Analysis of Clock Jitter in Continuous-Time Sigma-Delta Modulators

Transcription

1 Analysis of Clock Jitter in Continuous-Tie Siga-Delta Moulators V.Vasuevan, Meber, IEEE Abstract One of the factors liiting the perforance of continuous-tie siga-elta oulators CTSDM) is clock jitter. This jitter can be classifie as synchronous an accuulate/longter jitter. A clock that is erive fro a phase-lock loop PLL) contains both types of jitter. In this paper, we present a fraework that can be use to obtain the output spectru in the presence of jitter, either synchronous or accuulate, or a cobination of both. First, a general expression for the output power spectral ensity of the CTSDM in the presence of clock jitter is erive. Base on this, analytical expressions for the output power spectral ensity are obtaine for particular cases of synchronous an long-ter jitter. These are valiate against behavioural siulations. Inex Ters Clock jitter, siga-elta oulator I. INTRODUCTION The continuous-tie siga-elta-oulator CTSDM) has receive a lot of attention recently ue to its potential for obtaining high resolutions at low operating power. One proble liiting its perforance is clock jitter. This jitter can be classifie as synchronous an long-ter accuulate) jitter [1]. An analysis of jitter in phase-lock loops PLL) inicates that the jitter in the output is a cobination of correlate synchronous an long-ter jitter [], [1]. In the literature, the effect of white synchronous jitter on CTSDMs has been analyze extensively [3], [4], [5], [6]. Recently, the effect of synchronous correlate jitter has been stuie in [7]. In both cases, jitter is treate as a perturbation, which is possible because the variance of jitter is boune. There has been no systeatic stuy of the effects of accuulate jitter, although there is an epirical expression given in [3]. Perturbation oels cannot be use in the case of long-ter jitter since its variance is unboune. In this paper, we evelop a general theory for the analysis of the effects of clock jitter in CTSDMs, using nonlinear oels for jitter. Since nonlinear jitter oels are use, both synchronous an long-ter jitter can be analyze using a coon fraework. We first erive a general expression for the output power spectral ensity PSD) of the CTSDM in the presence of clock jitter. Base on this, we obtain analytical expressions for the output PSD for particular cases of synchronous an long-ter jitter. For the case of white synchronous jitter, we show that the expression for the output PSD is a slightly ore accurate version of the coonly use approxiation. For correlate synchronous an long-ter jitter, we erive analytical expressions for the output PSD that atch very well with siulations using behavioural oels. Using this fraework, it is also possible to obtain the output spectru if the clock has a cobination of synchronous an long-ter jitter. As a part of this analysis, we also obtain expressions for the effect of tiing jitter on igital to analog converters DACs). Throughout the analysis, we assue a non-return-to-zero DAC, which is usually the DAC use in CTSDMs, since it has better perforance than the return-tozero DAC in presence of clock jitter. The paper is organize as follows. Section II escribes the oel an soe of the assuptions use in the analysis. In section III, we erive expressions for the output PSD of the DAC in the presence of clock jitter. This is followe by an analysis of the oulator in section IV. In section V, we erive expressions for the output PSD in the presence of various types of jitter. Section VI contains coparisons with behavioural siulations an a iscussion of the results. Section VII conclues the paper. II. MODEL Figure 1a) shows the oel that is typically use to analyze a CTSDM with clock jitter. Clock jitter affects both the quantizer an the feeback DAC. However, within the signal ban, since the loop gain is large, jitter in the quantizer has a negligible effect on the output spectru. As a result, jitter inuce errors within the signal ban of the output spectru occur ainly ue to the feeback DAC. In Figure 1a), y j) [n] is the output igital signal an g c t) is the DAC output. With no clock jitter, g c t) = yct), j which is the continuous-tie version of y j [n]. With clock jitter, g ct) = yct+ξt)), j