In digital signal processing (DSP), Nyquist-rate sampling completely describes a signal by exploiting\nits bandlimitedness. Compressed Sensing (CS), also known as compressive sampling, is a DSP\ntechnique efficiently acquiring and reconstructing a signal completely from reduced number of\nmeasurements, by exploiting its compressibility. The measurements are not point samples but\nmore general linear functions of the signal. CS can capture and represent sparse signals at a rate\nsignificantly lower than ordinarily used in the Shannon�s sampling theorem. It is interesting to notice\nthat most signals in reality are sparse; especially when they are represented in some domain\n(such as the wavelet domain) where many coefficients are close to or equal to zero. A signal is\ncalled K-sparse, if it can be exactly represented by a basis, 1 N\ni i , and a set of coefficients k x ,\nwhere only K coefficients are nonzero. A signal is called approximately K-sparse, if it can be represented\nup to a certain accuracy using K non-zero coefficients. As an example, a K-sparse signal is\nthe class of signals that are the sum of K sinusoids chosen from the N harmonics of the observed\ntime interval. Taking the DFT of any such signal would render only K non-zero values k x . An example\nof approximately sparse signals is when the coefficients k x , sorted by magnitude, decrease\nfollowing a power law. In this case the sparse approximation constructed by choosing the K largest\ncoefficients is guaranteed to have an approximation error that decreases with the same power law\nas the coefficients. The main limitation of CS-based systems is that they are employing iterative\nalgorithms to recover the signal. The sealgorithms are slow and the hardware solution has become\ncrucial for higher performance and speed. This technique enables fewer data samples than\ntraditionally required when capturing a signal with relatively high bandwidth, but a low information\nrate. As a main feature of CS, efficient algorithms such as 1 -minimization can be used for\nrecovery. This paper gives a survey of both theoretical and numerical aspects of compressive sensing\ntechnique and its applications. The theory of CS has many potential applications in signal processing,\nwireless communication, cognitive radio and medical imaging.
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