# Perceptual Similarities between Artificial Reverberation Algorithms and Real Reverberation
## 2 Background
### Delay Network Reverberation
\begin{table}
\begin{tabular}{c c c c c} \hline
**Gardner** & **Filter** & **Delay (ms)** & **Delay Length** & **Gain** \\ \hline \multirow{8}{*}{Small-size-room} & Lowpass1 & & & 0.6 \\  & Delay & 24 & \(24\times 8\) & - \\  & DNA\_O & 35 & \(35\times 8\) & 0.3 \\  & DNA\_I1 & 22 & \(22\times 8\) & 0.4 \\  & DNA\_I2 & 8.3 & \(8.3\times 8\) & 0.6 \\  & SNA\_O & 66 & \(66\times 8\) & 0.1 \\  & SNA\_I & 30 & \(30\times 8\) & 0.4 \\  & Lowpass2 & & & 1/4200 \\  & GAIN & - & - & 0.5 \\ \hline \multirow{8}{*}{Medium-size-room} & DNA\_O & 35 & \(35\times 15\) & 0.3 \\  & DNA\_I1 & 8.3 & \(8.3\times 15\) & 0.7 \\  & DNA\_D2 & 22 & \(22\times 15\) & 0.5 \\  & Delay1 & 5 & \(5\times 15\) & - \\  & Allpass1 & 30 & \(30\times 15\) & 0.5 \\  & Delay2 & 67 & \(67\times 15\) & - \\  & Delay3 & 15 & \(15\times 15\) & - \\  & SNA\_O & 39 & \(39\times 15\) & 0.3 \\  & SNA\_I & 9.8 & \(9.8\times 15\) & 0.6 \\  & Delay4 & 108 & \(108\times 15\) & - \\  & Lowpass & & & 1/2500 \\  & GAIN & - & - & 0.5 \\ \hline \multirow{8}{*}{Large-size-room} & Lowpass1 & & & 0.6 \\  & Allpass1 & 8 & \(8\times 15\) & 0.3 \\  & Allpass2 & 12 & \(12\times 15\) & 0.3 \\  & Delay1 & 4 & \(4\times 15\) & - \\  & Delay2 & 17 & \(17\times 15\) & - \\  & SNA\_O & 87 & \(87\times 15\) & 0.5 \\  & SNA\_I & 62 & \(62\times 15\) & 0.25 \\  & Delay3 & 31 & \(31\times 15\) & - \\  & Delay4 & 3 & \(3\times 15\) & - \\  & DNA\_O & 120 & \(120\times 15\) & 0.5 \\  & DNA\_I1 & 76 & \(76\times 15\) & 0.25 \\  & DNA\_I2 & 30 & \(30\times 15\) & 0.25 \\  & Lowpass2 & & & 1/2600 \\  & GAIN & - & - & 0.34 \\ \hline \end{tabular} Note1: The DNA presents the double nested allpass filter, the SNA presents the signal nested allpass filter, and O presents the outer and I presents the inner. GAIN present all gain filters in each structure.  
\end{table} 
Table 5: The parameter values of the the late reverberation filters of the Gardner reverberation algorithm [7]. The input gain (\(b_{i}\)), output gain (\(c_{i}\)), and direct sound gain (\(d\)) are all set as 1 in this structure, and can be adjusted as required. The total minimum delay length \(M_{min}\) is defined as  
\[M\_min=\lceil 0.15\times Fs\times RT_{60}\rceil \tag{2}\]  
where \(\lceil\rceil\) indicates plus one for rounding; \(d_{i}\) is the preliminary estimate of the non-prime-number delay length for each delay line, and is expressed as  
\[d_{i}=\lceil M\_min/N+(i-1)\times M\_min\div(N\times 5)\rceil. \tag{3}\]  
\(ppwr_{i}\) is the index of the position of the prime number to be drawn from the set of prime numbers, given by  
\[ppwr_{i}=[log(d_{i})\div log(pri_{i})] \tag{4}\]  
where prime \(pri_{i}\) is 2, 3, 5, 7, 11, etc., \([]\) means rounding to the nearest whole number, and \(m_{i}\) is the delay of each delay line, which is a power of a distinct prime provided by  
\[m_{i}=pri_{i}^{ppwr_{i}}. \tag{5}\]  
\(g_{dB}\) is a per-sample attenuation gain, \(g_{lin}\) is a linear scale gain converted from a logarithmic scale gain, and \(g_{i}\) is the target gain of each delay line, given respectively as  
\[g_{dB}=\frac{-60}{RT_{60}\times Fs} \tag{6}\]  
\[g_{lin}=10^{\frac{g_{dB}}{20}} \tag{7}\]  
\[g_{i}=(g_{lin})^{m_{i}} \tag{8}\]  
[PERSON] published a topology structure for a reverberation algorithm in 1997 [9]. The implementation of this reverberation structure is composed of a pre-delay followed by a low-pass filter, a decorrelation stage, and a 'tank implementation', as shown in Figure 7[27]. The decorrelation stage includes four cascaded all-pass filters which can  
Figure 6: The structure of Feedback Delay Network (FDN) [10,41], where the part A represents the feedback coefficient matrix. perform a rapid build-up of echo density. The decorrelation stage does not have a feedback loop, and as such is straightforward; however, the 'tank' is more complex because of its recursive structure [27]. The 'tank implementation' consists of two cross-coupled symmetrical lines, which cause the infinite recirculation of the input signal.