Understanding Total Harmonic Distortion (THD): Measurement, Linearity, and Audible Thresholds
Master audio fidelity metrics: how Total Harmonic Distortion (THD) and THD+N are calculated, even vs odd order harmonics, and real-world audibility limits.
Safety First: Testing equipment distortion limits at maximum amplifier output can clip speakers. Increase volume gradually.
In high-fidelity audio specifications, Total Harmonic Distortion (THD) is the universal benchmark used to grade the linearity of audio components — from smartphone headphone jacks and professional DACs to vacuum tube amplifiers and studio subwoofers.
A theoretically perfect audio amplifier acts as a pure mathematical multiplier: $y(t) = A \cdot x(t)$. In physical analog reality, all electronic and mechanical systems exhibit non-linear transfer curves. When you pass a pure sine wave through a real-world component, the non-linearities distort the waveform, generating unwanted acoustic energy at integer multiples of the fundamental frequency.
1. How THD and THD+N are Calculated
To measure THD, an ultra-pure test sine wave at fundamental frequency $f_1$ (traditionally 1 kHz) is fed into the device under test. The output is analyzed with a notch filter or Fast Fourier Transform (FFT) to isolate the harmonic peaks.
THD Formula (Harmonics Only):
$$\text{THD} = \frac{\sqrt{V_2^2 + V_3^2 + V_4^2 + \dots + V_n^2}}{V_1} \times 100%$$
Where:
- $V_1$ is the voltage amplitude of the fundamental test frequency.
- $V_2, V_3, V_4, \dots$ are the amplitudes of the 2nd, 3rd, 4th, and higher harmonics.
THD+N Formula (Distortion plus Noise):
In practical bench testing (such as Audio Precision hardware), filtering each harmonic individually is slow. Instead, the analyzer notches out only the fundamental $f_1$, and measures everything remaining (all harmonics, AC mains hum, thermal resistor hiss, and RF interference):
$$\text{THD+N} = \frac{\sqrt{V_{\text{harmonics}}^2 + V_{\text{noise}}^2}}{V_{\text{total}}} \times 100%$$
Converting THD percentage to decibels (dB):
$$\text{THD (dB)} = 20 \log_{10}\left(\frac{\text{THD %}}{100}\right)$$
- 1.0% THD = -40 dB
- 0.1% THD = -60 dB
- 0.01% THD = -80 dB
- 0.001% THD = -100 dB (Common in modern solid-state DACs)
- 0.0001% THD = -120 dB (State-of-the-art reference equipment)
2. Even-Order vs. Odd-Order Harmonics: Musical Timbre
The human brain does not treat all harmonic distortion identically. The perceptual psychoacoustic profile depends heavily on whether the distortion is even-order or odd-order:
Even-Order Harmonics (2nd, 4th, 6th)
- Mathematical Multiples: 2 kHz, 4 kHz, 6 kHz (for a 1 kHz tone).
- Musical Interval: The 2nd harmonic is an exact musical octave higher. The 4th harmonic is two octaves higher.
- Perception: Even harmonics blend musically and harmoniously with the original signal. They thicken the sound, providing perceived “warmth,” fullness, and body. This is the characteristic sonic signature of single-ended triode (SET) vacuum tube amplifiers.
Odd-Order Harmonics (3rd, 5th, 7th)
- Mathematical Multiples: 3 kHz, 5 kHz, 7 kHz.
- Musical Interval: The 3rd harmonic is an octave plus a perfect fifth (a musical fifth above). Higher odd harmonics (7th, 9th, 11th) do not correspond to clean intervals in western 12-TET tuning.
- Perception: Odd harmonics sound dissonant, metallic, harsh, edgy, and abrasive. Symmetrical clipping in overdriven solid-state transistors produces intense odd-order harmonic spikes.
3. What are the Audible Thresholds of THD?
Audio marketing often touts minuscule differences in distortion (e.g. 0.0003% vs 0.0001%). But what can human ears actually resolve in blind listening tests?
Extensive psychoacoustic research (including studies by Geddes and Lee) establishes clear audibility limits:
| Audio Component | Typical Physical THD | Audible Threshold |
|---|---|---|
| Digital DACs / Preamps | 0.0001% – 0.005% | 0.1% (-60 dB) on pure sine tones; completely inaudible on music |
| Solid-State Power Amps | 0.01% – 0.08% | 0.5% – 1.0% (-46 to -40 dB) on complex orchestral transients |
| Tube Amplifiers | 0.5% – 3.0% | Audibly alters tonal balance (adds pleasant 2nd harmonic coloration) |
| High-End Headphones | 0.1% – 0.5% | 1.0% – 2.0% in the critical 1 kHz – 4 kHz speech band |
| Subwoofers (Sub-Bass <40 Hz) | 3.0% – 15.0% | 5.0% – 10.0% (-26 to -20 dB) due to the Fletcher-Munson equal-loudness curve |
Because human ears are remarkably insensitive to distortion in deep bass, a subwoofer producing 5% THD at 30 Hz sounds clean and punchy, whereas an amplifier producing 1% odd-order distortion at 3 kHz would sound unbearably harsh and fatiguing.