Multi-Carrier and Spread Spectrum Systems: From OFDM and MC ...

32 FundamentalsThe discrete length of the guard interval has to be⌈ ⌉τmax N cL g ≥T s(1.38)samples in order to prevent ISI. The sampled sequence with cyclic extended guard intervalresults inx v = 1 N∑c −1S n e j2πnv/N c, v =−L g ,...,N c − 1. (1.39)N cn=0This sequence is passed through a digital-to-analogue converter whose output ideallywould be the signal waveform x(t) with increased duration T s ′ . The signal is upconvertedand the RF signal is transmitted to the channel (see Chapter 4 regarding RF up/downconversion).The output of the channel, after RF down-conversion, is the received signal waveformy(t) obtained from convolution of x(t) with the channel impulse response h(τ, t) andaddition of a noise signal n(t), i.e.y(t) =∫ ∞−∞x(t − τ)h(τ,t) dτ + n(t). (1.40)The received signal y(t) is passed through an analogue-to-digital converter, whose outputsequence y v , v =−L g ,...,N c − 1, is the received signal y(t) sampled at rate 1/T d .Since ISI is only present in the first L g samples of the received sequence, these L g samplesare removed before multi-carrier demodulation. The ISI-free part v = 0,...,N c − 1,of y v is multi-carrier demodulated by inverse OFDM exploiting a DFT. The outputof the DFT is the multi-carrier demodulated sequence R n , n = 0,...,N c − 1, consistingof N c complex-valued symbolsR n =N∑c −1v=0y v e −j2πnv/N c, n = 0,...,N c − 1. (1.41)Since ICI can be avoided due to the guard interval, each sub-channel can be consideredseparately. Furthermore, when assuming that the fading on each sub-channel is flat and ISIis removed, a received symbol R n is obtained from the frequency domain representationaccording toR n = H n S n + N n , n = 0,...,N c − 1, (1.42)where H n is the flat fading factor and N n represents the noise of the nth sub-channel.The flat fading factor H n is the sample of the channel transfer function H n,i accordingto Equation (1.23) where the time index i is omitted for simplicity. The variance of thenoise is given byσ 2 = E{| N n | 2 }. (1.43)