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The Effects of Aperture Jitter and Clock Jitter in WidebandADCs
Michael Lohning and Gerhard FettweisDresden University of Technology
Vodafone Chair Mobile Communications SystemsD-01062 Dresden, Germany
Abstract
Designing leading-edge systems (e.g., communications systems) requires knowledge about the tech-nological limits. Jitter is the limiting effect in widebandADCs with a digitization bandwidth between1 MHz and 1 GHz. The effects of aperture jitter and clock jitter have been investigated previously.However, some very important aspects are still missing, in particular investigations on the spectraldistribution of the jitter induced error. This gap is filled by this article.
Keywords: wideband ADC, aperture jitter, clock jitter, signal-to-noise ratio, error spectrum
1 Introduction
Modern mobile communication receivers require high speed analog-to-digital converters which pro-vide a high resolution for a wide digitization bandwidth. This is particularly true for reconfigurablemultimode receivers. Due to technology dependent physicalerror effects today’s state-of-the-artwideband ADCs cannot cope with the enormous requirements regarding resolution, speed, and dig-itization bandwidth of such receivers. In [1]Walden identified the aperture jitter as the dominatingerror effect that limits the achievable signal-to-noise ratio (SNR) and therefore the resolution ofwideband ADCs (with a digitization bandwidth between 1 MHz and 1 GHz). As will be clarified inthis article, clock jitter influences the achievable SNR in asimilar way.
In the last few years different authors derived formulas to quantify the SNR limiting effect ofjitter in ADCs. WhileWalden used a worst case approach [1],Kobayashi presented an exact formulawhich allows to calculate the SNR in the presence of an aperture jitter [2]. The effect of the clockjitter was investigated byAwad [3], but only for the special case of sinusoidal input signals.
This article presents an explicit analysis of both the aperture jitter and the clock jitter effects forarbitrary stationary input signals. Additionally to the SNR formulas, also expressions for the corre-sponding error spectra at the ADC output are derived. The presented results enable developers ofnew receiver concepts to decide if and by which means jitter effects in the ADC can be compensated.Further, the analysis is useful for designers of future air interfaces (e.g. 250 Mbit/s WLAN) whohave to consider theoretical limits of ADC resolution and digitization bandwidth determined by thejitter parameters of current or future technologies.
2 Sampling with jitter
Consider a real periodic ADC input signal given by its Fourier series expansion
s(t) = c0 +¥â
k=1
Ac-k ã-ü2Π fkt + ck ã
ü2Π fktE= c0 +
¥âk=1
A|ck| ã-üjk ã-ü2Π fkt + |ck| ã
üjk ãü2Π fktE (1)
whereck, c-k denote the complex Fourier coefficients which can be represented by their magni-tudes|ck| and the corresponding phasesjk. In order to obtain general results independent of a spe-cial (possibly unknown) phase spectrum, the phasesjk shall be assumed as independent randomvariables uniformly distributed in the range(-Π, Π). With this assumption (1) can be written in theform
s(t) =¥â
i=-¥
si(t) =¥â
i=-¥
ci ãü2Π fit with E{cic
*k} =ìïíïî
|ci|2 if i = k
0 else(2)
which describes a wide-sense stationary (WSS) random process with the complex spectral compo-nentssi(t) [4]. The operator E{×} in equation (2) stands for the mean value.
