Fixed point saturation and round
I think this overly conservative. If you apply it to an IIR filter, you would conclude infinite bit growth which isn't the case.
For a filter with impulse response, $h[n]$, the &quot;worst case&quot; gain the filter can provide to the input is given as the absolute some the coefficients, i.e.
$$G_{max} = \sum |h[n]| $$
This means that if $x[n] &lt; x_{max}$ then $y[n] &lt; x_{max} \cdot G_{max} $ for all $n$.
You are guaranteed not to overflow as long as you accommodate this maximum gain.
Even that's too conservative for most applications. The input signal that will create this maximum gain is basically $sign(h[-n])$ which is extremely unlikely to occur. So in practice a statistical approach works better: Run a sizable number of representative input signals through the filter and look at the probability distribution function of the output.
At the end of the day this is a trade off between the noise from occasionally clipping an output sample vs the quantization/rounding noise you create on every output sample.