillators it somewhat easier to analyze the noise upconversion process. It is instructive to contrast phase noise production in Let’s use as an example a CMOS oscillator comprising a resonator-based oscillators with a cross-coupled differential pair sharply contrasting oscillator I negative resistor across an LC type, the relaxation oscillator. resonant circuit (Figure 10). Assume C DV F/F Out Generically this comprises a that the self-limited oscillation M1 M2 single reactance, almost always a amplitude is so large as to completely 2I capacitor, a regenerative memory switch the differential pair, M1-M2, M3 In element such as a flip-flop or while the current source M3 always Oscillator Mixer Schmitt trigger, and a means of Figure 11: Typical relaxation oscillator. remains in saturation. Then M1-M2 charging and discharging the capacitor (Figure 11). acts as a commutating mixer, which Figure 10: CMOS LC Oscillator. A simple mixer is shown to illustrate the inherent There are two fundamental differences between this and upconverts baseband flicker and mixing action in the oscillator. the harmonic oscillator discussed in the previous section. The white noise from M3 to fc , and downconverts white noise at single reactance is not frequency selective like the resonator, 2 fc in M3 to fc . If the noise bandwidth exceeds 2 fc and the and the regenerative element makes this into a discrete-time up- and down-conversion gains are equal, the net noise feedback loop. The two different means by which circuit noise delivered by M3 into the resonant circuit resembles its own converts into phase noise are now discussed. baseband noise spectrum, except with a flicker noise corner The frequency of oscillation is set by the charge/discharge frequency lowered by 2×. rate of the capacitor, and two separate trip points at the input The differential pair delivers noise to the resonant circuit of the regenerative element separated by some voltage DVt . over that fraction of the oscillation period when both M1 and Thus, the oscillation frequency is M2 are ON. This again consists of upconverted baseband I noise and downconverted noise from 2 fc . Otherwise, while fc = (5) 2C ◊ DVt only one of the pair FETs is ON it acts like a cascode on M3 Low-frequency noise in in the charging current, I, or vn on and contributes no net noise of its own. the reference voltages which set the regeneration trip points With complete commutation, the frequency translated will directly frequency modulate the oscillation to create close-in noise (at least the main component from M3) is once again phase noise [10]. Invoking textbook FM theory, if a modulating sinewave of frequency fm causes a peak deviation Df pk in the carrier frequency fc , then at a small modulation flicker noise remains unchanged as long as the flicker noise index the main sideband in the FM spectrum lies at fm with corner frequency lies below 1 f . 2 c strength Vssb = J1 FDf pkIª Df pk Phase Noise in Ring Oscillators A GHfm JK 2 fm (6) The ring oscillator at RF has raised interest among CMOS IC Thus, the SSB phase noise produced by a noise spectral designers because it is simple, fast, and readily yields output density S( in ) and S( vn ) is phases in quadrature. However, as is often true in engineering, the simplest circuits bundle so many nonlinear ( fm ) = FG IJ 2 and S( in ) fc ( fm ) = S( vn ) FG fc IJ 2 (7) effects that they are quite difficult to analyze. This seems to H K 2I 2 fm 2( DVt ) 2 H fm K be true in the matter of predicting phase noise in ring Flicker noise in either S( in ) or S( vn ) will produce, through oscillators. Time-domain [12] and frequency-domain [13] this straightforward process of FM, a phase noise spectrum of models for noise production in ring oscillators have been 1 / f 3 . High frequency components of S( in ) are filtered out by proposed. We present a new model here, which refines and the integrating capacitor, but the regenerative element reacts extends a previously published model [13] to more accurately to high frequencies in S( vn ) somewhat unusually. The capture the process of phase noise production. transition