Capacity of a dispersive channel and algorithms of power and bit loading

Chapter 1 Capacity of a dispersive channel 1.1 Capacity of a dispersive channel In general, a non ideal channel (linear dispersive channel) with additive Gaussian noise is characterized as a transmission medium with impulse response gCh(t) and additive Gaussian noise w(t) having power spectral density Pw(f). Let s(t) denote the channel input signal; then the receiver input signal is given by r(t) = s ∗ gCh(t) + w(t). In practice, the transmitted signal s(t) must satisfy a constraint on the statistical power expressed as ∫ +∞ 0 Ps(f) df ≤ VP (1.1) where Ps(f) is the power spectral density of s(t). The signal-to-noise ratio of the channel as a function of frequency is defined as ΓCh(f) = |GCh(f)|2 Pw(f) (1.2) The transmission passband B, is intuitively given by the interval of frequencies cha- racterized by large values of ΓCh(f). First, we want to determine an expression of the capacity C[b/s] that extends the expression valid for an ideal AWGN channel, to the case of a dispersive channel. To this end we divide the passband B, having measure B, into N subbands Bi, i = 1, . . . , N , of width ∆f = B/N , where ∆f is chosen sufficiently small so that Ps(f) and ΓCh(f) are, to a first approximation, equal to constants within the generic subband Bi, that is we assume ∃fi ∈ Bi such that Ps(f) = Ps(fi) and ΓCh(f) = ΓCh(fi), ∀f ∈ Bi. From Shannon equation (see (6.280) in [1]) the subchannel i has a capacity, C[b/s][i] = ∆f log2 [ 1 + ∆f Ps(fi) |GCh(fi)| 2 ∆f Pw(fi) ] (1.3) The total capacity C[b/s] is obtained by summing the terms C[b/s][i], that is N∑ i=1 C[b/s][i] = ∆f N∑ i=1 log2 [ 1 + ∆f Ps(fi) |GCh(fi)| 2 ∆f Pw(fi) ] , (1.4) 5