Sampling
Introduced| English | Meaning |
|---|---|
| Nyquist criterion | In sampling, the rate condition for band-limited signals; in feedback control, the stability test using a loop-frequency plot and its unstable-pole count. |
| aliasing/ˈeɪlɪəsɪŋ/ | In sampling, different continuous-time frequency components becoming indistinguishable as discrete-time samples. |
| anti-alias filter | A filter attenuating unwanted out-of-band components before sampling to reduce aliasing. |
| guard band | guard band |
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.
Sampling at f_s places copies of the signal's spectrum at every multiple of f_s.
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Multiplying by an impulse train in time convolves the spectrum with an impulse train in frequency, which copies it to each $kf_{s}$.
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.
A sensor signal has content up to 150 Hz. What sampling rate must be exceeded, in Hz?
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$f_{s} > 2f_{max} = 300$ Hz. In practice, sample comfortably above it to leave a guard band.
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.
A 4 kHz tone is sampled at 6 kHz. It appears as
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4 kHz is above $\dfrac{f_{s}}{2} = 3$ kHz, so it folds to $|4 - 6| = 2$ kHz. The samples are identical to a real 2 kHz tone.
A 7 kHz tone is sampled at 10 kHz. At what frequency does it appear, in kHz?
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7 kHz exceeds $\dfrac{f_{s}}{2} = 5$ kHz, so it folds to $10 - 7 = 3$ kHz.
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.
Where must the anti-alias filter go?
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An aliased tone is indistinguishable from a real one once sampled, so it must be removed before sampling.
Worked example: CD audio
- Content to 20 kHz, sampled at 44.1 kHz: $\dfrac{f_{s}}{2} = 22.05$ kHz. A 25 kHz interference tone would alias to $44.1 - 25 = 19.1$ 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.
Different waves can pass through the same samples
$f_s>2f_{max},\quad f_a=|f-kf_s|\le f_s/2$
The worked 4 kHz cosine sampled at 6 kHz shares every sample with a 2 kHz cosine. Change phase to see why a negative-frequency fold must reverse cosine phase. At the Nyquist boundary, a sine can vanish at every sample. The audio example uses a 25 kHz tone at 44.1 kHz, which aliases to 19.1 kHz; raising the rate to 96 kHz avoids folding that chosen tone.
Audio to 20 kHz, sampled at 96 kHz. How wide is the guard band?
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$\dfrac{96}{2} - 20 = 28$ kHz, against 2.05 kHz at 44.1 kHz. The anti-alias filter can be far gentler.
Three common slips
- Treating $f_{s} = 2f_{max}$ as safe. The inequality is strict; at equality a sine can be sampled at its zero crossings and vanish.
- Filtering after sampling. The alias is already inside the band.
- Expecting upsampling to restore detail. What was never sampled was never recorded.
A digital low-pass filter after the sampler can remove an alias.
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The alias lands inside the band, at a frequency real content may also have. No filter can separate the two once they share a frequency.
Recap
- Sampling copies the spectrum at multiples of $f_{s}$.
- Need $f_{s} > 2f_{max}$; above $\dfrac{f_{s}}{2}$, frequencies fold back as aliases.
- The anti-alias filter goes before the sampler, within the guard band.