The Q notation is a way to specify the parameters of a binary fixed point number format. Specifically, how many bits are allocated for the integer portion, how many for the fractional portion, and whether there is a sign-bit. For example, in Q notation, Q7.8 means that the signed fixed point numbers in this format have 7 bits for the integer part and 8 bits for the fraction part. One extra bit is implicitly added for signed numbers. Therefore, Q7.8 is a 16-bit word, with the most significant bit representing the two's complement sign bit.
There is an ARM variation of the Q notation that explicitly adds the sign bit to the integer part. In ARM Q notation, the above format would be called Q8.8. A number of other notations have been used for the same purpose.
Definition
General Format
U ⏟ u n s i g n e d Q m ⏟ i n t e g e r . n ⏟ f r a c t i o n {\displaystyle \underbrace {\mathrm {U} } _{\mathrm {\scriptscriptstyle unsigned} }\;\mathbf {Q} \;\underbrace {m} _{\mathrm {\scriptscriptstyle integer} }\;\;\mathbf {.} \;\underbrace {n} _{\mathrm {\scriptscriptstyle fraction} }}
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. As such, 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 word-size w = 16 bits in total, comprising the sign bit, three bits for the integer part, and 12 bits that are the fraction. This can be seen as a 16-bit signed (two's complement) integer, that is implicitly multiplied by the scaling factor 2 − 12 {\displaystyle 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 {\displaystyle 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 {\displaystyle 2^{-15}} , which range from 0.0 {\displaystyle 0.0} to ( 2 16 − 1 ) / 2 15 = + 1.999969482421875 {\displaystyle (2^{16}-1)/2^{15}=+1.999969482421875} .
ARM version A variant of the Q notation has been in use by ARM in which the m number also counts the sign bit. For example, a 16-bit signed integer which the TI variant denotes as Q15.0, would be Q16.0 in the ARM variant. Unsigned numbers are the same across both variants. While technically the sign-bit belongs just as much to the fractional part as the integer part, ARM's notation has the benefit that there are no implicit bits, so the size of the word is always m + n bits {\displaystyle m+n\ {\textrm {bits}}} .
Characteristics The resolution (difference between successive values) of a Qm.n or UQm.n format is always 2−n. The range of representable values depends on the notation used:
For example, a Q14.1 format number requires 14+1+1 = 16 bits, has resolution 2−1 = 0.5, and the representable values range from −214 = −16384.0 to +214 − 2−1 = +16383.5. In hexadecimal, the negative values range from 0x8000 to 0xFFFF followed by the non-negative ones from 0x0000 to 0x7FFF.
Math operations Q numbers are a ratio of two integers: the numerator is kept in storage, the denominator d {\displaystyle d} is equal to 2n. Consider the following example:
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