To simplify the discussion, assume initially that the channel matrix H is the identity matrix. Therefore, the received N samples, after removing the CP, can be written as
˘
y(`) =˘x(`) +w(`) +ρ(`)b(`)g(`). (6.10)
Since the deinterleaving process interleaves the vectors w(`),b(`) and g(`), it yields
y(`) =x(`) +w˘(`) + ˘ρ(`)˘b(`)˘g(`) (6.11)
where w˘ and ˘g are the interleaved AWGN and IN vectors, which have the same statistical properties as w and g. The burst and sample gating factors ˘ρ and b˘ have different properties from the original ρand b. For example, given that ρ(0) = 1 and
κ =Nt, then ˘ρ(`) = b0(`) = 1∀` ∈ {1,2, ...N}, i.e., the first sample of every OFDM
symbol after deinterleaving will be affected by an IN pulse, while all other samples in all symbols will be IN free. Consequently, the FFT output can be expressed as
rk(`) = κXN−1 n=0 yn(`)e −ωnk (6.12) = κ N−1 X n=0 h xn(`) + ˘wn(`) + ˘ρ(`)˘bn(`)˘gn(`)ie−ωnk, k = 0, 1, ..., N −(6.13)1
rk(`) = dk(`) +ψk(`) +uk(`) (6.14)
where the FFT of the AWGN ψk ∼ Nc 0, σ2w. The last term in (6.14) represents the FFT of the IN uk(`) = κ˘ρ(`)XN−1 n=0 ˘bn(`)˘gn(`)e −ωnk where uk ∼ Nc 0, σ2u , and σ2u =κ2˘ρ(`)σ2gXN−1 n=0 ˘bn(`) =κ2˘ρ(`)σ2gκ˘(`) (6.15)
where ˘κ6=κ is the number of nonzero elements in the vector˘b(`),i.e., the Hamming weight of ˘b(`).
It can be concluded from (6.14) and (6.15) that the deinterleaving process spreads the IN burst over most OFDM symbols within the deinterleaved block. The FFT process applied after the deinterleaving averages the IN pulses over all subcar- riers within a given OFDM symbol, which may cause the loss of up to κ OFDM symbols because σ2g σ2w.
An efficient solution to mitigate the effects of impulse noise is to apply blank- ing [12, 14], where the received samples with high amplitudes are set to zero. The contaminated samples are detected and suppressed by comparing the received sam- ples’ values with a particular threshold T1. Therefore, the output of the blanking nonlinearity is ˘qn = ˘yn ∀ |y˘n| ≤ T1 and 0 otherwise. We refer to this approach as sample-by-sample blanking. After blanking and deinterleaving, thenth sample of the FFT input qn = yn ∀ |yn| ≤ T1 and 0 otherwise, where yn = xn+wn +ρbngn. Therefore, the FFT output s(`) =Fq(`) and the kth subcarrier can be written as
sk(`) = κXN−1 n=0 qn(`)e −ωnk, k = 0, 1,..., N −1 =κX n∈Cyn(`)e −ωnk (6.16)
where C = {n ∈ {0,1, ..., N −1}|qn 6= 0}. The average error probability can be ex- pressed as
Pe =X
CP(e|C)P(C). (6.17)
However, evaluatingP(C) is quite difficult because it is a function of the bursts probability vector ρ, the bursts location vector n0, the bursts width vector κ, and the thresholdT1. Furthermore, it depends on the number and location of the blanked information samples.
The sample blanking thresholdT1 should be selected to minimize the BER, and it is a function of several variables such as signal-to-noise ratio (SNR), signal-to-IN ratio (SIR) and κ [12]. It is worth noting that the blanking process is not ideal in the sense that it will not necessarily blank all IN samples, and may blank some information samples, which is due to the high peak-to-average-power-ratio (PAPR) problem inherent in OFDM systems [28]. In addition, the sample-by-sample blanking is not feasible in frequency-selective channels due to the vast amplitude fluctuations that a signal may experience in such channels [16].
A possible remedy for these problems is to identify the contaminated symbols first, and then identify and blank the corrupted samples within that symbol. A simple technique to identify the contaminated symbols is to introduce an additional threshold T2 for the blanking process, whereT2 denotes the number of samples with amplitudes larger than T1. Then, all samples that have |yn| ≥ T1 will be blanked in a given received symbol, if and only if, the number of samples that will be blanked is larger than T2. Such blanking policy minimizes the chances to blank IN-free information samples from symbols other than the contaminated ones, and hence can improve the BER.
In a frequency-selective channel, despite the fact that detecting samples that are hit by IN, based on a threshold-based mechanism is impossible, detecting corrupted symbols based on the described technique is fairly easy and accurate. Hence, in frquency-selective channels, we simply blank the entire contaminated OFDM symbol. This approach is called symbol blanking for the rest of the chapter. The time diversity provided by the TDI interleaving mechanism allows symbol blanking without any
Based on the widely used assumption that σ2g σ2w, the symbol blanking process is expected to be highly accurate. By assuming a perfect burst detection process, the average probability of error over a block of N symbols can be expressed as
Pe=XN−1
=0 P(e|)P() (6.18)
wheredenotes the total number of IN bursts per block of N OFDM symbols, which has a binomial PDF P(=i) = N i pi(1−p)N−i. (6.19)
It is worth noting the symbol blanking substantially relaxes the system sensitivity to T1, which is difficult to estimate accurately in conventional OFDM systems [17, 29].