Efficient Phase Noise Modeling of a PLL-Based Frequency Synthesizer

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Proceedings of the 4th International Symposium on Communications, Control and Signal Processing, ISCCSP 2010, Limassol, Cyprus, 3-5 March 2010

Goulven Eynard, Noelle Lewis, Dominique Dallet and Bertrand Le Gal

Abstract-This paper proposes a computationaly-efficient procedure Section III presents the new efficient method to generate arbitrary dedicated to time-domain phase noise (PN) modeling of a phase PN spectrum. Finally, Section IV presents the performance evalu locked loop (PLL)-based frequency synthesizer in a radio-frequency (RF) ations of the proposed PN generation architecture. The results are context. The structure proposed is able to describe any kind of PN including thermal and flicker noise regions using time-domain equations. compared to the measured PN spectrum of an existing frequency The design process of the phase noise modeling process is detailed. An synthesizer. expression of the estimated mean square error (MSE) of the frequency response of the PN generating system is proposed and compared to II. PHASE NOISE IN A PLL-BASED OSCILLATOR a previously proposed PN generating system. The spectral behaviour of the simulated frequency synthesizer is compared to practical PN For an ideal oscillator operating at frequency fo, the voltage output measurements obtained with a spectral analyzer. can be expressed as v(t) = A cos(wot+4» where A is the amplitude, 4> is a fixed time-invariant phase reference, and Wo = 21rfo is the I. INTRODUCTION carrier pulsation, in rad.s-1. In a practical oscillator, both amplitude Phase noise (PN) is a crucial parameter that can limit the overall and phase are time-varying, which impacts the spectrum of the performance of contemporary wireless [1] and optical [2] transmis oscillator by spreading the power of the oscillator around the carrier sion systems. Efficient evaluation of the PN impact on the overall frequency. In most practical cases, the disturbance in the amplitude is performance of complete transmission systems are required. In this negligible [7] and therefore, only the random deviations of the phase context, simple and computationally efficient design methods of can be considered: system-level PN generators are of particular interest. Numerous methods exist in literature concerning PN modeling [3 v(t) = A cos(wot + 4>(t)) (1) 11]. One of the major difficulty is flicker noise modeling. Such non For small values of PN fluctuations (14)(t)1 « 1 rad.) equ. (1) can stationnary noise can be efficiently approximated from a Gaussian be simplified to : noise generator filtered through a suitably designed multirate filter bank. Practical realizations of flicker noise generators approximate v(t) ~ A cos(wot) - Ao4> (t )sin(wot) (2) the frequency response of the desired filter from a set of recursively defined one-order filtering cells. Perfect synthesis assumes an infinite The spectrum of 4>( t) is thus frequency translated around the carrier number of cells, but several approximations exists in the litterature frequency at ±wo. To quantify this PN, a commonly used spectral [6-11] in order to obtain practical systems. In [6] IIR and FIR measure is the single-sided band (SSB) spectral noise density L( ~w) filtering approximation methods are suggested to generate 1/fa (dBc/Hz): noise sequences. However, this solution turns out to be computa L(~w) = 1010 (noise power in I.-Hz BW @ Wo + ~w) (3) tionally complex for wideband colored noise generation, particularly g earner power for close-to-carrier noise modeling. In [9] a simplification of this method using multirate filter bank process is proposed, but the design where ~w is the carrier pulsation offset from the nominal carrier method of such multirate filters leads to non-straightforward design frequency pulsation Wo of the oscillator. A PN spectrum commonly methods. A computationally efficient and event-driven PN generation encountered at the output of a PLL-based frequency synthesizer is scheme were previously proposed in [7] for a free-running oscillator. plotted on Fig.l [12]. The PLL PN is usually majoritarily dominated However, this design method is not applied in the particular case of by the reference crystal oscillator (CO) below the loop bandwith PLL-based oscillators. WBW and by the voltage controlled oscillator (VCO) above WBW [5]. PN spectrum shapes of free-running and PLL-based oscillators The resulting PN spectrum observed at the output of the frequency are very different [5], and thus, the PN generation techniques can synthesizer on the frequency interval [fmin; f max] results in a differ slightly. The purpose of this paper is to develop a simple composite spectrum (see Fig.l), composed of three zones of 1/fa and computationally efficient design method of a PLL-based PN spectral noise shape, delimited by two corner pulsations WeI and W e2. generator. The results are asserted through a comparison between In the context of a PLL-based frequency synthesizer, Q is successively practical measurements on a PLL-based frequency synthesizer and equal to 3, 0 and 2 [5], [12]. the proposed PN generation scheme. III. NOISE GENERATION ARCHITECTURE The rest of the paper is organized as follow. In Section II, the general characteristics of PN in PLL-based oscillators are shortly A PN noise generator derived straightforwardly from [11] for adressed. PLL oscillators is plotted on Fig.2 (a). This architecture requires several independant Gaussian noise generators in order to obtain Goulven Eynard, Noelle Lewis, Dominique Dallet and Bertrand Le Gal are with a satisfying precision. It involves also a +3 dB imprecision error the IMS-Bordeaux 1 University - IPB ENSEIRB MATMECA, 351 Cours de la at each frequency corner of the spectrum of the generated PN, Liberation, 33405, Talence Cedex, France; e-mail: goulven.eynardwims-bordeaux.fr, due to the superposition of different spectral noise shape [11]. PN [email protected], [email protected], bertrand.legal@ims generator architectures proposed in [7], [9] are initially not supposed bordeaux.fr to include additionnal independant Gaussian noise generators, but

