Network Coding and Wireless Physical-layer ... - Jacobs University

10 Chapter 2: Introduction to Digital Communication Systems and Networks
where m x is the mean of the distribution, which is zero in this case, and σ 2 is the noise
variance.
The assumption of a Gaussian distribution is supported by the natural phenomenon
of thermal agitation of charge carriers inside an electrical conductor. The amplitude is
almost Gaussian distributed, and the frequency spectrum is almost white, thus the name
additive white Gaussian noise (AWGN) [30].
2.2.4 Rayleigh Fading Channel
The Rayleigh fading channel model is a physical abstraction of wireless channels, where
signal rays from the transmitter are scattered by obstacles in the environment, forming a
number of transmission paths to the receiver, who receives the summation of signals from
these paths. Therefore, the received band-pass signal may be expressed as follows [30].
r(t) = ∑ n
α n (t)s[t − τ n (t)], (2.6)
where α n (t) is the attenuation factor for the signal received on the n th path and τ n (t) is
the propagation delay of the n th path. Now, if the transmitted signal s(t) is expressed as
s(t) = R[s l (t)e j2πfct ], (2.7)
where s l (t) is the low-pass transmitted signal in complex base-band representation and f c
is the carrier frequency. The received low-pass signal in base-band can be written as
r l (t) = ∑ n
α n (t)e −j2πf cτ n (t) s l [t − τ n (t)]. (2.8)
According to (2.8), the equivalent low-pass channel in base-band can be described by
the following time-variant channel impulse response
c(τ; t) = ∑ n
α n (t)e −j2πf cτ n (t) δ[t − τ n (t)]. (2.9)