We assume an OFDM system adopted in high-rate wireless LAN standards [4, 5]. Figure 7.8 shows the subcarrier arrangement. The subcarriers from 0 to 63 are generated by the 64-point inverse fast Fourier transform (IFFT).
Among those, the 12 subcarriers from 0 to 5, 32, and 59 to 63 are called
‘‘virtual subcarriers,’’ which are not used for actual data transmission (in other words, the virtual subcarriers transmit 0). The four subcarriers of 11, 25, 39, and 53 are ‘‘pilot subcarriers,’’ which always transmit known symbols to adjust the frequency of the local oscillator at the receiver, and the remaining 48 subcarriers are ‘‘data subcarriers.’’ After the 48 data subcarriers are
Figure 7.7 Calculation of Euclidean distance in the symbol interleaved coded OFDM scheme.
Figure 7.8 Subcarrier arrangement of an OFDM scheme.
generated by means of 64-point IFFT, the 16-sample long guard interval is added to the generated waveform (also see Figure 5.26). Interleaving is done within one OFDM symbol. Table 7.1 shows the transmission parameters to demonstrate the BER performance.
Figure 7.9 shows the BER in a frequency nonselective Rayleigh fading channel, namely, where there is one path in the multipath delay profile.
The theoretical BER is given by [2]
PQ, coherent b, fading = ∑∞
d=dfree
dP(d) (7.27)
P(d) = 冕∞
0
P(␥b′)d d∑−1
k=0 冉d−k1+ k冊(1 − P(␥b′))kp(␥b′) d␥b′
(7.28)
Table 7.1
Transmission Parameters for BER Evaluation Number of data subcarriers 48 (64-point IFFT)
Guard interval length 16 [samples]
Modulation/demodulation CQPSK
Data burst length 10 [OFDM symbols]
Subcarrier recovery Perfect
Channel model Frequency nonselective Rayleigh fading, 2-path and 4-path i.i.d. frequency selective
Rayleigh fading (Delay of each path is uniformly distributed within the guard interval.)
Figure 7.9 BER in a frequency nonselective Rayleigh fading channel.
whereP(␥b′) is given by (4.11), namely, the BER of coherent QPSK when
␥b′ is given, and p(␥b′) is the p.d.f. of ␥b′, which is given by (4.18). In addition, {d} are the weighting coefficients calculated from the transfer function of the convolutional code. Table 7.2 shows the values of {d} for the convolutional code with the generators given by (7.1) and (7.2) [6]. In the calculation of (7.27), the summation was upper limited byd = dfree + 4 =14.
Table 7.2
{d} for the Convolutional Code Given by (7.1) and (7.2) d=dfree(=10) 36
d=dfree+1 0 d=dfree+2 211 d=dfree+3 0 d=dfree+4 1,404
152 Multicarrier Techniques for 4G Mobile Communications
The computer simulation result agrees well with the theoretical one.
For the frequency nonselective fading channel where all the subcarriers are subject to the same attenuation at a time, there is no diversity effect even if we employ channel coding. This is very clear from Figure 7.9, where the BER reduces by factor 10−1 when the averageEb′/N0 gains+10 dB.
Figures 7.10 and 7.11 show the BER in a 3-path i.i.d. frequency selective Rayleigh fading channel for symbol interleaving and bit interleaving, respectively. The theoretical lower bound is given by the BER expression for theLth order diversity (with L= 3) [2]:
PQ, coherent
b, fading = 冉4␥1b′冊L冉2LL−1冊 (7.29)
Figures 7.12 and 7.13 show the BER against the interleaving depth for symbol interleaving and bit interleaving, respectively. From Figures 7.10
Figure 7.10 BER of symbol interleaved coded OFDM scheme in a 3-path i.i.d. frequency selective Rayleigh fading channel.
TE AM FL Y
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Figure 7.11 BER of bit interleaved coded OFDM scheme in a 3-path i.i.d. frequency selective Rayleigh fading channel.
and 7.12, we can see that the interleaving depth of four symbols (× 12 symbols) is enough to obtain good BER performance. Even when we set the interleaving depth to more than four symbols, there is no significant performance gain obtained. On the other hand, from Figures 7.11 and 7.13, we can see that the interleaving depth of 8 bits (× 12 bits) is enough to obtain good BER performance.
Figure 7.14 compares the BER between a symbol interleaving depth of four symbols and a bit interleaving depth of 8 bits. The performance with 8-bit interleaving is superior to one with four-symbol interleaving. This is because in symbol interleaving, the upper bit and lower bit in one encoded output are transmitted over the same subcarrier, so pairwise errors tend to occur, whereas for bit interleaving, they are transmitted over different subcarriers, so pairwise errors do not occur.
Figures 7.15 and 7.16 show the BER in a 4-path i.i.d. frequency selective Rayleigh fading channel for symbol interleaving and bit interleaving, respectively. The theoretical lower bound is given by (7.29) with L = 4.
Figure 7.12 BER against interleaving depth for symbol interleaved coded OFDM scheme.
Figure 7.13 BER against interleaving depth for bit interleaved coded OFDM scheme.
Figure 7.14 BER comparison between symbol and bit interleaved coded OFDM schemes.
Figure 7.15 BER of symbol interleaved coded OFDM scheme in a 4-path i.i.d. frequency selective Rayleigh fading channel.
Figure 7.16 BER of bit interleaved coded OFDM scheme in a 4-path i.i.d. frequency selective Rayleigh fading channel.
Furthermore, Figures 7.17 and 7.18 show the BER against the interleaving depth for symbol interleaving and bit interleaving, respectively. From Figures 7.15 and 7.17, we can see that, even if we increase the symbol interleaving depth, we cannot much improve the BER performance. This may be because of pairwise errors. On the other hand, from Figures 7.16 and 7.18, we can see that the BER of the bit interleaved coded OFDM scheme with an appropriate bit interleaving depth is close to the lower bound, and that the BER is sensitive to the bit interleaving depth chosen and there is an optimum interleaving depth to minimize the BER for the given channel parameter setting.
Finally, Figure 7.19 compares the BER between a symbol interleaving depth of four symbols and a bit interleaving depth of 8 bits. The performance with 8-bit interleaving is much superior to that with 4-symbol interleaving.