# An Audio Distortion Dynamic Range Index for Evaluating the Performance of Audio Systems
## 2 Definition of the New Index
### Obtain the ADDR Plot with the Spectrum  
The input signal with the largest full-scale amplitude usually corresponds to the maximum distortion of the DUT, and the distortion behavior of the DUT can be extracted to the greatest extent. For a certain DUT, the ADDR equals to the SINAD and the SNR when the threshold equals to the maximum and the minimum of its range, respectively. Because the threshold in the ADDR definition is not fixed, the result of the ADDR index is a function of threshold, in which the ADDR changes monotonously. Moreover, the frequency response of the DUT is probably not flat, a more comprehensive measurement should be a function of frequency.  
Take the averaged spectrum of a DUT's output signal shown in Figure 1a as an example, the detailed steps of obtaining the corresponding ADDR plot illustrated in Figure 1b are as follows. Assume \(X(f)\) is the averaged spectrum of the DUT output at frequency \(f\).  
Step 1: Determine the span of the FFT bins of the fundamental signal of frequency \(f\);  
Step 2: Compute the power of the fundamental signal by the sum-of-squares of corresponding FFT bins: \(P_{s}\);  
Step 3: Set the minimum of threshold \(T_{min}\) as the magnitude of the highest noise spectral component and the maximum threshold \(T_{max}\) as the maximum magnitude of the distorted spectral component, respectively;  
Step 4: Determine the step size sequence of the moving threshold: \(T_{step}\);  
Step 5: Initialize threshold \(T\) as the maximum of distorted spectral components: \(T=T_{max}\);  
Step 6: Select the distorted spectral components that equal to or lower than threshold \(T\);  
Step 7: Compute the power summation of the selected distorted spectral components by the sum-of-squares of selected FFT bins: \(P_{distorted}\);  
Step 8: Compute the ADDR result corresponding to this specified threshold \(T\): \(ADDR=P_{s}/P_{distorted}\);  
Step 9: Decrease threshold \(T\) to \(T_{min}\) in steps of corresponding \(T_{step}\), and repeat steps 6, 7 and 8 after each decrement to obtain the ADDR plot as a function of threshold of the DUT.  
Figure 1a depicts that the range of the threshold is from the maximum of the harmonic, spurious components to the highest noise components. The ADDR of the DUT equals to the SINAD and the SNR when at its minimum and maximum, respectively.  
Figure 1b verifies that the ADDR index unifies the SNR and SINAD indexes when at its two extremes. Moreover, the ADDR plot depicts a subdivision process of the distortion between the SNR and the SINAD. The dotted line separates the ADDR plot into part 'A' and 'B', part 'A' depends on the noises of the DUT while part 'B' depends on the harmonic and spurious components. The ADDR  
Figure 1: (**a**) Averaged spectrum of DUT output. (**b**) ADDR plot, ADDR as a function of the threshold. Testing signal: 996.1 Hz, 2 V\({}_{\text{PP}}\); Band: DC-20 kHz; Sampling frequency: 40 kHz; Sampling points: 2048; Spectrum averaged times: 100.  
values between two ends are equivalent to multiple indexes similar to the SNR index. Thus, the ADDR plot is to a great extent a comprehensive depiction of the performance of the DUT.  
Please note that the definition of ADDR only considers the power of distorted spectral components rather than the different influences of different frequencies for a certain target application. If necessary, it can be balanced by a weighting function multiplied on the spectrum. For example, if DUTs are compared from the perspective of end-users' experience, psychoacoustic unevenness of distorted frequencies should be considered [22].