 T denotes transpose operation), D is an M X M matrix which performs M-point DFT operation, T is the N X M mapping matrix for sub-carrier assignment, and F performs NFFT point IFFT operation. After propagation through the multipath fading channel and addition of the Additive White Gaussian Noise (AWGN), removing the cyclic prefix and going through the NFFT point FFT module, the received signal vector in the frequency domain can be expressed as Z= HTDx + W where H is the diagonal matrix of channel response and W is the noise vector. Note that the maximum excess delay of the channel is assumed to be shorter than the cyclic prefix; therefore, the ISI can be mitigated by the cyclic prefix. The amplitude and phase distortion in the received signal due to the multipath channel is compensated by a Frequency Domain Equalizer (FDE) and the signal at the FDE output can be described as V = Cz where C = diag (c1, C2, CNFFT) is the diagonal matrix of FDE SECURITY 9.3 SC-FDMA and OFDMA principles sub-carrier. A particular sub-carrier may experience deep fading in a frequency selective fading channel. The IDFT de-spreading averages and spreads the fading effect, which results in a noise enhancement to all the QAM symbols. Hence, the IDFT de-spreading makes the DFT-S OFDM more sensitive to the noise. The sub-carrier mapping in SC-FDMA determines which part of the spectrum is used for trans- mission by inserting a suitable number of zeros (i.e., null sub-carriers inserted between the data sub- carriers) at the upper and/or lower end as shown in Figure 9-10. Between each DFT output sample L - 1 zeros are inserted. A mapping with L = 1 (as shown in Figure 9-10) corresponds to localized transmissions, i.e., transmissions where the DFT outputs are mapped to consecutive sub-carriers. There are two sub-carrier mapping schemes (localized and distributed) that could be used in the uplink. In order to demonstrate the similarities and differences between OFDMA and SC-FDMA opera- tion, let's assume that one wishes to transmit a sequence of 