ξt) is a rano process that oels the jitter. For typical oversapling ratios OSRs), the oulator can be analyze as a iscrete-tie syste. Therefore, saple versions of x c t) x [n]) an g c t) g [n]) are use an the loop filter is oelle as a iscrete-tie filter. For purposes of analysis, usually a linear oel of the analog-to-igital converter ADC) is use, the quantization error q[n], is oelle as uniforly istribute aitive white noise. In the literature, jitter is treate as a perturbation to the ieal an the feeback signal g [n] can be written as g [n] = y i) ξ[n] [n]+ T = y i) ) y i) [n] yi) [n 1] [n]+j[n] 1) Here, y i) [n] is the output of an ieal oulator i.e, a oulator with no clock jitter), j[n] is the voltage error corresponing to the tiing error ξ[n] an T is the clock perio. The negative feeback loop copensates for this error in the feeback signal at the input, resulting in a non-ieal output spectru. Since the jitter-inuce error is oelle as

2 xt) xt) e[n] nt nt + x [n] g [n] x [n] h s [n] h [n] nt h s [n] h [n] nt g ct) a) b) g ct) DAC DAC + q[n] nt + ξ[n] + q[n] nt + ξ[n] y j) [n] y [n] y a) [n] Fig. 1. Discrete-tie linearize oel of the oulator with clock jitter a) Actual oulator an b) A fictitious oulator with an inepenent process e[n] ae to the DAC output an aitive error at the input, the actual output spectru is the su of the ieal spectru an this error ultiplie by an appropriate transfer function), i.e, S y j) e jωt ) = S y i) e jωt )+S j e jωt H e jωt ) ) 1+H e jωt ) ) Here, S y j) e jωt ) is the output PSD of a oulator with jitter, S y i) e jωt ) is the PSD at the output of an ieal oulator, S j e jωt ) is the PSD of the voltage error j[n] an H e jωt ) is the iscrete-tie transfer function of the loop filter. The assuption here is that the error ue to jitter appears as an aitive error an the error ue to jitter j[n] is inepenent of the output voltage. An equivalent way to look at this oel is as shown in Figure 1b). It is a fictitious oulator in which we a an inepenent rano process e[n] in an attept to copensate for the error in the feeback signal. This is essentially the sae error that the negative feeback loop tries to copensate, which then appears at the output of the real oulator. If the jitter is truly aitive an the copensation is exact, then the output PSD of the oulator will be close to the ieal output. The loop equation for this fictitious oulator can be written as y [n]+g h )[n] = x h s )[n]+q[n] e h )[n] 3) In the equation, represents the convolution operator, h [n] is the ipulse response of the feeback path in the loop filter an h s [n] is the ipulse response of the signal path. We nee to solve for the PSD of the output, y [n], enote by S y e jωt ). If g [n] is given by equation 1), the loop equation in ters of the power spectral ensities can be written as: S y e jωt )+S j e jωt ) JTF = S x e jωt ) STF + S q e jωt ) NTF +S e e jωt ) JTF 4) STF = NTF = JTF = H s e jωt ) 1+H e jωt ) 1 1+H e jωt ) H e jωt ) 1+H e jωt ) 5) Here S x e jωt ), S q e jωt ) an S e e jωt ) are the PSDs of the inputx [n], the quantization errorq[n] an the rano process e[n]. H e jωt ) an H s e jωt ) are the Fourier transfors of the two ipulse responses h [n] an h s [n]. STF, NTF an JT F enote the signal, noise an the jitter transfer functions. An approxiate solution to the loop equation is given by S y e jωt ) S a) y e jωt ) = S x e jωt ) STF +S q e jωt ) NTF = S i) y e jωt ) S e e jωt ) = S j e jωt ) 6) In the above equation,s j e jωt ) can be obtaine fro equation 1) [3], [4]. This is an approxiate solution, since in general, S j e jωt ) is a function of S y e jωt ). To copute S j e jωt ), S y a) e jωt ) is use which in this case is the ieal output spectru). In the real oulator, the negative feeback loop will try to copensate for S j e jωt ) at the input resulting in an output spectral ensity given by: S j) y e jωt ) S a) y e jωt )+S e e jωt ) JTF 7) S y a) e jωt ) an S e e jωt ) are given in equation 6). The results using an aitive jitter error oel are foun to atch reasonably well with behavioural siulations an with experient. Therefore, in this paper, we continue to use this oel. However, we obtain ore accurate expressions for S y a) e jωt ) an S e e jωt ) base on a ore accurate nonlinear jitter oel. For various types of