Modulation Domain Analysis

A New Method for Measuring Jitter on Digital Devices

by Tom Carrico
Hewlett-Packard Company Santa Clara Division
[Ghost written by Mark Haas]

As the demand for faster and more efficient electronic hardware forces digital designers to push their circuits to the limit, small timing irregularities become magnified and can limit overall system performance. As a reulst of these higher clock and data rates, timing errors must be kept to an absolute minimum. Tracking down the source of these errors is the first step towards realizing an optimal digital system design.

These timing errors are known by many names: jitter, wander, unintended modulation, phase noise, and just plain fuzz on an oscilloscope. Unitl now, measuring jitter on clock or data signals has required costly, specialized test equipment that does not always provide a complete picture of the source of the jitter. While oscilloscopes and spectrum analyzers can reveal the overall magnitude of the jitter, they do not reveal the jitter's frequency components.

A new measurement approach introduced by Hewlett-Packard -- modulation domain analysis -- is the only technique available today that can directly display jitter. Using modulation domain analysis, designers of electronic and electro-mechanical products can more quickly and easily determine sources of jitter and phase noise at a lower cost than other currently available methods.

Traditional Measurement Techniques

The first piece of equipment a designer usually reaches for when a jitter problem is suspected is the oscilloscope. Scopes are an excellent tool for revealing the magnitude of the jitter. Typically, the clock signal is connected to the scope, a delay is set from the trigger, infinite persistence is enabled and the trace on the display gets fuzzy. After a while, markers can be placed to measure the width of the fuzz to get an idea of the total peak-to-peak jitter on the signal.

Figure 1 shows a scope measuring the jitter of a 155.52-MHz signal. Some high-performance scopes make the jitter measurement less subjective than the placement of markers. These scopes will accumulate the time of occurrence of a voltage level and place these times in a histogram format, showing how often each time interval occurs. From the histogram, the standard deviation can be calculated, indicating the RMS jitter on the signal. Figure lb shows the histogram and standard deviation of the signal in Figure la.

While an oscilloscope can show the magnitude of the jitter, it does not do a good job of revealing the source of the jitter. An oscilloscope samples voltage at a specified sample rate and then displays the measurements. The resulting display shows how the signal amplitude varies over time, but does not reveal how the frequency, time interval or phase varies over time.

An excellent tool for directly measuring the time-varying nature of frequency, phase and time interval -- and, therefore, finding the source of jitter -- is a Modulation Domain Analyzer (MDA). Although frequency vs time, phase vs time, and time interval vs time can be determined from data acquired by a scope, the large amount of memory and the sophisticated signal processing routines required to display these results make this method costly and impractical.

Measurements in the Modulation Domain

An MDA measures how a time interval, frequency or phase varies with time by recording when a signal edge occurs. An edge is commonly defined as either a positive- or negative-going voltage transition. Along with recording the time of each edge, an MDA continuously counts the number of edges. That is, there is no dead time between the time stamps of each edge. Figure 2 shows how an MDA makes continuous frequency measurements.

After a predefined number of measurements has occurred, the time stamps and edges are processed according to the requested measurement. For example, a time interval is calcualted as the difference between adjacent time stamps, and frequency is calcualted as the time between two time stamps divided by the edge count. Other measurements, such as phase or rise/fall times, can be made by manipulating the time and event counts.

By measuring time intervals (or frequency or phase) in a back-to-back or continuous fashion, a more complete picture of jitter on a signal is obtained. Examining the time-interval vs time results shows the magnitude and the time varying nature of the jitter. If an FFT (Fast Fourier Transform) is performed on this data, all the components of the jitter are revealed in the form of a jitter spectrum. This information greatly simplifies the process of determining the potential sources of the jitter.

Modulation domain analyzers are capable of measuring signals directly to 500 MHz (higher with downconversion or prescaling) with single shot resolution of 150 ps rms. Periodic jitter components of 10 ps can be resolved. As a result, this technique can be used on many types of signals. Clocks of any frequency can be examined. Jitter on random signals, like live digital telephone traffic or data from a disk drive, is also easily measured.

In a modulation domain analyzer display, the x-axis represents the time over which the measurement was taken, just like an oscilloscope. But unlike an oscilloscope, the y-axis of the MDA displays the frequency, time interval, or phase of the signal, not the amplitude of the voltage. Figure 3a shows an MDA view of a simple FM signal. This figure shows how the frequency of the signal (carrier) varies over time. The sine wave displayed, therefore, reveals the modulation on the carrier. The frequency of the sine wave is the FM rate, and the amplitude of the sine wave is the FM deviation.

Figure 3b shows an MDA view of a time deviation vs time measurement, the type of measurement that would be made to look at jitter on a clock. This function measures the time of occurrence of a clock edge and compares it with the time of occurrence of a "perfect" clock edge of the same frequency. The difference is then plotted as a function of time. For a very stable clock, these differences, or deviations, may be as small as 1 ns or a few hundred picoseconds. A perfect clock would be displayed as a straight, horizontal line because it has no time deviation. If any jitter or unwanted modulation is present, however, the deviations may be on the order of tens or hundreds of nanoseconds. Using this technique, time deviations as small as 200 ps can be measured.

