Introduction

Periodic timing signals are pervasive in modern electronics. Two common performance metrics to quantify their short term frequency stability are time interval error (TIE) and phase noise. Time interval error is a time domain measurement while phase noise is a frequency domain measurement.

Measurement noise, poor signal integrity, or a lack of understanding of the limitations of each metric frequently produce inaccurate TIE or phase noise measurements.

To facilitate making TIE measurements, most manufacturers of high performance digital sampling oscilloscopes offer software packages to compute the time interval error of its captured waveform. However, the measurement is only available for internal waveform captures, and the degree to which the measurement can be adjusted is limited and may not provide the flexibility required for an accurate measurement.

To provide time interval error measurements on both captured waveforms exported from a sampling oscilloscope and simulated waveforms, a C-based executable jitterhist is offered as an additional tool. It includes algorithms developed to allow for accurate TIE measurements in the presence of measurement noise, significant amounts of time interval error (greater than a single period), and offsets between the long term reference frequency and its estimated value. Both the TIE and phase noise of a periodic waveform are computed and provided in comma-separated variable file and graphical formats. jitterhist and its documentation may be downloaded in from the References section at the conclusion of this article.^6,7

Background

Time Interval Error

Definition

The time interval error of a periodic signal with long term frequency f_o is the difference in time between its threshold crossings and the corresponding threshold crossings of a reference waveform having the same long term frequency f_o as the signal.

Figure 1 illustrates the definition of time interval error using a phase modulated waveform as the periodic signal with a long term frequency f_o of 100 MHz. The x-axis is normalized to the period number of the ideal reference waveform, and the y-axis is relative to the switching threshold. The five positive edge based time interval errors are circled. There are two positive time interval errors, two negative time interval errors, and a single time interval error of approximately zero. Although not highlighted, there are also a set of five negative edge based time interval errors. The values of the negative and positive edge based TIE may not be the same nor have the same distributions.

TIE is expressed in units of time or in unit intervals (UI) where one UI is equal to the period of the reference waveform. 

Figure 1. Comparison of ideal and phase modulated 100 MHz square wave phase modulation: 0.20 UI 500 MHz bandlimited Gaussian noise trise = 5%, tfall = 25%, duty cycle = 50% 700 MHz lowpass filtered waveform.

Measuring Time Interval Error

To measure the positive edge based TIE at threshold crossing i, one locates the threshold value of the waveform on the i^th rising edge of the periodic signal and takes the difference between the time of its threshold crossing and the time of the ith rising edge of the reference signal. Although this appears to be a simple computation, it does possess some subtleties that require care. Three cases are discussed: the impact of high frequency components of phase modulation (f_m >> f_o); the impact of multi-unit interval amounts of low frequency phase modulation (f_m<< f_o); and the effect of an offset between the estimated long term average frequency and its true frequency. 

Impact of High Frequency Modulations

In some cases, high frequency modulation of a periodic signal can obscure its actual threshold crossing instant. This effect is most common when the source of the modulation is some type of random noise or when the signal is a measured waveform. It is illustrated in Figure 2 with a 100 MHz square wave amplitude modulated by 1 GHz bandlimited random uniform noise with a modulation index of 0.25. The threshold crossing of the waveform about the 41,930^th sample point is circled in Figure 2 and shown with greater detail in Figure 3 where each sample point is marked. Note that there are two positive edge threshold crossings over the course of a few sample points in the timing waveform where there is only a single positive edge threshold crossing in its ideal 100 MHz waveform. Therefore, if one were to compute the TIE, the TIE associated with the second positive edge threshold crossing near sample 41,930 will be about -1.0 UI since the next positive edge threshold crossing of the ideal 100 MHz waveform is at sample 42.930 (1000 samples in advance). Inspection of the waveform indicates that a TIE of -1.0 UI is not correct. Therefore, an accurate TIE measurement requires a methodology to detect and compensate for multiple threshold crossings due to noise.

Figure 2. Amplitude modulated 100 MHz square wave modulation index = 0.25, 1 GHz bandlimited random uniform noise. trise = 30%, tfall = 30%, duty cycle = 65%900 MHz. Lowpass filtered waveform, Sample rate = 100 GHz.

