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Several techniques have been proposed that attempt to reconstruct a sparse signal from fewer samples than the ones required by the Nyquist theorem. In this paper, an undersampling technique is presented that allows the reconstruction of the sparse information that is transmitted through Orthogonal Frequency Division Multiplexing (OFDM) modulation. The properties of the Discrete Fourier Transform (DFT) that is employed by the OFDM modulation, allow the estimation of several samples from others that have already been obtained on the side of the receiver, provided that special relations are valid between the original data values. The inherent sparseness of the original data, as well as the Forward Error Correction (FEC) techniques employed, can assist the information recovery from fewer samples. It will be shown that up to 1/4 of the samples can be omitted from the sampling process and substituted by others on the side of the receiver for the successful reconstruction of the original data. In this way, the size of the buffer memory used for sample storage, as well as the storage requirements of the Fast Fourier Transform (FFT) implementation at the receiver, may be reduced by up to 25%. The power consumption of the Analog Digital Converter on the side of the receiver is also reduced when a lower sampling rate is used.

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 [

Kalman filters [

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 [

A CS problem can be simply modeled as:

The vector _{d}_{d}_{d}

The _{p}

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),

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 [

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

The OFDM modulation is used in several telecommunication standards. The architecture of an environment that is based on OFDM modulation is shown in _{k}_{n}_{n}_{k}_{n}_{n}_{k}

Architecture of an OFDM system.

The OFDM modulation is an improvement of Frequency Division Multiplexing (FDM) and can be viewed as a transmission of the _{k}

where the ^{2} = −1.

Distribution of the parameters _{k}_{n}

The DFT transform has some interesting properties that can be exploited if the input data are sparse. Consider for example the matrix representation of a 16-point DFT displayed in _{k}_{k}_{k}_{1} _{15} would hold if for the real part of the twiddles we had:

Where _{1} = 0.382683, _{2} = 0.92388, _{3} = 0.707107 (the cosine or sine of the various twiddle angles). In a similar manner for the imaginary part of the twiddles we would get:

Combining Equations 5 and 6 in order to have _{1} _{15} we get the following conditions that must to be true:

There are several relations between the _{k}_{k}_{k}_{1} + _{7} = _{9} + _{15} _{3} + _{5} = _{11} + _{13} _{2} + _{6} = _{10} + _{14}

If the condition of Equation 9 holds, then the symbol _{15} can be used in the place of _{1} or _{n}_{n}_{n}_{k}

Substitutions like the one of _{1}_{15}_{n}_{2p + 1 + N/2}, with _{2t + 1} = _{2t + 1 + N/2} for

From Equations 10 and 11 it is obvious that the twiddle factors at even ^{k} = 1 if _{2t + 1} = _{2t + 1 + N/2} only if _{k}_{k}_{k}_{k + N/2} (=_{k + 2F}) are equal, the following equation is true:

Equation 12 focuses on two terms of the sum in Equation 15 and means that the pairs _{k}_{k + 2F} multiplied by the corresponding twiddle factors cancel each other in Equation 10 if they are equal. The same holds for Equation 11. If _{2t + 1} = _{2t + 1 + N/2} holds for almost all _{k}_{n}_{n}_{k}

In order to have _{k}_{k}_{cv}_{k}

The data rate _{k}_{cv}_{cv}_{k}_{cv}_{cv}

The interleaving method followed is not common to all standards and configurable interleavers are proposed to meet different requirements. An optimal interleaving strategy depends on many factors like, the data rate ^{−3} can be observed between different interleaving schemes for the same SNR. The interleaver in the method proposed in this paper is allowed to do any permutation separately in the data and the parity bit stream. It is also allowed to permute in a random way the data _{2}Q bits of each

Order of (

The encoding and interleaving scheme followed in this work has also to take into account the fact that pilot and padding symbols will be added before the IDFT transform. These pilot and padding symbols do not affect the proposed undersampling method if they are defined equal to the _{cv}

In the following, a Recursive Convolutional Systematic (RSC) encoder is used with a systematic and a parity output of data rate ^{2} + ^{3} and the feedback path that is described by 1 + ^{2}. A small 8-bit buffer at the output of the encoder groups the 4 data and 4 parity bits into a pair of 16-QAM symbols.