Figure 4 shows a time deviation vs time plot of the 155.52 MHz signal. Note that the time deviation varies sinusoidally with time. A feature of MDA is the ability to determine the rate and peak-to-peak deviation automatically. These values are pointed out in the figure. In this example, it appears that the signal varies 6.5 ns peak-to-peak at a 120 Hz rate. The 120 Hz value is indicative of some type of modulation due to the power supply.

The values on an MDA display can be scaled as needed. For example, telecommunications applications specify jitter in terms of unit intervals (UI), or one clock cycle. To convert from time deviation to unit intervals, each measurement is divided by the period of the signal. This is easily accomplished using the math capability built in to the MDA. The upper limit is virtually unlimited. In terms of unit intervals, deviations less than 0.001 UI on a 1.544-MHz carrier can be measured. Thousands of UI can be measured just as easily.

Jitter Spectrum Analysis

Close examination of figure 4 shows that the sine wave is not perfect. It is somewhat distorted, indicating that there may be other modulating frequencies affecting the signal. To determine if there are other modulating or jitter frequencies present, Fast Fourier Transform signal processing can be applied to the data to arrive at the jitter spectrum.

The jitter spectrum of a time deviation vs time measurement is analogous to a spectrum analyzer's view of a voltage vs time measurement. But unlike the spectrum analyzer, which shows all the frequency components of the original analog input signal, the jitter spectrum shows all the frequency components of the jitter (modulation) on the input signal.

The ability to show all sources of jitter is the major strength of jitter spectrum analysis. Figure 5a shows the results of the FFT performed on the signal shown in figure 4. Figure 5b shows the detail of a portion of figure 5a, revealing the main 120-Hz component. Figure 5c shows a different detailed portion of figure 5a, displaying a smaller jitter component at 34 kHz.

It is important to note that the graph in figure 5 is not the FFT of the original analog signal in figure 1, but the FFT of the time deviation vs time acquisition. The left edge of the x-axis (0 Hz) represents the carrier -- 155.52 MHz. The x-axis shows the jitter frequency offset from the carrier. The right edge represents the maximum offset frequency measured, or frequency span. In this measurement, the frequency span is 50 kHz. Spans as wide as 5 MHz can be measured. Below the value for the span, the width of each bin of the FFT is displayed, in this case 12.21 Hz. This bin width is the frequency resolution of the FFT. Using the FFT feature, it is easy to see jitter components as close as 0.01 Hz or less to the carrier. By dividing the frequency span by the bin width, the total number of bins in the FFT can be determined, which is exactly one-half the number of measurements collected by the MDA.

The y-axis shows the deviation from the signal at a particular offset frequency. In figure 5b, the largest component, at the left edge of the display, is at 120 Hz, and has an indicated deviation of -174 dBsec or 1.995 ns. The next largest component is at 34 kHz, with an amplitude of -216 dBsec or 15.84 ps rms. The jitter at 34 kHz may be due to a switching power supply modulating the clock. The 34-kHz component is 42 dB down from the 120 -z component, which is why it was not easy to detect in the time deviation vs time display.

Acquiring Large Amounts of Data with a Histogram

The source of jitter can often be found by examining the shape or distribution of data collected in a histogram. A histogram shows how often a particular time interval occurs, and is useful when a large amount of data needs to be gathered.

Over time, a clock with jitter will have many different periods. For example, to measure the probability of an errant time interval or metastable state, 10 or 100 billion measurements might need to be collected. If the probability of a metastable state is 10E-10, then, if 10E-11 measurements are collected in a histogram, some of the metastable events should show up as different time intervals.

The histogram function in an MDA can collect up to 2´110E-15 measurements as often as every 100 ns. By measuring millions of periods rapidly, it is easy to see the distribution of jitter, along with important statistics. Also, by making billions of measurements at a slower rate, long term drift of a clock can be characterized.

Figure 6 shows the histogram of a clock with a 1-ms period. Five million period measurements were made. In this case, the jitter distribution is close to Gaussian. Above the graph the statistics of the distribution can be seen.

Measuring Jitter on Live Data

There are times when the signal to be measured is not a clock but live data. That is, the signal can have runs of zeros between ones in a random pattern. For example, measurements may need to be made on live digital telephone signals, data read from a disk drive, or traffic on a local-area network. An MDA can be used to quickly measure the jitter on a live data signal.

Consider a 1.544 MHz data signal consisting of a one followed by a random string of zeros, then a one again, and so on. There will be a pattern of different time intervals between the ones. A one-zero-one will be 1294 ns, a one-zero-zero-one will be 1294+647 or 1941 ns, and so on. There will be different time intervals for each string of zeros, up to the maximum number of zeros allowed by the communications system being used. The high-speed histogram capability of the MDA will quickly show the maximum number of zeros and the jitter for each time interval, as well as the number of occurrences of a particular time interval.