Loop bandwidth -, ...... ~/ [i\J] . . I •• 1_ . I •• 1_ ...... ••••. : ..1:""..f- . I(a) Initial PN generator architecture '------~ w (log scale) W m in maw =wcl w "'IU"

Fig. I. Typical PLL oscillator phase noise spectrum. they are optionally proposed when a composite and multi-zones PN spectrum shape like the one plotted on Fig.l is required. The architecture initially proposed requires to adjust the power at the input of each filter, allowing the desired separation of each zone and finally the construction of the desired PN spectrum. Previously I(b) Proposed PN generator architecture I proposed design process of PN generating architecture requires an Fig. 2. Plot of initial (a) and proposed (b) time-domain phase noise generator optimization problem resolution to determine the gain at the input architectures. of each designed filter [9]. On [7] is proposed an arbitrary overall correction of 5.5 dB or 1.5 dB, depending if the different Gaussian noise sources are correlated or not. All of these techniques require approximated through a cascade of one-order integration cells [13] : an important design process. N From the time-domain flicker noise generation technique of [II], Happrox(P) = Ho II Hi(p) (6) a generalization is now proposed in this paper in order to generate i=l any composite PN spectrum in a computationally efficient way and with a facilited design process. where N is the number of elementary cells of the filter and : The most critical step for discrete-time noise modeling operation is the design of filters for 1/ r noise, where 0 < a < 2. Basically, and as suggested in [II], all cases where a > 2 can be derived from the (7) previous cases by using classical one-order integration. This is the strategy proposed in [II] for 1/ r PN modeling, reported on Fig.2 (a). The following part focuses on the generalization of the previously where Wi (resp. w;) are the poles (resp. zeros) of the approximated proposed filter design for any value of a on a desired frequency transfer function and 1 ::; i ::; N. N is the order of the resulting band [fmin ; f max]. This generalization leads to an easier composite digital filter and is chosen in order to obtain the optimal trade-off PN generation design model, as it will be shown in the second part. between the precision of the approximation of the function H (p) and The precision of the new model is also logically enhanced, as shown the overall computational complexity of the system. The expression of in the performance evaluation part. The non-integer order filtering N in term of the precision of the model (estimated mean square error operation is higlighted on Fig.2-(a) and (b) by a grayed-background (MSE)) is an important design parameter which will be quantified block. further. Once the parameters WL, WH and N are fixed, the design parameter "( is defined : A. 1/i" noise generation on [fmin ;fmax] N The aim is to approximate the transfer function of a filter on a given "( = (::r/ (8) frequency range [h j f H]. Basically, in order to choose the frequency band [h ;f H] in term of the wanted frequency band generation This parameter defines simultaneously the distance on a logaritmic [fmin ; fmax], the following values are choosen [II]: scale between each elementary cell and also the poles and the zeros position of each elementary cell. The general principle of 1/ f [hj fH] [fmin /lO; lO·fmax] (4) = generating noise is illustrated on Fig.3-(a). The recursively defined This can be obtained by a transfer function defined as : values of the poles and zero for this particular case, defined in [II] is reported here : (1 + P )" /2 W i = "(i-l)+1/4 WL (9)

