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

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

where

$\begin{matrix} \min (Q) = {\begin{matrix} 0 & unsigned, \\ - 2^{w s - 1} & signed, \end{matrix} \\ \max (Q) = {\begin{matrix} 2^{w s} - 1 & unsigned, \\ 2^{w s - 1} - 1 & signed . \end{matrix} \end{matrix}$

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

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

where

$\begin{matrix} \min (\tilde{V}) = {\begin{matrix} B & unsigned, \\ - F 2^{E} (2^{w s - 1}) + B & signed, \end{matrix} \\ \max (\tilde{V}) = {\begin{matrix} F 2^{E} (2^{w s} - 1) +B & unsigned, \\ F 2^{E} (2^{w s - 1} - 1) +B & signed . \end{matrix} \end{matrix}$

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