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Converting Phase Noise (dBc/Hz) to Phase Jitter (ps RMS) for Large Data Sets

This article discusses phase jitter calculations for large data sets, as usually obtained from test equipment where the raw output file contains a long list of (x,y) data for (Hz, dBc/Hz). The spreadsheet template discussed below therefore assumes the entered data is an accurate piecewise linear approximation of the actual data plotted on a linear-linear scale. This is what is meant by LARGE data sets. SMALL data sets must use the online phase-noise calculator, which assumes the entered data is an accurate piecewise linear approximation of the actual data plotted on a log-log scale (the math is necessarily different as a result). The calculator for small data sets is preferred, for example, if someone hands you a phase-noise plot (i.e. you don't have access to the raw data file) and asks you for the integrated phase jitter, or if you see a list of phase-noise data in a datasheet table and want to make a quick calculation. Continue reading below to work with large data sets.

Many industry standards specify jitter generation compliance with a maximum phase jitter value in units of RMS ps. One method to determine compliance is to measure phase-noise data and use it to calculate phase jitter. However, it is sometimes confusing how to do this conversion. An additional complication arises if the industry standard specifies jitter compliance numbers after a specific high-pass or band-pass filter has been applied.

This article clarifies the mechanics associated with such conversions using Excel spreadsheet examples. Two spreadsheets are provided to illustrate the conversion process for first and second-order filters, respectively. I'll arbitrarily choose XAUI Ethernet (156.25 MHz reference clock) for the high-pass filter example, and SONET OC48 (155.52 MHz reference clock) for the band-pass filter example. These spreadsheets may be used as templates to apply to additional standards. If an industry filter is not required for your application, either spreadsheet will do.

Let's begin by assuming that phase-noise has been measured and its raw data is available in columnar form as (frequency, dBc/Hz), for example, to facilitate using an Excel spreadsheet (of course, the mathematics won't change if different software is used). Download, then open either of the following files, and review the first two columns of example data (files were saved to support Excel 97 and later versions):

Spreadsheet for Phase-noise Conversion Using 1st-order Filter (XAUI Ethernet Reference Clock Example)
Spreadsheet for Phase-noise Conversion Using 2nd-order Filter (SONET OC48 Reference Clock Example)

For non-telecom applications with large spikes (i.e. spurs) present in the measured phase noise data, selectively remove them before starting the conversion process. After all, these spurs represent deterministic jitter, which will only create errors in estimating BER if they are included in the final RMS value. Smaller spurs (those that do not significantly influence the overall integration) need not be removed. Use another technique, such as histogram tail fitting using an TIA or oscilloscope, to measure deterministic jitter when estimating total jitter at a specified BER for non-telecom markets (telecom applications must include the spurs when calculating RMS phase jitter, although this is beyond the scope of this article).

Begin by taking the anti-log of the dBc/Hz data to assist integration. To keep things simple, apply the trapezoidal rule to integrate between data points. More accurate integration methods are possible, but the end result will not change assuming your instrument's frequency step size is sufficiently small (I recommend acquiring at least 20 data-points per decade, to make sure the sampled data is a piece-wise linear approximation of the actual phase-noise curve). For simplicity, the spreadsheets shown here represent only a limited data set. Your actual data file should contain much more data.

Complete the integration by summing the data in the trapezoidal rule column. Since the phase-noise measurement only accounts for a single-sideband, multiply this number by two to capture the other sideband. The resulting number is the variance. Take its square root to obtain the standard deviation of phase jitter in radians. Because the distribution's mean is zero, this is more commonly refered to as RMS phase jitter in practice. Divide by 2*pi*fc to obtain phase jitter in RMS seconds, where fc is the carrier frequency. Click on the Example Graph tab in Excel to view the example phase noise graphs.

Next we address conversions requiring industry-standard filters. Notice the industry-standard specified parameters (carrier and pole frequencies) located on the spreadsheet's right-hand side. Different standards may be input by changing these values as dictated by the standard. Standards requiring filters for jitter generation measurements will specify high-pass or pass-band filters by providing these zero and pole frequencies, as well as roll-off characteristics. From this information a transfer function of the filter can be modeled as a simple first or second order LaPlace function. Using simple math, the absolute magnitude of this function times its complex conjugate is the transfer function of this filter. The spreadsheet's Filter column expresses this value in dB. The filter is easily applied to the raw data by adding their respective columns in dB. To integrate this resulting phase-noise curve, first take its anti-log before applying the trapezoidal rule as done above. The conversion process is finished similarly as above.

Understanding and choosing the minimal steps required to get a job done is the art of maximizing an engineer's efficiency. I hope this article expands your tool chest in considering available options.

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