# An Under-Sampling Array Signal Processing Method Based on Improved Hadamard Matrix
## 3 Improved Hadamard Matrix
### Improved Hadamard Matrix  
Before introducing the improved Hadamard matrix, we first propose a method of compression zeroing (CZ) by introducing the concept of sparse representation:As shown in Equation (7), in the process of array compression sampling, the matrix expansion form under single snapshot data is as follows:  
\[\mathbf{y}= \begin{bmatrix}\phi_{11}&\phi_{12}&\cdots&\phi_{1N}\\ \phi_{21}&\phi_{22}&\cdots&\phi_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ \phi_{M1}&\phi_{M2}&\cdots&\phi_{MN}\end{bmatrix} \mathbf{x}, \tag{19}\]  
where \(\mathbf{y}=[\mathbf{y}_{1},\mathbf{y}_{2},\cdots,\mathbf{y}_{M}]^{T},\mathbf{x}=[\mathbf{x}_{1},\mathbf{x }_{2},\cdots,\mathbf{x}_{N}]^{T}\). According to the measurement matrix \(\Phi\), the compressed sampling data \(\mathbf{y}\) is obtained by the compressed sampling of the uniform line array data \(\mathbf{x}\). \[\mathbf{y}=x_{1} \begin{bmatrix}\phi_{11}\\ \phi_{21}\\ \vdots\\ \phi_{M1}\end{bmatrix} +x_{2} \begin{bmatrix}\phi_{12}\\ \phi_{22}\\ \vdots\\ \phi_{M2}\end{bmatrix} +\cdots+x_{N} \begin{bmatrix}\phi_{1N}\\ \phi_{2N}\\ \vdots\\ \phi_{MN}\end{bmatrix} =x_{1}\phi_{1}+x_{2}\phi_{2}+\cdots+x_{N}\phi_{N}, \tag{20}\]  
where \(\phi_{n}=[\phi_{1n},\phi_{1n}\cdots\cdot\phi_{Mn}]^{T}\). The compressed sampling data \(\mathbf{y}\) is also the received data of the non-uniform linear array obtained after the under-sampling. Equation (20) can be regarded as the conversion relationship between \(\mathbf{y}\) and \(\mathbf{x}\). It can be seen from Equation (20) that the compressed sampling data \(\mathbf{y}\) is the data received by each array element in ULA multiplied by the corresponding measurement matrix atom \(\phi_{n}\), It means that the M array after under-sampling compression can be considered to be obtained by clearing the data corresponding to the array elements that do not exist or do not want to be sampled in the original uniform array, that is, the actual received signal data under our under-sampling. However, in the actual array under-sampling compression process, it is not necessary to obtain the data of all the array elements, which means the elements in the atom \(\phi_{n}\) of the compressive sampling matrix \(\Phi\) at the corresponding positions of the non-existing array elements can be set to zero, which means that the process of obtaining the \(M\) elements compressed data \(\mathbf{Y}\) is to discard the corresponding several array elements data in the original ULA, i.e., all elements in the \(n\)th atom \(\phi_{n}\) corresponding to the \(n\)th non-existing sensor are set to zero, we call this particular method of zeroing as compression zeroing. We apply the CZ method to the partial Hadamard matrices above, and the improved Hadamard matrix can be obtained by the following equation:  
\[ \begin{array}{c}\text{\it select}\\ M\text{\it row}\\ \text{\it N column}\\ \text{\it H}_{U}\quad\Rightarrow\quad\left[\overline{\text{\it H}}_{U} \right]_{M\times N}=\overline{\Phi}_{M\times N},\end{array}  \tag{21}\]  
\[\overline{\Phi}_{M\times N}= \begin{bmatrix}1&1&1&\cdots&1\\ 1&-1&1&\cdots&-1\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&-1&-1&\cdots&1\end{bmatrix} _{M\times N}, \tag{22}\]  
\[\Phi= \begin{bmatrix}1&\cdots&0&\cdots&1\\ 1&\cdots&0&\cdots&-1\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&\cdots&0&\cdots&1\end{bmatrix} _{M\times N}. \tag{23}\]  
Based on the above partial Hadamard matrix construction, as shown in the above equation, the first \(M\) row and \(N\) column vectors are selected sequentially to form a new sub-matrix \(\overline{\Phi}\), and finally use the CZ method to replace all elements of \(\phi_{n}\) where does not exist with zero, and the measurement matrix \(\Phi\) is obtained, as shown in Figure 3, after under-sampling compression, the data and information received by the second array sensor that is not sampled need to use the CZ method to set all elements of \(\phi_{2}\) to zero. Therefore, we refer to this improved Hadamard matrix as the compression zeroing measurement matrix based on sequential partial Hadamard.