Bit accuracy improvement versus sample averaging

Supposing we have multiple samples of a scaled signal $x$ given by $y_i=a_ix+n_i$ where $n_i$ is AWGN how does the bit accuracy of estimation of $x$ improve with averaging of samples $y_i$ (number of bits per sample)? Assume we know $a_i$.

Say I want $x$ with an error affecting only $m$ bits does $\log m$ samples of $y_i$ suffice?

• Can you clarify more (mathematically), what you mean by bit accuracy and what you would average to increase accuracy? Do the $a_i$ change over time/measurements? Jan 8, 2017 at 9:19
• @MaximilianMatthé $a_i$ changes but we know $a_i$ and so for all purposes we can think of $a_i=1$ I think. Jan 8, 2017 at 9:57