Sampling, analogue-digital conversion and aliasing

Treat sampling and conversion as two operations

Sampling at rate fs selects values separated by interval Ts=1/fs. An ideal sampler can retain exact amplitudes at those instants while still losing information between them. Quantisation then assigns amplitude intervals to finite codes. More bits refine the amplitude grid; they do not shorten the sampling interval unless the time settings also change.

For a baseband signal known to contain no frequencies at or above an upper bound B, ideal reconstruction requires an appropriately chosen sampling rate greater than 2B. The assumptions include the signal restriction and ideal sampling/reconstruction. Sampling exactly twice a sinusoid’s frequency can produce an ambiguous sequence depending on phase, so “two points per cycle” is not a general reconstruction recipe.

Demonstrate an actual alias

The figure plots cosines at 20 Hz and 80 Hz sampled at 100 samples/s. At t=n/100, cos(2π·80n/100)=cos(2πn−2π·20n/100)=cos(2π·20n/100). Every displayed sample agrees although the continuous curves differ. The 80 Hz component folds to 20 Hz in this baseband representation.

A later digital low-pass filter cannot identify which of these two original cosines produced the identical sequence. An analogue filter before sampling can suppress unwanted input bands; its required attenuation depends on their amplitudes, useful band and allowed error. Practical filters need transition width and converters have input-bandwidth and timing limits. A nominal sample rate alone is not an anti-aliasing design.

Declare the quantiser’s interval convention

For our original 3-bit mid-rise quantiser over 0–4 V, use eight equal bins of width q=0.5 V. Code k represents input bin [0.5k,0.5(k+1)) V and reconstruction level 0.5k+0.25 V, for k=0,…,7. Treat the upper endpoint separately rather than silently inventing a ninth code.

With nearest bin-centre reconstruction and no clipping, the magnitude of quantisation error is at most q/2. Actual ADCs may use different transition/code conventions and also have offset, gain error, nonlinearity, reference noise and settling limitations. The conditional q/√12 rms error model requires suitable distribution/correlation assumptions; it is not a universal random-noise law for every deterministic sequence.

Keep time ambiguity separate from code spacing

At 100 samples/s, Ts=0.01 s. Our two unit-amplitude cosine sequences begin 1, 0.3090, −0.8090, −0.8090, 0.3090, 1 at n=0,…,5, despite their different continuous frequencies. These values follow the supplied trigonometric model, not a measured converter record.

In the separate 0–4 V quantiser, input 1.34 V falls in bin [1.0,1.5) V: code 2, binary 010, reconstructs 1.25 V. Reconstruction minus input is −0.09 V, within ±0.25 V. An input above the admitted range would clip or require another convention; the interior bound would not apply.

Increasing this same ideal span to 8 bits gives q=4/256=0.015625 V. That reduces the amplitude step by a factor of 32, while the 100 samples/s time grid remains unchanged. It cannot distinguish the two aliased sequences. A useful conversion record names rate, input filtering, span, code convention and relevant errors together.

Try the reasoning

1. Can increasing bit depth alone cure aliasing?

   Show the answer and reasoning

   No. Bit depth refines amplitude codes; aliasing concerns indistinguishable sampled time sequences.

2. Why must filtering happen before sampling to suppress an unwanted analogue alias?

   Show the answer and reasoning

   Once it has produced the same samples as a wanted component, ordinary downstream processing cannot separate their original identities without extra information.

3. Is q/√12 guaranteed for the error of the single 1.34 V example?

   Show the answer and reasoning

   No. That example has a particular −0.09 V error; the rms model depends on the distribution and correlation of many errors.

What to carry forward

Specify the time grid and amplitude grid separately. Prevent unwanted aliases before information becomes indistinguishable.