# An Audio Distortion Dynamic Range Index for Evaluating the Performance of Audio Systems
## 3 Critical Influence Factors
### Number of Sampling Points  
The impact of the sampling points \(N\) on the ADDR is mainly in four aspects. The first two aspects require \(N\) to be large enough, and the latter two aspects reflect the adverse effects of \(N\) being too large.  
(a) The ADDR comparison between DUTs must be with the same and large enough \(N\). The magnitude of the noise floor is affected by \(N\) due to the existence of processing gain [23]. For a wideband noise floor, \(PG\) can be calculated by  
\[RG(\text{dB})=10\text{log}_{10}\left(\frac{N}{2}\right) \tag{5}\]  
Large enough \(N\) can \"pulling out\" the harmonic and spurious from the background spectral noise floor so as to make the ADDR analysis meaningful [32].  
(b) \(N\) affects the subdivision ability of ADDR. Frequency resolution \(f_{0}\) is the basis of ADDR comparison, and decreasing \(f_{0}\) can be achieved by increasing \(N\).  
\[f_{0}=\frac{f_{s}}{N} \tag{6}\]  
\(f_{0}\) should be small enough to distinguish the fundamental, harmonic and spurious spectral components, which enables the depiction of all the detailed distorted behavior of the DUT.  
(c) Part 'A' in Figure 1b represents the SNR impact on the ADDR. The existence of \(PG\) makes the area of this part increase for a larger \(N\), which implies the amplification of the noise weight, therefore the effect of \(N\) on the ADDR is enhanced. This is reasonable because it happens to match its importance to audio systems. However, the enhancement should not be too large, a suitable \(N\) makes the noise floor about 10 dB to 20 dB lower than the smallest visible harmonic. Under such conditions, the result of the ADDR area comparison is more reasonable.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \(\beta\) & 0 & 1 & 3 & 5 & 7 & 9 & 12 & 15 & 18 & 21 & 27 & 33 \\ \hline \hline \(\text{CPG}\) (dB) & 0 & 0.65 & 3.3 & 5.28 & 6.67 & 7.71 & 8.93 & 9.87 & 10.65 & 11.31 & 12.39 & 13.26 \\ \hline \(\text{Scalo}pLoss\) (dB) & 3.92 & 3.64 & 2.5 & 1.75 & 1.33 & 1.07 & 0.82 & 0.67 & 0.56 & 0.49 & 0.38 & 0.32 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Correction Factors for Kaiser Windows with Parameter \(\beta\) (exact for large window lengths N).  
(d) It had been proved that \(N\) has a relationship with the accuracy of the test signal frequency [30], as are shown in (7) and (8).  
\[\frac{\Delta f}{f}\leq\frac{1}{4(N-1)},\ (M=1) \tag{7}\]  
\[\frac{\Delta f}{f}\leq\frac{1}{4(M-1)N},\ (M>1) \tag{8}\]  
where \(\Delta f\) is the maximum allowable error in the signal frequency, and \(M\) is the number of cycles in the data record. Equations (7) and (8) show that with larger values of \(N\), a fewer number of cycles will be required to obtain any given accuracy, but higher accuracy of the signal frequency will be required [30]. Signal generators with the characteristics of high output frequency accuracy and high adjustability of the fractional part can provide viable support for meeting the above-mentioned critical and strict constraints.  
In summary, without considering the calculation complexity, if the accuracy of the input signal is high enough, moderately increasing the number of sampling points is beneficial for the ADDR measurement. The best approach is to ensure that the noise floor drops in a reasonable range of positions, and use the largest value of \(N\) compatible with the frequency accuracy obtainable.