jitter, we write the loop equation in a for siilar to equation 4) an fin expressions for S y a) e jωt ) an S e e jωt ). These are then use to fin the output PSD using equation 7). The first step towars analysis of the oulator is to get an expression for the output of the feeback DAC in the presence of clock jitter. This is one in the next section. III. ANALYSIS OF THE DAC In this section, we obtain the PSD of the DAC output signal in the presence of clock jitter. The analysis in this section is base on work reporte in [8]. The input to the DAC is a uniforly space iscrete-tie signal, y [n]. Its PSD can be written as S y e jωt ) = 1 R y [k]e jkωt 8) f s In the above equation, R y [k] is the autocorrelation of the output signal, T is the clock perio an f s is the clock frequency. In ters of the unit step function ut), the continuous-tie signal y c t) corresponing to y [k] can be written as: y c t) = k k y [k]ut kt) ut k +1)T)) 9)

3 3 The power spectral ensity of y c t), S yc ω), is given by: S yc ω) = 1 e jωt ωt) S y e jωt ) 1) This is the spectru of the signal at the output of the DAC if the clock were jitter free. However, if the clock has jitter, the output of the DAC is the tie-istorte version of y c t), enote by g c t). Let the clock jitter be enote by ξt). The autocorrelation of the jittere wavefor, R gc t,τ), can be written as: R gc t,τ) = E{g c t+τ)g c t)} = E ξ {R yc τ +ξt+τ) ξt))} 11) E ξ ) is the expectation with respect to ξ an R yc t) is the autocorrelation of y c t) conitional to ξ. If αt,τ) = ξt + τ) ξt) an fαt,τ),τ) is the probability ensity function of αt,τ), we have R gc t,t+τ) = R yc τ +αt,τ))fαt,τ),τ)α 1) In ters of S yc ω), this autocorrelation can be written as: R gc t,t+τ) = 1 S yc η)fαt,τ),τ)e jητ+αt,τ)) ηα 13) Now, the characteristic function Mη,τ,t) of αt,τ) is given by: { Mη,τ,t) = E e jηαt,τ)} = Substituting for this in equation 13), we get R gc t,t+τ) = 1 fαt,τ),τ)e jηαt,τ) α 14) S yc η)mη,τ,t)e jητ η 15) The power spectral ensity of g c t), S gc ω,t) can obtaine as the Fourier transfor of the autocorrelation function. It is given by S gc ω,t) = 1 Q c η,ω η,t) = S yc η)q c η,ω η,t)η 16) Mη,τ,t)e jω η)τ τ 17) Q c η,ω η,t) is the Fourier transfor of the characteristic function Mη,τ,t) evaluate at ω η. It can be evaluate for various types of jitter as shown in the following subsections. A. Synchronous white jitter This cannot be evaluate for the DAC alone, since continuous tie white noise oes not exist. However, as will be seen later, it is not an issue when the DAC is use within the oulator, since we use a iscrete tie oel for the oulator. B. Synchronous correlate jitter Here we assue that ξt) is a correlate Gaussian process with autocorrelation given by R ξ τ) = σ e a τ 18) σ is the variance of ξt) an a is a easure of the correlation. The expression represents a siple first orer correlation. This for for the autocorrelation is otivate by the autocorrelation of jitter in the PLL []. Using this, the variance of αt,τ), σ α, can be written as: σ α = σ 1 e a τ ) 19) Therefore, characteristic function of ατ) is given by σ Mη,τ,t) = e η α = e η σ η σ ) l e la τ ) The Fourier transfor of the characteristic function is thus Q c η,ω η) = e η σ η σ ) l la la) +ω η) 1) C. Long ter jitter In this case, ξt) is a rano walk process. The characteristic function of αt,τ) is given by Mη,τ,t) = e η c τ ) c is the iffusion coefficient of the jitter. The Fourier transfor of the characteristic function can be written as η c Q c η,ω η) = 3) η c ) +ω η) IV. ANALYSIS OF THE MODULATOR Since the oulator is analyze as a iscrete-tie syste, we nee a saple version of the autocorrelation of the DAC output signal, R g [n,n+k]. It can be obtaine fro equation 15) as R g [n,n+k] = 1 This can equivalently be written as R g [n,n+k] = 1 S yc η)mη,kt,nt)e jηkt η 4) Mη ω s,kt,nt)e jηkt η 5) The corresponing PSD, S g e jωt ), is therefore S g e jωt ) = 1 Q η ω s,ω η,nt)η 6)

