Integrate Log-Spaced Phase-Noise Data Without Losing Units

A wrong hertz weight on a log-spaced phase-noise file makes the integrated jitter a different number even when every L(f) cell is copied correctly. Calculate each interval from its actual width in hertz before adding the pieces.

This sheet sits with the other RF engineering guides. The continuous integral, with both sidebands under the square root, is the form on the phase-noise guide. The discrete sum below is for a file of offsets, including a log-spaced file, and for a check that a denser split of one flat file does not move the sum.

Input or item	Value in the opened example	Why it matters
Offset frequency	Hertz. MT-008 Figure 5 runs from 0.01 MHz to 200 MHz.	Skyworks AN256 says the usual plot is log-log and the integration is linear-linear.
SSB level	-150 dBc/Hz, held flat in that figure.	The decibel number is not the height multiplied by hertz.
Linear density	10^(-150/10) = 1e-15 per hertz.	That density is the rectangle height for a flat bin.
Actual delta-f	199,990,000 Hz for the full span.	A log step and a hertz step are different weights.
Carrier	100 MHz, the frequency MT-008 chooses for this discussion.	Seconds come from radians divided by 2π times the carrier.
Printed result	About 1 ps in the tutorial, for that carrier and band.	Radians and seconds stay in separate cells.

What the log axis leaves unweighted

Skyworks AN256, Rev. 0.4, 21 September 2021 says phase-noise data are typically plotted on a log-log scale, and the integration should be performed on a linear-linear scale. Real data are not a continuous curve, so the note replaces the integral with a sum. It names a rectangular approximation, a trapezoidal approximation, and Simpson’s rule, and it calls the choice a personal preference. The archived text prints the rectangular step as area equals height times width. It does not print the trapezoidal weights or the Simpson weights. Leave those two methods blank until the formula is in front of you.

The same note explains why a captured file is sparse. A 1 Hz resolution bandwidth stepped across a wide offset range takes too long, so a wider resolution bandwidth is used. For the offset range from 10 to 100 MHz, a 1 MHz resolution bandwidth cuts the bin count from 90 million to 90. After the sweep, the note scales a bin to 1 Hz by subtracting 10 dB for every decade from the resolution bandwidth down to 1 Hz. Its worked scale factor is 10 × log10(1 MHz / 1 Hz) = 60 dB. Use that 60 dB figure for a 1 MHz resolution bandwidth. If a column is in dBc and the resolution bandwidth is absent, leave the dBc/Hz cell blank.

AN256’s picture with a carrier is a 1066 MHz example. On the expanded linear plot, each rectangle is 5 MHz wide, and the area of each rectangle is L(f) times 5 MHz. The vertical axis of that linear plot is already in picoseconds RMS. The note says each of those amplitudes is an RMS value, so the pieces combine as a root-sum-square. The archived text does not list the rectangle heights. Those cells stay blank. The log-plot axis ticks, from -40 dBc/Hz through -160 dBc/Hz, are axis labels. They are not sample values for the curve.

Takeaway: Multiply a linear density by a width in hertz. A missing resolution bandwidth leaves the per-hertz cell blank.

One flat file, counted two ways

Analog Devices MT-008, Rev. A, October 2008 works a single flat region in Figure 5. The carrier in the figure is 100 MHz, chosen for the discussion. The phase noise is -150 dBc/Hz. The integral runs from 0.01 MHz to about twice that carrier, and the figure marks the upper end as 200 MHz. The tutorial writes A = -150 dBc + 10 log10(200×10^6 − 0.01×10^6) = -150 dBc + 83 dB = -67 dBc.

The bandwidth term 10 log10(199,990,000) is 83.01 dB. The tutorial rounds it to 83 dB. The integrated power ratio of the same span is 10^(-150/10) × 199,990,000 = 1.9999e-7, which is -66.99 dBc. The two writings are that product, one of them rounded.

The figure’s frequency axis is marked at 10 kHz, 100 kHz, 1 MHz, 10 MHz, 100 MHz, and 1 GHz. The integration stops at 200 MHz, so the split below uses the marks that fall inside the integral and ends at the stated 200 MHz. Every level cell is the same -150 dBc/Hz.

Interval	f start (Hz)	f end (Hz)	L (dBc/Hz)	Linear density (1/Hz)	Delta-f (Hz)	Contribution
10 kHz to 100 kHz	10000	100000	-150	1e-15	90000	9.0e-11
100 kHz to 1 MHz	100000	1000000	-150	1e-15	900000	9.0e-10
1 MHz to 10 MHz	1000000	10000000	-150	1e-15	9000000	9.0e-9
10 MHz to 100 MHz	10000000	100000000	-150	1e-15	90000000	9.0e-8
100 MHz to 200 MHz	100000000	200000000	-150	1e-15	100000000	1.0e-7
Five-interval sum	10000	200000000	-150	1e-15	199990000	1.9999e-7
One interval	10000	200000000	-150	1e-15	199990000	1.9999e-7

The one-interval row and the five-interval sum match. They match because every L cell is -150 dBc/Hz and both versions use the hertz width. A denser split of this flat example does not move the integrated power.

The 10 kHz to 100 kHz interval is 90,000 / 199,990,000 of the hertz, which is 0.000450. Counting each of the five intervals as an equal share gives that row 0.2. Keep the 0.000450 weight with the sum. The 0.2 share counts rows. This page does not turn that share into a jitter figure.

