# Relation between acoustical parameters Definition and Clarity [closed]

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.

• "help me understand this a bit better" is not a question. Are you asking for the mathematical relation b/w the two? Please ask a (conceptual) question. Thanks. Jan 13 at 0:28
• i want to know how to write $C_{50}$ in terms of $D_{50}$ Jan 13 at 0:31
• This is not the kind of question suitable for this site. There is no conceptual issue but just a request for help with math. If you write the integral from $0$ to $50ms$ as $A$, the integral from $50ms$ to $\infty$ as $B$, the integral from $0$ to $\infty$ as $A+B$, starting from the definition of $C_{50}$ and the formula you want to check, it should not be too difficult to verify their identity. Jan 13 at 9:26

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.