OFDM System and Channel Models

6.2.1

OFDM system model

Consider an OFDM system withN subcarriers modulated by a sequence ofN complex data symbolsd= [d₀,d₁,....,d_N−1]T. The data symbols are selected uniformly from a particular constellation such as quadrature amplitude modulation (QAM). The modulation process can be implemented efficiently using N-points IFFT, where the IFFT output during the`th OFDM blockx(`) =FHd(`). MatrixFis the normalized FFT matrix whose elements are defined as F_i,k = κe−ωik, κ = 1/√N, ω = j2π/N, j =√−1 and{i, k}= 0, 1,...,N−1 denote the row and column numbers, respectively. SinceFis a unitary matrix, thenFH =F−1. Consequently, thenth sample in xcan be expressed as

x_n(`) = κXN−1 i=0 di(`)e

ωin_{, n = 0, 1, ..., N −1. (6.1)}

To eliminate the inter-symbol-interference (ISI) between consecutive OFDM symbols in frequency-selective multipath fading channels, a cyclic prefix (CP) of length ¯N samples no less than the normalized channel delay spread (L_h) is formed by copy- ing the last ¯N samples of x and appending them at the front of x to compose the transmission symbol with a total length N_t = N + ¯N samples and a duration of T_t seconds. Hence, the complex baseband symbol transmitted during the `th signaling period can be expressed as

˜

x(`) = [x<sub>N−N</sub>¯(`), x_N−N+1¯ (`),..., xN−1(`), x0(`) , x1(`),. . ., xN−1(`)]T. (6.2)

Consequently, the ith transmitted sample ˜x_i(`) = x<sub>hi−N¯i</sub>(`), i = 0, 1, ..., N_t −1, where hii_,i mod N. Then, the sequence ˜xis upsampled, filtered and up-converted to a radio frequency centered at f_c.

composed of L_h+ 1 independent multipath components each of which has a gain h_m and delay m×T_s, where m ∈ {0, 1,..., ¯N}. The channel taps are assumed to be constant overN OFDM symbols, which corresponds to quasi-static multipath fading channels. The received sequence ˜y = ˜_H(`)˜x(`)+˜z(`), where the channel matrix ˜_H is an N_t ×N_t Toeplitz matrix with h₀ on the principal diagonal and h₁,..., h_L

h on

the minor diagonals, respectively [24] and ˜z(`) represents the overall additive noise that includes AWGN and IN. For notations’ simplicity, we omit the subscript ` in the remaining parts of this section. Given that L_h<N¯, the nth sample of ˜y can be expressed as ˜y_n=PL_h

i=0hix_hn−i−N¯i+zn. Subsequently, the receiver should identify

and extract the sequence y = [˜y_N¯, ˜y_N¯+1, . . . ,y˜_{N+ ¯N−1}] and discard the first ¯N CP samples [˜y₀, ˜y₁, . . . ,y˜_N¯−1]. Therefore,

y_n =XLh

i=0hixhn−ii+zn+ ¯N, 0≤n ≤N −1. (6.3)

In vector notation, the sequence y can be expressed as

y=_Hx+z (6.4)

wherez= [z_N¯,z_N¯+1,· · ·,z_N¯+N−1]T, and the channel matrix_His circulant [24], [25].

6.2.2

Impulsive noise model

The IN is usually characterized by bursts of one or more short pulses whose ampli- tude g, duration T_I, and time of occurrence are all random parameters. The main models used for the amplitude distribution are the Middleton class-A [26], exponen- tial [18] and Gaussian [27]. The common distributions of the bursts’ arrival process are partitioned Markov chains (PMC) [18], Gated Bernoulli-Gaussian (GBG) [27], and Poisson [19]. The burst width distribution is usually modeled using the PMC or as the sum of two lognormal random variables [13].

To enable analytical performance evaluation over IN channels, the GBG model is widely adopted in the literature where the IN is modeled as a sequence of indepen-

dent, Bernoulli distributed bursts with equal and fixed width, which is equal to the duration of a single time-domain OFDM sample [15, 16, 27]. Hence, the overall noise samples at the receiver frontend z_n=w_n+ρ_ng_n, where the AWGNw_n ∼ N_c 0, σ2_w and σ2_w =E|wn|2 = N0/2. The IN component ρngn is modeled as a gated Gaus-

sian process, where ρ_n ∼ B {0,1} where p=P{ρ_n = 1}, and g_n ∼ N_c 0, σ2_g where σ2_g σ2_w [15].

In this work, the GBG model is adopted to represent the IN, which is modeled as a sequence of independent bursts each of which consists of 1≤_κ≤N+ ¯N pulses. In general, the burst width_κis a random variable whose distribution can be modeled as a uniform or lognormal distribution, or by using the PMC [13]. Therefore, the overall received noise z=w+ρb g, where z= [z₀, z₁,..., z_{N+ ¯N−1}]. The vector b is used to specify the position as well as the width of the IN burst with respect to the OFDM symbol. Given that n₀ is the index of the first OFDM sample that is affected by a noise burst, then the elements of b are

b_n= ( 1, n₀ ≤n < n₀+_κ 0, otherwise , (6.5) where n₀ ∼ U 0, N + ¯N −_κ

. Therefore, the location of the noise burst will be random and uniformly distributed over the OFDM symbol period. Thus, any IN burst can affect a maximum of one OFDM symbol.