H(p) = n; ( WH) 0< /2' (5) w; = "(i-l)+3/4 WL , (to) I+ L WL where 1 ::; i ::; N. The generalization to any kind of slope can be where H 0 is the overall gain of the filter, WH = 21rf Hand obtained by following the principle that the distance between each WL = 21rh. The transfer function of this a /2 order filter can be cell stay the same, whereas the distance between the pole and the IHi(jw )l (dB ) A Estimated Mea n Square error 0 .0030 , . \ ' , , ... .\ l !~.i~!~I.~~.g~."!~~~j!~I1.~~~~!~~c::~~~!li . TI 0 .0025 \\ i / i i 1 1 1 ...... \ . ~ ~ : : ; j . 0.0020 " , , ... L- ~

W + W'HI (log scale) i 1 \ i i 1 1 1 00015 ,·,,·,·,··:'·\···,,·:"" Proiiosed·P;.{iiene'raiion' arcii iiecture (H) (;) (H i) (H) (il (i+l) Y W L Y WL Y WL Y WL Y WL Y WL i \, i/! 1 1 1 ...... ~ ~'~ : : ; j . :(8) 1/fphase noise generation II (b) 1/Fphase noise generation I 0 .00 10 :/' ", : : : :

Fig. 3. 1/ f phase noise generation principle and extension to 1/ f a noise oooos~ >••• >i:,",..~c" ,.:: ~ : :: ::l:~ : ~~ : ~ generation principle illustration.

0 .0000 1 zero of each cell is modified. Following this principle, the following equations are straightforwardly obtained : Fig. 4. Precision and computational complexity of the initial and the proposed PN generation architecture in term of the number of cells N. Wi = ,(i- l )+( 1/2-a/4) WL (11)

I _ (i - l )+ ( I/2+a/4 ) W i -, WL · (12) Discretization of the filter is carried out using the Tustin transforma- The general principle of 1/t " generating noise is illustrated on Fig.3 tion. : p = T2 z-1z+ I ' wereh T's IS th e samp l' mg peniod .Th us : (b). On the following part and from the proposed generalization, an s efficient design method for composite PN spectrum is now proposed. N + I -1 ""' ai aiz Hd(Z) = CO + ~ b, + b;Z-1 (17) B. Composite PN generation spectrum with Considering a composite PN spectrum as proposed on Fig.I, the optimal transfer function is now defined as : a, = a; = c.T; (18)

bi W iTs 2 (19) P = + 1+3/ 2 ( 1+ p) b; = W iTs - 2 (20) H(p) = Ho Wel WH (13) ( 1+.E.... ) 1+..E.... The resulting time domain equation can be written : WL Wc2 N The approximation function from Fig.2-(b) can be defined as : y[n] = co x[n] + L Ji[n] (21) i = 1

(14) with: (22)

1 + ...E..-~ 3 / 2 The design of the proposed PN generator leads to a similar archi where I1~1 H}b)(p) approximate ~ . It can be predicted ~ tecture as the one proposed in [7]. Similarly, a multi-rate process that for the same number of cells ,th;L SE of the architecture of and anti-aliasing filter can be efficiently used. In the next part, the Fig.2-(b) is reduced compared to the architecture of Fig.2-(a), since performance of the current noise generation process is compared to the interval of approximation is [Jmin ;fel], compared to the interval the original process proposed in [11]. [fmin ; fma x] for he same values of cells N. This results is asserted through simulations. IV. PERFORMANCE EVALUATION A. Frequency response MSE C. Discrete-time noise generation An estimation of the MSE frequency response J(N) between the ideal filter and its practical implantation in term of [t., fH and N The obtained analog transfer function has to be decomposed in can be written: a sum of one order cells in order to decrease the overall numerical error in the computation of the filter coefficients. The decomposed J(N) = ~E fJ.IIH!~ ;,t;!o x(JW(k»)W (23) version of the filter equ. (6) is of the form : IIH(JW(k»)I-

N where w(k) is in a discrete interval !21l"h;21l"fH] com (a ,b) () _ ""' c, Happrox p - CO + L.J ~+ (15) posed of M logarithmically-spaced samples, Ha~~r o x (jw(k)) (resp. i= 1 ' P H~~p r o x (jw( k)) is the transfer function of architecture plotted on with Fig.2-(a) (resp. Fig.2-(b» . As an illustrative example, the measured spectrum of an Agilent 8662A using a spectrum analyzer using (16) [i. = 100Hz, fH = 1MH Z is plotted on Fig.6. The parameter of the desired frequency response (Jel = 3kHz, [ ez = 1MH z ) are ·80 ·90

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Fig. 5. Agilent 8662A frequency synthesizer measured phase noise spectrum Fig. 6. Power spectral density of the generated phase noise spectrum. (/0 = 3.109 Hz).