The tutorial prints the RMS phase jitter for A = -67 dBc as 6.32e-4 radians. That printed number matches the square root of 2 × 10^(A/10). The product 2 × 10^(A/10) without the root is 3.99e-7, and that product is not the printed radian result. Use the square-root form. The square root of twice the unrounded sum 1.9999e-7 is 6.324e-4 radians. Divide by 2π × 100 MHz and the time figure is 1.007e-12 s. Dividing the tutorial’s printed 6.32e-4 radians by the same carrier gives 1.006e-12 s. The tutorial prints about 1 ps.

The factor of 2 under the root is the tutorial’s both-sidebands step. The carrier in the denominator is the 100 MHz of this figure. The same radian result on another carrier is a different number of seconds. The carrier in your own file is the source output frequency. How frequency synthesizers work covers how a PLL sets that output. The microwave synthesizer guide treats loop bandwidth as the split between which noise source you inherit at which offset, which is why the start and stop frequencies have their own cells.

Takeaway: On this flat file, one hertz-weighted bin and five hertz-weighted bins sum to 1.9999e-7. An equal share per log row is a different weight.

The record to copy

Copy one block per file. The filled column repeats the MT-008 Figure 5 arithmetic above. The empty column is the file in front of you. Leave a cell empty when that file does not state it.

Field	MT-008 Figure 5	Your file
Carrier (Hz)	100000000
Integration start (Hz)	10000
Integration stop (Hz)	200000000
Level unit	dBc/Hz
Resolution bandwidth (Hz), if the level is not already per hertz	Already dBc/Hz in the example
Sideband factor under the square root	2
Interpolation	One flat level across each hertz width
Integrated power ratio	1.9999e-7
Integrated power (dBc)	-66.99 unrounded; -67 printed
RMS phase (radians)	6.324e-4 unrounded; 6.32e-4 printed
RMS time	1.007e-12 s unrounded; about 1 ps printed

For a file that is not flat, keep one row per adjacent pair. Three blank rows are the start of that list.

Row	f start (Hz)	f end (Hz)	SSB L (dBc/Hz)	Linear density (1/Hz)	Delta-f (Hz)	Contribution
1
2
3

Linear density is 10^(L/10) once L is in dBc/Hz. Delta-f is the end frequency minus the start frequency, both in hertz. The contribution is the linear density times delta-f. Add the contributions, convert the sum with 10 log10, take the square root of 2 times 10^(A/10) for radians, and divide by 2π times the carrier for seconds. A single L held across a wide interval is the rectangular step AN256 prints. The note also names a trapezoidal sum and does not print that formula in the archived text, so a sloping interval stays on the rectangular row you can write, or the cells stay empty.

Takeaway: The printed 1 ps belongs to the 100 MHz carrier and the 0.01 MHz to 200 MHz span in MT-008 Figure 5. Another carrier, or another span, gets its own row.

When this does not apply

AN256 treats a spur as a separate sinusoidal term and then root-sum-square combines it with the noise integral. In the 1066 MHz example, a 5 MHz rectangle on the spur at 30 MHz exaggerates that spur. The note says narrowing the rectangle to the drawn curve still treats the spur power as uncorrelated noise. The archived text does not give that spur’s level in dBc. This sheet has no spur row. Jitter that folds spurs into the total is outside this page.

The tutorial’s upper limit of about twice the sampling frequency applies to an ADC sampling clock with no filter between the oscillator and the converter input. Write the stop frequency your band uses. The tutorial’s lower-limit comments are separate: specifications are generally not given below about 10 Hz, 100 Hz is a reasonable lower limit when that point exists, and otherwise the 1 kHz or the 10 kHz point is the one it tells you to use. The flat example starts at 0.01 MHz.

Analog Devices AN-1067 shows a sparse-sampling case on an FFT, which is a different record from a phase-noise CSV. Example 1 states that a 4k transform of a 32 MSPS record has a frequency resolution of 8 kHz, and that a 2 kHz modulation term combines with the fundamental and the surrounding bins. A 65k transform of that setup, the note says, resolves to 500 Hz and shows two symmetric terms. A wide bin can swallow a nearby tone. Those two resolutions stay with that FFT example. They are not a correction added to the MT-008 sum.

The 1066 MHz rectangles in AN256 remain without heights. Other jitter figures later in MT-008, including the Wenzel levels and the ADF4360 line-segment result, stay in that tutorial. They are not rows here.

Takeaway: A spur level the file does not state stays off the noise sum. The 8 kHz and 500 Hz figures stay with the AN-1067 FFT example.

Method and sources

The numbers above are compiled from public notes archived on 11 October 2026. AN256 supplies the linear-axis requirement, the rectangular area, the 60 dB scale factor, the drop from 90 million bins to 90, the 5 MHz rectangles on the 1066 MHz example, and the separate spur treatment. MT-008 Figure 5 supplies the -150 dBc/Hz flat region, the 100 MHz discussion carrier, the 0.01 MHz to 200 MHz limits, and the printed -67 dBc, 6.32e-4 radian, and 1 ps results. AN-1067 Example 1 supplies the 8 kHz and 500 Hz FFT resolutions. No oscillator was measured for this page. Corrections follow our editorial policy.