EC-7.6 Sampling, Aliasing and Reconstruction

What this is and why it exists

This is the result that lets digital electronics handle the physical world at all. Everything stored, transmitted or processed as numbers got there by being sampled: audio, images, radio, medical signals, sensor readings. The reason none of it has to lose anything is the theorem in this topic.

It is also the topic most often reduced to one memorised inequality. Sample at more than twice the highest frequency. Correct, and almost useless on its own.

The inequality is not the interesting part. The interesting part is a picture of the spectrum repeating. Once you hold that picture you can derive the inequality rather than recall it. The same picture answers what aliasing does and why a filter must sit in front of the converter. It also answers why recovery is possible at all, and why real hardware needs a correction the theory never mentions.

The vocabulary

• Sampling — taking the value of a continuous signal at regular instants.
• Sampling interval — the time between consecutive samples.
• Sampling rate — the number of samples per second; the reciprocal of the interval.
• Impulse train — a regularly spaced set of unit impulses, used to model sampling.
• Band-limited — having no energy above some highest frequency.
• Nyquist rate — twice the highest frequency present; the lowest rate that works.
• Aliasing — high frequency energy appearing as a lower frequency after sampling, indistinguishably.
• Anti-alias filter — an analog low-pass filter placed before the sampler.
• Reconstruction — recovering the continuous signal from its samples.
• Zero-order hold — holding each sample value for a whole interval, which is what real hardware does.

The mental model

Model sampling as a multiplication. Take the continuous signal and multiply it by an impulse train: a set of impulses spaced by the sampling interval. Where an impulse sits, its area is scaled by the value of the signal there. Between impulses, nothing.

That model looks like an odd way to describe reading a voltmeter, and its whole purpose is what happens next. Multiplying two signals in time convolves their transforms in frequency, and the transform of an impulse train is another impulse train. Convolving a spectrum with a set of evenly spaced impulses places a copy of the spectrum at each impulse.

So here is the picture, and it is the topic:

The spectrum of a sampled signal is the original spectrum, repeated over and over, spaced by the sampling rate.

Draw it. A triangle centred on the origin, reaching out to the highest frequency in the signal on each side. Then the same triangle again, centred at the sampling rate. Then again at twice the sampling rate, and so on in both directions.

Now everything follows from looking at the drawing.

Recovery is possible whenever one copy can be separated from its neighbours, because that copy is the original spectrum untouched. Isolate it and transform back and you have the original signal exactly. So the question becomes: when do the copies stay apart?

Each copy extends the highest frequency to either side of its centre, and the centres are spaced by the sampling rate. The copies stay apart when the spacing exceeds twice the highest frequency. That is the sampling theorem, read off the picture rather than memorised, and the boundary rate is the Nyquist rate.

Sample too slowly and the copies overlap. In the overlap region, energy from the high frequencies of one copy lands on top of the low frequencies of the next. The two add. Once added they cannot be separated, because nothing in the sum records where each part came from. A high frequency has come back as a lower one, wearing its identity. That is aliasing, and the well-known picture of it is a wheel filmed at the wrong rate appearing to turn slowly backwards.

The consequence is severe and worth stating plainly. Aliasing is not a distortion that later processing reduces. It is a loss of information, and no amount of clever work on the samples undoes it. The only cure is prevention.

Which gives the anti-alias filter its job. The overlap has to be prevented while the signal is still continuous. So an analog low-pass filter goes in front of every sampler, removing energy above half the sampling rate before any sampling happens. There is no digital substitute, because by the time the signal is digital the damage is done. This is why a converter chip is never used alone.

Reconstruction is the last piece. In principle, isolating one copy means passing the samples through an ideal low-pass filter that keeps everything below half the sampling rate and nothing above. In the time domain that filter interpolates smoothly between the samples, and the interpolation is exact.

But the previous topic already showed that the ideal sharp filter cannot be built. So real hardware does something cruder: it holds each sample value steady for one whole interval, producing a staircase. That staircase is a decent first approximation, and its cost is known. There is a gentle droop in the magnitude response across the band, plus energy left at high frequencies. Both are dealt with afterwards, by a correction in the response and a smoothing filter at the output. The theory says exact recovery is possible; the hardware gets close and then corrects for the difference.

One caution to finish. The theorem assumes the signal is band-limited, and real signals are not, quite. There is always a little energy above any frequency you name, from noise if nothing else. In practice, band-limited means the energy above the limit has been pushed far enough down to be irrelevant. That is exactly what the anti-alias filter is for. The assumption is made true rather than found true.

What you should now be able to explain or do

• Model sampling as multiplication by an impulse train and say why that model is worth using.
• Draw the spectrum of a sampled signal as repeated copies and state their spacing.
• Derive the sampling theorem from that drawing rather than recalling the inequality.
• Explain what aliasing does and why no later processing can undo it.
• Say why the anti-alias filter must be analog and must precede the sampler.
• Describe ideal reconstruction, explain why it cannot be built, and say what real hardware does instead and at what cost.
• Explain in what sense a real signal is band-limited.

Check yourself