jitter model is studied and simulated in high-frequency ... There are several ways to model the jitter ... that the same amount of jitter in time sense has much.
EFFICIENT JITTER MODEL FOR HIGH-FREQUENCY BANDPASS SAMPLING Ville Syrjälä, Mikko Valkama, Markku Renfors Tampere University of Technology Institute of Communications Engineering P.O Box 553, FIN-33101 Tampere, Finland ABSTRACT
2. I/Q-CARRIER JITTER MODEL
In this paper, the applicability of a simple I/Q-carrier jitter model is studied and simulated in high-frequency bandpass sampling context. In the model, sampling jitter is applied only to the carrier components of an arbitrary I/Q modulated signal. The model is found accurate especially when considering the spectral characteristics of the output. Still, some minor systematic imprecision can be seen when viewing constellations of, for example, linearly modulated signal. The overall accuracy of the model is anyway impressing considering its implementation simplicity. Due to its simplicity, the implementation of the model in a simulator is a very straight-forward process.
Usually, the sampling jitter is modeled by using basic ways to model phase noise. In effect, jitter can be seen as type of phase noise in a signal. If we have a sinusoidal signal s (t ) = A0 cos(2π f 0 t ) , the jitter changes the signal to form:
1. INTRODUCTION
slp (t ) = sI (t ) + j sQ (t )
As the need for high-speed sampling of communications signals becomes more and more interesting, the sampling process stands out as a performance-limiting factor. Jitter noise stemming from unintentional inaccuracies in the sample instants is one of the main noise sources in a sampling process of high-speed sampling communications system [1], [6]. Thus in simulations, the jitter phenomenon must be accurately and efficiently modelled. There are several ways to model the jitter phenomenon. This paper introduces one of those. Naturally, the first modeling approach that comes into mind is to model the jitter with the knowledge of the mathematical properties of the signals at hand. Such approaches have been used in studies, for example, in [2] and [5]. Naturally, our model is also based on this approach but it takes a more general look into the problem from arbitrary communications band-pass waveforms point of view. Some simplifications or possibilities for performance enhancements can be gained by deeper understanding.
Here j is the imaginary unit, and sI (t ) and sQ (t ) are two real signals composing the complex signal. Due to the fact that communications takes place out of the baseband frequencies, the communications signals are usually presented in band-pass form that leads to the presentation of signals in following form:
s j (t ) = A0 cos(2π f 0 (t + j (t ))) = A0 cos(2π f 0 t + 2π f 0 j (t )) = A0 cos(2π f 0 t + Φ (t )) in which Φ (t ) = 2π f 0 j (t ) is the phase shift, and j (t ) is the jitter term. So basically jitter is frequency dependent phase shift. The frequency dependency of the jitter gives us a way to approximate the behaviour of the jitter. An arbitrary complex communications basebandequivalent signal slp (t ) can be presented as follows:
s (t ) = sI (t ) cos(2π f c t ) − sQ (t ) sin(2π f c t )
This presentation has the sine and cosine terms for the frequency shift. This is in general called I/Q modulation, in which f c denotes the carrier frequency [4]. The 90-degree delayed signal (−sine) presents the imaginary component of the complex signal, and both real and imaginary parts of the signal can thus be reconstructed on the receiver side. In the ideal model, the jitter affects both the lowfrequency and the carrier components of the signal. The signal in ideal jitter model reads as follows:
s (t + j (t )) = sI (t + j (t )) cos(ωc (t + j (t ))) − sQ (t + j (t )) sin(ωc (t + j (t )))
In general, as the signal frequencies increase, the jitter causes greater and greater voltage error [3]. This means that the same amount of jitter in time sense has much heavier effect on high-frequency signals. This is also consistent with the jitter equations derived by Shinagawa et al. in [6]. Now if the jitter values j (t ) are small compared to the bandwidth of the signal and thus to the rate of change of sI (t ) and sQ (t ) , we can model the jitter by adding it only to the carrier components of the I/Q-signal. The signal model with jitter can then be written as follows: sIQC (t ) = sI (t ) cos(2π f c (t + j (t ))) − sQ (t ) sin(2π f c (t + j (t )))
This approach should work well under the previously mentioned assumptions. This is because the jitter noise due to jitter in the carrier components is heavily dominating compared to the jitter power generated by adding the jitter to the message signal. This approach to jitter modeling is very simple and illustrative. We were not able to find any previous publications in which this method was used though. We call the model “I/Q-carrier (IQC) jitter model”. The model is usable in simulator in which jitter must be applied to the system. In signal creation phase, applying ideal jitter to the system can be very heavy operation. With IQC model, the jitter can be applied only to the carrier components, and thus the performance of the simulations can be greatly increased. Naturally, with deterministic signals, using the ideal model is also possible but it is by no means easy or efficient as very complex deterministic formulae for the signal must be used. 3. BASIS FOR SIMULATIONS OF I/Q-CARRIER JITTER MODEL