times in a relaxation oscillator are known to be set Without loss of generality, consider a ring oscillator by the first crossing of a voltage ramp across a noisy comprising four threshold, which irreversibly triggers the regenerative element differential delay stages Delay Delay Delay Delay Stage Stage Stage Stage to change state. The ramp cannot distinguish between an (Figure 13). A short ring intersection with high frequency will always be the case or low frequency noise (Figure for oscillation at RF. The td td Noisy DV 12). For instance, the effect will steady-state oscillation is be the same if the noise on the very close to a large sinewave, with an Model threshold is of frequency fm , Figure 12: Regenerative element samples noise amplitude set by full fc - fm , or fc + fm . In other on thresholds. switching of Oscillator. each Figure 13: CMOS Ring function. A model to calculate the additive noise transfer words, the relaxation oscillator samples wideband noise on the differential pair current threshold [11]. This causes the noise spectrum of S( vn ) at into the loads. The oscillator derives its frequency from the frequencies above 1 f to alias back to the Nyquist band from cumulative delay in the stages making up the ring. It follows 2 c by symmetry that if all the stages are identical, then as the 0 to 1 fc and raise the white noise floor appearing in the 2 sinewave traverses each stage of the ring its amplitude remains second equation in (7). However, the phase noise caused by unchanged, and it experiences a phase lag of 45°. For ease of visualization, assume that only one of the delay So far the analysis applies to high frequency noise entering stages in the ring is noisy, and the others are noiseless. Then the oscillating loop. We now note that the ring oscillator is for frequencies around fc , the ring oscillator may be modelled also akin to the relaxation oscillator, in that low frequency noise as a single noisy differential pair with negative feedback from in the charge/discharge currents will modulate the delay of the output to the input via an ideal delay line, td (Figure 13). each stage, and therefore the td . Again, there is a linear The unity gain delay line models the other three noiseless dependence between the noise currents which stages because its gain is one (specifically at the oscillation charge/discharge the output capacitor in each delay stage and frequency), and we lump into it the delay of the entire ring, the frequency of oscillation. Suppose the deviation coefficient i.e. td = 1 / ( 2 fc ) . is K v1 Hz/A for the tail current and K v2 Hz/A for the load We now draw from the preceding analyses of phase noise current; the differential pair does not contribute any in harmonic and relaxation oscillators to understand the ring substantial low frequency noise. Then, using FM theory as oscillator. In that the ring oscillator comprises a continuous- before, the spectral density of phase noise is: time feedback loop with a delay line resonator, it is similar to 2 K v1S( in1 ) 2 K v2S( in2 ) the harmonic oscillator. Noise propagates around this loop, ( fm ) = + (10) fm2 fm2 and Leeson’s model applies to how additive and upconverted The ring oscillator, in conclusion, is a hybrid of the noise converts into phase noise. The noise transfer function is: harmonic oscillator and the relaxation oscillator. Sv / A2 e jwt d / 2 Sf = = Sv / A2 (8) The Spectral Linewidth of an Oscillation - jwt d 2cos(wtd / 2 ) 1+ e As in the harmonic oscillator, noise at the oscillation The foregoing analysis concerns itself with noise spectra at frequency experiences infinite gain, and the phase noise frequency offsets from the carrier frequency. In all cases, the spectrum for small offset frequencies fm from fc is: spectral density becomes infinite at the oscillation frequency. However, in practice the spectral density of an oscillation is ( fm ) = 1 Sv =1 v S fc FG IJ 2 (9) never infinite, even if one were to measure it on a spectrum 2 2 A (w mtd )2 2 A2p 2 fm H K analyzer with almost zero resolution bandwidth. Over any By comparing this with the expression for an