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 Incoming links: High-resolution quantization approximation

Past exam of the physics A course of the University of Cambridge / 2011 / Paper 2 / Section A / 5 / Solution 2026-10-20

In the high-resolution quantization approximation, a smooth input distribution varies little across one conversion interval, so rounding to the nearest level gives an approximately uniform quantization error

ϵ \in [- Δ V /2, Δ V /2]

. Therefore

⟨ ϵ ⟩ = 0, ⟨ ϵ^{2} ⟩ = \frac{1}{Δ V} \int_{- Δ V /2}^{Δ V /2} ϵ^{2} d ϵ = \frac{Δ V ^{2}}{12}, ϵ_{rms} = \frac{Δ V}{12} .

The input must stay within the analogue-to-digital converter's range. A large standard deviation alone does not guarantee uniform quantization error: an input concentrated exactly on widely separated conversion levels has zero error. Truncation without a half-step correction also has a different mean and root mean square.

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