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

`$\mathrm{max}\left(Q\right)-\mathrm{min}\left(Q\right),$`

where

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

`$\mathrm{max}\left(\stackrel{˜}{V}\right)-\mathrm{min}\left(\stackrel{˜}{V}\right),$`

where

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