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Fixed-Point Quantization

Module by: Douglas L. Jones. E-mail the author

Summary: Finite word lengths introduce quantization error in fixed-point systems. Truncation quantization causes a larger maximum error and a negative bias compared to rounding, but is easier to implement in hardware. Similarly, wraparound overflow is typically worse than saturation, but also requires more hardware.

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 =Q⁢x X Q Q x is our notation for quantization. e=Q⁢x−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 .

Figure 1
(a) (b)
Figure 1(a) (subfig1aFixed-PointQuant.png)Figure 1(b) (subfig1bFixed-PointQuant.png)

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

Figure 2
(a) (b)
Figure 2(a) (subfig2aFixed-PointQuant.png)Figure 2(b) (subfig2bFixed-PointQuant.png)
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.

Figure 3
wraparoundsaturation
(a) (b)
wraparound (subfig3aFixed-PointQuant.png)saturation (subfig3bFixed-PointQuant.png)
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 -1 1 -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.

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