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is insigni cant compared to the offsets contributed by the local oscillators. Since the offsets contributed by the local oscillators are symmetric in both uplink and downlink, the mobiles can make use of the downlink pilot to estimate the frequency offset and correct for the offset locally at the mobile transmitter. In any case, because of the more challenging nature in the uplink synchronization problem, we shall focus our attention on the uplink direction of the multiuser OFDMA systems. 10.5.2 Signal Model
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In this subsection, we shall elaborate on the uplink signal model of the OFDMA systems incorporating the effect of frequency and timing offsets from the K active users. The transmitter (mobile) and receiver (base station) block diagrams are shown in Figures 10.11 and 10.12, respectively. Each mobile transmitter transmits on the assigned subcarriers. Let Bk denote the set of assigned subcarriers for user k. In the kth mobile transmitter, the datastream {c(k)} is serial-to-parallel-converted and partitioned into a i block of data c(k) = [c(k) (0), . . . , c(k) (Lm - 1)]T of length Lm = |Bk|, where |Bk| m m m denotes the cardinality of set Bk and ( )T denotes transpose. The vector c(k) is m extended with the insertion of nf - Lm zeros to produce the IDFT inputs B(k) = [B(k) (0), . . . , B(k) (nf - 1)]T, where the component of B(k) is de ned as m m m m
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( k) c m (l ), if n = Bk (l ) ( Bmk ) (n) = otherwise 0,
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Figure 10.11. Mobile transmitter of the OFDMA system.
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Figure 10.12. Base station receiver of the OFDMA system.
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where Bk(l) denotes the lth element of the set Bk. For example, if nf = 64 and Bk = {1, 5, 64} during the mth block, then there are only three nonzero 64IDFT inputs of the kth mobile transmitter: c(k) (1), c(k) (2), c(k) (3) mapping to m m m subcarrier 1, subcarrier 5, and subcarrier 64, respectively. The other subcarrier inputs are all zero. The continuous time-domain signal samples during the mth OFDM symbol after IDFT, cyclic pre x of length Ncyclic and D/A conversion is given by
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(k ) if t [0, n f T0 ) xm (t )(l ), (k xm ) (t ) = (k ) xm (t + n f T0 ), if t [- N cyclicT0 , 0)
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where T0 is the OFDM sample duration given by 1/W and x(k) (t) is the timem domain signal before cyclic pre x given by
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k x (m ) (t ) =
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k) m
j 2pnWt exp nf
Hence, the transmitted signal from the kth mobile after compensation for the frequency and timing offsets is given by x (k ) (t ) = exp(- j 2pD k t )
m =-
(k xm ) (t - t k )g(t - mT - t k )
where T = (nf + Ncyclic)T0 is the OFDM symbol duration; D k and t k are the compensation for frequency offset and timing offset of user k, respectively; and g(t) is the windowing pulse for subcarrier spectral shaping, which is nonzero only for t [-NcyclicT0, nfT0). For simplicity, we assume rectangular pulse for g(t). At the receiver, the signal from the kth mobile suffers from a frequency offset of Dk and a timing offset of tk. The frequency offset is contributed by the Doppler shift and the local frequency offsets at the transmitter and the receiver. While the receiver frequency offset at the base station is common for all users, the Doppler shift and the transmitter frequency offset are different for each user. Similarly, the timing offset is contributed by the propagation delay and the timing errors at the mobile transmitter D/A and the base station receiver A/D sampling. While the base station receiver A/D sampling timing offset is common for all users, the rst two components are different for each user. Let k = Dk - D k and ek = tk - t k be the residual frequency offset and the residual timing offset of user k respectively. In addition, decompose the residual timing offset ek into an integral part and a fractional part as e k = (m k + d k )T0 (10.48)
where mk = [ek/T0] is the integral part of the residual timing offset and dk [0, 1) is the fractional part of the residual timing offset. The received signal after A/D (sampling at t = nT0) is given by