into account the channel and noise, the discrete frequency-domain signal can be obtained from Eq. (1.3) by (1.6) which is in fact the DFTof{y,y1,...,Yn-1}.The subcarrier orthogonality in Eq. (1.2) also applies to the discrete-time representation: (1.7) such that Yk=Xk. In summary, the OFDM modulation and demodulation can be realized by IDFT and DFT, respectively, which is very simple by virtue of the computationally efficient inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT) algorithms. 1.2.3 OFDM in Multipath Channel When taking into account the multipath channel h=[ho,h1,...,L-1],where = L is the num- ber of channel taps, the orthogonality of subcarriers will be unfortunately broken, as shown in Figure 1.2. A CP can be used to avoid ISI when an OFDM signal is transmitted over the multipath channel, where the CP is essentially an identical copy of the last portion of the OFDM symbol. This CP preserves the orthogonality of the subcarriers and prevents ISI between successive OFDM symbols. Specifically, the CP of length > L, which is com- posed of the last M samples of the OFDM symbol, namely {XN-," ,XN-1}, is appended to the beginning of the OFDM symbol. After adding the CP, the new time-domain OFDM symbol yields (1.8) The effect of CP is illustrated in Figure 1.2, which shows that subcarrier orthogonality is maintained provided that the length of CP is no less than the channel delay spread. Delay OFDM symbol Figure 1.2 The multipath and CP effects. 1.2 OFDM Principle After passing through the channel, the received signal without noise is (1.9) where < n - k N denotes n - k modulo N. This means that by appending a CP to the chan- nel input, the linear convolution associated with the channel impulse response becomes a cyclic convolution. Let us take another look at the relation in Eq. (1.9) via matrix representation. The channel output including the CP can be expressed in matrix form: (1.10) The channel matrix of dimensions (N (N + 1)1) in Eq. (1.10) is a Toeplitz matrix. Removing the CP leads 