THD, THD+N, SINAD and ENOB: what each number is actually measuring

Distortion specifications are where datasheets do the most quiet reinterpretation. The same converter can be quoted at −110 dB of THD+N, 112 dB of SNR, 111 dB of SINAD and 18.3 effective bits, and all four numbers are correct because each measures something slightly different. Distortion is not a single quantity, and a figure without its definition and its measurement bandwidth is not a specification.

The analyser below derives all of them from one tone. It generates a signal with harmonics you set yourself, so you can check the measurement against arithmetic it has not seen.

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The five numbers, and how they relate

• THD — the harmonic content only, as a ratio to the fundamental: √(V₂² + V₃² + ...) / V₁. Noise is excluded entirely, so THD can be excellent while the device is hissing.
• THD+N — everything that is not the fundamental, harmonics and noise together, as a ratio to the fundamental. It is the honest total, and it is what a notch-filter measurement produces.
• SNR — the fundamental against the noise only, with the harmonics taken out of the residual. This is why SNR is always better than THD+N.
• SINAD — signal to noise and distortion, which is exactly the reciprocal of THD+N. In decibels, SINAD = −THD+N. Two names for one number, and quoting both as if they were independent is a common way to make a device look better than it is.
• ENOB — the effective number of bits, from SINAD: (SINAD − 1.76) / 6.02. It answers "how many bits would an ideal converter need to be this clean?".

The relationships are worth stating as a chain, because only two of the five are independent:

In the tool, setting noise to zero makes THD+N collapse onto THD — with the defaults it comes out at 1.1225% against 1.1225% for THD — and adding noise pulls THD+N away while THD stays put. That single experiment is the whole distinction.

Harmonics add as powers, not as percentages

If a device has 1% second harmonic and 1% third harmonic, its THD is not 2%. The amplitudes add in quadrature, because each harmonic is a separate frequency and their energies are independent:

The tool reproduces that exactly: set H2 and H3 to 1% and zero everything else and it reports 1.4142%. Linear addition of distortion percentages is one of the most common arithmetic errors in audio, and it always overstates the total.

It is worth knowing the reason the error goes unnoticed: for small numbers, √(a²+b²) and a+b differ by less than 30% even when a = b, and pundits rarely check. But it also means a device with 1% at five different harmonics has 2.24% of THD rather than 5% — which matters when you are deciding whether something is broken.

Measuring a harmonic correctly is not reading one bin

A windowed tone is not a single spectral line. A Hann window spreads a bin-centred tone across three bins in the ratio 0.5, 1.0, 0.5 of amplitude, and a tone between bins spreads across more. Reading the tallest bin therefore understates the component by up to 1.42 dB, which is the scalloping loss described in the FFT article.

The usual fix is to sum the power across the main lobe. Doing that correctly hides a trap: the amplitude spectrum is normalised by the window's coherent gain, which assumes the tone sits in exactly one bin. Summing those amplitudes over a lobe therefore over-counts by the window's ENBW — a factor of exactly 1.5 for Hann, or 1.8 dB. The analyser divides by ENBW to restore the energy normalisation, under which the bin powers sum to the record's mean square. That is also why the fundamental is reported at −6.02 dBFS for a 0.5 amplitude tone rather than the −4.26 dBFS the naive sum produces.

There is a second reason the harmonic sum is not a one-bin job: a harmonic's offset from the nearest bin is the fundamental's offset multiplied by the harmonic order. A fundamental half a bin away from centre puts its tenth harmonic five bins away, so a search that looks only at the nearest bin will miss it entirely. The tool sums a lobe, so it does not care.

The bandwidth caveat, and the folding harmonic

THD+N integrates everything from DC to the measurement bandwidth, so the same device measured to 20 kHz and to 100 kHz gives two different numbers — the wider bandwidth always looks worse, because it collects more noise. Every THD+N figure needs its bandwidth stated, and this analyser's bandwidth is F_s/2, which it reports.

Then there is the harmonic that folds. A 5 kHz fundamental has its fifth harmonic at 25 kHz; in a 48 kHz record that component is above Nyquist and comes back at 23 kHz. It is still real distortion, it still raises THD+N, and it is no longer at a harmonic frequency, so a harmonic table built from multiples of the fundamental will not list it. The tool stops its harmonic list at Nyquist for exactly that reason, and reports the count so you can see it happen: at a 5 kHz fundamental with a 48 kHz rate, only H2, H3 and H4 are listed.

What the numbers mean in practice

• Below 0.01% (−80 dB) THD — a good modern DAC or headphone amplifier, and better than the transducers on either end of it.
• 0.1% (−60 dB) — a respectable analogue amplifier, and about the point where loudspeaker distortion becomes the limit.
• 1% (−40 dB) — audible on a pure tone, hard to hear on music with two or more simultaneous notes, and routinely exceeded by loudspeakers at ordinary levels.
• 10% (−20 dB) — audible as a change in timbre, and the region where valve amplifiers are often described as "warm" rather than as distorting.

The converse reading matters too: an SNR of 100 dB with 0.1% THD gives a SINAD of about −60 dB from the distortion, and ENOB of 9.7 bits — even though the datasheet's SNR suggests 16 bits. Whichever number is quoted, find out which of the five it is.

Five mistakes worth avoiding

1. Adding harmonic percentages linearly. They add in quadrature; 1% + 1% is 1.414%.
2. Quoting THD+N without its bandwidth. The figure is meaningless until you say where the integration stops.
3. Reading a single bin for a tone that is not bin-centred. Up to 1.42 dB low with Hann, and worse for the higher harmonics.
4. Treating THD and THD+N as interchangeable. One excludes noise; the other is dominated by it whenever the device hisses.
5. Ignoring harmonics above Nyquist. They fold into the band and inflate THD+N while vanishing from a harmonic-order table.

Summary

Use THD when you care about nonlinearity, THD+N when you want the honest total, and SINAD or ENOB when you are comparing against a converter's resolution. Compute each harmonic by summing its main lobe with the energy normalisation, state the bandwidth, and remember that the fifth harmonic of a high-frequency tone may not be where you left it.

Values shown are engineering aids rather than measurements; see the disclaimer and the tool index.