Abstract
In digital signal processing (DSP), Nyquistrate sampling completely describes a signal by exploiting its bandlimitedness. Compressed Sensing (CS), also known as compressive sampling, is a DSP technique efficiently acquiring and reconstructing a signal completely from reduced number of measurements, by exploiting its compressibility. The measurements are not point samples but more general linear functions of the signal. CS can capture and represent sparse signals at a rate significantly lower than ordinarily used in the Shannon’s sampling theorem. It is interesting to notice that most signals in reality are sparse; especially when they are represented in some domain (such as the wavelet domain) where many coefficients are close to or equal to zero. A signal is called K-sparse, if it can be exactly represented by a basis, , and a set of coefficients , where only K coefficients are nonzero. A signal is called approximately K-sparse, if it can be represented up to a certain accuracy using K non-zero coefficients. As an example, a K-sparse signal is the class of signals that are the sum of K sinusoids chosen from the N harmonics of the observed time interval. Taking the DFT of any such signal would render only K non-zero values . An example of approximately sparse signals is when the coefficients , sorted by magnitude, decrease following a power law. In this case the sparse approximation constructed by choosing the K largest coefficients is guaranteed to have an approximation error that decreases with the same power law as the coefficients. The main limitation of CS-based systems is that they are employing iterative algorithms to recover the signal. The sealgorithms are slow and the hardware solution has become crucial for higher performance and speed. This technique enables fewer data samples than traditionally required when capturing a signal with relatively high bandwidth, but a low information rate. As a main feature of CS, efficient algorithms such as -minimization can be used for recovery. This paper gives a survey of both theoretical and numerical aspects of compressive sensing technique and its applications. The theory of CS has many potential applications in signal processing, wireless communication, cognitive radio and medical imaging.
Highlights
The traditional approach of reconstructing signals or images from measured data follows the well-known Shannon sampling theorem, which states that the sampling rate must be twice the highest frequency
It is for this reason that the theory of compressive sensing uses a lot of tools from probability theory
Coherence is in some sense a natural property in the compressed sensing framework, for if two columns are closely correlated, it will be impossible in general to distinguish whether the energy in the signal comes from one or the other
Summary
The traditional approach of reconstructing signals or images from measured data follows the well-known Shannon sampling theorem, which states that the sampling rate must be twice the highest frequency. The fundamental theorem of linear algebra suggests that the number of collected samples (measurements) of a discrete finite-dimensional signal should be at least as large as its length (its dimension) in order to ensure reconstruction. This principle underlies most devices of current technology, such as analog to digital conversion and medical imaging. The first naive approach to a reconstruction algorithm consists in searching for the sparsest vector that is consistent with the linear measurements This leads to the combinatorial 0 -problem, which is Non-deterministic Polynomial-time hard (NP-hard) in general. It will be convenient to introduce some notations and concepts that we will use throughout the remainder of the paper
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