In the ADCs(t) is sampled at the time instantstn = nT+Jn with the nominal sampling periodT . Jn
are the random sampling time variations due to aperture and clock jitter. For a block ofN samplingpoints the mean error power caused by the random jitter process can be calculated as
PJ =1N
N-1ân=0
E{e(nT )e*(nT )} (3)
where
e(nT ) = s(nT+Jn) - s(nT ) (4)
=¥â
i=-¥
[si(nT+Jn) - si(nT )] (5)
=¥â
i=-¥
ei(nT ) (6)
stands for then-th error sample. As one can show, using the orthogonality property of the spectralcomponentssi(t) of the ADC input processs(t) (see (2)), the sampling errorsei(nT ) generated fordifferent componentssi(t) are also orthogonal, i.e. the corresponding error powers accumulate:
PJ =1N
N-1ân=0
¥âi=-¥
E9 |ei(nT )|2 = (7)
The derivation of the mean sampling error power E9|ei(nT )|2= caused by jitter for a single spectralcomponentsi(t) is straightforward. Using the relationship E9eüΜJn= = E9e-üΜJn=, which holds for everyzero mean jitter process with symmetrical probability density functions (PDFs), one gets
E9|ei(nT )|2= = 2 |ci|2I1- E9eü2Π fiJn=M (8)
and finally
PJ =1N
N-1ân=0
¥âi=-¥
2 |ci|2I1- E9eü2Π fiJn=M (9)
where E9eü2Π fiJn= is the characteristic function of the overall jitter process.So far only periodic input signals have been considered. Theresults can easily be extended to
non-periodic signals. Substituting the definition of the Fourier coefficients
ci =1T0
T0/2
à-T0/2
s(t)ã-ü2Πi f0tât (10)
with the input signal periodT0 = f -10 and fi = i f0 into (9) and taking the limit forT0 ® ¥ yields
PJ =1N
N-1ân=0
2
¥
à-¥
Sss( f ) I1- E9eü2Π f Jn=Mâ f (11)
whereSss( f ) is the power spectral density (PSD) ofs(t). Finally, the jitter dependent SNR (in deci-bels) is given by
SNRJ = 10 log
æçççççççççççè
Ù ¥-¥ Sss( f )â f
2N
N-1Ún=0
¥Ù-¥
Sss( f ) I1- E9eü2Π f Jn=Mâ f
ö÷÷÷÷÷÷÷÷÷÷÷ø
dB (12)
It is not sufficient to consider the mean sampling error poweronly when aiming at a completedescription of the effects of aperture jitter and clock jitter in ADCs. Also its spectral distribution,i.e. the corresponding error power spectrumSeeIeü2Π f T M at the ADC output, has to be analyzed. Thisanalysis is particularly useful in order to decide if, by which methods, and to which extent the jit-ter dependent error effects in ADCs can be compensated.SeeIeü2Π f T M is defined as the discrete-timeFourier transform (DTFT) of the error auto-correlation functionsee(nT , mT ) = E{e(nT )e*(mT )}. Us-ing the stochastic input signal model described by (2), the following expression can be derived forthe error auto-correlation function at the ADC output.
see(nT , mT ) =¥â
i=-¥
B|ci|2ãü2Π fi(n-m)T × B1+ E9ãü2Π fi(Jn-Jm)= - E9ãü2Π fiJn= - E9ã-ü2Π fiJm=FF (13)
The extension of (13) to non-periodic signals yields
see(nT , mT ) =
¥
à-¥
Sss( f )ãü2Π f (n-m)T × B1+ E9ãü2Π f (Jn-Jm)= - E9ãü2Π f Jn= - E9ã-ü2Π f Jm=Fâ f (14)
As can be seen, the error auto-correlation function dependson the characteristic functions E9ãü2Π f Jn=and E9ã-ü2Π f Jm= of the overall jitter process at the sampling time instantsnT , mT and of the char-acteristic function E9ãü2Π f (Jn-Jm)= of the differenceJn - Jm. Before transforming (13) or (14), i.e.calculating the corresponding error power spectrum, it is useful to analyze E9ãü2Π f Jn=, E9ã-ü2Π f Jm=,and E9ãü2Π f (Jn-Jm)= for the different kinds of jitter. This will be done in the next section.
