Audio bit depth and dither: 16-bit, 24-bit, 32-bit float

Bit depth sets how finely each audio sample's level is stored. It fixes the quantization noise floor at roughly 6 dB per bit: about 96 dB of range for 16-bit and 144 dB for 24-bit. Dither is a small, deliberate noise added when you cut the bit depth, so the lost bits turn into a smooth hiss instead of distortion. Apply it once, at the final reduction, for example 24-bit to 16-bit for CD.

What bit depth controls, and what it doesn't

Every PCM sample is a number. With N bits, a fixed-point sample can take 2^N values: 65,536 for 16-bit and 16,777,216 for 24-bit. Bit depth has nothing to do with frequency response; that comes from the sample rate. It sets how accurately each sample's amplitude is stored, and therefore how far below full scale the noise floor sits.

The standard rule comes from the ideal-converter analysis in our ADC quantization guide. The ratio of full scale to one step is 20·log₁₀(2^N) = 6.02N dB. The ratio of a full-scale sine to the RMS rounding noise is 6.02N + 1.76 dB.

The familiar "96 dB" and "144 dB" are the 6.02N figures. Both are ceilings for an ideal quantizer with no dither, and both describe the storage format, not any real device.

Why 24-bit converters never reach 144 dB

A 24-bit word can represent 144 dB of range, but the analog circuit feeding the converter cannot deliver it. Resistors, op-amps, the voltage reference and the converter's own modulator all produce noise well above a 24-bit LSB.

A quick calculation shows the limit. Johnson (thermal) noise in a resistor is √(4kTRB). For a 1 kΩ source at 300 K over a 20 kHz audio band:

If full scale is a 2 V RMS sine, that single resistor sets a ceiling of 20·log₁₀(2 / 0.58×10⁻⁶) ≈ 131 dB, and a real input stage has many such noise sources. Datasheets agree. TI lists its PCM4222 audio converter at 123 dB A-weighted dynamic range (121 dB unweighted over 20 kHz) for 24-bit PCM output. Sound Devices' technical notes put the best single audio converters at roughly 130 dB. At about 6 dB per bit, 123 dB is roughly 20 bits; the lowest bits of a 24-bit recording are analog noise.

What 32-bit float gives you

A 32-bit IEEE 754 float stores a sign, an 8-bit exponent and a 24-bit significand (23 bits stored plus an implied leading 1). Audio software treats ±1.0 as 0 dBFS. Two properties follow.

• Constant relative precision. Each value carries a 24-bit significand, so rounding error stays roughly 2⁻²⁴ of the value, around 144 dB below the signal, whether it is loud or very quiet.
• Enormous range. The largest normal value is about 3.4×10³⁸ and the smallest about 1.18×10⁻³⁸, a span of about 1,529 dB. About 770 dB of that sits above 0 dBFS. Inside a float mix, a bus that peaks at +12 dBFS is not clipped; pull the master fader down and the waveform is intact.

The headroom only exists inside the float chain. A DAC, a 24-bit export or a fixed-point plug-in still clips anything above 0 dBFS. Recorders that write 32-bit float files do not capture 1,500 dB of real range. Sound Devices describes combining multiple converter stages to reach over 142 dB at the input. The float file just guarantees that this range is never clipped or lost by a gain setting.

Truncation distortion at low levels

Reducing word length means throwing bits away. The crude way is to truncate (or round) each sample to the nearest 16-bit value. For loud, complex music the resulting error looks like random noise. For quiet material it does not.

This is the sound of undithered reverb tails, fade-outs and quiet piano decays: a grainy buzz that changes with the music. Sound On Sound's technical editor points out that truncation distortion can be subtle at normal levels but is exposed in fades. The DSP Guide describes the same failure for slowly changing signals.

Dither types: RPDF, TPDF and noise shaping

Dither is random noise added just before the word length is reduced. It shifts each sample's rounding decision at random, so the average output follows the input even below one LSB. Lipshitz, Wannamaker and Vanderkooy's 1992 survey in the Journal of the Audio Engineering Society set out what each dither type actually achieves.

RPDF: rectangular, 1 LSB peak to peak

RPDF dither is uniform noise spanning ±½ LSB. It makes the average error zero for every input, which removes the distortion. But the power of the error still depends on the signal, so the hiss changes level as the music moves. The survey calls this noise modulation and judges it audibly undesirable.

TPDF: triangular, 2 LSB peak to peak

Add two independent RPDF sources and you get a triangular distribution spanning ±1 LSB. TPDF makes both the mean and the variance of the error independent of the signal, so what remains sounds like steady white hiss. The cost is noise: the total error variance becomes Δ²/4 instead of Δ²/12, three times the power, or 10·log₁₀3 ≈ 4.8 dB. A dithered 16-bit channel therefore has about 98.1 − 4.8 ≈ 93.3 dB SNR for a full-scale sine. The survey identifies TPDF as the lowest-noise non-subtractive dither that fully removes both distortion and noise modulation, which is why it is the default.

Noise-shaped dither

Noise shaping feeds the requantization error back through a filter so the noise spectrum is tilted: less noise in the midrange, where hearing is most sensitive, and more near the top of the audio band. Total noise power is not reduced, but the noise is harder to hear. Commercial mastering tools offer many proprietary curves. Because the shaped noise is concentrated at high frequencies, it should be the very last process. Further gain changes, sample rate conversion or lossy encoding can disturb the curve, and the added top-end noise can then be the most audible noise left in the file.

When to dither, and when not to

• Dither whenever word length is reduced and the error would otherwise be correlated with the signal. The classic case is the final 24-bit or 32-bit float master going to a 16-bit CD or 16-bit streaming file.
• Dither once. Every dither stage adds its own noise. Keep the whole mix and mastering chain at 24-bit or float and reduce to 16-bit as the final export step.
• Going from 32-bit float to 24-bit technically reduces word length too. The rounding error there is more than 140 dB below full scale, far under any converter's analog noise, so dither is optional; applying TPDF is harmless.
• Do not dither when increasing bit depth (16 to 24) or when the format stays float. No information is discarded, so there is nothing to linearize.

File size math

Uncompressed PCM size is simple multiplication:

Divide by 8 to convert bits to bytes. WAV and AIFF add a small header; lossless compression such as FLAC reduces the stored size without changing the decoded samples.

To hear and see truncation steps, error signals and SQNR change as you lower the bit depth, open the quantization playground.