Calculation of sound level in dB

Question

I was never good in math, so I need help how to convert linear value to dB.

First I show you how I calculated the linear value from -24dB.

var dB, linearLevel: integer;
begin
dB := -24;
linearLevel := Round( 1/exp( 2.30258509299 * (abs(dB)/20) ) *32767);
end;

Where 1/exp( 2.30258509299 * (abs(-24)/20) ) ; is normalized value.

For conversion to sample value I use 1/exp(...)*32767

My problem is here: conversion back

I was told to use formula 20*log10(linearLevel). So I tried to create it but my result is 18 instead 24 (or -24) dB.

linearValue := 2067; // the result of the formula above    
db := round( exp( 20*2.30258509299*(linearLevel / 32767) ) );

How to calculate the dB?

Answer 1

If

linearLevel := Round( 1/exp( 2.30258509299 * (abs(dB)/20) ) * 32767);

we lose information due to the rounding (in the language of mathematics, the Round function is not injective). So given a linearLevel value, it is impossible to get back the original dB value. Therefore, let us consider

linearLevel := 1/exp( 2.30258509299 * (abs(dB)/20) ) * 32767;

instead. This implies

linearLevel / 32767 := 1/exp( 2.30258509299 * (abs(dB)/20) )

and

32767 / linearLevel := exp( 2.30258509299 * (abs(dB)/20) )

and

ln(32767 / linearLevel) := 2.30258509299 * (abs(dB)/20)

and

ln(32767 / linearLevel) / 2.30258509299 := abs(dB)/20

and

20 * ln(32767 / linearLevel) / 2.30258509299 := abs(dB).

Here we again have an issue, since the absolute value function is not injective. If abs(dB) is 7, we cannot possibly tell if dB is 7 or -7.

But if we assume that dB is non-positive, we finally have

dB = -20 * ln(32767 / linearLevel) / 2.30258509299.

Simplifications

Since 2.30258509299 is ln(10), this is

dB = -20 * ln(32767 / linearLevel) / ln(10).

But log10(x) = ln(x) / ln(10), so we can write

dB = -20 * Log10(32767 / linearLevel)

where the Log10 function is found in the Math unit.

Also, using the law a log(b) = log(b^a) in the case a = -1, we can even write

dB = 20 * Log10(linearLevel / 32767).