3 Jitter models
Aperture jitter stands for the random sampling time variations in ADCs whichare caused by broad-band noise in the sample-&-hold circuit. It is commonly modeled as a stationary white Gaussianprocess [1, 2, 3], i.e. the corresponding sampling time variationsJap
n = tn - nT are assumed tobe independent identically distributed (i.i.d.) Gaussianrandom variables with zero mean and thevarianceΣ2
ap. Hence, the characteristic functions ofJapn andJap
n - Japm are
E9ã±ü2Π f Jnap= = E9ã±ü2Π f Jm
ap= = ã-2Π2 f 2Σ2ap (15)
and
E9ãü2Π f (Jnap-Jm
ap)= = ìïíïî
1 if n = mAã-2Π2 f 2Σ2apE2 if n ¹ m
(16)
Since the aperture jitter process is stationary its characteristic functions do not depend on the absolutesampling time instantnT , i.e they are time-invariant. Usually the aperture jitter process of an ADC
is characterized byΣap =1Σ2
ap which is the so-called root mean square (rms) aperture jitter. Rmsaperture jitter values of state-of-the-art high performance ADCs lie between 0.25 ps and 0.1 ps [5, 6].
Clock jitter is a property of the clock generator that feeds the ADC with the clock signal. It iscaused by the phase noise of the oscillator and generates additional sampling time errors in the ADC.In [7] it is shown that the phase noise of free running oscillators can be modeled as a Wiener process,i.e. a continuous-time non-stationary random process withindependent Gaussian increments. Time-discretization of the Wiener process yields the model of accumulated timing-jitter commonly usedfor the clock jitter [3, 8], where the sampling time variationsJacc
n are modeled as
Jacc0 U 0 and Jacc
n =nâ
i=1
∆i (17)
The jitter increments∆i are i.i.d. Gaussian random variables with zero mean and the vari-anceΣ2
∆i = cT . Therefore each sampleJaccn of the clock jitter process is a zero mean Gaussian random
variable with the varianceΣ2
acc(nT ) = E 9|Jaccn |
2= = cnT (18)
Herec stands for the phase noise constant of the oscillator (defined in [7]) andT for the nominalclock period. The productcT of the phase noise constant and the nominal clock period is the so-called cycle jitter variance. Its square root
0cT is a typical parameter of clock generators known as
the rms cycle jitter.Using the accumulated jitter model the following expressions for the characteristic jitter functions
can be derivedE9ã±ü2Π f Jn
acc= = ã-2Π2 f 2Σ2acc(nT ) = ã-2Π2 f 2cnT (19)
andE9ãü2Π f (Jn
acc-Jmacc)= = ã-2Π2 f 2cT |n-m| (20)
It should be noted that in the case of clock jitter the characteristic functions strongly depend on theabsolute sampling time. This time-varying behavior is caused by the non-stationarity of the clockjitter process. The accumulated clock jitter of a free running oscillator is a highly non-stationaryprocess. Its variance increases linearly with the samplingtime (see (18)) and its standard deviation(rms jitter)Σacc(nT ) =
0Σ2
acc(nT ) =0
cnT increases with the square root of the sampling time.
Hence, the longer the sampling block is, the larger are the sampling time variations caused by theclock jitter. The highest sampling time variations occur atthe end of the sampling block. Theyare characterized by the maximum rms jitter which reaches a value of
0cNT for a block of the
lengthNT .The difference should be noted between the constant cycle jitter parameters (cycle jitter vari-
ancecT , rms cycle jitter0
cT ) on one side, which are used to characterize a clock generator, and thetime-varying parameters of the clock jitter process on the other side.
As can be seen from equation (18) the strength of the clock jitter process of a clock generatordepends on the phase noise constantc which characterizes the quality of the oscillator used in theclock generator. The lower the value ofc is, the lower is the phase noise of the oscillator andthe higher is its quality. Typical phase noise constants of state-of-the-art integrated oscillators liebetween 10-19 s and 10-21 s [9]. For ultra-stable temperature compensated or oven controlled crystaloscillators (TCXOs, OCXOs) phase noise constants from 10-22 s down to 10-25 s can be reachedwith today’s technologies (e.g. see [10, 11, 12, 13])1.