Figure 7 shows a histogram of 1 million time intervals of a 1.544-MHz data signal with jitter. With an MDA, this measurement took less than 1 second. This particular data stream has a maximum of 13 zeros between ones. Note that there are 13 different distributions, corresponding to the 13 time intervals.

In figure 7, markers have been placed around one of the distributions for the purpose of extracting statistics about that time interval. Above the graph are the minimum time interval in that distribution, the maximum time interval, and the mean and standard deviations. The markers can be placed anywhere on the screen, allowing statistics to be measured on any of the distributions.

A data signal typically is not periodic. If the time deviation vs time or the FFT were made on a data signal, the results would have little meaning because the samples would not occur at uniform time intervals. It would be impossible to determine the jitter. However, because the MDA records the time of each edge and the frequency of the clock that is associated with the data is usually known, the time deviation vs time graph can be reconstructed and jitter on a data signal can be measured just as on a clock. This is accomplished by fitting a curve to the data using a cubic spline fit. The new wave form based on this fit is then re-sampled uniformly in time and plotted on the display. The re-sampling allows the FFT to be taken on data that is now sampled uniformly in time,

Why, then does the scope view in figure 1 indicated a smaller amount of jitter? This is a result of the way the measurement is made with a scope. Using a scope, jitter is examined on an edge at a specified delay from a trigger edge. If the delay is small compared to the jitter frequency, the scope will not be able to detect the jitter. The delay time is effectively a high-pass filter. In order to see low-frequency jitter using a scope, the delay must be set long enough for the jitter to have a chance to move the edge under observation. Not having an oven-controlled crystal timebase, however, the scope adds its own jitter for long delay times.

For the purposes of measuring wideband jitter, an oscilloscope is not the best instrument. However, it is an excellent tool for measuring cycle-to-cycle litter with very high (picosecond) resolution.

Jitter Transfer Function

As a signal propogates through a system, jitter on the signal may be amplified or attenuated as it passes through amplifiers, delay lines, repeaters and other devices. The ratio of the output jitter to the input jitter at any particular frequency is called the jitter gain, and is usually expressed in decibels (dB). If the jitter gain is calculated for many frequencies across a desired bandwidth, the jitter transfer function is obtained for the device under test. It is important to characterize the jitter gain or transfer function for individual components that may end up cascaded in a system in order to keep the overall jitter at acceptable levels.

A jitter transfer function measurement is made at a particular carrier frequency across a user determined bandwidth. For example, a phase-locked loop with a center frequency of 20 MHz may have a jitter bandwidth of 1 MHz. To make this measurement, a source that can generate 20 MHz and be FM modulated at rates up to 1 MHz is required. A block diagram of the measurement setup is shown in figure 8. Although the block diagram shows a computer, the jitter transfer function can be measured by hand.

A Sample jitter transfer function would proceed as follows: The signal generator is set to 20 MHz, and the modulation rate is set to the low end of the jitter bandwidth. In this example, use 100 Hz. When setting the deviation, it is important to remember to use a value near the amount of jitter that the device can normally expect to encounter. Too much deviation may cause the circuit to behave in an unpredictable manner.

Determine the peak deviation and set the generator accordingly. Note that most generators indicate deviation in Hz. To convert to time deviation use the following formula:

Time Deviation p-p = FM Deviation / (pi * Deviation Rate * Input Frequency)

To generate 50 ns of time deviation at a 100 Hz rate, therefore, set the FM deviation to 60 Hz.

Another way to look at the deviation is in terms of phase deviation. This is the same as time deviation, except that the result is scaled by 2*pi*input frequency. It is sometimes easier to think of keeping the phase deviation constant throughout the measurement. For example, during the course of the transfer function measurement, the input jitter is always 0.2 radian. To convert to phase deviation from FM deviation and rate use the following formula:

Phase Deviation p-p = 2-FM Deviation / Deviation Rate

To get 0.2 radian of deviation at a 100 Hz rate, set the FM deviation to 10 Hz.

Now that the rate and deviation are properly set, a time deviation measurement at the input of the device using the MDA is made. The FFT is performed and the amplitude of the spur at the jitter frequency is measured. This measurement is then repeated at the output of the device. The difference between the output and the input is the jitter gain in dB. The whole procedure is repeated to build up a plot of jitter gain vs jitter frequency.

This procedure can be easily automated. Figure 9 shows the jitter transfer function for a phase-locked loop. The input frequency is 10 MHz and the maximum jitter frequency is about 2 MHz. Often, for phase-locked loops, the jitter transfer function is called the phase-locked loop transfer function.

The jitter transfer function shows the jitter bandwidth of the device under test. In Figure 10, the -3 dB point is at about 800 kHz. Note that there is very little positive jitter gain. In cascaded systems it is important to keep positive jitter gain to a minimum.

Summary

Modulation domain analyzers are very good tools for uncovering the sources of jitter. An MDA's ability to measure live data signals means jitter can be measured anywhere in a digital system without a clock signal. Wideband jitter components from below 1 Hz to 5 MHz are accurately and easily measured. By examining jitter vs time, the FFT of jitter vs time, or the histogram of jitter, it is much easier to determine the source of unwanted jitter.

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