4 4 Q η ω s,ω η,nt) = k Mη ω s,kt,nt) e jω η)kt 7) It is the iscrete Fourier transfor of the saple characteristic function. As explaine in section II, we first analyze a oulator that has an inepenent process e[n] ae to the output of the DAC. The loop equation of this oulator is given by equation 3). In ters of PSDs, this loop equation can be written as: S y e jωt )+S g e jωt ) H e jωt ) +S y g e jωt )H e jωt ) +S g y e jωt )H e jωt ) = S x e jωt ) H s e jωt ) +S q e jωt )+S e e jωt ) H e jωt ) 8) S y g e jωt ) is the cross-spectral ensity between the output an the feeback signal. As entione, the ai is to put the above equation in a for siilar to equation 4). We can then obtain expressions for S y a) e jωt ) an S e e jωt ) an use the to fin the output PSD of the real oulator. Equation 8) inicates that to fin the output power spectral ensity, we nee to fin the cross-spectral ensity. To o this, we first copute the cross-correlation ters R y g [n,k] by sapling the corresponing continuous-tie cross-correlation R ycg c t,t+τ). Now, R ycg c t,t+τ) = E{y c t+τ)g ct)} = E ξ {R yc τ +ξt))} 9) Since the autocorrelation R yc t) can be written as the inverse transfor of the power spectral ensity, we have R ycg c t,t+τ) = 1 S yc η)e ξ {e jηξt) }e jητ η 3) The saple cross-correlation can therefore be written as R y g [n,n+k] = 1 E ξ {e jη ωs)ξnt) }e jηkt η 31) The corresponing cross-spectral ensity is given by: S y g e jωt,nt) = 1 πf s E ξ {e jη ωs)ξnt) } e )η jω η)kt 3) R g y [n,n+k] an S g y e jωt,nt) can be foun in a siilar anner. This analysis inicates that to fin the power spectral ensity of the jittere wavefor, we nee to first fin the Fourier transfor of the saple characteristic function of jitter. This can then be use to obtain an expression for S g e jωt ) using equation 6). We also nee to evaluate the cross-spectral ensities for various types of jitter. This is one in the following subsections. k A. White synchronous jitter In this case, jitter is regare as a perturbation with uncorrelate tiing errors. Therefore, the characteristic function can be written as: Mη ω s,kt,nt) = e η ωs) σ, k = 1, k = 33) σ is the variance of the clock jitter. Note that the characteristic function is inepenent of n. The iscrete Fourier transfor of this function at ω η is obtaine as: Q η ω s,ω η) = ) e η ωs) σ δω η) Substituting this in equation 6), we get S g e jωt ) = 1 f s +1 e η ωs) σ ) 34) e ω ωs) σ S yc ω ω s ) + 1 e η ωs) σ ) η 35) Now, the first ter in the above equation can be written as: S y e jωt ) e ω ωs) σ sinc ω ω s )T/) 36) This expression can be use irectly in the equation 8) an the resulting loop equation can be solve to obtain an expression for S y a) e jωt ). The isavantage of oing this is that the series su has to be evaluate for each jitter variance. Instea, we can get a very goo approxiation by noting that, for all practical values of jitter, the exponential ter is close to one an it ies own ore slowly than the sinc function. Therefore, to a very goo approxiation, its effects on aliasing can be ignore an the ter 36) can be approxiate as: e ω σ S y e jωt ) sinc ω ω s )T/) 37) With in the range ± an 1% jitter, the error ue to this approxiation is only about.6% at f s / an is saller at lower frequencies. If there is 5% jitter, the error in this approxiation is about 6% at f s /. We can ake siilar approxiations for the secon ter. Denoting the series su of sinc square functions as Sω), the secon ter in equation 35) can be approxiate as: 1 Sη)S y e jηt ) 1 e η σ ) η 38) The resulting error in the signal-to-noise ratio SNR) epens on the shape an the high frequency gain of the NTF. However, in all the cases siulate, this approxiation gives an SNR to within..3b of the siulate value, with four aliasing siebans 4 4). With only one sieban inclue, the SNR atches to within 1B. If the range of is increase to a very large value, the approxiation will overestiate the aount of aliasing, but the SNR reains within.5b of the

5 5 siulate value. This is true even for 1% clock jitter the error in the approxiation is larger. In orer to obtain the cross-correlation, we nee to fin E ξ {e jηξnt) } an E ξ {e jηξn+k)t) }. In this case, both are equal to e η σ /. Making siilar approxiations as for S g e jωt ), the expressions for the cross-spectral ensities are given by: S y g e jωt ) = S g y e jωt ) e ω σ / S y e jωt )Sω) 39) B. Correlate synchronous jitter In this case, jitter is regare as a perturbation, but the tiing errors are correlate. Here, we assue an autocorrelation of the for given in equation 18). In this case, the characteristic function can be written as: Mη ω s,kt) = e η