Next we illustrate the applicability and accuracy of the previous simple jitter model. The simulation cases are 16QAM modulated signal with 30 MHz bandwidth and band-pass noise signal with 150 MHz bandwidth, both at navigation E2-L1-E1 band centered at 1575.42 MHz. In 16QAM case, the model performances are studied by observing the constellations of the signals. For symbols, constellation diagrams, mean-square-errors (MSE) and histograms are studied. Root-mean-square (RMS) sampling-jitter values of 15 ps and 30 ps are used. Even the lower, 15 ps, sampling jitter value is quite high, but since the
differences in models is what we are interested in, this high lower value is justified. 30 ps sampling jitter value represents a very high jitter, whose contribution to the signal is also very interesting to see. In certain spread spectrum systems, like navigation systems, the sampling of band-pass noise type signals is also of interest. In band-pass noise case, the observations are naturally restricted to the frequency-domain spectral characteristics. At the beginning of the process, the system creates a complex baseband waveform. After the complex signal is created, the system implements the I/Q modulation. This signal is the signal we are interested in. For practical reasons, internal sampling rates of 4 GHz and 4.2 GHz are used for 16QAM and bandpass noise cases respectively. In 16QAM case the jitter models are implemented by altering the complex baseband waveform creation phase and adding the jitter to the carrier components of the I/Q modulator. Constellation diagrams are obtained by using a receiver model. First, bandpass sampling by decimation is used. After this, there is still need for a small frequency down-conversion to get the signal to the baseband. After the down-conversion, the signal is filtered so that the resulting signal has only in-band jitter noise left in it. After this, we have a baseband signal that can be displayed as a constellation. In bandpass noise case, the jitter models are implemented by using oversampling, and by altering the timing of the I/Q modulator. The signal is then transferred near the baseband using bandpass sampling by decimation, and the figures for the frequency domain spectra can thus be drawn. Both cases are explained in detail later on. 4. SIMULATIONS AND DISCUSSION
Simulations are run in few phases. First, constellations of 16QAM signal with 15 ps and 30 ps sampling jitter are studied. Finally, band-pass noise signal cases are studied with both jitter values. 4.1. 16QAM Signal Studies with Constellations
In 16QAM case, the jitter models are used accurately as already presented. The ideal jitter model is based directly on the pulse shaped symbols. The 16QAM symbols are pulse shaped with raised cosine filter as usual, but the values of the raised cosine are calculated in the sampling moments affected by jitter. This gives us an ideally jittered baseband signal. The baseband signal is then I/Q modulated with the same jitter added to the carrier components of the I/Q modulator. In IQC
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Figure 1: Symbol constellation of 16QAM signal (left: ideal jitter; right: IQC jitter) (15 ps jitter, 250 MHz subsampling rate, 30 MHz BW, at E2-L1-E1.)
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Figure 2: Symbol constellation of 16QAM signal (left: ideal jitter; right: IQC jitter) (30 ps jitter, 250 MHz subsampling rate, 30 MHz BW, at E2-L1-E1.)
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Figure 3: Histograms of 16QAM constellations (upper = phase, lower = absolute value) with 15 ps jitter.
model, the baseband 16QAM signal is created by pulse shaping the symbols with ideal raised cosine without the jitter. The I/Q modulation is done in same way as in ideal jitter model. In Figure 1 and Figure 2, the constellation diagrams of 16QAM signal with ideal and IQC jitter model are compared in 15 ps and 30 ps sampling jitter cases respectively. The diagrams give only rough information about the symbol distribution but it can be seen that the
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Figure 4: Histograms of 16QAM constellations (upper = phase, lower = absolute value) with 30 ps jitter.
constellations of the IQC jitter model and ideal model are very much alike with both RMS jitter values. The ideal model generated slightly more randomness around the symbol values but the change is hardly visible. When the same information is compared with histograms of the absolute value and of the phase, a clearer conclusion can be made. This is illustrated next. In Figure 3 and Figure 4, histograms of the absolute value and the phase of the signal are presented in 15 ps
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Figure 5: Band-pass noise signals with jitter (left pair) and the resulting error signals (right pair). Signal bandwidth 150 MHz, center frequency 1575.42 MHz, bandpass sampling with 420 MHz sampling rate, 15 ps sampling jitter. Ideal jitter model (left subplot), IQC jitter model (right subplot).