4 Aperture jitter versus clock jitter effects
By means of the jitter models presented above the error effects caused by aperture jitter and by clockjitter shall be compared. The overall SNR as well as the spectral distribution of the mean error powerwill be considered.
Comparison of the overall SNR
Substituting the characteristic jitter functions given by(15) and (19) into (12) yields the followingSNR formulas
SNRap = 10 log
æçççççççççççè
Ù ¥-¥ Sss( f )â f
2¥Ù-¥
Sss( f ) I1- ã-2Π2 f 2Σ2apMâ f
ö÷÷÷÷÷÷÷÷÷÷÷ø
dB (21)
SNRacc = 10 log
æçççççççççççè
Ù ¥-¥ Sss( f )â f
2N
N-1Ún=0
¥Ù-¥
Sss( f ) I1- ã-2Π2 f 2cnT Mâ f
ö÷÷÷÷÷÷÷÷÷÷÷ø
dB (22)
As can be seen, in the case of aperture jitter the signal-to-noise ratio is independent of the numberof sampling pointsN (and therefore of the sampling block lengthNT ), while in the case of clockjitter the SNR strongly depends on it. This fact is illustrated by the curves in Fig. 1, which show thecorresponding SNRs calculated for different sampling block lengths assuming a 50 MHz sine waveas an input signal, a sampling frequency of 100 MHz, and the given jitter parameters.
In Fig. 2 the SNR formulas are evaluated for a fixed sampling block length of 1 ms. The samplingfrequency and the jitter parameters are the same as in Fig. 1.The solid curves represent the results for
1In data sheets oscillators are usually characterized by parameters like the cycle jitter variance, the rms cycle jitter, orby measurement data of the phase noise spectrum. Assuming the model of a free running oscillator an equivalent phasenoise constantc can be calculated from these parameters [7]. The main advantage of using the phase noise constant tocharacterize the quality of oscillators is thatc on the contrary to all the other parameters does not depend onthe oscillatorfrequency. Hence, the quality of oscillators designed for different oscillator frequencies can be directly compared bymeans of their phase noise constants.
10
10
0
−8
10
10
2
−6
10
10
4
−4
10
10
6
−2
10
10
8
0
10
10
10
2
10
10
12
4
10
10
14
6
GSM time slot UMTS frame
slope: −10 dB/decade
−3 dB
0
20
40
60
80
100
120
aperture jitter
clock jitter
0.01 ps100 MHz
0.25 ps
50 MHz
rms cycle jitter (clock jitter)sampling frequency
rms aperture jitter
input frequency
block length in seconds
number of sampling points
SN
R in
dB
Figure 1: Comparison of the mean SNR caused by aperture jitter and clock jitter while sampling a50 MHz sine wave using different sampling block lengths
a sinusoidal input signal with the specified frequencies. The dashed curves are calculated for a band-limited real-valued white noise input with different cut-off frequencies2. One can see that the SNRtrends are, in principle, the same for both kinds of jitter aswell as for both kinds of input signals. TheSNR decreases with about 6 dB per doubling the input frequency or bandwidth (which is equivalentto a loss of 1bit of ADC resolution) and converges to-3 dB if the highest input frequency exceedsthe inverse rms (root mean square) jitter valuesΣ-1
ap and 1/0
cNT . The following can be concluded:
1. The SNR is mainly determined by the highest frequency components of the input signal andsecondarily by its bandwidth.
2. Although the absolute SNR values strongly depend on the jitter parameters of the ADC and theclock generator in connection with the specific sampling block length, the effect of aperturejitter and clock jitter on the overall SNR is, in principle, the same.
2In the case of the band-limited white noise input the values given at the horizontal axis in Fig. 2 are equivalentto the positive cut-off frequencyfc of the noise PSD. The PSD of the input noise is constant for allfrequencies in theinterval[- fc, fc] and zero for all other frequencies. Hence, the noise bandwidth is 2fc.
