Recent Patents on Signal Processing, 2012, 2, 127-139 127 3:99/8346/12 $100.00+.00 © 2012 Bentham Science Publishers Compressive Sampling Methods and their Implementation Issues Nikos Petrellis * Computer Science and Telecommunications Dept., TEI of Larisa, TEI Campus, Larisa 41110, Greece Received: 28 April 2012; Revised: 22 June, 2012; Accepted: 22 June 2012 Abstract: The ultra wideband telecommunication systems require nowadays mixed signal front end modules like Analog- to-Digital Converters with ever increasing cost in terms of speed, power consumption, die area and complexity if the Nyquist rate is respected. Since the signals in various applications are inherently sparse or compressible, a great attention has been recently given to systems that use under-sampling methods without significant information loss. These methods are covered by a signal processing branch known as Compressed (or Compressive) Sensing or Compressive Sampling. A Compressive Sampling approach is useful if it leads to a lower cost solution compared to the cost of the architecture that would be required if Nyquist sampling was adopted or if the signal frequency is so high that no appropriate Nyquist sampler is available at all. A number of recent Compressive Sampling patents will be reviewed in this article focusing on the feasibility of their implementation since the mathematical modeling that is often adopted is based on discrete input values and cannot be directly applied to the real world analog signals. Moreover, computational intensive optimization problems require to be solved in order to fully reconstruct the original input signal from a small number of samples. Keywords: Compressive sampling (sensing), undersampling, analog digital conversion, VLSI, DSP. 1. INTRODUCTION The Compressive Sampling (CS) is formally defined as a method for finding solutions to underdetermined linear systems i.e., to linear systems with fewer equations than the unknown variables. In signal processing this method is used basically for the reconstruction of original signals with fewer samples than the ones required by the Nyquist rate. The Nyquist sampling rate requires a sampling frequency that is at least twice as high as the highest frequency component of the input signal. In the underdetermined linear system mentioned above, the vector with the measured values can be expressed as a product of an appropriate matrix with the vector containing the Nyquist input samples. The selection of the matrix that multiplies the input vector is a key point for the success of every CS approach. Moving to higher communication throughputs and carrier frequencies in ultra wideband systems, the requirements posed to Analog-to-Digital Converters (ADCs) are ever getting stricter. More specifically, if an ADC has to operate in a multi-GS/s sampling rate, expensive architectures like Flash ADCs have to be used. These architectures require large die area and consume extremely high power even for resolutions as low as 7-bits. The CS is a method that can often relax the sampling frequency requirements of an ADC since it would be forced to sample the input only at the time points that are necessary for the input signal reconstruction. Nevertheless, the sampling intervals are not always regular (non-uniform sampling) and the original signal reconstruction is difficult since it cannot be carried out using low cost hardware. *Address correspondence to this author at the Computer Science and Telecommunications Dept., TEI of Larisa, TEI Campus, Larisa 41110, Greece; Tel: +302410684542; Fax: +302410684573; E-mail: npetrellis@teilar.gr The input signal has to be sparse (either in the time or the frequency domain) or compressible in order to apply CS techniques. A signal is considered to be K-sparse if its N samples retrieved at Nyquist rate can be represented by a vector of size N that contains K non-zero elements (K<<N). If a signal is not sparse but can be described by less than N values (e.g., as a sequence of vectors) then it is considered as compressible. For example, a straight line in an image can be defined by the coordinates of its endpoints instead of storing all of its pixels. Similarly, a circle can be defined only by the coordinates of its center and the length of its radius, a triangle can be defined only by the coordinates of its vertices, etc. If a signal has a lot of near-zero values, they can all be considered as zero without significant information loss in many CS applications. This process is often called “sparsification”. Such signals may represent images like the ones produced by Magnetic Resonance Imaging (MRI), radar imaging, Optical Character Recognition (OCR), tele- surgery, multi-player online games, speech recognition, occasional sensor sampling, etc. The fundamental theory for Compressive Sampling is described in [1] by Donoho. A more abstract introduction to CS is given in [2]. In [3-10] implementable CS methods are proposed for general applications. In [11] CS techniques are discussed for the channel estimation of telecommunication systems that use Orthogonal Frequency Division Multiplexing (OFDM). The application of CS in sensor networks like GPS is shown in [12-16] while its use in imaging applications like MRI is shown in patents [17] and [18]. As already mentioned the architecture of ADCs that can support compressive sampling is a very important issue for the feasibility of CS methods. Appropriate ADC implementations and architectures are described in [19, 20]. In Section 2 the mathematical modeling of the CS problem is given as well as a number of optimization targets that have been proposed in the referenced approaches for the