# A Psychoacoustic-Based Multiple Audio Object Coding Approach via Intra-Object Sparsity
## 2 Proposed Compression Framework
### Downmix Signal Compressing by SQVH  
SQVH is a kind of efficient transform coding method which is used in fixed bitrate codec [26; 27; 28]. In this section, SQVH with variable bitrate for encoding downmix signal is designed and described as follows.  
For the \(n^{\text{th}}\) frame, the downmix signal \(\mathbf{d}_{n}\) attained in Algorithm 2 can be expressed as:  
\[\mathbf{d}_{n}\equiv[d_{n}(1),d_{n}(2),\cdots,d_{n}(L)] \tag{10}\]  
\(\mathbf{d}_{n}\) need to be divided into 51 sub-bands, each sub-band contains 20 TF instants, respectively (without considering the last 4 instants). The sub-band power (spectrum energy) is determined for each of the 51 regions and it is defined as root-mean-square (_rms_) value of coterminous 20 MDCT coefficients computed as:  
\[R_{rms}(r)=\sqrt{\frac{1}{20}{\sum\limits_{l=1}^{20}{d_{n}^{2}(20(r-1)+l)}}} \tag{11}\]  
where \(r\) is region index, \(r\) = 0, 1, \(\ldots\), 50. The region power is then quantized with a logarithmic quantizer, \(2^{(i/2+1)}\) are set to be quantization values, where \(i\) is an integer in the range [8; 31]. \(R_{rms}(0)\) is the lowest frequency region, which is quantized with 5 bits and transmitted directly in transmission channel. The quantization indices of the remaining 50 regions, which are differentially coded against  
Figure 2: Example of TF (Time-Frequency) instants extraction and de-mixing procedure with eight unique simultaneously occurring sources.  
the last highest-numbered region and then Huffman coded with variable bitrates. In each sub-band, the Quantized Index (\(Ql\)) value can be given by:  
\[Ql_{r}(l)=\min\Big{\{}\ \ \Big{|}\frac{d_{n}(20\cdot(r-1)+l)|}{R_{rms}(r) \times d_{stepsize}}+b\Big{|},\ \ \ MAX\ \ \Big{\}} \tag{12}\]  
where \(q_{stepsize}\) is quantization steps, \(b\) is an offset value according to different categories, \(\lfloor\ \rfloor\) denotes a round-up operation, \(MAX\) is maximum of MDCT coefficients corresponding to that **category** and \(l\) represents the \(l^{\text{th}}\) vector in the region \(r\). There are several **categories** designed in SQVH coding. The **category** assigned to a region defines the quantization and coding parameters such as quantization step size, offset, vector dimension \(v_{d}\) and an expected total number of bits. The coding parameters for different category is given in Table 1.  
As is depicted in Table 1, four categories are selected in this work. Category 0 has the smallest quantization step size and uses the most bits, but not vice-versa. The set of scalar values, \(Ql_{r}(l)\), correspond to a unique vector is identified by an index as follows:  
\[v_{index}(i)=\sum_{j=0}^{v_{d}-1}Ql_{r}(i\times v_{d}+j)(MAX+1)^{[v_{d}-(j+1)]} \tag{13}\]  
where \(i\) represents the \(i^{\text{th}}\) vector in region \(r\) and \(j\) is the index to the \(j^{\text{th}}\) value of \(Ql_{r}(l)\) in a given vector. Then, all vector indices are Huffman coded with variable bit-length code for that region. Three types  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Categories** & \(q_{stepsize}\) & \(b\) & \(MAX\) & \(v_{d}\) & **Bit Count** \\ \hline
0 & \(2^{-1.5}\) & 0.3 & 13 & 2 & 52 \\
1 & \(2^{-1.0}\) & 0.33 & 9 & 2 & 47 \\
2 & \(2^{-0.5}\) & 0.36 & 6 & 2 & 43 \\
3 & \(2^{0.0}\) & 0.39 & 4 & 4 & 37 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The coding parameters for different category.  
of bit-stream distributions are given in the proposed method, whose performance is evaluated in next section.