Page 277 - Antennas for Base Stations in Wireless Communications

250 Chapter Seven

scatterers within the propagation environment or the motion of the
transmitter and/or receiver. The time delay relative to the excitation time
t is represented by t. It is assumed that the input response is finite, i.e.,
h P = 0 for t >t 0 and that h P remains constant over a time interval t 0
so that the physical channel can be treated as a linear, time-invariant
system over a single transmission. The input signal x A (t) creates the
field x P (t, q T , f T ) radiated from the transmit array, where (q T , f T ) denote
the elevation and azimuth angles. At the receive array, the field distribu-
tion is expressed as the convolution:

y t,θ R ,φ R ) = ∫ 0 2 π ∫ 0 π −∞ ∫ ∞ h t, ,θ R ,φ R ,θ T ,φ T x ) p p t ( − τ ,θ T ,φ T )sin(θ T ) d dθ T dφ (7.4)
τ
τ
(
(
p
T
p
The element in the receive array samples this field and generates
N × 1 signal vector, y′ (t), at the array terminals. The noise from
R
A
the propagation channel and receiver front-end electronics (thermal
noise) is lumped as a N × 1 vector g(t) and is injected at the receive
R
antenna terminals. The resulting signal-plus-noise vector, y (t), is then
A
downconverted to produce N × 1 baseband output vector y(t), which is
R
finally passed through a matched filter whose output is sampled once
(k)
per symbol to produce y , after which the space-time decoder produces
ˆ( )k
estimates b of the originally transmitted symbols.
The nature of the MIMO channel is important in the design of efficient
communication algorithms and understanding its performance limits.
For a system with N transmitting antennas and N R receiving antennas,
T
and assuming frequency-flat fading over the bandwidth of interest, the
MIMO channel at a given time instant can be written as
 H H .... H 
,
,
 H 1 1 H 1 2 .... H 1, N T 
,
T
H =   2 1 , 2 2 2, N   (7.5)
 H H .... H 
  N 1 N ,2 N , N T  
,
R R
R
R
th
where H m,n is the channel gain between the m receiving antenna
th
and n transmitting antenna pair. A full-rank transfer matrix results
in optimal MIMO system performance, which is achievable when the
correlation between the different antennas is low. Under ideal condi-
tions when the channel elements are totally decorrelated, H m,n (m =
1,2,…., N R , n = 1,2,…., N T ) ∼ i.i.d. N(0,1). Hence, for an independent and
identically distributed Rayleigh fading MIMO channel, H=H w and the
spatial diversity order is equal to N T N R . However, with the increasing