105
106
107
108
109
1010
1011
1012
1013
−3 dB
slope: −6 dB/octave
1014
0
20
−20
40clock jitter
4.77 dB
aperture jitter
60
80
0.01 ps
1 ms
100 MHz
0.25 ps
100000
rms cycle jitter (clock jitter)
block length
sampling frequency
rms aperture jitter
number of sampling points
bandlimited white noisesine as input signal
100
highest input frequency in Hz
SN
R in
dB
120
140
Figure 2: Comparison of the SNR caused by aperture jitter andclock jitter for a sinusoidal inputsignal and a band-limited real white noise input with different signal/cut-off frequencies
Comparison of the error power spectra
In order to obtain expressions for the error power spectra caused by the different jitter processes,the corresponding error auto-correlation functionsseeap andseeacc will be derived and transformed bymeans of the DTFT.
Using the definitionk = n - m and substituting (15) and (16) into (13) yields the error auto-correlation function for the case ofaperture jitter
seeap(kT ) =¥â
i=-¥
|ci|2 ãü2Π fikT × g2
i +¥â
i=-¥
|ci|2 × [2gi - g2
i ] × ∆Kr(k) (23)
Heregi = I1- ã-2Π2 f 2
i Σ2apM (24)
denotes a gain coefficient specific for each spectral component of the input signal and
∆Kr(k) =ìïíïî
1 if k = 0
0 else(25)
stands for the Kronecker impulse. As one can see from (23) theerror auto-correlation func-tion seeap(kT ) is composed of two terms, the first one accounting for a periodic part and the secondone accounting for an aperiodic part. By transformingseeap(kT ) to the frequency domain one getsthe corresponding error power spectrum
SeeapIeü2Π f T M = 1T
¥âl=-¥
¥âi=-¥
|ci|2 ∆I f - fi -
lTM × g2
i +¥â
i=-¥
|ci|2 × [2gi - g2
i ] (26)
The transformation of the periodic part ofseeap(kT ) yields a periodic line spectrum comprising thespectral components of the input PSD weighted with the squares of the spectral gainsgi. The trans-formation of the aperiodic error auto-correlation term yields a constant component in the error power
spectrum expressed by the second term in (26). In the majority of applications the maximum inputsignal frequencyfimax is very low compared to the inverse of the rms jitter valueΣap. Hence, thegainsgi get very small so that the error power spectrum can be approximated by
SeeapIeü2Π f T M » ¥âi=-¥
2|ci|2 × gi = const (27)
Eq. (27) proves the common assumption that the mean error power caused by aperture jitter (inmost cases) is equally distributed over the whole digitization band(-1/ [2T ] < f < 1/ [2T ]), whichmotivates the approach to increase the jitter dependent SNRin a given frequency band by means ofoversampling and filtering.
RecentlyDa Dalt in [14] derived an expression for the error power spectrum caused by stationarysampling jitter in an ADC. The derivation is based on a linearapproximation of the sampling processand considers both white as well as colored jitter processes. Due to the linearizationDa Dalt’sformula only predicts the constant part of the error power spectrum in the case of white aperturejitter. It does not predict the first term in (26), i.e. the periodic line spectrum, which characterizes thecomponents of the error signal that are correlated with the input signal. This part of the error powerspectrum has to be considered if e.g. a (narrowband) high frequency component signal is sampledwith a low quality sample-&-hold circuit (i.e. with strong aperture jitter) and if it is intended toreduce the resulting error power by means of oversampling and filtering. Since only the error powercontained in the white part of the error power spectrum can bereduced by this method, but not theerror power concentrated in the line spectrum, the latter determines the upper limit for the achievableSNR. Hence, oversampling can increase the signal-to-noiseratio to a certain extent only.
