, the subcarrier pulse used for transmission is rectangular. This shape is chosen so that the pulse-forming and modulation can be performed by a simple inverse discrete Fourier transform (IDFT). The IDFT can be efficiently implemented by an inverse fast Fourier transform (IFFT). At the receiver, a fast Fourier transform (FFT) is performed to reverse the operation. The IFFT leads to a sin(x)/x type of spectrum of the subcarriers, shown in Fig. 7-2. Although the spectra of the subcarriers overlap, the information transmitted over the sub- carriers can still be separated due to the orthogonality relation. Orthogonality is the ability to distinguish every subcarrier at the receiver without any interference from other subcarriers. The mathematical property of orthogonality states that the integral of the product of any two subcarriers is zero. By using an IFFT for modulation, the spacing of the subcarriers is chosen such that at the frequency where the received signal is evaluated, all other signals are zero. However, to preserve this orthogonality, the following conditions must be met: The receiver and transmitter must be perfectly synchronized. This means they must both assume exactly the same modulation frequency and the same timescale for transmission. This requires a sophisticated receiver. The analog components of the transmitter and receiver must be of very high quality. There should be no multipath channel. The last point is particularly disappointing because this approach was chosen specifically to combat the multipath channel (increasingly problematic below 11 GHz). Fortunately, there is a solution to this problem in the artificial prolongation of the OFDM symbol. This is done by periodically repeating the "tail" of the symbol and preceding the symbol with this Chapter 7 Wireless scheduling and Quality of Service optimization techniques 239 FIGURE 7-2 IFFT to implement OFDM efficiently. IFFT, inverse fast Fourier transform; OFDM, orthogonal frequency division multiplexing. repetition. At the receiver, this s