ωs) σ 1 e a k T ) As in the case of white synchronous jitter, the value of Mη,kT) is close to one up to ω s / an ies own slowly after that. Therefore, we can ake siilar approxiations as for the white synchronous jitter case an S g e jωt ) can be written as: S g e jωt ) 1 Sη)S y e jηt )Q η,ω η)η 4) Q η,ω η) = e η σ η σ ) l [ 1 e lat ] 1+e lat e lat cosω η)/f s ) 41) The error in the approxiation is larger for larger values of a relatively uncorrelate jitter). The upper boun for this error is the error incurre with white synchronous jitter. If substitute irectly in the loop equation equation 8)), it becoes ifficult to obtain even the approxiate solution S y a) e jωt ). As entione previously, the ai is to write the loop equation in the for of equation 4), i.e. in a for suitable to the aitive jitter oel. To that en, we write this integral as a iscrete su over frequency bins of with f i. Obviously, as the bin size becoes saller, the integral is represente ore accurately. The bin with neee to get goo estiates of the SNR epens on the OSR an the shape of the NTF, but typically we have foun that aroun hunre bins within the signal banwith is sufficient. A siilar nuber of bins is require to get estiates of the SNR fro behavioural siulations. The ain proble in writing it as a iscrete su is that Q η,η ω) has a peak at η = ω an falls very rapily on both sies, especially for sall values of a large correlation ties). Since η σ is typically very sall in the signal banwith, this axiu value can becoe very large. So, nuerical estiation in a finite bin size becoes ifficult. However, since the jitter power containe in a bin is liite, a nuerically ore robust estiate can be obtaine by fining the approxiate jitter power in each bin. On oing this, we obtain S g e jωt ) i Sf i )S y f i )e fiσ) f i ) σ ) l P c ff i ) 4) P c f f i) is the jitter power in each frequency bin an is given by P c ff i ) = f i+ f i f i f i 1 e lat 1+e lat e lat cosf ν)/f s ) 43) It is possible to obtain an analytical expression for the above integral [9] an it is given by ν f s Pff c i ) =, l =,f i f = 1 [ { )} 1+A πf tan 1 π 1 A tan fi +.5 f i ) f { s )}] 1+A πf tan 1 1 A tan fi.5 f i ), else A = e lat. The iscrete su can then be written as a su of two coponents, naely, the PSD at f = ω, spectru is esire, an the su of the PSDs at the other frequencies. Therefore, S g e jωt f) σ ) l ) Sf)S y f)e fσ) P c ff) f s + f Sf i )S y f i )e fiσ) i ) σ ) l P ff c i ) f i f l=1 45) Since E ξ {e jηξnt) } an E ξ {e jηξn+k)t) } are both equal to e η σ /, the cross-spectral ensities are given by equation 39) C. Long-ter jitter The iscrete tie version of the characteristic function given by equation ) can be written as 44) Mη ω s,kt,nt) = e k η ωs) σ 46) In the above equation, σ = ct. In this case, it is not possible to ake the sae approxiations as for the previous two cases. The iscrete Fourier transfor of Mη,kT) is given by: Q η,ω η) = 1 e η σ 1+e η σ e η σ cosω η)/f s ) 47)

6 6 In this case, we can write S g e jωt ) as S g e jωt ) = 1 Q η ω s,ω η)η 48) The function Q η,ω η) has a relatively narrow peak at η = ω an falls own quite rapily on both sies. Since its effect is localize, aliasing fro the other frequency bans is quite sall. In all the siulations perfore, we have seen that copletely neglecting aliasing gives an error of about 1 B in the SNR. If one sieban is inclue, the error in the SNR is of the orer of.3.5b. As in the case of correlate jitter, we write the integral as a iscrete su. On oing this, we obtain S g e jωt ) S yc f)p l ff)+ f i fs yc f)p l ff i ) 49) It is convenient in this case to expan the series su, with the range of f i epening on the nuber of aliasing frequency bans. P l f f i) is the jitter power ue to long-ter jitter) in the i th frequency bin, given by: P l ff i ) = f i+ f f i f 1 A i ν 1+A i A icosf ν)/f s ) f s = 1 [ { )} tan 1 1+Ai πf fi +.5 f i ) tan π 1 A i f { s )}] tan 1 1+Ai πf fi.5 f i ) tan 1 A i f s 5) Unlike the correlate jitter case, A i is not a constant within the frequency bin. To obtain an analytical approxiation for the integral, we assue A i takes on the value at the ipoint of the bin. HenceA i = e f i ) σ. This is a goo approxiation since the bin size is usually quite sall. The nuber of bins use is siilar to that in the