mohatta2015.pdf

MORE MATH 21
1.4.2 Channel
The model used in the previous section is extended to allow for multiple transmit
and receive antennas. The received vector (N r receive antennas) can be modeled
as
N,. L-l
r W H Σ Σ 8^ι (0 (ί - re) + n(í), (1.45)
¿=i *=o
where g¿ is a vector of medium response coefficients, one per receive antenna.
Also, unless otherwise indicated, all vectors are column vectors.
In general, the medium responses from transmit antennas in different locations
will have different path delays. We can handle this case by modeling all possible
path delays and setting some of the coefficient vectors to zero.
By substituting (1.33) into (1.45), we obtain the following model for the received
signal:
where
N t K-\ , . oo
KOhÊEV^W Σ ^m{t-rnT)sf{m)+n{t), (1.46)
ΐ=1 fc=0 m=-oo
is the channel response.
e=o
(1.47)
1.4.2.1 Noise and interference models Here the noise model is extended for multiple
receive antennas, and more general noise models are considered. We will still
assume the noise has zero mean, i.e.,
m n (i) 4 E{n(i)} = 0, (1.48)
where boldface is used for column vectors. All vectors are JV r xl.
The noise may be colored, meaning that there may be correlation from one time
instance to another as well as from one antenna to another, and the covariance
function may be a function of time. For multiple receive antennas, the correlation
is defined as
C„(íi,í 2 ) 4 E{[nft!) - m„(ti)][n(ta) - m n (i 2 )] H }, (1.49)
where superscript "H" denotes conjugate transpose (Hermitian transpose). If t\ =
t 2 + T and the correlation depends on both t 2 and τ, then it is considered nonstationary.
If it only depends on τ, it is stationary and is then written as C n (r).
We will still assume the noise is proper, also known as circular. With circular
Gaussian noise, the I and Q components of n(i) are uncorrelated and have the same
autocorrelation function, i.e.,
E{n r (Í!)n;(í 2 )} = E{n i (í 1 )n*(í 2 )} = (0.5)C n (í 1 ,í 2 ) (1.50)
E{n,.(tiK(t 2 )} = E{ni(iiK(t 2 )} = 0. (1-51)