Is there an expansion for element-wise scaled convolution?

Is there an expansion for element-wise scaled convolution?

If $x = a\cdot b$ is used to indicate $x_i = a_i\cdot b_i$,
$y = a / b$ denotes $y_i = a_i / b_i$, and $a*b$ denotes convolution,
then is there a simplification for this expression:
$$ \frac{(a\cdot b)*(c\cdot d)}{b*d} $$
It was tempting to assume this $=a*c$, but I couldn't find any property
for it (the Wikipedia page for convolution only shows scalar
multiplication, but not anything more).
The reason is that I want to compute the expression shown, but some
elements of $b$ and $d$ are very close to zero.
Thanks a lot for any advice you can provide.