# An Audio Distortion Dynamic Range Index for Evaluating the Performance of Audio Systems
## 3 Critical Influence Factors
### Leakage  
For an \(N\) points finite sequence sampled at a sampling frequency of \(f_{s}\). The DFT processing generates an \(N\) point transformation, and the frequency corresponding to the output sequence is  
\[F(m)=m\left(\frac{f_{s}}{N}\right),\ (m=0,1,2,3,\ldots,N-1) \tag{2}\]  
The condition that DFT can correctly generate amplitude information is that the energy of the input data sequence is accurately included in the frequencies given by (2) [23]. In other words, the frequency of sampled signal \(f_{i}\) is at an integer multiple of the frequency resolution. For a single sine signal input, the output of the DUT includes not only the fundamental signal, but also the unwanted harmonic, spurious and noise components. It is rather difficult to ensure that all signals do not have spectral leakage, but its influence on the ADDR index is different for each spectral component. Generally, the higher the power of the FFT bin is, the more obvious the effect of the FFT bin is. Because the proposed ADDR index uses a single pure sinusoidal test signal, the output power of the DUT is mainly concentrated at the input signal frequency and its harmonics, and the power of the noises is relatively much lower. Therefore, the leakage caused by the fundamental signal and its harmonics should be the focus of attention, and the leakage caused by the noises is negligible [24]. The leakage can be reduced by various omethods [25; 26; 27; 28; 29], which are mostly based on the estimation of fundamental frequency. However, the optimal measurement condition is to make the sampling process satisfy the requirement of coherent sampling for both the fundamental frequency and its harmonic frequencies simultaneously, which significantly increases the spectral resolution of the FFT and creates an ideal environment for critically ADDR performance evaluation of the DUT. The frequencies of the set of harmonics \(f_{h}\) are the integer multiples of the fundamental frequency \(f_{i}\). Therefore, the condition making the fundamental wave satisfy coherent sampling also ensures that the harmonics meet the coherent sampling condition. The coherent sampling of a pure single sine test signal occurs when the number of cycles in the sampling window is an integer. \[\frac{f_{i}}{f_{s}}=\frac{M}{N} \tag{3}\]  
where \(M\) is the integer number of cycles in the data records. Although the coherence relationship is applicable to any \(M\) and \(N\), the optimized choice of \(M\) and \(N\) can further improve the accuracy and the computational complexity. First, the optimized value of \(N\) is a power of 2 because of the inherent periodicity of FFT. Secondly, the optimized value of \(M\) is an odd or prime, which can eliminate part even all the common factors of \(N\). This is because the common factors between \(M\) and \(N\) lead to different harmonics of \(f_{i}\) in the same frequency bin during the FFT after aliasing [23]. The magnitude accuracy of the fundamental signal is highly related to the accuracy of the ADDR plot. Figure 3 shows the leakage effect on the magnitude error percentage for a shift in \(M\) from \(M-0.5\) to \(M+0.5\). In practice, these shifts of \(M\) represent an input frequency range from about 986 Hz to about 1006 Hz. The vertical axis error percentage is the result of the corresponding horizontal axis changing from \(M-0.5\) to \(M+0.5\). Since the signal magnitude is known, the optimal condition is to achieve 0% magnitude error. Indeed, the minimum of the magnitude error achieves 0.0002% only when the frequency is close to 996.1 Hz. The maximum percentage of the magnitude error is more than 35%, and only the samples of \(M\) between the very slim region formed by these two dotted lines achieves a magnitude error percentage smaller than 0.1%. In summary, the ADDR index is very sensitive to the leakage.