RMS Jitter From Phase Noise

Hi Grant,

The basic issue we are wrestling with, I believe, is the accuracy of using a logarithmic scale to approximate a linear scale to estimate the value of a definite integral.

In your specific example, it appears you have 129 points per decade - which is quite a few. When you use the trapezoidal rule to perform an integration, you are approximating the area of each segment by taking the average y value formed by (x1,y1) and (x2,y2) - i.e. (y1+y2)/2 - and multiplying it by the difference in x (x2-x1) to estimate the area. If you are using a logarithmic y scale, such as your formula is using, then you are assuming the logarithmic value of y1 is average y value (-38.1251812 dBc/Hz in your example). If the linear difference between y2 and y1 is very small, then the log of the average y value is approximately log y1 or log y2 since they are quite close. However, if the linear difference in the magnitude of y2 and y1 is significant, then the average value is not well approximated by log y1 or log y2. For example, if y1 >> y2, then the average linear value is about (y1)/2 - or about 6 dB less than just using log y1. 

In my opinion, if you do not want to be concerned with the possible approximation of using a logarithmic scale for linear scale in numerical integration, I would convert the phase noise to a linear scale and then perform the integration. This is the methodology I always use.

> Am I trying for too fine of detail?  

My thoughts....It is very unusual to measure phase noise at an offset frequency of 1 Hz unless you are designing a rather specialized type of oscillator - maybe you are! In fact, technically, the term describing the impact of phase noise below a 10 Hz offset is "wander" and not "jitter" and the phase noise measured at this offset frequency will be very noisy in most common oscillators. Wander measurements are rather specialized with names such as MTIE and TDEV. More conventional offset frequencies might start at 1 or 10 kHz or, perhaps, 100 Hz. A more typical number of points per decade to provide a reasonable estimate of jitter due to a phase noise characteristic is 20. The use of more points per decade might be warranted if there are a lot of spurs in your phase noise due to supply noise or deterministic jitter (for example, reference clock spurs in a frequency synthesizer). In this case, the finer frequency bins provide a more accurate estimate the impact of spurs on the jitter. With a coarse frequency interval, the spurs will artificially inflate the resulting jitter computation.

Let me know your thoughts Grant. I hope my explanation made some sense!

Shawn

Hi Grant,

The basic issue we are wrestling with, I believe, is the accuracy of using a logarithmic scale to approximate a linear scale to estimate the value of a definite integral.

In your specific example, it appears you have 129 points per decade - which is quite a few. When you use the trapezoidal rule to perform an integration, you are approximating the area of each segment by taking the average y value formed by (x1,y1) and (x2,y2) - i.e. (y1+y2)/2 - and multiplying it by the difference in x (x2-x1) to estimate the area. If you are using a logarithmic y scale, such as your formula is using, then you are assuming the logarithmic value of y1 is average y value (-38.1251812 dBc/Hz in your example). If the linear difference between y2 and y1 is very small, then the log of the average y value is approximately log y1 or log y2 since they are quite close. However, if the linear difference in the magnitude of y2 and y1 is significant, then the average value is not well approximated by log y1 or log y2. For example, if y1 >> y2, then the average linear value is about (y1)/2 - or about 6 dB less than just using log y1. 

In my opinion, if you do not want to be concerned with the possible approximation of using a logarithmic scale for linear scale in numerical integration, I would convert the phase noise to a linear scale and then perform the integration. This is the methodology I always use.

> Am I trying for too fine of detail?  

My thoughts....It is very unusual to measure phase noise at an offset frequency of 1 Hz unless you are designing a rather specialized type of oscillator - maybe you are! In fact, technically, the term describing the impact of phase noise below a 10 Hz offset is "wander" and not "jitter" and the phase noise measured at this offset frequency will be very noisy in most common oscillators. Wander measurements are rather specialized with names such as MTIE and TDEV. More conventional offset frequencies might start at 1 or 10 kHz or, perhaps, 100 Hz. A more typical number of points per decade to provide a reasonable estimate of jitter due to a phase noise characteristic is 20. The use of more points per decade might be warranted if there are a lot of spurs in your phase noise due to supply noise or deterministic jitter (for example, reference clock spurs in a frequency synthesizer). In this case, the finer frequency bins provide a more accurate estimate the impact of spurs on the jitter. With a coarse frequency interval, the spurs will artificially inflate the resulting jitter computation.

Let me know your thoughts Grant. I hope my explanation made some sense!

Shawn