# A Psychoacoustic-Based Multiple Audio Object Coding Approach via Intra-Object Sparsity
## Appendix A Sparsity Analysis of Audio Signal in the MDCT Domain
### Statistical Analysis Results  
To reveal the superior properties of MDCT, in each frame, 315 mono audio recordings selected from University of Iowa Music Instrument Samples (Iowa-MIS) audio database [37] sampled at 44.1 kHz and 100 mono speech recordings selected from Nippon Telegraph & Telephone (NTT) database are chosen as the test data. The selected audio recordings contain 7 types of tonal solo instruments. In this statistics work, a 2048-point STFT and MDCT basis with 50% overlapping is applied to form the time-frequency instants. Meanwhile, a KBD window with the size of 2048 points is used as the window function to meet the demand of overlap-add. A statistical analysis of \(N\!PTF\) is taken with the \(F\!EPR\) ranged from 98% to 80%. Results are shown in Figure 11 with 95% confidence intervals. Note that STFT-domain descriptions corresponding to instruments or speech are respectively denoted by 'Flute-STFT', 'Violin-STFT', 'Sax-STFT', 'Oboe-STFT', 'Trombone-STFT', 'Trumpet-STFT', 'Horn-STFT' and 'Speech-STFT'. In contrast, MDCT-domain representations are respectively regarded as 'Flute-MDCT', 'Violin-MDCT', 'Sax-MDCT', 'Oboe-MDCT', 'Trombone-MDCT', 'Trumpet-MDCT' and 'Speech-MDCT'.  
Figure 16 indicates that by decreasing \(FEPR\), the averaged \(NPTF\) degrades as well. More precisely, \(NPTF\) is a convex function as \(FEPR\) decreases uniformly in terms of all test instruments and speech, that is, audio object or speech signal are sparse both in STFT and MDCT domain. Furthermore, it shows that there exists a noticeable difference between adjacent light color and dark color bars, in other words, the averaged \(NPTF\) in the MDCT domain is much lower than that in the STFT domain for each instrument and speech with a certain \(FEPR\).  
While the energy compaction property of MDCT is fairly intuitive, it becomes agnostic as the \(FEPR\) changes. To measure the disparity between the averaged \(NPTF\) for MDCT coefficients and STFT coefficients of audio signal with a known \(FEPR\), a Normalized Relative Difference Ratio (NRDR) is defined as (\(k\) is \(NPTF\) and \(r_{FEPR}\) is \(FEPR\)):  
\[NRDR(r_{FEPR})=\frac{k(r_{FEPR})_{\text{STFT}}-k(r_{FEPR})_{\text{MDCT}}}{k(r_{ FEPR})_{\text{STFT}}} \tag{10}\]  
where \(k(r_{FEPR})_{\text{STFT}}\) and \(k(r_{FEPR})_{\text{MDCT}}\) are the averaged \(NPTF\) for an audio signal in the STFT and MDCT domain with a certain \(FEPR\), respectively. NRDR is the difference between them. The larger the NRDR is, means that the less \(NPTF\) needed in the MDCT domain. Then, a statistical bar graph is presented which reflects the relationship between NRDR and \(FEPR\).  
Results are shown in Figure 17 with different NRDR at \(r_{FEPR}\) = 98-80%. It can be observe that the NRDR of all tested audio signals are non-negative, which means that the averaged \(NPTF\) in the MDCT domain is higher than that in the STFT domain. This result testifies that the performance of MDCT is absolutely dominant for all of the tested 8 items.  
Interestingly, we find that NRDR is gradually increasing as \(r_{FEPR}\) uniformly decrease from 98% to 88%. When 80% \(\leq r_{FEPR}\leq\) 88%, the NRDR maintains at the same level or slightly grow. Videlicet, with the decrement of \(FEPR\), the superiority of MDCT is becoming increasingly obvious.  
The next phenomenon needs to be noted is that the sparsity of violin and trumpet is particularly evident in the MDCT domain, because their NRDR can reach up to 60% when \(r_{FEPR}\) = 80% whilst other instruments can only achieve roughly 45%-55%. Besides, the sparseness of selected speech signals is weaker than all instruments in the MDCT domain but maintain consistency as far as the global regularity.  
Hence, the results in Figure 17 confirm that, for all tested signals, MDCT has a better energy compaction capability than STFT to the great extent. It means that audio or speech signal is more sparse in the MDCT domain than in the STFT domain.