mohatta2015.pdf

14 INTRODUCTION
is the "channel" response, which includes the symbol waveform at the transmitter
as well as the medium response.
1.3.2.1 Noise and interference models The term n(t) models noise. Here we will
assume this noise is additive, white Gaussian noise (AWGN). Such noise is implicitly
assumed to have zero mean, i.e.,
m n (i) = E{n(i)}=0. (1.25)
The term "white" noise means two things. First, it means that different samples
of the noise are uncorrelated. It also means that its moments are not a function of
time. That is, the covariance function is given by
C n (h,h) â Ε{[η(ίι) - m n {h)\[n*{t 2 ) - m* n {t 2 )\) = N a 6 D (U - h), (1.26)
where 5r>{r) denotes the Dirac delta function (a unity-area impulse at τ = 0).
Another implicit assumption with AWGN is that it is proper, also referred to
as circular. This has to do with the relation between the real and imaginary
parts of an arbitrary noise sample n(i () ) = n = n r + jn,. With circular noise,
the real and imaginary components of n(io) are uncorrelated and have the same
distribution. With AWGN, this distribution is assumed to be Gaussian, which is a
good model for thermal noise. A circular, complex Gaussian random variable (r.v.)
has probability density function (PDF)
where m n is the mean, assumed to be zero, and TVo is the one-sided power spectral
density of the original radio signal (noise on the I and Q components has variance
σ 2 = 7V()/2). If we write n — n r + jrii, where n r and n¿ are real random variables,
then n r is Gaussian with PDF
1 J-(x m, 12
/„„ (x) = -== exp
V7r7V 0 L Λ/ 0 J
and has cumulative distribution function (CDF)
v
Tl
(1.28)
F nr (x)àp T {n r