LC tuned finite observation period, the oscillation frequency will waver circuit, we may ascribe an effective Q of p / 2 to the delay-line around a mean value due to drifts in temperature, vibration, resonator. The additive noise entering the oscillator is almost etc. One may think of this wavering as a residual FM caused all frequency-converted noise. For instance, the tail current by environmental factors. Associated with each source of flicker noise is upconverted by the switching differential pair. modulation, such as temperature or vibration, is a deviation The large sinusoidal bias current in the load FETs upconverts constant. The mean-square value of the oscillator output their flicker noise. Similarly the oscillating bias in the voltage (or current) will then spread over this usually quite differential pairs upconverts their flicker noise. small frequency range of wavering, which, borrowing a term into AM and PM phasors (Figure 14). The two types of from optics, is called the spectral linewidth. modulation will produce superimposed phase noise sidebands Calculations of the spectrum of a carrier frequency at fm which are indistinguishable on a spectrum analyzer. modulated by Gaussian noise identify this spectral linewidth. In practice the random AM may be removed by a limiter If a wideband white Gaussian noise voltage with spectral or when the oscillator output is applied to a commutating 2 density Sv V /Hz modulates the frequency of a sinewave with mixer. The PM will remain and produce the net phase noise. deviation constant K Hz/V, then the resulting frequency Therefore, it is of interest to measure the random PM 3 spectrum is : separately from the AM. It is possible to do this by mixing the A2 p 2 K 2Sv noisy oscillation with an appropriately delayed version of ( fm ) = (11) itself. If the delay is exactly a quarter period, then the 2 2 2 ( p K Sv )2 + ( 2pfm )2 oscillation will self-downconvert to zero, and any amplitude The spectral density is almost constant over the offset modulations will also downconvert to zero. However, phase modulations will downconvert to their respective offset frequency range e0,pK 2Sv / 2j which defines the spectral frequency. linewidth. [ A + a(t )]sin(wt + f(t )) ¥ [ A + a(t - T )]cos(wt + f(t - T )) 4 4 (12) AM Noise, FM Noise, and True Measurement of Phase Noise A2[f(t ) - f(t - T )] 4 Noise in an oscillator produces fluctuations not only in phase, using the small angle approximation, and assuming no but also in amplitude. Consider, for correlation between f(t ) and f(t - T ) . fm fm 4 instance, noise in the reference trip points fm An automated phase noise measurement system drives a of a relaxation oscillator. This modulates fm variable delay element with the amplitude of the capacitor waveform, FM a feedback loop to hold the and also the frequency. As another average DC at the mixer example, consider a random phasor at an AM Oscillator output to zero [15]. A Spectrum Figure 14: Additive noise creates Analyzer offset frequency fm adding to the FM and AM components. spectral analysis of the Pwr Split oscillation phasor. This may be resolved output fluctuations at frequencies offset from DC Figure 15: Delay discriminator method to accurately meas- will directly read out ( f m ) . ure phase noise. As the measurement system automatically tracks out the 3 oscillation frequency, it is insensitive to slow drifts in the This expression was derived from first principles by Masoud Djafari of UCLA, and corrects for an error in an original calculation by Stewart [14]. frequency4. This makes it convenient to accurately measure phase noise in an oscillator with a large spectral linewidth, Conclusions such as in a relaxation or ring oscillator. The large drifts in Measured phase noise -70 these oscillators make it difficult to measure noise sidebands Phase Noise, dBc/Hz characteristics of RF-CMOS Fli ck er -80 No on a spectrum analyzer, but a phase noise measurement LC (Figure 16) and ring ise Do mi -90 na instrument will produce repeatable results. oscillators (Figure 17) show nt -100 that flicker noise dominates at -110 the frequency