Figure 3. Amplitude modulated 100 MHz square wave modulation index = 0.25, 1 GHz bandlimited random uniform noise. trise = 30%, tfall = 30%, duty cycle = 65% 900 MHz. Lowpass filtered waveform, Sample rate = 100 GHz (expanded view).

Impact of Multi-Unit Interval, Low Frequency Phase Modulations

A second TIE measurement issue that requires care occurs when a periodic signal of long term average frequency f_o is phase modulated by a lower frequency f_m whose modulation amplitude exceeds a unit interval.

The manner in which multi-unit interval TIE is computed requires "bookkeeping". To illustrate its computation, a zero-mean sinusoidal signal with sinusoidal phase modulation is considered.

Equation 1 describes a sinusoid at a reference frequency f_o with an initial phasethat is sinusoidally phase modulated at frequency f_m with a modulation amplitude A_m UI. Equation 2 represents the reference frequency from which the time interval error is computed.

If the magnitude of exceeds 1.0, the phase modulation at some time will add or subtract 2π radians from the phase ofand will cause the N^th rising or falling edge ofto cross the (N ± 1)^th rising or falling edge of

(1)

(2)

Therefore, to compute the TIE accurately, one must record both the number of each positive and negative edge threshold crossing event and its corresponding time for the reference and signal waveforms. It is not sufficient to take the difference between threshold crossing times of their adjacent edges when the amplitude of the phase modulation is equal to or greater than 1 UI.

Estimating the Long Term Average Frequency 

When the long term average frequency value of the ideal or reference waveform is not known, it may be estimated as the average frequency, f_ave, of the periodic signal using Equation 3.

(3)

Impact of Reference Frequency Offset on TIE

If the periodic signal contains phase modulation, the difference in time between its threshold crossings will not be constant. As a result, depending on both the time span of the sample and its modulation frequency components, the average frequency will vary from the true long term average frequency f_o.

Since the TIE of a waveform is computed relative to the long term average frequency, when the long term average frequency is incorrect, the accuracy of the TIE measurement is compromised.

Equation 4 defines the time interval error TIE (i) for the i^th threshold crossing of the periodic signal and its reference signal at its long term average frequency . Equation 5 provides insight into the impact of using a reference signal at frequency f_ave where  f_ave differs from f_o by fΔ . When f_ave is used to estimate the long term frequency f_o and there is a small difference between the two frequencies, the i^th TIE measurement will include a linear term in i whose slope is proportional to  fΔ / f_o. The slope will increase the peak-to-peak TIE when compared to using the exact long term frequency to compute the TIE.

(4)

Hence, an accurate estimate of the peak-to-peak or rms TIE requires an algorithm to correct the TIE for any offset between the average frequency of the periodic signal and its true long term average frequency.

Phase Noise Background and Limitation

In the frequency domain, measures of the frequency stability include the spectral density of frequency fluctuationsand the spectral density of phase fluctuations. The latter is the mean square phase variation in a 1 Hz bandwidth at an offset frequency f from the carrier with units (radians)²/Hz.

The expression for the ratio of the power in one phase noise sideband on a per Hertz basis in the units of dBc/Hz,, is defined by Equation 6 and represents the IEEE preferred means of characterizing phase noise.^2,3

(6)

When attempting to estimaterom the sidebands of its carrier to compute L(f), for the case where the integral ofbetween frequencies f and greater is much less than 1 radian, one may use the fact that the Bessel function coefficients for narrow band phase modulationare approximately 0. In this case, is approximately the double-sideband phase noise-to-carrier ratio. To minimize the error of this approximation, it is suggested that reliablevalues be less than approximately -20 dBc/Hz.⁴

jitterhist TIE Measurement Algorithms

Impact of High Frequency Modulation 

Several possible approaches were considered to accommodate periodic signals whose threshold voltage crossings are modulated by high frequency noise. The method found to be most robust and computationally efficient includes a metric to detect if the threshold crossings are impacted by high-frequency noise and a moving average algorithm to estimate the threshold crossing in the presence of a “noisy” threshold crossing.