previous case. For long-ter jitter, the asyptotic cross-correlations are given by li E ξ{e jηξnt) } = li cnt = 51) n n e η Siilarly, E ξ {e jηξ[n+k]t) } also tens to zero asyptotically. This tells us that the correlation between the output signal an the jittere version of the output signal tens to zero. Although this result is a little counter-intuitive, it can be explaine as follows. The continuous tie version of the feeback signal is g c t) = y c t+ξt)). In the case of long-ter jitter, the variance of ξt) grows with tie. When this happens, the saple paths of the feeback signal will be wiely isperse in tie. Therefore, asyptotically, the enseble average E{y c t)g ct+τ)} will go to zero. Consequently, the cross-spectral ensities will also ten to zero. This is a consequence of the oel that we use. As explaine in section II, we neglect the effect of jitter errors ue to the quantizer, since it is iinishe by the loop gain. While this is a goo approxiation in the signal ban, it ay not be as goo an approxiation at higher frequencies. So the quantizer output or the oulator output) will not aintain perfect tie as is assue in the oel, but will contain soe high frequency jitter power. This in turn will lea to soe correlation between the output an the feeback signal at higher frequencies. However, as entione, within the signal ban, zero correlation is a very goo approxiation. V. POWER SPECTRAL DENSITY OF THE OUTPUT SIGNAL OF THE MODULATOR Using the results of the previous section, we obtain an equation for the output power spectral ensity for various types of jitter. A. Synchronous white jitter Using equations 35), 37), 38) an 39) in equation 8). we have S y e jωt )+S j e jωt ) H e jωt ) = S x e jωt ) D H se jωt ) +S q e jωt ) 1 D D +S ee jωt ) H e jωt ) D 5) D is given by [ D = 1+Sω) e ω σ H e jωt ) an + e ω σ / H e jωt )+H e jωt ) )] 53) S j e jωt ) = 1 Sη)S y e jηt ) 1 e η σ ) η 54) Therefore, the approxiate solution to the loop equation is S y a) e jωt ) = S x e jωt ) H se jωt ) D S e e jωt ) = 1 Sη)S a) y e jηt ) +S q e jωt ) 1 D 1 e η σ ) η 55) As iscusse in section II, the power spectral ensity at the output of the actual oulator is given by S j) y e jωt ) = S a) y e jωt )+S e e jωt ) H e jωt ) D 56) In a siga-elta converter, the sapling frequency is typically uch larger that the signal banwith an σ is a fraction of the clock perio. Therefore η σ is close to zero, giving 1 e η σ η σ 57) Also, e ω σ,sω) 1, which eans D 1+H e jωt ). In this case, S a) y e jωt ) is the sae as the ieal output

7 7 spectru, S i) y e jωt ). Further, if all aliasing fro other frequency bans is neglecte in the expression for S e e jωt ) i.e. Sη) sinc ηt/), we get S e e jωt ) = 1 σ T ) πfs 1 e jηt S i) y η)η 58) This is the approxiation that is coonly use in jitter calculations. It results in an error of about 1.5 B in the preicte SNR, which has also been reporte [4]. B. Synchronous correlate jitter Using equations 45) an 39) in the loop equation an following a proceure siilar to the one use for synchronous white jitter, we can write approxiate solution to the loop equation as: S y a) e jωt ) = S x e jωt ) H se jωt ) +S q e jωt ) 1 D D S e e jωt ) Sf i )S y a) f i )e fiσ) f i f f i ) σ ) l P c ff i ) 59) D = 1+Sω) e ω σ H e jωt ) ω σ ) l P c ff)+e ω σ / [ H e jωt )+He jωt ) ]) 6) The output power spectral ensity of the actual oulator, S y j) e jωt ), can be copute using equation 56), using appropriate values for S y a) e jωt ) an S e e jωt ). In all the cases siulate, the SNR obtaine using these expressions with four siebans, are within..3b of siulate values. As the correlation tie increases i.e. a becoes saller, the characteristic function is ore nearly equal to one, as is assue in the theory. In this case, the siulate an theoretical values of the SNR atch to within.1b. As for the case of white synchronous jitter, if ω σ << 1, which is the case for typical OSRs, S y a) e jωt ) can be approxiate well by the ieal output spectru. A reasonable estiate of the SNR can be obtaine by using the ieal spectru an ignoring aliasing in the expression for S e e jωt ). C. Long-ter jitter Using equation 49) an the fact that the cross-spectral ensities are zero in the loop equation, an approxiate solution to the loop equation can be written as: y e jωt ) = H