Frequency analysis without the fog

A spectrum splits the variability of the beat series by rhythm: how much sits in slow waves, how much in the breathing wave. Making one takes four decisions, resampling, detrending detrending Detrending Removing the slow drift from a series of beat intervals before a spectrum is calculated, so the drift does not pile up as false power in the lowest frequencies. More about Detrending , the window and the method, and each one moves the numbers, which is why two programs disagree about the same recording. This lesson walks the chain on the teaching recording and then shows the one confound that matters most: breathing rate breathing rate Breathing rate How many breaths you take a minute. It moves HRV on its own: slower, deeper breathing makes the beat-to-beat swing larger, and below about 9 breaths a minute it shifts from the HF band into LF. More about Breathing rate .

The teaching recording was built for this. Its 341 intervals contain a breathing wave at 0.25 Hz (15 breaths a minute, 35 ms tall), a blood-pressure wave at 0.10 Hz (15 ms), a slow drift at 0.02 Hz (10 ms) and 6 ms of random beat-to-beat noise. A good analysis should find those three waves in those three bands.

What a spectrum is

Any wobbling series can be written as a sum of sine waves of different frequencies. The spectrum reports how much variance each frequency carries, in ms² per Hz. Add up the spectrum between two frequencies and you get the power of that band, in ms²:

$$P_{\mathrm{band}} = \int_{f_1}^{f_2} S(f)\,\mathrm{d}f$$

Read it as: the area under the spectrum between the band’s lower and upper limits. Add up every band and you get, apart from the parts the detrender removed, the variance of the series, SDNN SDNN SDNN The standard deviation of all NN intervals in a recording, in milliseconds. It measures overall variability, and its value depends strongly on how long the recording lasts. More about SDNN squared. The Task Force (1996) fixed the bands at ULF ULF ULF power The variance of the heartbeat intervals below 0.003 Hz, in ms²: rhythms slower than one cycle every 5 minutes, including the day-night cycle. It can only be measured from a 24-hour recording. More about ULF power below 0.003 Hz, VLF VLF VLF power The variance of the heartbeat intervals in the 0.003 to 0.04 Hz band, in ms²: rhythms repeating every 25 seconds to 5 minutes. It needs at least a 5-minute recording and is best read over 24 hours. More about VLF power 0.003 to 0.04 Hz, LF LF LF power The variance of the heartbeat intervals that falls in the 0.04 to 0.15 Hz band, in ms²: rhythms repeating every 7 to 25 seconds. Once read as sympathetic activity, which is now disputed. More about LF power 0.04 to 0.15 Hz and HF HF HF power The variance of the heartbeat intervals that falls in the 0.15 to 0.40 Hz band, in ms²: the range of normal breathing. It mainly reflects vagal activity when breathing is 9 to 24 breaths a minute. More about HF power 0.15 to 0.4 Hz, as reproduced by Shaffer and Ginsberg (2017), who write the VLF floor as 0.0033 Hz. Kubios uses the same LF and HF limits by default and starts VLF at 0.

The chain, decision by decision

Resampling. The NN series has one value per beat, so its points are unevenly spaced in time. The fast Fourier transform needs values on a regular clock. The standard fix fits a cubic spline through the beats and reads it off at 4 Hz, which is Kubios’s default (Kubios HRV User’s Guide, 2021); Quigley and colleagues (2024) call 4 Hz “generally preferred”, and ask for the method to be reported.

Detrending. A slow drift, from settling into a chair or a change in temperature, is not a rhythm, but the transform will count it as very-low-frequency power and spread it into neighbouring bands. A detrender removes it. The simplest subtracts a straight line. Kubios’s default is the smoothness-priors method of Tarvainen and colleagues (2002), a time-varying high-pass filter whose smoothness is set by a parameter λ; the guide says its default setting removes most of what lies below 0.04 Hz, which is the VLF band (Kubios HRV User’s Guide, 2021). This site’s library uses λ = 500. Every program detrends differently, and some not at all.

Stationarity Stationarity Stationarity The assumption that the rhythms in a recording stay the same from start to finish. Frequency measures such as LF and HF need it; a recording that changes partway through breaks it. More about Stationarity . Spectral methods assume the series’ mean and variance do not change during the window. Heartbeats never quite oblige. Quigley and colleagues (2024) say detrending can handle a drifting mean but a changing variance cannot be removed the same way, and offer short epochs as one remedy; they also note that for the ordinary uses of LF and HF, even large violations may not matter much. Catai and colleagues (2020) ask for the stationarity criterion to be reported. Laborde and colleagues (2017) advise analysing a 24-hour recording as a run of 5-minute pieces rather than as one window, which would break the stationarity assumption.

Window and method. Three estimators are in common use.

