Signal Processing Applications of Compressed Sensing

Compressed sensing represents a revolutionary signal processing technique that breaks through the limitations of traditional Nyquist sampling theorem. It enables the recovery of original signals through optimization algorithms even when the sampling rate is significantly lower than the Nyquist frequency. This technology is particularly suitable for processing sparse signals or signals that have sparse representations in certain transform domains.

In compressed sensing, signal reconstruction is one of the key steps. The Orthogonal Matching Pursuit (OMP) algorithm, as a greedy algorithm, is widely used for signal recovery and reconstruction. The core concept of OMP algorithm involves iteratively selecting atoms (columns of the measurement matrix) that exhibit the highest correlation with the current residual, gradually approximating the original signal.

The basic workflow of OMP algorithm includes the following steps: First, initialize the residual as the observed signal and select an empty support set. Then, in each iteration step, the algorithm calculates the inner products between the current residual and each column of the measurement matrix, selecting the column with the maximum correlation to add to the support set. Subsequently, update the signal estimate on the support set using least squares method and compute the new residual. This process repeats until reaching the preset iteration count or when the residual becomes sufficiently small.

The advantage of Orthogonal Matching Pursuit algorithm lies in its relatively high computational efficiency, making it suitable for medium-scale signal reconstruction problems. However, its performance depends on the sparsity of the signal and the coherence of the measurement matrix. In practical applications, to improve reconstruction accuracy, algorithm enhancements are typically required, such as introducing regularization terms or optimizing stopping conditions.

The combination of compressed sensing and OMP algorithm demonstrates strong potential in multiple fields, including medical imaging (such as MRI), wireless communication, and radar signal processing. These application scenarios share the common characteristic that the signals themselves possess sparsity or can achieve sparse representation through certain transformations (like wavelet transform). Through compressed sensing technology, not only can sampling costs be reduced, but signal processing efficiency can also be improved, opening new possibilities for modern signal processing.