For the case ofclock jitter the error auto-correlation function can be derived by substituting (19)and (20) into (13). This yields
seeacc(kT , nT ) =¥â
i=-¥
|ci|2 ãü2Π fikT A1+ ã-2Π2 f 2
i c|k|T - ã-2Π2 f 2i cnT - ã-2Π2 f 2
i c(n-k)T E (28)
In contrast to the aperture jitter caseseeacc does not only depend on the sampling time differ-encekT = (n - m)T but also on the absolute sampling time instants. Only forn ® ¥, when the lasttwo terms in (28) vanish,seeacc becomes independent ofn andm and can be discrete-time Fouriertransformed in the common way
SeeaccIeü2Π f T , nT MÄÄÄÄÄn®¥ = limn®¥
DTFTk 9seeacc(kT , nT )= (29)
= DTFTk ;limn®¥
seeacc(kT , nT )? (30)
= DTFTìïíïî
¥âi=-¥
|ci|2 ãü2Π fikT A1+ ã-2Π2 f 2
i c|k|T Eüïýïþ
(31)
=1T
¥âl=-¥
∆I f - lTM * B ¥â
i=-¥
|ci|2 ∆( f - fi) + |ci|
2 f 2i c
Π2 f 4i c2 + ( f - fi)2
«¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬®
Lorentzian spectrum
F (32)
Here the indexk at the DTFT-operator indicates that the DTFT is calculated over the time differ-encekT and not over the absolute sampling timenT . The symbol* denotes the convolution operator.As can be seen from (32) forn ® ¥ the error power spectrum consists of the spectral components
of the input PSD, each overlayed by a Lorentzian spectrum. The convolution with the Dirac impulsetrainÚ¥l=-¥ ∆I f - l
T M describes the spectral repetition of the baseband spectrumat integer multiplesof 1/T which is the result of the sampling operation.
In practical applications the sampling block length is limited(n < ¥). In order to model the errorpower spectrum in this case one can use the following approach. For each sampling time instantnT a short-time error power spectrum is calculated. The observation period is determined by thesampling block lengthNT , which should not be to small in order to reduce windowing effects. Theresulting spectra can be interpreted as a time-varying power spectrumSeeacc Ieü2Π f T , nT M in the senseof a discrete-time Rihaczek spectrum, which is a special kind of a time-frequency representation fornon-stationary signals and processes [15, 16, 17]. Hence, the time average
Seeacc Ieü2Π f T M = 1N
N-1ân=0
Seeacc Ieü2Π f T , nT M (33)
is supposed to be a meaningful measure of the spectral distribution of the mean error power. For suf-ficient long observation intervalsNT the time-averaged error power spectrum can be approximatedby
SeeaccIeü2Π f T M » 1T
¥âl=-¥
∆I f - lTM * C ¥â
i=-¥
|ci|2 ∆( f - fi) × gi +
¥âi=-¥
|ci|2 f 2
i c
Π2 f 4i c2 + ( f - fi)2
×Hw( f - fi, N, T )G(34)
with the spectral gain coefficients
gi =1N
N-1ân=0
I1- ã-2Π2 f 2i cnT M (35)
which are specific for each spectral component of the input signal, and with the spectral weightingfunction
Hw(D f , N, T ) = 1-sin(ΠD f NT )N sin(ΠD f T )
ã-üΠD f (N-1)T (36)
The spectral weighting withHw(D f , N, T ) is a result of the (time-varying) windowing in the timedomain caused by the finite observation interval. As can be seen from (36),Hw(D f , N, T ) con-tains an oscillating complex-valued term. Consequently, the time-averaged error power spec-trum SeeaccIeü2Π f T M will contain oscillating complex-valued components as well, which cannot beinterpreted as proper components of a common error power spectrum.
Such components are known as statistical cross-terms. Theyare a typical feature of time-frequency representations of non-stationary signals and processes and indicate time-frequency cross-correlations[15, 17]. The cross-terms tend to mask the so-called auto terms which build the real(time-varying) power spectrum and characterize the mean distribution of the signal or processpower [18]. Time-averaging reduces the statistical cross term. They decrease as the averagingintervalNT increases and vanish if the averaging interval becomes infinite.
