## 1 Introduction  
Sampling a signal at a frequency below the Nyquist rate is often called undersampling and is used when information that is changing at a relatively low rate is transmitted over a high frequency carrier. In these cases, the sampling on the side of the receiver is performed at a rate that is twice the information bandwidth instead of the peak frequency. A common practice is to sample an Intermediate Frequency (IF) rather than the baseband one [1]. Two popular techniques for the reconstruction of a signal from fewer samples than the ones required by the Nyquist rate, are the Compressed Sensing and the Kalman filters. Kalman filters [2; 3; 4; 5] are used in a variety of applications ranging from economics to vehicle navigation, especially for tracking the value of dynamic parameters in noisy environments. They can also be used for the reconstruction of a signal with fewer samples in real time by applying a two step recursive process: (a) prediction, where the current state variables are estimated, and (b) update of the estimated values, when the next measurement is available. Real time signal reconstruction approaches based on Very Large Scale Integration (VLSI) implementations have been proposed in the past, based on Kalman filters [6]. A VLSI implementation of a Kalman filter, where multiple parallel operations can be performed in a single iteration, is described in [7]. The Compressed (or Compressive) Sensing or Compressed Sampling techniques are also based on undersampling of sparse, or in more general terms, compressible information. One of the first approaches in this field has been published in [8]. A Compressed Sensing (CS) technique that is more appropriate for hardware implementation has been proposed in [9] for general applications. In these Compressed Sensing approaches, the original information recovery is performed by only a small number of samples, much smaller the number required by the Nyquist rate. By the term sparse it is meant that most of the original data values must have a trivial common value like zero (or near zero). If the original data values are compressible, although they may not be sparse, they can be expressed with fewer values, e.g., a piece of line in an image can be defined only by two points and there is no need to store the coordinates of all the intermediate points. Sometimes, the differences between successive samples are zero instead of the absolute values and these sparse differences are taken into consideration in order to apply CS techniques. Formally, an \(M\times N\) matrix is considered to be \(S\)-sparse if only \(S\) of its values are non-zero with \(S<<M\times N\). In this paper, the sparse information will also be expressed as the fraction of non-trivial values, e.g., in a signal with 1% sparseness, only 1% of its values are non-zero. A CS problem can be simply modeled as:  
\[y=Ax+e \tag{1}\]  
The vector \(x\) (size: \(N_{d}\)) in Equation 1 is the original input, \(y\) is the actual measurement (size: \(M<<N_{d}\)), and \(e\) is an \(M\)-size vector representing noise or the error from the input recovery. The matrix \(A\) (size: \(M\)x\(N_{d}\)) can represent the digitization process. As Equation 1 does not have a single solution, the adopted CS method tries to find a vector \(x\) that fits Equation 1 with an acceptable approximation error \(e\). The following optimization target has been proposed in various CS approaches in order to locate such an appropriate solution for vector \(x\):  
\[\min\left|Ax\right|\right|_{\!\!1}\text{ subject to }\left|y-Ax\right|_{\!\! \!0}\leq th \tag{2}\]  
The \(L_{p}\) norm (\(p=2\) in most implementations) is used to restrict the difference between \(y\) and \(Ax\) below the threshold \(th\). The matrix \(A\) can be expressed as a product of other matrices with special features (e.g., partial Fourier or Hamadad matrices) and the minimization (2) can be applied to the product of \(x\) with one of these elementary matrices.