# A Psychoacoustic-Based Multiple Audio Object Coding Approach via Intra-Object Sparsity
## Appendix A Sparsity Analysis of Audio Signal in the MDCT Domain
### Measuring the Sparsity of Audio Signal  
A time-frequency representations of an audio signal can be obtained by a linear transform. Specifically, for a general dictionary of atoms \(D=\{\varphi_{l}\}\), the linear representation of an audio signal \(s_{n}(m)\) in \(n^{\text{th}}\) frame can be defined by:  
\[S(n,l)=\sum_{m=1}^{M}s_{n}(m)\varphi_{l}(m) \tag{1}\]  
where \(n\), \(m\) and \(l\) represent frame number, time index and frequency index, respectively. \(M\) is the length of each frame. Short-time Fourier Transform (STFT) basis functions and Discrete Cosine Transform (DCT) basis functions are ordinarily used as time-frequency atoms in speech and audio signal compression. DCT is widely used in audio coding mainly because of its energy compaction feature. Nevertheless, due to the blocking effect caused by the different quantitative level between frames, the processed signal cannot be perfectly reconstructed by IDCT. Evolved from DCT, MDCT has emerged as an efficaciously tool in high quality audio coding over the last decade because it helps to mitigate the blocking artifacts that deteriorate the reconstruction of transform audio coders with non-overlapped transforms [34]. It should be noted that MDCT can be taken as a filterbank with 50% overlapped window, hence, Time Domain Aliasing Cancellation (TDAC) must be exploited in the practical processing. Meanwhile, the chosen window function must satisfy the TDAC requirement. In this work, a Kaiser-Bessel derived (KBD) window [35] is chosen to meet the computing needs of TDAC and overlap-add algorithms. Particularly, for a finite-length audio signal whose MDCT coefficients are densely concentrated at low indices than the STFT (Short Time Fourier Transform) does, which is called \"energy compaction\" property [36]. With this prerequisite, a detailed comparative study and analyses of energy compaction feature (a.k.a. sparsity) of different audio objects in the STFT domain and MDCT domain is implemented. To measure and explore sparsity of audio signal in the time-frequency domain, a measurement addressed as Frame Energy Preservation Ratio (_FEPR_) and the Number of Preserved TF bins (_NPTF_) was proposed in [19]. Specifically, the sparse approximation signal of \(S(n,l)\), referred as \(S^{\prime}(n,l)\), containsthe maximum \(K^{*}\) TF instants by preserving the portion of TF instants according to their amplitude of \(S(n,l)\) while setting the other TF instants to zero, which can be expressed by:  
\[S(n,l)=\left\{ \begin{array}{ll}S(n,l),&\mbox{ if }l\in\mathcal{L}\\ 0,&\mbox{ otherwise}\end{array} \right. \tag{10}\]  
where \(\mathcal{L}\triangleq\{l_{1},l_{2},\cdots,l_{K^{*}}\}\), is the set of \(K^{*}\) frequency indices corresponding to the maximum \(K^{*}\) time-frequency instants. Thus, \(S^{\prime}(n,l)\) is a \(K^{*}\)-sparse signal. Suppose \(\Theta_{n}\equiv[S(n,1),\cdots,S(n,L)]\) is the \(L\)-dimensional vector denotes the TF representation of the audio object signal in \(n^{\mbox{th}}\) frame, \(\Theta^{\prime}_{n}\equiv[S(n,1),\cdots,S(n,L)]\) is sparse approximation vector of \(\Theta_{n}\). Then, the Frame Energy Preservation Ratio (\(F\!EPR\)) can be given by:  
\[r_{F\!EPR}(n)=\frac{\left\|\Theta^{\prime}_{n}\right\|_{1}}{\left\|\Theta_{n} \right\|_{1}} \tag{11}\]  
where \(\left\|\cdot\right\|_{p}\) denotes the \(l_{p}\)-norm. Afterwards, for arbitrary given \(r_{F\!EPR}^{*}\), if there exists a series of subset \(\mathcal{L}_{i}\subset\{1,2,\cdots,L\}\), \(i=1,2\),..., such that the corresponding sparse signal vector \(\Theta^{\prime}_{n,i}\equiv[S^{\prime}_{i}(n,1),\cdots,S^{\prime}_{i}(n,L)]\). The Number of Preserved TF instants (\(N\!PTF\)), written as \(k\), is defined as a function of \(r_{F\!EPR}^{*}\):  
\[k(r_{F\!EPR}^{*})=\inf\Biggl{\{}\left\|\Theta^{\prime}_{n,i}\right\|_{0}\left| \frac{\left\|\ \Theta^{\prime}_{n,i}\right\|_{1}}{\left\|\ \Theta_{n}\right\|_{1}}\ \geq r_{F\!EPR}^{*}\,\ i=1,2, \cdots,\Biggr{\}} \tag{12}\]  
where \(\inf\{\cdot\}\) represents the infimum.