12.1 Fixed-point quantization

The fractional $B$ -bit two's complement number representation evenly distributes $2^B$ quantization levels between $-1$ and $1-2^{-(B-1)}$ . The spacing between quantization levels is then $$(\frac{2}{2^B}=2^{-(B-1)}, {}_B)$$ Any signal value falling between two levels is assigned to one of the two levels.

$X_Q=Q(x)$ is our notation for quantization. $e=Q(x)-x$ is then the quantization error.

One method of quantization is rounding , which assigns the signal value to the nearest level. The maximum error is thus $\frac{{}_B}{2}=2^{-B}$ .

Another common scheme, which is often easier to implement in hardware, is truncation . $Q(x)$ assigns $x$ to the next lowest level.

Overflow is the other problem. There are two common types: two's complement (or wraparound ) overflow, or saturation overflow.