## 4 Normalization of Measurement Results
### Approximate Normalization of Reverberation Time Measurements  
As an alternative to adjusting the entire impulse response, it is possible to perform relatively simple corrections of the reverberation time parameters EDT, \(T_{20}\) and \(T_{30}\) and normalize to standard conditions. This is a simplified method that assumes nearly linear decay curves and applies approximation for the air attenuation described in Section 3.  
Figure 7 shows a generic decay curve. The time is \(t=0\) at the arrival of the direct sound, which is when decay starts. For the EDT, the time \(t_{2}\), when the decay is 10 dB down, is needed. For \(T_{20}\), the times \(t_{1}\) and \(t_{3}\), when the decay is 5 dB and 25 dB down, are needed, respectively. Similarly, for \(T_{30}\), \(t_{1}\) and \(t_{4}\), when the decay is 5 dB and 35 dB down, are needed, respectively. The most challenging part of the approximation method is to find the point where the decay curve is 5 dB down. This is achieved by assuming a straight line for the first part of the decay curve; then, \(t_{1}\) is simply \(t_{2}\) divided by two. It is shown in Table 2 how the various times can be derived from the measured reverberation time parameters.  
Figure 7: Generic decay curve with times related to reverberation time parameters. Below is an alternative axis for the distance of sound propagation.  
Since the speed of sound depends on the temperature, and the measurements can be made at other temperatures than 20 \({}^{\circ}\)C, it is necessary to change from the time domain to the distance domain using Equation (2). Generic decay curves before and after the normalization are displayed in the distance domain in Figure 8. The decay drop \(D_{i}\) is the same before and after the correction, but the distance \(r_{i}\) is changed to \(r_{i,\text{corr}}\).  
With the symbols in Figure 8, a geometrical analysis yields Equations (11) and (12).  
\[X_{i}+\Delta L_{i}=D_{i}+\Delta L_{0} \tag{11}\]  
\[\frac{r_{i}-d}{X_{i}}=\frac{r_{i,\text{corr}}-d}{D_{i}} \tag{12}\]  
The correction term \(\Delta L_{i}\) is the difference between the octave-band attenuation at measurement conditions and at standard conditions calculated with the approximation method, Equations (9) and (10). Similarly, \(\Delta L_{0}\) is the correction term in the distance \(d\). After the correction, the distance \(r_{i}\) is changed to  
\[r_{i,\text{corr}}=c_{0}\ t_{i,corr}+d \tag{13}\]  
where \(c_{0}\) is the speed of sound in standard conditions (343.2 m/s). Then, the corrected time \(t_{i,\text{corr}}\) is calculated with Equation (14).  
\[t_{i,\text{corr}}=\frac{c}{c_{0}}t_{i}\frac{D_{i}}{D_{i}+\Delta L_{0}-\Delta L _{i}} \tag{14}\]  
This is applied for \(i=1\) through 4. Finally, the equations for calculating the normalized reverberation times are as follows:  
\[\text{EDT}_{\text{corr}}=6\ t_{2,\text{corr}} \tag{15a}\] \[T_{20,\text{corr}}=3(t_{3,\text{corr}}-t_{1,\text{corr}})\] (15b) \[T_{30,\text{corr}}=2(t_{4,\text{corr}}-t_{1,\text{corr}}) \tag{15c}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Time \(t_{i}\)** & **Level, dB** & **Equation for \(t_{i}\)** \\ \hline \(t_{1}\) & \(-5\) & \(\
icefrac{{\mathrm{\
u}}}{{4}}\)\(t_{2}\) \\ \(t_{2}\) & \(-10\) & EDT/6 \\ \(t_{3}\) & \(-25\) & \(T_{20}/3+t_{1}\) \\ \(t_{4}\) & \(-35\) & \(T_{30}/2+t_{1}\) \\ \hline \end{tabular}
\end{table}
Table 2: Times and levels on the decay curve, and their relation to reverberation parameters.  
Figure 8: Generic decay curve before the correction (full line) and after the correction (dotted line). The distance axis is used instead of the time axis.  
The method described in this section considers the change in the speed of sound due to different temperatures and the change of air attenuation due to temperature and relative humidity. It is noted that the distance \(d\) from source to receiver should be known for accurate results at high frequencies. However, in some cases, the distance can be neglected, namely when the distance to the knee point \(r_{\rm k}\) (8a) is longer than any of the distances \(r_{i}\).