Sampling

Wheels that turn backwards

• In films, a car's wheels sometimes seem to spin slowly backwards. The camera takes 24 frames a second, and the wheel turns almost a full spoke between frames.
• A digital audio channel samples sound the same way. This lesson shows when samples capture a signal exactly and when they lie.

Sampling copies the spectrum

• Sampling multiplies $x(t)$ by a train of impulses every $T$ seconds. In frequency, that copies the spectrum to every multiple of $f_{s} = \dfrac{1}{T}$.
• If the copies do not overlap, a low-pass filter can pick out the original and rebuild $x(t)$ exactly.

The Nyquist criterion

• A signal with no content above $f_{max}$ is recoverable if $f_{s} > 2f_{max}$: the Nyquist criterion 奈奎斯特准则.
• Below that rate the copies overlap. A frequency above $\dfrac{f_{s}}{2}$ folds back into the band as a false lower one: aliasing 混叠.

Worked example: the 4 kHz alias

• A 4 kHz tone sampled at 6 kHz: $4 > \dfrac{6}{2} = 3$, so it aliases to $|4 - 6| = 2$ kHz.
• Check the samples: $\cos\left(2\pi \cdot \tfrac{2}{3}n\right) = \cos\left(2\pi \cdot \left(-\tfrac{1}{3}\right)n\right) = \cos\left(2\pi \cdot \tfrac{1}{3}n\right)$. The two tones give identical samples.

The anti-alias filter

• Once a tone has folded, no later processing can tell it from a real low tone. So an anti-alias filter 抗混叠滤波器 must remove everything above $\dfrac{f_{s}}{2}$ before the sampler.
• The guard band 保护频带 between $f_{max}$ and $\dfrac{f_{s}}{2}$ is the room the filter has to roll off in.

Worked example: CD audio

• Content to 20 kHz, sampled at 44.1 kHz: $\dfrac{f_{s}}{2} = 22.05$ kHz. A 23 kHz interference tone would alias to $44.1 - 23 = 21.1$ kHz, inside the audio band.
• The guard band is only 2.05 kHz, which needs a steep filter. Studios sample at 96 kHz (a 28 kHz guard band) so a gentle filter will do.

Three common slips

Recap