The fractional BB-bit two's complement number representation evenly distributes 2B 2 B quantization levels between -1 -1 and 1-2-B-1 1 2 B 1 . The spacing between quantization levels is then 22B=2-B-1≐ Δ B ≐ 2 2 B 2 B 1 Δ B Any signal value falling between two levels is assigned to one of the two levels.

X Q =Qx X Q Q x is our notation for quantization. e=Qx-x 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 Δ B 2=2-B Δ B 2 2 B .

Another common scheme, which is often easier to implement in hardware, is truncation. Qx Q x assigns xx to the next lowest level. The worst-case error with truncation is Δ=2-B-1 Δ 2 B 1 , which is twice as large as with rounding. Also, the error is always negative, so on average it may have a non-zero mean (i.e., a bias component).

Overflow is the other problem. There are two common types: two's complement (or wraparound) overflow, or saturation overflow. Obviously, overflow errors are bad because they are typically large; two's complement (or wraparound) overflow introduces more error than saturation, but is easier to implement in hardware. It also has the advantage that if the sum of several numbers is between -11 -1 1 , the final answer will be correct even if intermediate sums overflow! However, wraparound overflow leaves IIR systems susceptible to zero-input large-scale limit cycles, as discussed in another module. As usual, there are many tradeoffs to evaluate, and no one right answer for all applications.