• Welch’s periodogram (FFT): cut the resampled series into overlapping segments, shape each with a window such as a Hann, transform each, and average the squared magnitudes. Kubios defaults to a 300-second window with 50% overlap. This site’s library uses 64-second segments with 50% overlap, which smooths the spectrum more and is one small reason its numbers differ from Kubios’s. The result is a jagged but honest estimate with no model to choose.
• Autoregressive (AR) modelling: fit a model that predicts each sample from the previous ones, then read the spectrum off the model. It gives a smooth curve with clean peaks, which is why Laborde and colleagues (2017) recommend it for HF, with a model order of at least 16 for short recordings; Kubios’s default order is 16, and Catai and colleagues (2020) give 8 to 20 as the usual range. The order is a choice, and the spectrum depends on it. This site describes AR spectra but does not compute them.
• Lomb-Scargle periodogram: works on the uneven beat times directly, with no resampling and no detrending step, at the cost of a different kind of smoothing. Kubios offers it as an alternative.

Quigley and colleagues (2024) report that FFT and AR spectra “usually lead to essentially equivalent results”, that different estimates of the breathing-linked variability correlate above 0.80 with each other, and that the choice rests on practical and theoretical grounds. Here is the teaching recording through four chains from this site’s library:

The first row is the default chain, smoothness-priors detrending before Welch; the Lomb-Scargle row skips resampling and detrending altogether. All four find the breathing peak at 0.25 Hz and the blood-pressure wave near 0.10 Hz. LF and HF agree within a few percent, so the ratios do too: LF/HF LF/HF LF/HF ratio LF power divided by HF power, with no unit. It was proposed as a gauge of balance between the sympathetic and parasympathetic systems, but research shows it cannot measure that reliably. More about LF/HF ratio is 0.19 for the three Welch chains and 0.20 for Lomb-Scargle, and the normalised units normalised units Normalised units LF or HF power as a percentage of their sum (LF + HF), rather than in ms². The two always add up to 100, so they carry one piece of information between them. More about Normalised units come out the same in every chain, LF nu 16% and HF nu 84%. VLF is where the chains part: the smoothness-priors detrender removes almost all of it, the straight line and the Lomb method keep it. On a short, steady recording like this one the method matters less than the settings; on a drifting one, the detrender and the window decide the slow bands. That is the plain reason two programs disagree.

Take the rhythm apart

Five minutes of the teaching recording

Power at each frequency, Hz Breathing

VLFLFHF

0 0.1 0.2 0.3 0.4 0.5

ULF

needs 24 h

VLF

3 ms²

LF

120 ms²

HF

635 ms²

Total

758 ms²

LF nu

15.9

HF nu

84.1

LF/HF

0.19

Peak

0.250 Hz, HF

Rested, 15 breaths a minute (0.250 Hz), 5 minutes, Welch's method after detrending. The breathing peak sits at 0.250 Hz in the HF band, with the ~0.1 Hz blood-pressure wave as a smaller LF peak. LF/HF is 0.19.

Rested Stressed Drifting Ectopic beats

Breathing rate 15 breaths a minute Recording length 5 minutes

Welch (FFT) Lomb–Scargle

Detrend first (smoothness priors, λ = 500)

The controls work when JavaScript is on.

A line chart of power against frequency from 0 to 0.5 Hz with the VLF (0.003–0.04 Hz), LF (0.04–0.15 Hz) and HF (0.15–0.4 Hz) bands shaded and a marker at the breathing frequency. Readouts give ULF, VLF, LF, HF and total power in ms squared, LF and HF in normalised units, the LF/HF ratio and the peak frequency, followed by a sentence explaining what changed. Controls: a breathing-rate slider from 5 to 20 breaths a minute, a recording-length slider, a detrending checkbox, Welch or Lomb–Scargle buttons, and four preset buttons for a rested, stressed, drifting or ectopic recording.

The bands and what they are claimed to mean

The minimum lengths and the claimed sources follow Shaffer and Ginsberg (2017). Behind the lengths is the Task Force (1996) rule that a recording should hold about ten cycles of the slowest frequency in the band, which Laborde and colleagues (2017) and Quigley and colleagues (2024) restate. Quigley and colleagues add that typical adults with intervals of 1000 ms or less can get a usable HF from 30 seconds. Heathers (2014) is stricter about LF: ten cycles of 0.04 Hz take 250 seconds, so he argues that anything built on LF, normalised units and LF/HF included, needs at least 5 minutes, and he notes that the ten-cycle rule itself appears never to have been derived analytically.

Normalised units divide LF and HF by their sum, so each becomes a share of the two bands together:

$$\mathrm{LF\,nu} = \frac{\mathrm{LF}}{\mathrm{LF}+\mathrm{HF}}\times 100, \qquad \mathrm{HF\,nu} = \frac{\mathrm{HF}}{\mathrm{LF}+\mathrm{HF}}\times 100, \qquad \frac{\mathrm{LF}}{\mathrm{HF}} = \frac{\mathrm{LF\,nu}}{\mathrm{HF\,nu}}$$

Read it as: LF’s share of LF-plus-HF, HF’s share, and the ratio between them. The three are one fact: LF nu is 100 minus HF nu, and LF/HF = 0.6 is the same as LF nu = 37.5% (Heathers, 2014). A paper that reports two of them as separate findings has reported one finding twice. Among the 2012 studies Heathers surveyed that used normalised units or LF/HF, 59% did exactly that, and the LF/HF ratio had worse precision than either normalised unit. Some programs, Kubios among them, define nu against total power total power Total power The sum of the power in every band of the spectrum, in ms²: the variance of the heartbeat intervals. For a 5-minute recording it is VLF + LF + HF; over 24 hours ULF is added. More about Total power minus VLF instead, which gives different numbers; a report should say which. The recording’s LF nu is 16% and its LF/HF is 0.19, which tells you that the breathing wave dominates, and nothing about which nerve did what.

