Q-Format Converter

The three rules of Q arithmetic

A Qn number stores $x\cdot 2^n$ in an integer, covering $[-1,1-2^{-n}]$ with resolution $2^{-n}$. The rules that prevent 90% of fixed-point bugs:

1. Addition — same format in, same format out, but the result can overflow: adding two Q15 values needs either saturation or one guard bit. 2. Multiplication — formats add: $Q15\times Q15=Q30$, so the int32 product must be shifted right by 15 (and rounded: add $2^{14}$ before the shift) to come back to Q15. 3. The asymmetry — −1.0 is representable but +1.0 is not, and $(-1)\times (-1)$ overflows: the only multiply that saturates in hardware MAC units. That's why CMSIS-DSP's __SSAT exists.