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Figure 6: Band-pass noise signals with jitter (left pair) and the resulting error signals (right pair). Signal bandwidth 150 MHz, center frequency 1575.42 MHz, bandpass sampling with 420 MHz sampling rate, 30 ps sampling jitter. Ideal jitter model (left subplot), IQC jitter model (right subplot).
and 30 ps sampling jitter cases respectively. The differences between the models are clearly very small when observing the phase. There are naturally some minor random differences but clearly no deterministic or systematic behaviour is visible. By studying the histogram of the absolute value, some minor differences are visible though. The same kind of effect that was hardly visible in the constellation diagram can also be seen here little easier. It can be seen that some samples from the lower amplitude values move randomly to a little higher amplitude values if we compare IQC model to the ideal jitter model. Anyway, with this visible but very small difference, accuracy of IQC jitter model is still excellent. To further verify these results, MSE values are calculated for the symbol constellations. MSE values are obtained by comparing jitter modeled signals to the ideal non-jittered signals one by one. For both, 15 ps and 30 ps, sampling jitter values, the MSE values of ideal jitter model are less than 3 percent higher than those of the IQC jitter model. The small differences that
were visible in histogram of absolute value contribute to this difference in the MSEs. 4.2. Band-Pass Noise Signal Studies with Frequency Domain Spectra
In band-pass noise case, first a low-pass noise signal is generated by filtering a white complex Gaussian noise signal to proper bandwidth. Then I/Q mixing with highfrequency carrier is used to get the signal to high frequencies as a band-pass noise signal. The “ideal” case is got by using oversampling of 81 on the created band-pass signal. Then, the corresponding jittered samples are acquired by selecting the nearest corresponding samples. It is noticeable that this is by no means exactly “ideal” jitter but it is anyway a good estimate and accurate enough for visual observations. In IQC jitter model, jitter is applied only to the highfrequency carrier components before mixing. Frequency domain spectra of band-pass noise signal can be seen in Figure 5 and Figure 6 for 15 ps
and 30 ps sampling jitter respectively. For both cases, the differences are practically invisible. The spectra are drawn using logarithmic amplitude scale. When observing frequency spectra of band-pass noise signal, the output of the IQC jitter model cannot be distinguished from that of the ideal jitter model. 5. CONCLUSIONS
Performance of IQC jitter model was studied in a few realistic example cases. 15 ps and 30 ps sampling jitter values were under study, as were 16QAM constellations and band-pass noise spectra. Simulations show that accuracy of IQC jitter model is very high. Despite of the relatively high RMS sampling jitter values used, the accuracy of the IQC model was excellent and only slight differences were visible in the study of 16QAM constellations. In frequency spectra, no differences were distinguishable. It is not very trivial to apply jitter to a complex modulated signal in general. With IQC model, the jitter can simply be added to the carrier of the signal and thus the performance of the process can be greatly improved. The implementation of the jitter is thus a very simple process. 6. REFERENCES [1]
Amin B., and Dempster A. G. “Sampling and Jitter Considerationg for GNSS Software Radio Receivers,” in IGNSS Symposium 2006. 17 – 21 July 2006.
[2]
Brannon B., “Aperture Uncertainty and ADC System Performance,” Application Note AN-501, Analog Devices.
[3]
Kobayashi H., Morimura M., Kobayashi K., and Onaya Y., “Aperture Jitter Effects in Wideband Sampling Systems,” Instrumentation and Measurement Technology Conference, 1999. IMTC/99. Proceedings of the 16th IEEE, Vol. 2.
[4]
Potter C., “Digital Modulation,” The IEE Measurement, Sensors, Instrumentation and NDT Professional Network, Cambridge RF Ltd.
[5]
Shimanouchi M., “An Approach to Consistent Jitter Modelling for Various Jitter Aspects and Measurement Methods,” in ITC International Test Conference. IEEE 2001.
[6]
Shinagawa M, Akazawa Y., and Wakimoto T., “Jitter Analysis of High-Speed Sampling Systems,” IEEE Journal of Solid-State Circuits, Vol. 25, No. 1, February 1990.
