Numerical Formats

Integer/Fractional Relationship

The same two's complement N-bit word may represent an integer value or a fractional value. The same arithmetic and logic hardware may be used with either format. We just interpret the results differently.

(1) x_FRACTION=x_INTEGER/2^N−1

(2) x_INTEGER=x_FRACTION∗2^N−1

For N = 16 bits, 2^N-1 = 32768

1.15(Q15) Signed Fractional Format

To understand how Q15 numbers are handled, let's assume we have a 16-bit integer in 2's complement format where s is a sign bit and a_n is the digit at bit n.

(3) a = <sa₁₄a₁₃...a₁a₀>

For a positive integer a, the value of the number a is:

(4) a = a₁₄2¹⁴+a₁₃2¹³+...+a₁2¹+a₀2⁰

For a Q15 number, the weightings of each bit change. Let b = a / 2¹⁵, then the value of a positive Q15 number b is:

(5) b = a₁₄2⁻¹+a₁₃2⁻²+...+a₁2⁻¹⁴+a₀2⁻¹⁵

The maximum negative value that can be handled is 0x8000 = -1.000000000000 and the maximum positive value is 0x7FFF = 0.999969482422. Zero is represented by 0x0000.