Lesson 14 covered why LF/HF is not a measure of “sympathovagal balance” (Billman, 2013). Quigley and colleagues (2024) go further: the construct of a bipolar balance “is no longer tenable”, and neither LF/HF nor LF should be used as a measure of sympathetic sympathetic Sympathetic nervous system The "fight or flight" branch of the autonomic nervous system. It speeds up the heart and makes it pump harder, but it acts too slowly to drive beat-to-beat HRV. More about Sympathetic nervous system activity. Report the absolute powers; use the ratio, if at all, as a description of the spectrum’s shape.

The breathing confound

The HF band assumes you breathe between 9 and 24 times a minute, because 0.15 to 0.4 Hz is 9 to 24 cycles a minute (Shaffer and Ginsberg, 2017). Breathe slower than 9 a minute and the breathing wave crosses into LF. The vagal drive has not changed; the band it lands in has.

The teaching recording makes this exact. The explorer above rebuilds it with the breathing wave at any rate you choose, every other component and the noise unchanged. Its model also does what slow breathing does in a real person: the breathing swing grows as the pace nears 6 a minute, from 35 ms at 15 a minute to 91 ms at 6. Set the explorer to 15 and then to 6 and it shows these numbers:

The breathing wave now reads as LF, HF has nearly vanished, HF nu has dropped from 84% to 0.4%, and LF/HF has gone from 0.19 to 252.7, a number any “balance” score would call extreme sympathetic dominance. The wider swing is not what flips the ratio. Hold the swing at 35 ms and breathe at 6 a minute, and LF/HF still reaches 69.3 while RMSSD falls to 19.5 ms, because a slow wave changes little from one beat to the next. The band the wave lands in moves the ratio; the size of the swing only decides by how much.

This is the mechanism behind lesson 13’s rule not to pace your breathing during a baseline baseline Baseline Your own usual HRV level and range, built from weeks of readings taken the same way. Comparing a reading with your baseline says more than comparing it with other people. More about Baseline reading, and behind the resonance breathing resonance breathing HRV biofeedback A breathing practice in which you watch your heart rate live and breathe slowly, often near 6 breaths a minute, to make the heart's rise and fall with each breath as large as possible. More about HRV biofeedback you will meet in biofeedback: a wave at about 0.1 Hz is what the LF band is built to catch. Try it below; the slider sets the pace, the strip shows the swing, and the readout gives RMSSD for the last 30 seconds.

Breathe and watch your heart

A teaching model, not a measurement

Breathed in

Heart rate

60 bpm

RMSSD

46 ms

Last 30 seconds

Step

Heartbeats Breathing in

Gap before each beat 0.85–1.15 s, longer is higher

Breathing pace 6 breaths a minute

This picture moves when JavaScript is on.

A breathing circle grows on each in-breath and shrinks on each out-breath. A strip of heartbeats below it shows the beats bunching closer together during the in-breath, which is shaded, and spreading apart during the out-breath. A line under the strip plots the gap before each beat: it dips on the in-breath and rises on the out-breath. A slider sets the pace from 4 to 15 breaths a minute, and readouts give heart rate and RMSSD for the last 30 seconds. The swing is largest at about 6 breaths a minute.

Laborde and colleagues (2017) recommend measuring breathing rather than pacing it, and checking that the rate stays inside 9 to 24 a minute whenever a vagal conclusion is drawn. Quigley and colleagues (2024) go further for within-person designs and ask for respiration-adjusted and unadjusted results side by side. Many wearables now estimate breathing rate from the beat series itself, which is the HF peak HF peak Peak frequency The frequency, in Hz, at which a band of the HRV spectrum has its highest power. The HF peak is usually the breathing rate; the LF peak is usually near 0.1 Hz. More about Peak frequency frequency read backwards: on the recording, 0.25 Hz is 15 breaths a minute.

Go deeper What a reproducible spectrum report states

Quigley and colleagues (2024), Catai and colleagues (2020) and the GRAPH checklist (lesson 20) converge on one list: the resampling method and rate; the detrending method; the window type, length and overlap for Welch spectra, or the model order and fit for AR spectra; the band limits; whether nu uses LF + HF or total power minus VLF; the units (ms², or their log, which is often needed because powers are skewed); the epoch length and how stationarity was handled; and the breathing rate, measured or estimated. Given those, two analysts should get the same LF from the same beats. Without them, a difference between two papers may be a difference between two settings.

Lesson 19 asks a different question of the same beats: not how much variability there is at each rhythm, but how ordered it is.