Digital Sampling

How the model works

Sampling at rate f_s keeps only the values at times n/f_s. The Nyquist–Shannon sampling theorem says a signal containing only frequencies below f_s / 2 (the Nyquist frequency) can be rebuilt exactly from its samples by sinc interpolation:

x(t) = Σₙ x[n] · sinc(f_s t − n), sinc(u) = sin(πu) / (πu)

A frequency above f_s/2 is not lost but disguised: its samples are identical to those of a lower alias,

f_alias = | f − k f_s | for the whole number k that brings it into 0 … f_s/2

The spectrum chart shows why: sampling copies the signal's spectrum around every multiple of f_s, and only the copy inside the shaded band 0 … f_s/2 is reconstructed. Real converters therefore put an anti-aliasing filter before the sampler.

A zero-order hold (a staircase that holds each value until the next sample) is how simple converters output samples; it is not an exact reconstruction. Quantization rounds each sample to one of 2ⁿ levels; for a full-scale sine wave the signal-to-quantization-noise ratio is about 6.02 n + 1.76 dB.

Frequencies here are in hertz on a slow time scale so the waves are visible; the same rules apply to audio at kilohertz. Sinc reconstruction uses a finite stretch of samples, so it is very slightly imperfect near the edges.

Guided experiments

1. Plenty of samples. A 3 Hz wave sampled at 20 Hz. Prediction: 3 Hz is well below the Nyquist frequency of 10 Hz, so the sinc reconstruction lies on top of the original. 3 Hz at 20 Hz
2. Aliasing. A 9 Hz wave sampled at 10 Hz. Prediction: the samples trace a slow 1 Hz wave (|9 − 10| = 1); the reconstruction is a convincing but wrong 1 Hz signal, like a wagon wheel turning backwards in a film. 9 Hz at 10 Hz
3. Exactly at Nyquist. 5 Hz sampled at 10 Hz, phase 0°, then 90°. Prediction: with phase 0° every sample falls on a zero crossing and the signal disappears; with phase 90° every sample hits a peak. At exactly f_s/2 the samples cannot tell amplitude from phase. Phase 0° Phase 90°
4. Few bits. 3-bit quantization. Prediction: the samples snap to 8 levels and the reconstruction is visibly distorted; the theory gives about 19.8 dB SQNR, against about 98 dB for 16 bits (CD audio). 3 bits

Challenge

CD audio samples at 44.1 kHz. What is the highest frequency it can represent? If a 30 kHz ultrasonic tone reached the converter without filtering, at what frequency would it be heard?

References

• Shannon, C. E. (1949), "Communication in the presence of noise", Proceedings of the IRE 37(1): 10–21.
• Oppenheim, A. V. and Schafer, R. W., Discrete-Time Signal Processing, 3rd ed. (2010), ch. 4.
• Smith, S. W., The Scientist and Engineer's Guide to Digital Signal Processing (1997), ch. 3.