This is also true for the presented case. For large averagingintervalsNT the oscillating termin (36) converges to a periodic Kronecker impulse train withthe period 1/T . Thus, the weightingfunctionHw(D f , N, T ) is zero for integer multiples of 1/T and approximately one for all other valuesof D f . Substituting this approximation to (34), it can be stated that the error power spectrum causedby accumulated clock jitter consists of two significantly different parts. (Similarly to the aperturejitter case) the first part comprises the lines of the input PSD weighted with the spectral gains ˜gi. The
second part is composed by Lorentzian shaped spectra centered around the frequenciesfi of the inputsignal components. This means that the corresponding errorpower is strongly concentrated aroundthe frequency components of the input signal. Moreover, as can be seen from (34), the error powerconcentrated in a certain line component at the frequencyfi is determined by the power|ci|
2 of thecorresponding input signal component multiplied with the spectral gain ˜gi. Since the spectral gains ˜gi
significantly increase as the frequenciesfi increase (see (35)), higher frequency signal componentsare more disturbed than lower ones. This is a significant difference to the aperture jitter effect wherethe greatest part of the error power is equally distributed over the whole digitization band.
It can be concluded that the error noise caused by clock jitter is highly correlated. Consequently,the clock jitter dependent SNR cannot be increased by oversampling techniques.
5 Simulation results
In order to confirm the above analysis, some simulation results are presented.Fig. 3 shows the error power spectrum simulated for an input signal that comprises two equal-
power sine waves with the frequencies of 10 MHz and 100 MHz. The overall input signal powerwas normalized to one, i.e. the power of each of the two sine wave components is 0.5. The signal wassampled with a sampling frequency of 400 MHz and white Gaussian aperture jitter with a standarddeviation (rms aperture jitter)Σap = 0.25 ps. As predicted by the analytical results forfimax � Σ
-1ap ,
the error power spectrum is white.
−300
−250
−200
−150
−100
−50
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x 108
Frequency in Hz
Pow
er S
pect
ral D
ensi
ty in
dB
SNR: 79.0 dB
10 MHz, 100 MHz
0.25 pssampling frequencyrms aperture jitter
signal frequencies400 MHz
Figure 3: Mean error PSD caused byaperture jitter while sampling a mix of two equal power sinewaves whose frequencies arevery low compared to the inverse rms jitter value
The same input signal was used to obtain the error power spectrum in Fig. 4. However, it wassampled with accumulated Gaussian clock jitter. The sketched error PSD is the average of 100independent Monte Carlo simulations. The chosen phase noise constant and the sampling blocklength ensure that the conditionfimax � 1/
0cNT holds. As expected, the error power spectrum
shows narrow peaks at±10 MHz and±100 MHz surrounded by Lorentzian shaped spectra. The fact,that the peak PSD values differ, confirms the statement that higher frequency signal components aresignificantly more disturbed than lower frequency components.
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x 108
−200
−180
−160
−140
−120
−100
−80
−60
−40
−20
Frequency in Hz
Pow
er S
pect
ral D
ensi
ty in
dB
SNR: 69.5 dB
block length 0.1 msnumber of sampling points 40000
400 MHz1e−20 s
rms cycle jittersampling frequencyphase noise constant
0.005 ps
Figure 4: Mean error PSD caused byclock jitter while sampling a mix of two equal-power sinewaves whose frequencies arevery low compared to the inverse rms jitter value 1/
0cNT
Figs. 5 and 6 show the error power spectra generated by aperture jitter and clock jitter, respectively,for a sine wave input signal whose frequency equalsΣ-1
ap or exceeds 1/0
cNT , respectively. Thesignal amplitude was normalized to one. In the aperture jitter case the error PSD comprises thetypical white noise floor and in addition the two spectral lines of the input PSD. This corresponds tothe theoretically derived expression in (26). The error PSDin Fig. 6 generated by clock jitter for alarge number of sampling points is similar to that determined by (32). It shows the two lines of theinput PSD surrounded by widely spread Lorentzian spectra. Due to the very high signal frequencies(which results from the assumption of state-of-art jitter parameters) the simulation results presentedin Figs. 5 and 6 are of less practical but of great theoreticalinterest. They impressively confirm thecorrectness of the analytically derived expressions for the error power spectra caused by the differentjitter processes.