The 16-QAM data symbols can be reordered by a channel interleaver and then placed at odd positions. Similarly, the 16-QAM parity symbols can be potentially reordered and placed at even positions. Such channel interleavers will be tested in the framework of different _{cv}_{cv}_{cv}_{cv}

At the side of the receiver, the sampling of the _{2t + 1} symbols by the ADC may be omitted as they can be substituted by the _{2t + 1 + N/2} or vice versa. The original and the substituted _{n}_{n}_{n}_{2t + 1 + N/2} symbols are substituted (e.g., the _{2t + 1 + N/2 + N/4}) in order to achieve a better BER, the lower ADC sampling rate will be applied to the 25% of the time interval that is needed to receive all the _{n}

A full advantage of the reduced ADC power, the buffering memory and the simpler FFT implementation can be taken if only sparse data are exchanged,

The DFT required in OFDM environments is implemented efficiently using Decimation In Time (DIT) FFT as was proposed in the mid-sixties by Cooley and Turkey [^{2} operations required by the initial _{2}

Implementation of an 8-point DFT as a radix-2 DIT FFT (the red arrows indicate zero butterfly outputs).

The _{i0}_{i1}

The twiddle factors _{n}

Generally an

Each of the terms in Equation 14 represents one of the

Although, some alternatives of the proposed undersampling technique may cooperate more efficiently with high-radix FFTs we will focus on the affect of the proposed scheme to the implementation of a radix-2 FFT like the one shown in

The _{n}_{2p + 1} and _{2p + 1 + N/2} that are expected to be equal (denoted by the same type of dashed lines in _{n}_{n}_{c}

As can be shown from _{n}_{n}_{n}

The FFT requires Random Access Memory (RAM) memory for storing both its input and its intermediate results. The twiddle factor coefficients are usually stored in Read Only Memory (ROM). In VLSI FFT implementations with single memory, the butterflies are using the same memory for storing their inputs and their results requiring lower memory but they are slower since reading the input and writing the output values cannot be carried out concurrently. In dual memory architectures the butterfly results are stored in a separate memory concurrently with the reading of the input values. Pipeline architectures consist of multiple stages with intermediate storage requirements, thus, they have the highest memory requirements but they also can achieve the highest speed.

Several more advanced FFT implementations have been proposed, with low memory requirements without compromising speed. In [

Original FFT implementation of [

In this paper, we assumed that in the general case all the input values are stored initially in a RAM buffer and the 25% reduction that can be achieved is referring to this buffer as is the case in [

In

In this section, the simulation results from the encoding, the interleaving and the sample substitute policy described in the previous sections will be presented for a diversity of data sparseness degrees and substituted samples. The system has been described in MATLAB and includes all the OFDM stages that have been described by

The implementation of the Digital Control of Sampling Speed and Sample Replacement unit could have been based on a simple Finite State Machine that would deactivate the ADC for as long as the padding symbols are received (assuming its size is known) or sample them and let the Guard Removal module of the receiver (see

At the output of the DFT, the pilot symbols are removed by the Pilot Symbol Removal unit and a 16-QAM demodulation takes place. The resulting bit stream is de-interleaved and decoded either using Viterbi or Reed Solomon (RS) algorithm by the module FEC Decoder of

BER

As can be seen from ^{−2}. This is owed mainly to the high sparseness of the data. For example, if the sparseness degree is less than 2% this means that the 1536-bit packets have less than 30 bits with non zero value. When SNR is high enough (e.g., 10 dB or higher) the errors are caused by the inversion of some of these non-zero bits or the inversion of their adjacent zero bits. Thus, the error combinations are limited and therefore the BER values tend to be discrete.

In ^{−6} for 0.5% sparseness when the channel SNR is adequately high. Nevertheless, there is no significant improvement when the sparseness is 2% or 4%. In

In

BER

In

BER

An undersampling method that can be implemented with low complexity hardware in an OFDM environment was presented in this paper. It can reduce the samples needed at the receiver by up to 1/4 reducing the power consumption, the size of the memory needed to store the ADC samples and the complexity of the FFT. The proposed method can recover the original sparse data with zero or very low error assisted by the employed FEC scheme without the use of time consuming iterative processes.

Future work will focus on the investigation of the efficiency of different QAM modulations, FFT sizes and implementations, FEC methods, such as LDPC and Turbo Codes, interleaving schemes,

This work is protected under the provisional patents with application No. 20130100014 and 20140100069 (Greek Patent Office (OBI)).

The author declares no conflict of interest.