## 3 The Proposed Undersampling Method
\begin{split}& X_{k}e^{i2\pi(2t+1)(4F)}+X_{k+4F/2}e^{i2\pi(k+4F/2)(2t+ 1)(4F)}\\ &=X_{k}e^{i\pi t/F}e^{i\pi k(2F)}-X_{k+2F}e^{i\pi t/F}e^{i\pi k(2 F)}=0\end{split}  \tag{12}\]  
Equation 12 focuses on two terms of the sum in Equation 15 and means that the pairs \(X_{k}\) and \(X_{k\ +\ 2F}\) multiplied by the corresponding twiddle factors cancel each other in Equation 10 if they are equal. The same holds for Equation 11. If \(X_{2t\ +\ 1}=X_{2t\ +\ 1}=N2\) holds for almost all \(t<F\), then all the \(X_{k}\) symbols with odd \(k\) add up to a zero or near zero value and this can be achieved if these symbols are derived by sparse data. In this case, half of the \(x_{n}\) symbols at the odd positions can replace the rest of the \(x_{n}\) symbols with odd \(n\). Thus, the receiver can avoid sampling (or storing) up to \(F\) values since they can be estimated by other already available values. The number of the samples that can be substituted by others can be further increased if additional restrictions apply to the relation between the \(X_{k}\) values as was shown by the Equations 5-9. In order to have \(X_{k}\) symbols with identical value placed at odd \(k\)-positions, an appropriate interleaving and FEC encoding should be employed. Even in this case, a few \(X_{k}\) symbols will not be equal to the common value (denoted \(X_{cv}\) henceforth) of the rest of the \(X_{k}\) symbols, since the input data are sparse but not all zero. A perfect reconstruction of the input data is not always feasible due to this fact but the employed FEC method helps in achieving a low enough bit error rate. The data rate \(R\) of the employed FEC encoder, should be greater or equal to 1/2. This is because if \(R=1/2\), the data and the parity bits can be grouped separately by the \(Q\)-QAM modulator of Figure 1. In this case, half of the \(X_{k}\)\(Q\)-QAM symbols will correspond to data and most of them will have the same trivial value \(X_{cv}\) due to the sparseness of the original data. These \(X_{cv}\) symbols should be placed by the interleaver at the odd positions. If \(R>1/2\), the proposed method is still applicable as there will be additional data \(X_{k}\) symbols (most of them will be equal to \(X_{cv}\)) that will be placed at even positions. However, \(R<1/2\) is not recommended because some parity symbols will have to be placed at odd positions increasing the number of the symbols with different value than \(X_{cv}\) and, thus, the efficiency of the proposed method will be reduced. The transmission capacity is not confined since the proposed method is not suggested only for cases where the data rate \(R\) is low. On the contrary it is preferably applied when \(R\geq 2\), _i.e._, when most of the transmitted bits are data and not parity. 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 \(R\), the decoding algorithm used, its randomness, the minimum distance where a value should be moved, _etc._ In [16] an appropriate interleaving scheme is proposed for Turbo Code decoders that use the min-sum algorithm. In the simulation results of this work it is shown that the higher the data rate \(R\) is, the higher is the BER. Moreover, a difference in the BER in the order of \(10^{-3}\) 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 \(Q\)-QAM symbols between odd positions and the parity \(Q\)-QAM symbols between even positions but it is not allowed to move data symbols to even positions and parity symbols to odd positions. The interleaving scheme followed in this paper includes a pseudo random permutation in the log\({}_{2}\)Q bits of each \(Q\)-QAM symbol.