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<title>Background and Analysis</title>
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<h3>Background and Analysis</h3>
<h4>Table of Contents</h4>
<ul>
<li class="inline"><a href="#intro">Introduction</a>
<li class="inline"><a href="#allan">Allan Deviation</a>
<li class="inline"><a href="#phase">Phase and Frequency Offset Characteristics</a>
<li class="inline"><a href="#avg">The Effects of Averaging Interval</a>
<li class="inline"><a href="#ref">References</a>
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<h4 id="#intro">Introduction</h4>
<p>In order to understand how the new kernel algorithms operate to improve the accuracy of the system clock, it is necessary to examine in detail the behavior of an undisciplined clock oscillator. The accuracy attainable with NTP, or any other protocol that provides periodic offset measurements, depends strongly on the stability of the oscillator and its adjustment mechanism. Typical oscillators utilize a quartz or surface acoustic wave (SAW) resonator without specific temperature, mechanical or drive stabilization. The temperature dependency of these oscillators is typically in the order of a part-per-million (PPM) in frequency per degree Celsius temperature change.</p>
<p>In order to reduce the residual time errors to the order of nanoseconds, it is necessary to use a precision source, such as a cesium oscillator or GPS timing receiver, and a PPS interface. For the results described in this presentation, a cesium oscillator was used as the precision source and the PPS interface used a pin of either a serial or parallel port interface. The experiment plan involved the collection of sample offsets measured at the interface with the clock discipline disabled and the system clock allowed to free-run. These data were then saved in a file and used by the <tt>kern</tt> simulator described on the <a href="util.htm">Utility Programs</a> page to generate the plots.</p>
<h4 id="#allan">Allan Deviation</h4>
<p>As shown in [1], it is possible to characterize the behavior of a typical quartz oscillator in terms of its Allan deviation as a function of averaging interval. A typical plot can be approximated by two intersecting straight lines in log-log scales. In general, white phase noise (jitter), which is represented by a straight line with slope -1 on the plot, dominates at the smaller intervals, while random-walk frequency noise (wander), which is represented by a straight line with slope +0.5, dominates at the larger intervals. The intersection of the two lines is called the Allan intercept, which serves to characterize the particular synchronization source and clock oscillator.</p>
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<td align="center"><img src="pic/allan.gif" alt="gif"></td>
<td align="center"><img src="pic/phk.gif" alt="gif"></td>
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<td align="center">Figure 1. Allan Deviation for Nanosecond and Microsecond Kernels</td>
<td align="center">Figure 2. Allan Deviation for Normal and SMI-Enabled Kernel</td>
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</table>
<p>Figure 1 shows the Allan deviation characteristics for four experiments distinguished with reference to the leftmost end of each trace. The uppermost trace (1) represents a typical computer oscillator and microsecond kernel, as described previously. The next lower trace (2) represents another computer oscillator and the nanosecond kernel, as described in this presentation. In both of these cases a cesium oscillator and PPS interface were used as the source and the computer oscillator allowed to free-run over periods ranging from 1.5 days to 10 days. The next lower trace (3) represents a synthetic model with phase noise limited only by the resolution of the microsecond kernel clock. The bottom trace (4) represents a synthetic model with phase noise limited only by the resolution of the nanosecond kernel clock.</p>
<p>Figure 1 clearly shows the Allan deviation for each case and how it is affected by the intrinsic jitter and wander. At the smaller averaging intervals with the cesium oscillator and PPS interface, the characteristic is dominated by sample jitter and interrupt latencies. Typical sample jitter contributions include noise pickup on the interface cable, incorrect cable impedance terminations and the reading resolution of the system clock itself. However, case (2) achieves lower jitter than case (1), primarily because the reading resolution of the nanosecond kernel is much lower than the microsecond kernel. While cases (1) and (2) involve real signals and kernels, the remaining cases involve synthetic data representing the best that can be done with either the microsecond kernel (3) or nanosecond kernel (4). For these cases the wander parameter was set to coincide with case (2). Obviously, there is room for improvement; however, further refinement is beyond the scope of this presentation.</p>
<p>At the larger averaging intervals, the characteristic is dominated by wander due to the individual oscillator design, crystal drive level, temperature variation and power supply regulation. Obviously, case (1) achieves lower wander than case (2) by a significant factor. This was due to the fact that, in case (1) the ambient room temperature was held to a narrow range less than one degree Celsius, while in case (2) the temperature varied over a much wider range on a hot Summer day in Denmark.</p>
<p>The main motivation for constructing the Allan deviation plot is to determine the Allan intercept for each combination of signal source and clock oscillator. The intercept represents the optimum averaging interval for the best clock accuracy and stability. If the averaging interval is less than the intercept, errors due to jitter dominate, while if greater than the intercept, errors due to wander dominate. The curves shown in Table 1 show intercepts that vary from 2 s for case (4) to 2000 s for case (1). For the best performance, it is necessary to tune the averaging intervals to match the intercept. Notwithstanding these observations, it is probably better to err on the high side of the intercept, since the slope of the wander characteristic is half that of the jitter characteristic. For the remainder of this presentation and unless otherwise noted, a compromise averaging interval of 128 s will be assumed.</p>
<table width="60%" cols="5">
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<td align="center">Case</td>
<td align="center">Kernel</td>
<td align="center">Time Interval</td>
<td align="center">Deviation</td>
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<td align="center">1</td>
<td align="center">microsecond</td>
<td align="center">2000 s</td>
<td align="center">10<sup>-2</sup> s</td>
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<tr>
<td align="center">2</td>
<td align="center">nanosecond</td>
<td align="center">50</td>
<td align="center">3x10<sup>-2</sup></td>
</tr>
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<td align="center">3</td>
<td align="center">synthetic microsecond</td>
<td align="center">200</td>
<td align="center">5x10<sup>-3</sup></td>
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<td align="center">4</td>
<td align="center">synthetic nanosecond</td>
<td align="center">2</td>
<td align="center">5x10<sup>-4</sup></td>
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</table>
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