Determine the Range of Fixed-Point Numbers

R2026b

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, Range, and Precision, 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.

In MATLAB^®, you can use the fixed.extractNumericType function to extract the numeric type from an input, then use the lowerbound and upperbound functions to determine the minimum and maximum finite representable values of the numeric type. The input can be specified in a number of ways, such as:

u = numerictype(1,8,3)
u = fixdt(0,5,1)
u = fi(pi,1,6)
u = 'sfix4_En2'
u = int8(4)
u = pi
u = half(3)
u = 'single'

For this example, consider the input u = 'sfix5_En1'.

u = 'sfix5_En1';
nt = fixed.extractNumericType(u);
minFinRep = lowerbound(nt)
maxFinRep = upperbound(nt)

minFinRep = 

    -8

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 5
        FractionLength: 1

maxFinRep = 

    7.5000

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 5
        FractionLength: 1

Alternatively, use the range function to determine the minimum and maximum finite representable values of the numeric type.

minMaxFinRep = range(nt)

minMaxFinRep = 

   -8.0000    7.5000

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 5
        FractionLength: 1

Use the stripscaling function to determine the minimum and maximum stored integer values of the numeric type.

siMinMaxFinRep = stripscaling(range(nt))

siMinMaxFinRep = 

   -16    15

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 5
        FractionLength: 0