# How do I calculate the bits of resolution in my derived value?

It might be a simple question but I couldn't find it via google, so I might not use the right terms.

Say I have two measurement values $$P$$ and $$Q$$. $$P$$ has 7 bits of resolution and $$Q$$ has 6 bits. So P can have all values from 0 up to $$2^7 -1$$, and Q can have all values from 0 up to $$2^6 -1$$.

Now I calculate a derived value $$R$$ through $$R = aP + bQ + c$$, a, b and c are constants.

How many bits of resolution does $$R$$ have? $$\min(6,7)$$, $$\max(6,7)$$ or $$6 + 7$$?

I know that $$R$$ can have $$2^{13}$$ different values, but are the least significant bits really... significant? Can I actually make decisions based on those bits?

That depends largely on the constants and how you implement the multiplication. If you multiply two signed fixed point numbers with lengths of $$B_1$$ and $$B_2$$ the result will be a signed fixed point number with a length of $$B_1+B_2=1$$ bits.
If you just sum $$P$$ and $$Q$$ the resulting resolution would be
$$B_{sum} = \log_2\left(2^{B_P-1} + 2^{B_Q-1} \right) + 1 = 7.585$$