Finally, it should be noted that also all simulated SNR values are in accordance with the analyti-cally predicted results.
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0
x 1012
Frequency in Hz
Pow
er S
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ensi
ty in
dB
SNR: −2.9 dB
8−8 −6 −4 −2 0 2 4 6
4000 GHz
0.25 pssampling frequencyrms aperture jitter
signal frequency16000 GHz
Figure 5: Mean error PSD caused byaperture jitter while sampling a sine wave whose frequencyequals the inverse rms jitter value
−100
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Frequency in Hz
Pow
er S
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ty in
dB
SNR: −3.01 dB
−8 −6 −4 −2 0 2 4 6 8
block length 0.1 msnumber of sampling points 1600000
16 GHzphase noise constant 1e−11 s
rms cycle jittersampling frequency
25 ps
Figure 6: Mean error PSD caused byclock jitter while sampling a sine waves whose frequencyisgreater than the inverse rms jitter value 1/
0cNT
6 Conclusions
The similarities and differences of the error effects caused by aperture and clock jitter in widebandADCs have been presented. As an extension to previous publications not only the overall SNR wasconsidered but also the spectral distribution of the generated error power. By means of analyticallyderived expressions (confirmed by simulations) it has been shown that the SNR limiting effect ofaperture jitter and clock jitter is, in principle, the same,but the resulting error power spectra aresignificantly different. In the case of aperture jitter the mean error power is uniformly distributedover the whole digitization band, so that the jitter dependent SNR in a given frequency band can
be increased by oversampling techniques. In the case of clock jitter the error power is concentratedaround the frequencies of the input signal components. Thus, oversampling does not help to increasethe SNR.
Acknowledgements
This work is a contribution to the project RMS (Reconfigurable Mobile Communication Systems)funded by the German Ministry for Education and Research (BMBF) and by the Alcatel SEL AG,Stuttgart, Germany.
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Michael Lohning ([email protected]) received his MSc/Dipl.-Ing. degree in electrical engineering from the Dresden University of Tech-nology, Germany, in 1999. Since April 1999 he has been with the VodafoneChair Mobile Communications Systems at the Dresden University of Tech-nology, Germany, working towards his PhD. He has been involved in differentresearch projects which intend to develop software radio based front-end ar-chitectures for reconfigurable mobile communications systems. His currentresearch interests include software radio, specifically the design and inves-tigation of digital signal processing algorithms for reconfigurable front-endsand the analysis of the performance limiting effects in wideband analog-to-digital interfaces.
Gerhard Fettweis ([email protected]) received his MSc/Dipl.-Ing. and PhD. degree in electrical engineering from the Aachen University ofTechnology (RWTH), Germany, in 1986 and 1990, respectively. From 1990to 1991 he was a Visiting Scientist at the IBM Almaden Research Center inSan Jose, CA, working on signal processing for disk drives. From 1991 to1994 he was Scientist with TCSI, Berkeley, CA, responsible for signal pro-cessor developments for mobile phones. Since September 1994 he holds theVodafone Chair Mobile Communications Systems at the Dresden Universityof Technology, Germany. He is an elected member of the SSC Society’s Ad-ministrative Committee since 1999, and of IEEE ComSoc Boardof governorsfrom 1998 to 2000, respectively. He has been associate editor for IEEE Trans-actions on Circuits and Systems II, and associate editor forIEEE Journal onSelected Areas in Communications wireless series. Over theyears he has or-ganized and been on the program committee of numerous IEEE workshopsand conferences.
