Fixed Point Arithmetic Routines

Introduction

This library implements fixed point arithmetic in C, along with vectors and quaternions. The format for fixed point is the "Q format".

Check out the Doxyren reference at https://jcarrano.github.io/fixed_point_arith/

Acknowledgements

Most of this code was written between March and July 2013 by Juan I. Carrano, with contributions from Andrés Calcabrini, Juan I. Ubeira and Nicolás Venturo.

Fractional (fixed point) numbers and Q format

Fixed point binary numbers are able to represent numbers of the form

     N-1
x = ± Σ a(i)*2**(i+r)
     i=0

Where r is generally less than 0. The a(i)'s are either 0 or 1.

One possibility for a computer representation would be to form a binary number out of the a(i)'s plus a sign bit. A better alternative is to use two's complement.

For all N bits binary numbers with bits `b(N-1) ... b2 b1 b0` and two constants p and q, such that `p+q = N`, the Qp.q format tells us how to interpret those bits to find out which number ther represent. Note the distinction between the binary number itself and the value which it represents. For example, if we have a 32 bit value we can think of it as a signed integer, unsigned integer, a IEEE floating point number or just a bit mask.

The Qp.q format can be defined in terms of the two's complement representation of the number, which we call B. Then, in the Qp.q format, the value is `B/(2**q)`. We can see that p and q have an interpretation: q is the number of fractional bits and p of integer bits (plus the 'sign' bit).

Signed integers can be seen as a special case of Q numbers: `QN.0`.

Fixed point representations have some advantages over floating point:

• The resolution is constant.
• Many CPUs, especially microcontrollers, do not have hardware floating point support. Soft fixed point is much faster than soft floats.
• The addition and substraction operations are exactly the same as for integers.
• If the format can only represent values in the range (-1,1), the the product of two such numbers will always be in-range.

Operations on fixed point numbers

The C programming language does not natively support fractional types. Thus, we must define fractional operations in terms of integer operations. Earlier we mentioned that we could use an integer B to represent a value of B/(2**q). Note, however, that from the compiler's perspective, this object will always be an integer. It will behave as a fractional type only if we apply fractional operations to it.

Addition

Substracion

The same as addition

Product (fractional by fractional -> fractional)

Product (fractional by integer -> integer)

Quotient

dfsdf

The quotient operations is not implemented in this library at the moment.

Quaternions

References