## 1 Introduction
Iterative and/or greedy algorithms have been proposed in order to solve the optimization problem defined above like: basis pursuit, gradient project, gradient pursuit, non-convex projection, Iteratively Reweighted Least Squares (IRLS), Orthogonal Matching Pursuit (OMP), Compressive Sampling Matching Pursuit (CoSaMP), Subspace Pursuit (SP), Iterative Hard Thresholding (IHT), _etc._ However, all of these iterative or recursive algorithms require hardware implementations of high complexity and thus the gain by the use of a lower cost/speed/power Analog Digital Converters (ADC) is often cancelled. Surveillance applications can take advantage of CS techniques as described in [10]. Sensor networks can also benefit from CS algorithms [11]. The authors of [12] describe a higher level CS approach incorporated in a wireless multimedia sensor network in which the images from multiple cameras are used. In [12], the differences between successive frames are used in order to get a CS modeling where only a few pixels are non-zero. The CS and the Kalman filter signal reconstruction approaches that were referenced above, are iteratively solved, e.g., by greedy algorithms. Kalman filters are more appropriate for hardware implementations with reusable resources although the iterative procedure required to update the state variables takes time that may be critical for real time applications. This can be solved with pipeline architectures but additional hardware resources will be required. The CS approaches have not been used for the reconstruction of the original data in Orthogonal Frequency Division Multiplexing (OFDM) environments because even if they are sparse, the encoding, the interleaving and the IDFT applied at the transmitter, cancel their sparseness. However, several CS techniques have been applied in order to reduce the number of pilot symbols required for channel estimation and thus exploit more efficiently the available bandwidth [13; 14; 15]. An undersampling approach that is not based on difficult optimization problems is presented in this paper, which can be implemented in real time environments with low cost/complexity hardware. The prerequisite is to have a high degree of sparseness in the original information. If this condition is true, the properties of the Fourier transform can be exploited in order to estimate the values of several samples by others that have already been retrieved. There are also some restrictions that have to be obeyed concerning the employed Interleaving and Forward Error Correction (FEC) methods. The complexity of the Discrete Fourier Transform (DFT) implementation can also be reduced using the proposed method if for example a Digital Signal Processor (DSP) or Column recursive radix-2 Fast Fourier Transform (FFT) implementation is employed. The number of samples required by the proposed technique, for the original information recovery, is higher compared to the number of samples required by a CS approach. However, it can be implemented with very low cost and high speed hardware since it is based on a non-iterative procedure with fixed steps. The proposed method includes: (a) an appropriate bit and/or channel interleaving/deinterleaving procedure, (b) the use of a FEC encoding scheme with a minimum data rate of 1/2, (c) Analog Digital Converter (ADC) sampling control at the side of the receiver and (d) the estimation of samples that have not been retrieved by the ADC from other available samples by simply copying their value. The aforementioned four steps are incorporated in the modeling of an OFDM system in MATLAB. The results from the simulation of this model show that in many cases, the Bit Error Rate (BER) is zero or at least very low (below \(10^{-4}\)) using the proposed method. The architecture of an OFDM system is presented in Section 2. The proposed undersampling method and the architecture of the corresponding OFDM transceiver are described in Section 3. The effect of the proposed method to the FFT implementation is discussed in Section 4.