se jωt ) S x e jωt )+ 1 D D S qe jωt ) S e e jωt ) sinc πf i T)S y a) f i )Pff l i ) 61) f i f S a) As entione, the range of f i epens on the nuber of aliasing siebans inclue. Once again, the output power spectral ensity can be copute using equation 56). As a consequence of the zero correlation between the output an the feeback signal in this case, the rano process e[n] cannot copensate well for the errors ue to jitter. Therefore, unlike the synchronous jitter case, S y a) e jωt ) cannot be approxiate well by the ieal spectru. VI. RESULTS AND DISCUSSIONS The results obtaine using analytical expressions were copare with behavioural siulations for first, secon an thir orer oulators. The secon orer oulator ha a stanar feeback architecture with STF = z 1 an NTF = 1 z 1 ). The thir orer oulator was a feeforwar oulator with a Butterworth loop filter. The out-of-ban gain was 1.4. A tie step of.5σ was use to integrate the state equations of the loop filter uring behavioural siulations. Correlate rano nubers were obtaine using the etho suggeste in [1]. Results were obtaine for a one-bit an two-bit quantizer as well as for a linear oel with aitive uniforly istribute quantization error. The output spectru is the average power spectru obtaine fro five siulations. A Hann winow was use to fin the iscrete Fourier transfor. The progras were written in Python an run on a Linux syste. Soe representative results are now iscusse. All the results are plotte as a function of the noralize frequency f/f s ). A. White synchronous jitter Figure shows the results obtaine for white synchronous jitter for two ifferent signal aplitues. The analytical oel clearly preicts the noise ue to jitter very well. The SNR atches to about.3b. As expecte fro the theory, noise level oes not epen on the signal aplitue. Power Spectru B) - -9 Ap:.6 Ap:.6 Ap: Noralize Frequency D = 1+ H e jωt ) sinc πft)p l ff) 6) Fig.. Output spectru of a secon orer oulator with an OSR of 18 an 1% white clock jitter. The quantizer is a two-bit quantizer.

8 8 B. Correlate synchronous jitter Figure 3 shows a coparison of theoretical an siulation results for the case of correlate synchronous jitter, with two ifferent jitter variances. The results are seen to atch well. The ifference between the theoretical an siulate values of the SNR is less than.1b, with four aliasing siebans. Power Spectru B) - -9 Silation Theoretical Noralize Frequency Fig. 3. Output spectru of a secon orer oulator with an OSR of 3 an a correlation length of 1T with 1% clock jitter signal aplitue is.6v). The quantizer is a two bit quantizer. Figure 4) contains plots of the output spectru for a fixe σ an ifferent correlation ties. It is seen that the noise levels ue to jitter are lower for longer correlation ties. A siilar tren is observe in Nyquist rate ADCs [11],[1]. Power spectru B) - white.1t T 1 T Noralize frequency Fig. 4. Output spectru of a secon orer oulator with an OSR of 3 an 1% jitter for various correlation ties. The quantizer is a two bit quantizer. In the case of synchronous jitter, the SNR obtaine fro siulations an analytical results atche very well within.3b, usually better). As expecte, the atch is better for longer correlation ties. C. Long-ter jitter Figure 5 shows the results obtaine for a thir orer oulator an an OSR of 18 with long-ter clock jitter. Once again, the results of the analytical expressions atch very well with behavioural siulations. It is seen fro the figure that with long-ter jitter, there is a sprea in the signal spectru at the output. Both this sprea an the noise level ue to jitter epen significantly on the signal aplitue. Power Spectru B) - Aplitue:.1 Aplitue:.1 Aplitue:.8 Aplitue: Noralize Frequency Fig. 5. Output spectru of a thir orer oulator with an OSR of 18 an 1% long ter clock jitter for two ifferent input aplitues,.1 an.8v. A one bit quantizer is use. Power spectru B) - Theoretical One bit unifor rano Noralize frequency Fig. 6. Output spectru of a secon orer oulator with an OSR of 3 an 1% long ter clock jitter. If the output spectru has a strong haronic content, the true spectru iffers fro the spectru obtaine using the linear oel, even if there is no jitter. This eviation is typically not very significant within the signal ban. Since the jitter analysis is also one using the linear oel, one woul

