Relation between acoustical parameters Definition and Clarity [closed]

Question

In room acoustics, clarity ($C_{50}$) is defined as $$ C_{50} = 10*\log_{10}{\frac{\int_0^{50ms}p² (t)dt}{\int_{50}^{\infty}p² (t)dt}} $$

and definition ($D_{50}$) is $$ D_{50} = \frac{\int_0^{50ms}p² (t)dt}{\int_{0}^{\infty}p² (t)dt} * 100$$

So clarity is defined in deciblels, while definition is just a percentage. A lot of books say the relation between the two is: $$ C_{50} = 10 * \log_{10}{\frac{D_{50}}{1 - D_{50}}} $$

I tried a lot of different substitutions and using the property that $\int_{0}^{\infty}p² (t)dt$ = $\int_{0}^{50}p² (t)dt + \int_{50}^{\infty}p² (t)dt$, but I can't seem to derive the relation between these two parameters. I'd be glad if someone could help me understand this a bit better.

Answer 1

This is simply down to the confused way different sources are treating the $D_{50}$ quantity.

If you omit the multiplication by $100$ in the formula you gave for $D_{50}$ and substitute that into the $C_{50}$ formula you get the desired result.

It's the factor of $100$ to make a percentage that's causing the issue. It's only there to be "human friendly" and give a percentage and shouldn't be there from a mathematical point of view.