# 2.1 Fixed-point quantization

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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 $B$ -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.

The worst-case error with truncation is $=2^{-(B-1)}$ , which is twice as large as with rounding. Also, the error is always negative, so on average it may have a non-zeromean (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 easierto implement in hardware. It also has the advantage that if the sum of several numbers is between $\left[-1 , 1\right)$ , the final answer will be correct even if intermediate sums overflow! However, wraparound overflow leaves IIR systemssusceptible to zero-input large-scale limit cycles, as discussed in another module. As usual, there are many tradeoffs to evaluate, andno one right answer for all applications.

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