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Q (number format)

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A number of other notations have been used for the same purpose.

Definition

Texas Instruments version

The Q notation, as defined by Texas Instruments, consists of the letter Q followed by a pair of numbers m.n, where m is the number of bits used for the integer part of the value, and n is the number of fraction bits.

By default, the notation describes signed binary fixed point format, with the unscaled integer being stored in two's complement format, used in most binary processors. The first bit always gives the sign of the value(1 = negative, 0 = non-negative), and it is not counted in the m parameter. Thus, the total number w of bits used is 1 + m + n.

For example, the specification Q3.12 describes a signed binary fixed-point number with a w = 16 bits in total, comprising the sign bit, three bits for the integer part, and 12 bits that are the fraction. That is, a 16-bit signed (two's complement) integer, that is implicitly multiplied by the scaling factor 2−12

In particular, when n is zero, the numbers are just integers. If m is zero, all bits except the sign bit are fraction bits; then the range of the stored number is from −1.0 (inclusive) to +1.0 (exclusive).

The m and the dot may be omitted, in which case they are inferred from the size of the variable or register where the value is stored. Thus, Q12 means a signed integer with any number of bits, that is implicitly multiplied by 2−12.

The letter U can be prefixed to the Q to denote an unsigned binary fixed-point format. For example, UQ1.15 describes values represented as unsigned 16-bit integers with an implicit scaling factor of 2−15, which range from 0.0 to (216−1)/215 = +1.999969482421875.

ARM version

A variant of the Q notation has been in use by ARM. In this variant, the m number includes the sign bit. For example, a 16-bit signed integer would be denoted Q15.0 in the TI variant, but Q16.0 in the ARM variant.

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