Sound as Data

Sampling rate and the Nyquist limit

The sampling rate $f_s$ is how many times per second we record the signal's value, measured in hertz (Hz). CD audio uses $f_s = 44{,}100$ Hz, 44,100 readings every second.

How high does $f_s$ need to be? The Nyquist–Shannon sampling theorem gives an exact answer: to represent a signal without loss, the sampling rate must be at least twice the highest frequency present in it.

$f_s \geq 2\, f_{\max}$

Turned around, a given sampling rate faithfully captures frequencies only up to the Nyquist frequency, half the sampling rate:

$f_{\text{Nyquist}} = \frac{f_s}{2}$

Human hearing tops out around 20 kHz, so capturing every audible frequency needs $f_s \geq 40$ kHz; CD's 44.1 kHz leaves a small margin for the anti-aliasing filter. Speech energy, by contrast, lives mostly below 8 kHz, so telephone and many speech-recognition pipelines sample at 16 kHz, enough to make speech intelligible while storing far less data.

Bit depth and quantization

Sampling decides when we read the signal; bit depth decides how precisely each reading is stored. With $n$ bits per sample there are $2^n$ distinct amplitude levels available:

$\text{levels} = 2^{n}$

So 8-bit audio has $2^{8} = 256$ levels, while 16-bit audio (the CD standard) has $2^{16} = 65{,}536$. Because a real amplitude almost never lands exactly on a level, each sample is rounded to the nearest one, introducing a small quantization error. More bits mean smaller steps and a better signal-to-noise ratio; a good rule of thumb is roughly 6 dB of headroom per bit:

$\text{SNR} \approx 6.02\,n + 1.76 \text{ dB}$

At 16 bits that is about 96 dB, comfortably beyond what the ear resolves, which is why 16-bit is the consumer standard. Production and mastering often use 24-bit to leave room for editing before the final render.

Worked example: how big is one second of sound?

The raw (uncompressed) data rate follows directly from the three dials, rate, depth, and number of channels:

data rate = sample_rate x (bit_depth / 8 bytes) x channels

CD audio, stereo, 1 second:
  44,100 samples/s x 2 bytes x 2 channels = 176,400 bytes/s
  -> about 172 KiB per second, ~10.1 MiB per minute

16 kHz mono speech, 1 second:
  16,000 samples/s x 2 bytes x 1 channel = 32,000 bytes/s
  -> about 31 KiB per second

Music-grade CD audio costs roughly 5.5x the bytes of 16 kHz mono speech per second, the reason ASR pipelines standardize on the lower rate is not just history but bandwidth and compute. (These figures are for uncompressed PCM; codecs like MP3, Opus, and AAC shrink them dramatically by discarding perceptually irrelevant detail.)

Further reading

• Nyquist–Shannon sampling theorem, the precise statement and its proof sketch.
• The Scientist and Engineer's Guide to Digital Signal Processing, Steven W. Smith's free textbook; chapters on ADC, sampling, and quantization.
• Xiph.org: 24/192 Music Downloads … and why they make no sense, a rigorous, accessible treatment of sampling and bit depth for playback.
• librosa documentation, the standard Python library for loading, resampling, and analyzing audio.