Detection Mechanism

In Figure 3, the rising edge of the periodic signal crosses the voltage threshold, falls below the threshold on the following sample, and then resumes its rising slope and crosses the threshold a third time. As a result, the on-time between these two rising edge threshold crossings is the difference between the second and first threshold crossings or the sample time of 10 ps. This creates a duty cycle measurement of 10 ps divided by the long term period of the 100 MHz periodic signal or 0.001%. If the noisy threshold crossings were to occur in the falling edge of the periodic signal, the off-time would be 10 ps and the resultant duty cycle 99.999%.

jitterhist measures the duty cycle of each period of the periodic signal and records both the maximum and minimum duty cycle values for the entire waveform sample. When the minimum duty cycle falls below 5% or the maximum duty cycle exceeds 95%, jitterhist assumes the threshold crossings occur too quickly and must be re-analyzed to estimate the actual threshold crossings.

Estimating the Threshold Crossing Time

jitterhist optionally applies a moving average algorithm to the data samples to create a waveform from which the threshold crossings are computed using linear interpolation. When the number of moving average samples, S, is set to zero, the moving average algorithm is not used to estimate threshold crossings, and all threshold crossings are linear interpolated between the data samples. With the moving average input parameter set to S and with N input data samples, the moving average waveform will have N - 2S samples.

Figure 4 is an expanded view of the timing waveform shown in Figure 2 and includes the moving average waveform created by jitterhist with S set to 12. The moving average waveform crosses the switching threshold smoothly with a single switching threshold crossing event while the actual data samples have multiple threshold crossings between adjacent data samples.

Figure 4. Amplitude modulated 100 MHz square wave samples and moving average (n = 12) modulation index = 0.25, 1 GHz bandlimited random uniform noise. trise = 30%, tfall = 30%, duty cycle = 65% 900 MHz lowpass filtered waveform, sample rate = 100 GHz.

Combining the Detection and Threshold Crossing Algorithms

If the number of moving average samples is set to a non-zero value, jitterhist combines the detection and the threshold crossing algorithm in a single iterative feedback loop. A moving average waveform using 2S samples is created from the data samples and its TIE is computed. If the maximum duty cycle exceeds 95% or the minimum duty cycle is less than 5%, the number of moving averages is increased from S to S + 1 and the TIE and maximum/minimum duty cycles are re-computed.

For a non-zero moving average value of S and N input data samples, S is increased until any one of the following conditions is detected.

1. The number of moving averages is S + 20 
2. Both the maximum and minimum duty cycle fall below 95% and above 5% respectively
3. If S becomes 10% of the output data sample size or

Low Frequency, Multi-Unit Interval Modulation

jitterhist allows for TIE amplitudes that spans multi-unit intervals by using separate transition edge arrays for the reference signal and the periodic signal. The index of each array for a specific threshold crossing time corresponds to its threshold crossing number. Arrays exist for both positive and negative edge transitions. As a result, the TIE of the i^th positive edge of the periodic signal with respect to its ideal waveform is the difference between the i^th element of its positive edge transition time array and the element of its ideal waveform positive edge transition time array. This avoids the potential for choosing the incorrect transition time of the periodic signal or its reference waveform when the phase modulation exceeds one unit interval.

Long-Term Average Frequency Offsets

jitterhist includes an optional algorithm to address an offset between the long term average frequency estimate and its actual value. To compensate for an offset between the ideal waveform’s average frequency and its true long term average, jitterhist computes the slope and intercept of the TIE samples and removes the linear term from the TIE.

