Quantization — Data Conversion

Along the way

1
Finite codebook. Each sample voltage maps onto \(2^N\) codes. \(\mathrm{LSB}=\mathrm{FSR}/2^N\). Smaller LSB → more bits or a smaller range.
2
Peak error is geometric. In range (no clip), \(|e|\le\mathrm{LSB}/2\). That is the vertical gap from the stair to the ideal line — you can see it.
3
Why \(\mathrm{LSB}/\sqrt{12}\). Model \(e\) uniform on \([-\Delta/2,\Delta/2]\). Variance is \(\Delta^2/12\). Part 4 puts that noise power against a full-scale sine.
4
Busy signals make the model honest. A slow DC or a tiny sine can make \(e\) look like a sawtooth, not white noise. Dither or a busy input is why the textbook formula works on the bench.
5
Clipping is a different animal. Overdrive the sine and the error is no longer a small sawtooth — you get flat-top harmonics (Part 6), not extra quantization noise.

DNL/INL (Part 7), missing codes, and mid-tread vs mid-rise endpoints. Those change stair geometry, not the \(\Delta/2^N\) scaling story.

Next: Part 4 — turn \(\sigma_e^2=\Delta^2/12\) into \(6.02N+1.76\,\mathrm{dB}\).

Transfer curve & quantization error

Left: ideal line vs mid-rise staircase · Right: sine in → quantized out + error \(e=q-x\). Error axis is fixed volts (\(\pm\mathrm{FSR}/4\)), not zoomed to LSB.

Bits \(N\) FSR (V) Sine amplitude (V peak)

Ideal \(q=v_{\mathrm{in}}\) Quantizer

Input sine Quantized Error e

Levels —

LSB —

Peak |e| —

RMS e (ideal) —

\[ \mathrm{LSB} = \dfrac{\mathrm{FSR}}{2^N} \]

= —

\[ |e|_{\max} = \mathrm{LSB}/2 \]

= —

\[ \sigma_e = \mathrm{LSB}/\sqrt{12} \]

= —

Mid-rise uniform quantizer model. \(\sigma_e=\mathrm{LSB}/\sqrt{12}\) assumes error uniform on \([-\mathrm{LSB}/2,\mathrm{LSB}/2]\) — the bridge into Part 4’s SNR formula.

Try it: drop to 3 bits — fat stairs, large rose error. Raise to 8 bits — the blue quantized sine hugs the gray input and the error shrinks like \(2^{-N}\).

Analog Intuition

Quantization staircase

Transfer curve & quantization error