9 9 expect soe ifferences fro theoretical estiates. This is seen in Figure 6. However, as can be seen fro the figure, the atch with behavioural siulations is still very goo within the signal ban. This was foun to be true in all the cases siulate, own to an OSR of 16. It is seen fro Figure 6 that the theoretical results atch well with behavioural siulations if a linear quantizer oel is use. The two results also atch very well if a two-bit quantizer is use. Power spectru B) - Long ter jitter Noralize frequency Fig. 7. Output spectru of a secon orer oulator with an OSR of 3 an 1% clock jitter. The jitter is a cobination of synchronous an long-ter jitter. For reference, results for long-ter jitter alone is inclue. A two bit quantizer is use. It is also possible to use this analysis when the total jitter is a cobination of various types of jitter. For exaple, at the output of the PLL, the total jitter is a cobination of synchronous an long-ter jitter [1], []. Figure 7 shows the results obtaine if a cobination of long-ter jitter an correlate synchronous jitter is use. In this siulation, half the jitter variance is ue to synchronous correlate jitter. For reference, the spectru for long-ter jitter alone is also inclue. It clearly shows there is a reuction in the jitter error as expecte). The Lorentzian sprea is also reuce. In the case of long-ter jitter, the aitive jitter oel itself is not as goo an approxiation as for the synchronous jitter case. Even here, the atch is generally very goo up to about f s /1 an always very goo up to the Nyquist frequency. sall. In this case, the nuber of points require to resolve the jitter in behavioural siulations becoes very large. In this paper, we continue to use the iscrete-tie oel of the oulator. This works well for typical OSRs use. However, a better oel woul be to have the sapling one after the loop filter. ACKNOWLEDGEMENT The author thanks anonyous reviewer 3 for coents that le to better approxiations in the analysis an reviewer 1 for coents for iproveent in the presentation of the paper. REFERENCES [1] K.Kunert, Preicting the phase noise of PLL-Base frequency synthesizers. Nov. 3. [] A.Mehrotra, Noise analysis of phase-locke loops, IEEE Trans. Circuits Syst.-I, vol. 49, pp , Sept.. [3] J.A.Cherry an W.M.Snelgrove, Clock jitter an quantizer etastability in continuous-tie elta-siga oulators, IEEE Trans. Circuits Syst.- II, vol. 46, pp , June [4] H.Tao,L.Toth an J.M.Khouri, Analysis of tiing jitter in banpass siga-elta oulators, IEEE Trans. Circuits Syst.-II, vol. 46, pp , Aug [5] O.Oliaei, State-space analysis of clock jitter in continuous-tie oversapling ata converters, IEEE Trans. Circuits Syst.-II, vol. 5, pp , Jan. 3. [6] K.Doris et al., A general analysis of tiing jitter in D/A converters, in Proc. IEEE ISCAS, pp ,. [7] Y.Chang et al., An analytical approach for quantifying clock jitter effects in continuous-tie siga-elta oulator, IEEE Trans. Circuits Syst.-I, vol. 53, pp , Sept. 6. [8] W.M.Brown an C.J.Palero, Syste perforance in the presence of stochastic elays, IEEE Trans. Info. theory, vol. 8, pp. 6 14, Sept [9] K.F.Riley, M.P.Hobson an S.J.Bence, Matheatical ethos for Physics an Engineering. Cabrige University Press, [1] N.J.Kasin, Discrete siulations of colore noise an stochastic processes an 1/f α power law noise generati on, Proc. IEEE, vol. 83, pp. 8 87, May [11] A.V.Balakrishnan, On the proble of tie jitter in sapling, IRE Transactions on Inforation Theory, pp. 6 36, April 196. [1] V.Vasuevan, Siulation of the effects of tiing jitter in track-an-hol an saple-an-hol circuits, in Proc. 4 n Design Autoation Conf., pp. pp 397 4, 5. VII. CONCLUSION In this paper, we have presente a fraework to analyze clock jitter in continuous tie siga-elta oulators. The analysis was one for synchronous as well as long-ter jitter. The results obtaine using the analytical expressions were foun to atch very well with behavioural siulations. Using the sae fraework, we have shown that is possible to obtain the output spectru, when the clock jitter is a cobination of synchronous an long-ter jitter. This is typically what woul be obtaine at the output of the PLL. These analytical expressions are especially useful when jitter variances are