Determine the Range of Fixed-Point Numbers

Fixed-point variables have a limited range for the same reason they have limited precision — because digital systems represent numbers with a finite number of bits. As a general example, consider the case where an integer is represented as a fixed-point word of size ws. The range for signed and unsigned words is given by

$$\max \left(Q\right)-\min \left(Q\right),$$

where

$$\begin{array}{c}\min \left(Q\right)=\{\begin{array}{cc}\text{0 }& \text{ unsigned,}\\ -2^{ws-1}& \text{ signed,}\end{array}\\ \max \left(Q\right)=\{\begin{array}{cc}{2}^{ws}-1& \text{ unsigned,}\\ 2^{ws-1}-1& \text{signed}\text{. }\end{array}\end{array}$$

Using the general [Slope Bias] encoding scheme described in Scaling, the approximate real-world value has the range

$$\max \left(\tilde{V}\right)-\min \left(\tilde{V}\right),$$

where

$$\begin{array}{c}\min \left(\tilde{V}\right)=\{\begin{array}{cc}\text{B }& \text{ unsigned,}\\ -F{2}^E\left(2^{ws-1}\right)+B& \text{ signed,}\end{array}\\ \max \left(\tilde{V}\right)=\{\begin{array}{cc}F{2}^E\left(2^{ws}-1\right)\text{+B }& \text{ unsigned,}\\ F{2}^E\left(2^{ws-1}-1\right)\text{+B }& \text{signed}\text{. }\end{array}\end{array}$$

If the real-world value exceeds the limited range of the approximate value, then the accuracy of the representation can become significantly worse.