PN measurements of a RF frequency synthesizer and the generated extracted from theses measurements. The MSE obtained for various PN was proposed to validate the procedure. values of cells N are reported , assuming, is = io /100 = 30MH z, ACKNOWLEDGEMENT and M 800. The results compared to the initial architecture = The work is supported by the DGA - SCERNE project. proposed on [11] are compared on FigA . From this figure, it can be noted that the estimation precision of REFERENCES the architecture plotted on Fig.2-(a) is biased. Below a certain value [1) D. Petrovic, W. Rave, and G. Fettweis, "Effect of phase noise on ofdm of N, adding new cells will not increase the precision of the filter, systems with and without pll : Characterization and compensation," IEEE while still increasing linearly the computational complexity of the Trans. on Communications, vol. 55, no. 8, pp. 1607-1616, August 2007. [2) A. Demir, "Nonlinear phase noise in optical-fiber-communication sys system . This bias can be partly explained by the +3 dB error at tems," IEEE Journal on Lightwave technology, vol. 25, no. 8, pp. 2002 each frequency comer. On the other part, the obtained MSE of the 2032, August 2007. proposed architecture is lower and exhibits no bias. [3) A. Demir, A. Mehrotra, and J. Roychowdhury, "Phase noise in oscilla tors: an unifying theory and numerical methods for characterization," B. Experimental Results IEEE Trans. on Circuits and Systems - I: Fundamental Theory and Applications, vol. 47, no. 5, pp. 655-674, May 2000. From the same measured PN spectrum (see Fig.5), using the previ [4) A. Demir, "Computing timing jitter from phase noise spectra for oscil ously designe filter response, the frequency results are analyzed in the lators and phase-locked loops with white and lIf noise," IEEE Trans. on frequency-domain with a periodogram. The periodogram performs an Circuits and Systems > I: Regular Papers, vol. 53, no. 9, pp. 1869-1884, average on several sub-windows of length M = 8192 , distributed September 2006. [5) K. Kundert, "Modeling and simulation of jitter in pll frequency synthe over the discrete-time signal obtained at the output, with an over sizers," Cadence Design Systems, 2001. lapping between each consecutive window of 50%. Considering the [6) N. 1. Kasdin, "Discrete simulation of colored noise and stochastic effect of a rectangular windowing on the periodogram, the theoretical process and (1/ f O<) power law noise generation," proc. IEEE, vol. 8, and the measured PSD are related by : no. 5, pp. 802-827, May 1995. [7) R. B. Staszewski, C. Fernando, and P. T. Balsara, "Event-driven simu 2M lation and modelling of phase noise of an rf oscillator," IEEE Trans. on psdmeas = PSdtheo + 10 log (Ts ) (24) Circuits and Systems, vol. 52, no. 4, pp. 723-733, April 2005. [8) M. Guglielmi, " lIf signal synthesis with precision control," Signal The spectrum of the noise obtained is plotted on Fig.6. The asymp processing. , vol. 86, pp. 2548-2553, October 2006. totic values are plotted on Fig.5 and Fig.6 in order to make an efficient [9) J. Park, K. Muhammad, and R. Kaushik, "Efficient modeling of 1/ r noise using multirate process," IEEE Trans. on Computer-aided design comparison. The simulated PN well matches the measured results . of integrated circuits and systems, vol. 55, no. 7, pp. 1247-1256, July 2006. V. CONCLUSION [10) G. W.Wornell, "Wavelet-based representation for the lIffamily of fractal On this paper, a computationally-efficient time-domain PN gen processes," Proc. IEEE, vol. 81, no. 10, pp. 1428-1450, Oct. 1993. [Il) N. Lewis, G. Monnerie, L. Lewis, J. Sabatier, and P. Melchior, "Auto eration model has been proposed. A generalization of [II] design matic procedure generating noise models for discrete-time applications," technique was given and from this technique , an efficient time domain in Proc. of ISCAS '06, Island of Kos, 2006, pp. 5720-5720. PN generation architecture was proposed . The design process of [12) Phase Noise Profile Aid System Testing. Maxim Application Note 3822, the new proposed PN modeling architecture was clearly explicited. June 2006. [13) A. Oustaloup, F. Levron, F. Nanot, and B. Mathieu, "Frequency band The results are asserted through evaluation of the estimated MSE complex non integer differenciator : characterization and synthesis," of the frequency response of the proposed filter, compared to the IEEE Trans. on Circuits and Systems I : Fundamental Theory an previously proposed architecture. Finally, a comparison between the Applications, vol. 47, no. 1, pp. 25-40, 2000.