offsets of interest. The design of -120 10 10 4 2 3 4 5 6 78 5 2 3 4 Offset Frequency, Hz oscillators to certain phase noise specifications will require Figure 16: Measured Phase Noise in 900 MHz RF CMOS LC Oscillator (Q=3 to 4). a priori modelling of FET flicker noise, an art which is still at an early empirical stage [16, 17]. Although the various frequency conversion effects may be qualitatively anticipated, it only seems feasible to model the various conversion gains and the noise modulations caused by large oscillations in the bias currents and voltages through recently developed numerical techniques [18]. Circuit designers have 20 L(fm), dBc/Hz successfully used hand 0 -20 calculations to predict -40 Flic amplifier noise, and even the -60 ker No ise phase noise caused by -80 -100 additive noise in nearly -120 linear oscillators, but it 10 2 3 4 10 2 3 4 10 2 3 4 10 2 3 4 10 2 3 4 5 6 seems that the prediction of Offset Frequency, Hz CMOS oscillator phase Figure 17: Measured Phase Noise in 800 MHz 4-stage noise, when it is finally CMOS Ring Oscillator. possible, will heavily rely on advanced CAD tools. 4 As long as the rate of drift is within the tracking bandwidth of the delay-adjustment feedback. References [14] J. L. Stewart, “The Power Spectrum of a Carrier Frequency Modulated by Gaussian Noise,” Proc. of the IRE, vol. 42, no. 10, [1] I. L. Berstein, “On Fluctuations in the Neighborhood of Periodic pp. 1539-1542, 1954. Motion of an Auto-Oscillating System,” Doklad. Akad. Nauk., vol. [15] C. Schiebold, “Theory and Design of the Delay Line 20, no. 1, pp. 11, 1938. Discriminator for Phase Noise Measurements,” in Microwave J., [2] W. A. Edson, “Noise in Oscillators,” Proc. of IRE, vol. 48, no. 8, vol. 26, no. 12, pp. 102-112, December 1983. pp. 1454-1466, 1960. [16] A. A. Abidi, J. Chang, C. R. Viswanathan, J. Wikstrom, and J. Wu, [3] J. A. Mullen, “Background Noise in Nonlinear Oscillators,” Proc. “Uniformity in Flicker Noise Characteristics of CMOS IC of IRE, vol. 48, no. 8, pp. 1467-1473, 1960. Technologies,” presented at Ninth Int'l Conf. on Noise in Physical [4] D. B. Leeson, “A Simple Model of Feedback Oscillator Noise Systems, Montreal, pp. 469-472, 1987. Spectrum,” Proc. of IEEE, vol. 54, no. 2, pp. 329-330, 1966. [17] J. Chang, A. A. Abidi, and C. R. Viswanathan, “Flicker Noise in [5] W. P. Robins, Phase Noise in Signal Sources. London: Peter CMOS Transistors from Subthreshold to Strong Inversion at Peregrinus, 1982. Various Temperatures,” IEEE Trans. on Electron Devices, vol. 41, [6] A. Demir and A. Sangiovanni-Vincentelli, “Simulation and no. 11, pp. 1965-1971, 1994. Modeling of Phase Noise in Open-Loop Oscillators,” presented at [18] R. Telichevsky, K. Kundert, and J. White, “Receiver Custom IC Conf., San Diego, pp. 453-456, 1996. Characterization using Periodic Small-Signal Analysis,” presented [7] R. A. Pucel and J. Curtis, “Near-Carrier Noise in FET at Custom IC Conf., San Diego, CA, pp. 449-452, 1996. Oscillators,” presented at MTT Int'l Microwave Symp., Boston, pp. 282-284, 1983. [8] B. T. Debney and J. S. Joshi, “A Theory of Noise in GaAs FET Microwave Oscillators and Its Experimental Verification,” IEEE Trans. on Electron Devices, vol. ED-30, no. 7, pp. 769-776, 1983. [9] H. J. Siweris and B. Schiek, “Analysis of Noise Upconversion in Microwave FET Oscillators,” IEEE Trans. on Microwave Theory & Tech., vol. MTT-33, no. 3, pp. 233-242, 1985. [10] J. G. Sneep and C. J. M. Verhoeven, “A New Low-Noise 100- MHz Balanced Relaxation Oscillator,” IEEE J. of Solid-State Circuits, vol. 25, no. 3, pp. 692-698, 1990. [11] C. A. M. Boon, I. W. J. M. Rutten, and E. H. Nordholt, “Modeling the Phase Noise of RC Multivibrators,” presented at Midwest Symp. on Circuits and Systems, Morgantown, WV, pp. 421- 424, 1984. [12] T. C. Weigandt, B. Kim, and P. R. Gray, “Analysis of Timing Jitter in Ring Oscillators,” presented at Int'l Symp. on Circuits & Systems, London, pp. 27-30, 1994. [13] B. Razavi, “A Study of Phase Noise in CMOS Oscillators,” IEEE J. of Solid-State Circuits, vol. 31, no. 3, pp. 331-343, 1996.

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