Bins, Aliases, and What Sampling Cannot Know

Two Patterns Can Share the Same Samples

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.

highlighted = computed this step

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