Although this solution appears to remove the error term due to an offset frequency, it is not robust with respect to the time span and modulation frequency content of the input data. If the time span of the samples is not sufficient to include a number of periods of its low frequency phase modulation, the average value of the TIE will be non-zero and forms a linear term. When the linear term is removed from the TIE to compensate for an offset between the ideal waveform’s average frequency and its true long term average, this erroneously increases the peak-to-peak TIE over the time span of the input samples.

jitterhist includes an optimization step following the initial slope correction. The initial estimate of the slope is used as the basis for the range of a two‑step search that determines a linear correction polynomial that minimizes the peak-to-peak TIE while compensating for any frequency offset between the average and true frequency.

jitterhist Methodology to Compute Phase Noise

To avoid the conversion errors fromtoat larger amplitudes of phase modulation that occur when using demodulated sidebands to estimate, jitterhist derives the phase noise directly from the power spectral density of the TIE (a sampled version of the phase).

The single sided phase noisein dBc/Hz is computed from the single sided power spectral density of the TIE in UI²/Hz using Equation 7.

(7)

The power spectral density algorithm is based on Welch's algorithm⁵ and uses the last N samples of the M time interval samples with a single segment where

Example of jitterhist TIE and Phase Noise Results using Captured Waveform

To illustrate the TIE and phase noise results using jitterhist, the program was used to analyze a 2 GHz current-mode logic (CML) waveform amplitude modulated at 2.1 GHz. The differential signal was applied to the 50 ohm ports of a sampling oscilloscope and sampled every 6.25 ps (160 GHz) for a total sample time of 13.107 us (2,097,152 sample points). jitterhist was used to compute the TIE and phase noise. Figure 5 illustrates a portion of the data as a function of time. The positive and negative edge based TIE and their respective phase noise are provided in Figure 6 and Figure 7. The 2.1 GHz amplitude modulation produces the 100 MHz sideband evident in Figure 7. The temporal rms values shown in Figure 6 and the integrated rms values in Figure 7 correlate well.

Figure 5. Captured 2 GHz CML waveform with 2.1 GHz amplitude modulation. 2,097,152 samples at 160 GHz (total sample time of 13.107 us).

Figure 6. Negative and positive time interval error versus time. Time interval error computed relative to 2.000000 GHz. Sample frequency = 160.0000 GHz, number of moving average samples = 0.

Figure 7. Phase noise of negative and positive time interval errors. Time interval error computed relative to 2.000000 GHz. Sample frequency = 160.0000 GHz, number of moving average samples = 0.

Summary

Making accurate time interval error (TIE) or phase noise measurements of a periodic signal presents a number of challenges. The program jitterhist includes algorithms and settings to provide accurate TIE measurement results for periodic waveforms with significant amounts of high-frequency measurement/modulation noise, an offset between its estimated and true long term average frequency, and for waveforms having multiple unit intervals of low frequency modulation when the time span of the samples includes more than one period of the modulation. jitterhist computes phase noise directly from TIE measurement results and avoids the approximation error associated with using a demodulated spectrum.

The program provides an additional tool for TIE and phase noise measurements of measured or simulated waveforms. In the case of measured waveforms, it complements TIE analysis packages offered by sampling oscilloscope manufacturers.

jitterhist and its documentation and installation instructions are available for download.^6,7

REFERENCES

1.  D. B. Leeson, "Oscillator Phase Noise: A 50-Year Review," in IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 63, no. 8, pp. 1208-1225, Aug. 2016 "IEEE Standard for Jitter and Phase Noise," in IEEE Std 2414-2020 , vol., no., pp.1-42, February 26, 2021, doi: 10.1109/IEEESTD.2021.9364950. 
2.  "IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology--Random Instabilities," in IEEE Std 1139-2008 (Revision of IEEE Std 1139-1999) , vol., no., pp.1-50, February 27, 2009, doi: 10.1109/IEEESTD.2009.6581834.
3.  T. E. Parker, "Characteristics and Sources of Phase Noise in Stable Oscillators," 41st Annual Symposium on Frequency Control, 1987, pp. 99-110
4.  P. Welch, "The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms," in IEEE Transactions on Audio and Electroacoustics, vol. 15, no. 2, pp. 70-73, June 1967.
5. jitterhist Program for Download, OneDrive, August 2024.
6. S. Logan, "Description, Installation, and Use of jitterhist: A Program to Compute and Analyze Time Interval Error," OneDrive, August 8, 2024.