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

Chapter 2: Introduction to Digital Communication Systems and Networks 11
When the number of paths is large enough, the central limit theorem holds that the
channel impulse response behaves according to a zero-mean, complex-valued, Gaussian
distribution. If we denote the random variables of the real part, the imaginary part, and
the envelope of the impulse response by C R , C I , and C E , respectively, it follows that
C E =
√
C 2 R + C2 I . (2.10)
If C R and C I are independent Gaussian-distributed random variables with zero mean
and a variance of σ 2 , the probability density function of C E can be expressed as
p CE (r) = r σ 2 exp −r2 /2σ 2 . (2.11)
This distribution is called Rayleigh, giving rise to the name Rayleigh fading channel.
Note that when there are fixed scatterers or lines of sight in the channel, the zero-mean
assumption does not hold and the distribution is therefore not Rayleigh-distributed but
Ricean distributed. We, however, will not discuss the Ricean distribution in detail.
2.3 Routing in Communication Networks
In more complicated networks such as the Internet, there are more channels and other elements
involved in transmission. These elements are situated between the transmitter and
the receiver and are often called intermediate nodes. Traditionally, there are three important
functions that these nodes may choose to perform to help packets reach the destination.
These are store-and-forward, amplify-and-forward, and decode-and-forward. After
the emergence of network coding, the last two functions can be modified into amplify-andforward
with network coding and decode-and-forward with network coding, respectively.
No matter how sophisticated the signal processing techniques are at the intermediate
nodes before packets are forwarded, the most important thing is that packets must be
forwarded in the right direction. Otherwise, they will not reach the destination or might