The measurement matrix is a key technology in compressive sensing processing. Different measurement matrices will affect the actual reconstruction of the target estimate, it is very important to design a compressed sampling matrix. This part mainly focuses on the process of designing such a deterministic measurement matrix and explains how this design can reduce redundant data introduced in the array.
3.1. Hadamard Matrix
The elements of the Hadamard matrix are 1 or −1, and the construction process of the Hadamard matrix is as follows:
It can be seen from the above construction process that this construction method firstly generates a Hadamard orthogonal matrix of size , where . Second, based on the above Hadamard matrix, intercepting N columns of sub-matrices to obtain a partial Hadamard measurement matrix with low coherence and good partial orthogonality. However, the above Hadamard matrix is a random measurement matrix, which is difficult to achieve in the project, and when the number of matrix sizes is greater than 2, the number of matrix sizes must be multiple of 4.
3.2. 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:
where
,
.
According to the measurement matrix
, the compressed sampling data
is obtained by the compressed sampling of the uniform line array data
.
where
.
The compressed sampling data 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 and . It can be seen from Equation (20) that the compressed sampling data is the data received by each array element in ULA multiplied by the corresponding measurement matrix atom , 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 of the compressive sampling matrix 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 is to discard the corresponding several array elements data in the original ULA, i.e., all elements in the nth atom corresponding to the nth 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:
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
, and finally use the CZ method to replace all elements of
where does not exist with zero, and the measurement matrix
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
to zero. Therefore, we refer to this improved Hadamard matrix as the compression zeroing measurement matrix based on sequential partial Hadamard.
The improved Hadamard matrix is a deterministic measurement matrix. On the one hand, in the array signal processing, if there are several sensors error receiving or failure, we can use CZ method to zero the corresponding column vector element in the measurement matrix to remove invalid or useless data and itself interference from the array; On the other hand, in the process of compressing and under-sampling the array, as shown in
Figure 3, the relationship between the array signals before and after under-sampling compression is established through the improved Hadamard matrix, the unsampled array data and redundant information are removed by the method of CZ, to accelerate the reconstruction speed and enhance the performance of target estimation.
After the above-mentioned array compression zeroing method, there is still irrelevance between the columns of the measurement matrix , and when the number N is close to the number U, it can still basically satisfy the RIP.
However, the selection of U and N in the improved Hadamard matrix proposed above still has certain application conditions. The size of its construction dimension U must satisfy an integer multiple of 2, that is, , therefore, our reconstructed N needs to be as close to U as possible. From the Hadamard matrix of , and through the above introduction method, after selecting a sub-matrix of size composed of M rows and N columns, which still has strong non-correlation and partial orthogonality.