Bins, Aliases, and What Sampling Cannot Know

Two Patterns Can Share the Same Samples

Two different patterns can agree at sparse sample positions, so the same recorded samples may not identify the underlying signal. The ambiguity and separation checks are exact only for the declared positions and toy patterns; real sampling adds jitter, anti-alias filtering, finite bandwidth, noise, and model-selection uncertainty.

A finite sample set can be exactly ambiguous until denser samples are taken. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

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Sparse samples can leave two patterns ambiguous

Sampling positions 0, 2, 4, 6 gives the same output for both patterns, so ambiguous bit is 1.

(1,1,1,1)=(1,1,1,1)⇒1(1,1,1,1)=(1,1,1,1)\Rightarrow 1
Sparse alias ledgerThe ambiguity is finite and explicitly checked.patternA=1,1,1,1,1,1,1,1patternB=1,-1,1,-1,1,-1,1,-1sparsePositions=0,2,4,6densePositions=0,1,2,3,4,5,6,7sparseOutput=1,1,1,1|1,1,1,1denseOutput=1,1,1,1,1,1,1,1|1,-1,1,-1,1,-1,1,-1ambiguousBit=1 bitseparatedBit=1 bit

Denser samples separate the patterns

Sampling all 8 positions gives different dense outputs, so separated bit is 1.

dense A≠dense B⇒1\text{dense A}\ne\text{dense B}\Rightarrow 1
Dense separation ledgerThis is exact finite ambiguity, not the full Nyquist theorem.patternA=1,1,1,1,1,1,1,1patternB=1,-1,1,-1,1,-1,1,-1sparsePositions=0,2,4,6densePositions=0,1,2,3,4,5,6,7sparseOutput=1,1,1,1|1,1,1,1denseOutput=1,1,1,1,1,1,1,1|1,-1,1,-1,1,-1,1,-1ambiguousBit=1 bitseparatedBit=1 bit