Wireless CommunicationChapter: Analog and Digital
Transmission

Various definitions of BER's are relevant to a system designer: the instantaneous BER B_{0} of an individual subcarrier with a given amplitude, and the localmean BER B_{1}, thus averaged over all channels. We compute B_{0} as the BER for a given b _{n,n}, but otherwise averaged over all channels, i.e., averaged over b _{m,n }(m ¹ n). That is, B_{0} can be interpreted as the expected value of the BER if only the subcarrier amplitude is known (or estimated) from measurements, but without any knowledge about the instantaneous value of the ICI. A typical OFDM receiver would forward such side information to the error correction decoder.
We consider a quasistationary radio link, in which channel variations cause ICI, but the power P_{0} = b _{n,nb n,n}*/2 for each subcarier is reasonably constant during an OFDM frame. Formally these two assumptions conflict, but for small Doppler shifts they may be reasonably accurate [1011]. For OFDM, the instantaneous signaltonoiseplus ICI ratio g equals
(10)
We calculate BER's for BPSK, however the usual QAM can be expressed with similar formulas. In a Rayleigh channel, b _{m,n} results from the addition of many independent waves, so it is a complex Gaussian random variable [17]. The contributions to the ICI become Gaussian, so the BER B_{0} = erfc(Ö g ). Moreover, since b _{n,n }is complex Gaussian, b _{n,nb n,n}* has an exponential distribution with mean P_{0}. Therefor, the SNR g has the probability density fg (g )= g _{lm}^{ 1} exp{g _{lm}^{1} g } with localmean SNR
g _{lm} = P_{0} T_{s} /( N_{0} + T_{s} S _{(D¹ 0)} P_{D} ).
After averaging, the localmean BER for BPSK modulation becomes (see computation for BER for Rayleigh fading)
(11)
In the denominator, the summing is over all integer D within the range of active subcarriers, thus including D = 0.
See also localmean SER for QAM in Rayleigh fading channel.
For engineering applications with small Doppler spreads, a rule of thumb can be derived by considering only ICI for adjacent subcarriers and by making a firstorder approximation for the sinc. For arguments near zero, we take sinc(f f_{s}^{1}) » 1, so we find that P_{0} » P_{T}. For D = 1 (and for f << f_{s}), we approximate sinc(1 + f f_{s}^{1}) » sinc(1) + (1 + f f_{s}^{1 } 1) sinc'(1) = f f_{s}^{1}. Moreover we observe that P_{1} = P_{1}. Inserting these, and using eq. 2.272.3 in [20], we find
(12)
This leads to the localmean BER for small l
(13)
The effect of the ICI dominates over the AWGN for vehicle speeds
v > f_{s} f_{c}^{1}c Ö (2N_{0}/P_{T} T_{s}). (14)
Figure 2: Received power P_{0} (green), and the variances P_{1}, P_{2}, and P_{2} of the ICI (red) versus the normalized Doppler spread l for p_{T} = 1.
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