## 4 Proposed Method: Variable Embedding Locations Exploration
### Solution 2: Adaptive Hiding with Distortion Estimation  
The primary goal of distortion estimation is to identify DCT coefficients that are less susceptible to perceptual degradation when used for data embedding. This section describes the specific distortion estimation policy employed in our method.  
Stego DCT coefficients, denoted as \(X_{n}^{\prime}\), are generated by expanding the original coefficients \(X_{n}\) and embedding a binary bit \(B\):  
\[X_{n}^{\prime}=2X_{n}+B,\quad 1\leq n\leq N,\quad B\in\{0,1\} \tag{3}\]  
To effectively select DCT coefficients for embedding, we leverage a statistical approach to estimate the potential distortion introduced by embedding a bit into each expanded coefficient. Assuming that each \(X_{n}\) is constant, we can approximate the distortion by calculating the mean squared error (MSE) between the stego and original data. Theoretically, minimizing the MSE by selecting coefficients with lower estimated distortion should lead to less perceptible degradation.  
Building upon our previous work [5], we extend the concept by considering variable embedding locations and evaluating their impact on audio quality.  
Let \(E_{n}\) represent the estimated distortion (MSE) for the \(n\)-th DCT coefficient after embedding a bit \(B\):  
\[E_{n}=E[(X_{n}^{\prime}-X_{n})^{2}] \tag{4}\]  
Given that \(P(B=0)=P(B=1)=0.5\), we can calculate \(E_{n}\) as follows:  
\[E_{n} = E[(2X_{n}+B-X_{n})^{2}]\] \[= E[X_{n}^{2}+2X_{n}B+B^{2}]\] \[= X_{n}^{2}+2X_{n}E[B]+E[B^{2}]\] \[= X_{n}^{2}+X_{n}+\frac{1}{2}\]  
Equation (5) reveals that the distortion of each coefficient closely follows the magnitude of \(X_{n}\).  
To facilitate efficient embedding, we divide the \(N\) DCT coefficients into \(M\) blocks of equal size. The average estimated distortion for the \(i\)-th block, \(\overline{E_{i}}\), is computed as:  
\[\overline{E_{i}}=\frac{M}{N}\sum_{n=(i-1)N/M+1}^{iN/M}E_{n},\quad 1\leq i\leq M \tag{6}\]  
The blocks are then sorted in ascending order of their average distortion, resulting in a sorted index list \(ind\). Coefficients in blocks with lower distortion indices (e.g., \(ind\leq 4\)) are selected for embedding.  
The embedding process is as follows.  
1. **DCT Transformation**: Divide the original audio data into frames of length \(N\) and apply the integer DCT-IV transform to each frame, yielding DCT coefficients \(X_{n}\).
2. **Distortion Estimation**: Calculate the estimated distortion \(E_{n}\) for each DCT coefficient using Equation (5).
3. **Block-Based Sorting**: Segment the \(N\) DCT coefficients into \(M\) blocks, compute the average distortion \(\overline{E_{i}}\) for each block, and sort them in ascending order.
4. **Coefficient Expansion and Embedding**: For blocks with \(ind\leq 4\), expand the selected DCT coefficients \(X_{n}\) and embed the data bits: \(X_{n}^{\prime}=2X_{n}+B\).
5. **Table Embedding**: Embed the sorted block index table \(T_{i}\) into the last \(M\) DCT coefficients.  
Inverse DCT Transform: Apply the inverse integer DCT-IV transform to the modified DCT coefficients \(X_{n}^{\prime}\) to obtain the stego audio data.  
The extraction and reconstruction process is as follows.  
1. **DCT Transformation**: Transform the stego audio data into DCT coefficients \(X_{n}^{\prime}\).
2. **Table Extraction**: Extract the sorted block index table \(T_{i}\) from the last \(M\) DCT coefficients.
3. **Coefficient Extraction and Data Recovery**: For blocks with \(T_{i}=1\), extract the embedded data bits \(B\) from the expanded coefficients \(X_{n}^{\prime}\) and reconstruct the original DCT coefficients \(X_{n}\).
4. **Inverse DCT Transform**: Apply the inverse integer DCT-IV transform to the reconstructed DCT coefficients \(X_{n}\) to obtain the original audio data.