## 2 The Architecture of an OFDM System  
The OFDM modulation is used in several telecommunication standards. The architecture of an environment that is based on OFDM modulation is shown in Figure 1. The input data are encoded at the transmitter and along with the generated parity bits are interleaved and modulated using \(Q\)-Quadrature Amplitude Modulation (\(Q\)-QAM). \"Pilot\" symbols are inserted assisting the receiver in the estimation of the channel condition (e.g., multipaths, fading, _etc._). This sequence of \(Q\)-QAM symbols \(X_{k}\) (data, parity, pilots) may be padded with an appropriate value forming an input set of values for the Inverse Discrete Fourier Transform (IDFT) that generates the time symbols \(x_{n}\). Both \(x_{n}\) and \(X_{k}\) symbols are complex numbers. The time symbols \(x_{n}\) and are transmitted over the channel either in baseband mode or by mixing them with a high frequency carrier. Cyclic Prefix (CP) is added to avoid Inter-Symbol Interference (ISI) and Inter-Carrier Interference (ICI). A Digital to Analog Converter (DAC) at the transmitter converts the arithmetic values of the IDFT output and the CP symbols into the analog signal that will be transmitted over the communication channel. A pair of DACs may be used to convert separately the real and imaginary parts of these symbols. Although this analog signal may be transmitted over a high frequency carrier as stated earlier, the details of this case will not be discussed since they do not affect the proposed undersampling method. The channel adds Additive White Gaussian Noise (AWGN). Channel fading, multipaths, _etc._, are additional noise sources that should be taken into consideration for more accurate modeling of OFDM channels but for simplicity reasons the proposed undersampling technique will be evaluated using only a broad range of AWGN noise at the channel. The receiver can sample the distorted by the channel noise \(x_{n}\) real and imaginary parts by a pair of ADCs after ignoring the CP part. The symbols \(X_{k}\) are recovered at the output of a DFT. The padding symbols and the pilots are removed and the remaining \(Q\)-QAM data and parity symbols are mapped-back to data/parity bits, deinterleaved and feed the input of the FEC decoder. The Cyclic Prefix, CP Removal, and Guard Removal blocks are placed in dotted boxes to indicate that they will not play an important role in the simulations that will be described in Section 5.  
Figure 1: 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 \(N\), \(X_{k}\) symbols over \(N\) different subcarriers. Although these subcarriers have a small frequency difference, they are orthogonal to each other and, thus, their interference is minimal. The DFT and the Inverse DFT (IDFT) transforms are defined by the Equations 3 and 4 respectively:  
\[X_{k}=\sum_{n=0}^{N-1}x_{n}w_{N}^{kn}\,0\leq\mathrm{k}<\mathrm{N}-1 \tag{3}\]  
\[x_{n}=\sum_{k=0}^{N-1}X_{k}W_{N}^{kn}\,0\leq\mathrm{n}<\mathrm{N}-1 \tag{4}\]  
where the \(W_{N}^{kn}=e^{j\frac{2\pi kn}{N}}\) and \(W_{N}^{kn}=e^{-j\frac{2\pi kn}{N}}\) are the twiddle factors and \(j^{2}=-1\).