Learning · Step 5

Truncation & DC offset

Every time you drop low bits — after a multiply, when you store a wide result, when you scale a signal down — you quantize. How you drop those bits decides whether the leftover error is harmless noise or a permanent DC offset.

In two's complement, plain truncation is an arithmetic shift right — it is floor, rounding toward −∞. The truncated value is always less than or equal to the true value, so the error always lands in (−1, 0] LSB — it is never positive. Average that one-sided error over a signal and the mean is −½ LSB: a constant bias, the same on every sample.

A fixed value subtracted from every single sample is a DC component, by definition. Truncate a zero-mean sine and its average is no longer zero — the whole waveform sags by ½ LSB. Round-to-nearest instead spreads the error symmetrically over [−½, +½] LSB; the mean is ≈ 0, so the error is zero-mean quantization noise, not a DC offset. Noise you can live with; DC you usually can't.

Half an LSB sounds tiny, and once it is harmless. But it wastes headroom, it thumps on mute/unmute, it stacks up when truncations cascade — and it is catastrophic through an integrator: a CIC or an IIR accumulator turns a constant bias into a ramp. The fix is nearly free: round instead of truncate — one add of ½ LSB before the shift.

Try it

A zero-mean sine, quantized. Drag the quantizer coarseness and flip the mode — watch the red mean line leave zero under truncation and snap back under rounding.

Quantizer bits 3

ideal sine quantized quantized mean (DC)

What to notice

Flip Truncate → Round — the red mean line jumps from −0.5 LSB straight onto zero. That snap is the fix: one extra add.
Make the quantizer coarser (fewer bits) — the staircase turns blocky and the DC offset grows. It is always −0.5 LSB, so a bigger LSB means a bigger offset.
Under Truncate, every step of the blue curve sits on or below the ideal sine — the error is never positive. That one-sidedness is exactly where the DC comes from.
Under Round the error is still there — but it falls above and below the curve in equal measure. Zero-mean noise, no DC.

Key rules

Round, don't truncate, whenever the result still carries a signal — truncation's −0.5 LSB mean is a DC offset, not noise.
Rounding is nearly free — one add of ½ LSB before the shift. There is no good reason to ship the DC.
DC is worst through integrators — a CIC or IIR accumulator turns a constant bias into an ever-growing ramp. Never feed a truncated signal into one.
Truncation bias cascades — each truncated stage adds its own −0.5 LSB; down a long chain the offsets simply sum.
Convergent rounding (round-half-to-even) removes even the tiny tie bias that plain round-half-up leaves — reach for it in long integrators and metering.

Step exam

Answer all 3 questions correctly to complete this step.

Plain two's-complement truncation rounds every value toward:
The average (mean) error of truncation is approximately:
A constant -0.5 LSB error on every sample appears in the signal as: