A. Paulraj, et. Al. “Array Processing for Mobile Communications.”
2000 CRC Press LLC. <http://www.engnetbase.com>.
ArrayProcessingforMobile
Communications
A.Paulraj
StanfordUniversity
C.B.Papadias
StanfordUniversity
68.1IntroductionandMotivation
68.2VectorChannelModel
PropagationLossandFading
•
MultipathEffects
•
Typical
Channels
•
SignalModel
•
Co-ChannelInterference
•
Signal-
Plus-InterferenceModel
•
BlockSignalModel
•
Spatialand
TemporalStructure
68.3AlgorithmsforSTP
Single-UserST-MLandST-MMSE
•

theapplicationsofSTPincellularnetworks.Finally,weconcludewithasummaryinSection68.5.
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68.2 Vector Channel Model
Channel effects in a cellular radio link arise from multipath propagation and user motion. These
create special challenges for STP. A thorough understanding of channel characteristics is the key to
developing successful STP algorithms. The main features of a mobile wireless channel are described
below.
68.2.1 Propagation Loss and Fading
The signal radiated by the mobile loses strength as it travels to the base station. These losses arise
from the mean propagation loss and from slow and fast fading. The mean propagation loss comes
from square law spreading, absorption by foliage, and the effect of vertical multipath. A number of
models exist for characterizing the mean propagation loss [22, 30], which is usually around 40 dB
per decade. Slow fading results from shadowing by buildings and natural features and is usually
characterized by a log-normal distribution with standard deviation agreed to 8 dB. Fast fading results
from multipath scattering in the vicinity of the moving mobile. It is usually Rayleigh distributed.
However, if there is a direct path component present, the fading will be Rician distributed.
68.2.2 Multipath Effects
Multipath propagation plays a central role in determining the nature of the channel. By channel we
mean the impulse, or frequency response, of the radio channel from the mobile to the output of the
antenna array. We refer to it as a vector channel, because we have multiple antennas and, therefore,
we have a collection of channels. The mobile radiates omnidirectionally in azimuth using a vertical
E-field antenna. The transmitted signal then undergoes scattering, reflection, or diffraction before
reaching the base station, where it arrives from different paths, each with its own fading, propagation
delay, andangle-of-arrival. Thismultipathpropagation, inconjunctionwithusermotion, determines
the behavior of the wireless channel. Multipath scattering arises from three sources (see Fig. 68.1).
There are scatterers local to the mobile, remote dominant scatterers, and scatterers local to the base.
We will now describe these three scattering mechanisms and their effect on the channel.
FIGURE 68.1: Multipath propagation has three distinct classes, each of which gives rise to different

FIGURE68.3: Channel frequencyresponseat fourdifferentantennas forGSMin atypicalhilly terrain
channel at 1800 MHz. Mobile speed is 100 KPH. The response is plotted at four time instances spaced
66 µsecs apart.
TABLE 68.1 Typical Delay, Angle and Doppler Spreads in Cellular
Applications
Environment Delay spread (
µ
sec) Angle spread (
deg
) Doppler spread (Hz)
Flat rural (Macro) 0.5 1 190
Urban (Macro) 5 20 120
Hilly (Macro) 20 30 190
Microcell (Mall) 0.3 120 10
Picocell (Indoors) 0.1 360 5
A multipath channel structure is illustrated in Fig. 68.2. Typical path power and delay statistics
can be obtained from the GSM
1
standard. Angle-of-arrival statistics have been less well studied but
several results have been reported (see [1, 2, 3]). The resulting channel is shown in Fig. 68.3.We
show a frequency response at each antenna for a GSM system. Since the channel bandwidth is 200
KHz, it is highly frequency-selective in a hilly terrain environment. Also, the large angle spread causes
variations of the channel from antenna to antenna. The channel variation in time depends on the
Doppler spread. Notice that since GSM uses a short time slot, the channel variation during the time
slot is negligible.
68.2.4 Signal Model
We study the case when a single user transmits and is received at a base station with multiple antennas.
The noiseless baseband signal x
i
(t) received by the base station at the ith element of an m element

, α
R
l
(t) is the complex path fading, τ
l
is the path delay, and u(·) is the transmitted
signal that depends on the modulation waveform and the information data stream. In the IS-54
TDMA standard, u(·) is a π/4 shifted DQPSK, gray-coded signal that is modulated using a pulse with
square-root raised cosine spectrum with excess bandwidth of 0.35. In GSM, a Gaussian Minimum
Shift Keying (GMSK) modulation is used. See [12, 30, 55] for more details. For a linear modulation
(e.g., BPSK), we can write
u(t) =

k
g(t − kT )s(k)
(68.2)
where g(·) is the pulse shaping waveform and s(k)represents the information bits.
In the above model, we have assumed that the inverse signal bandwidth is large compared to the
travel time across the array. For example, in GSM the inverse signal bandwidth is 5 µs, whereas
the travel time across the array is, at most, a few ns. This is the narrowband assumption in array
processing. The signal bandwidth is a sum of the modulation bandwidth and the Doppler spread,
with the latter being comparatively negligible. Therefore, the complex envelope of the signal received
by different antennas from a given path are identical except for phase and amplitude differences that
depend on the path angle-of-arrival, array geometry, and the element pattern. This angle-of-arrival
dependent phase and amplitude response at the ith element is a
i
(θ
l
) [37].
We collect all the element responses to a path arriving from angle θ

)α
R
l
(t)u(t − τ
l
)
(68.3)
where
x(t) =[x
1
(t) x
2
(t) ... x
m
(t)]
T
and x(t) and a(θ
l
) are m-dimensional complex vectors. The fade amplitude |α
R
(t)| is Rayleigh or
Rician distributed depending on the propagation model.
The channel model described above uses physical path parameters such as path gain, delay, and
angle of arrival. When the received signal is sampled at the receiver at symbol (or higher) rate, such
a model may be inconvenient to use. For linear modulation schemes, it is more convenient to use a
“symbol response” channel model.
Such a discrete-time signal model can be obtained easily as follows. Let the continuous-time
output from the receive antenna array x(t) be sampled at the symbol rate at instants t = t
o
+ kT .

L

l=1
a
i
(θ
l
)α
R
l
g((M
d
+  − j)T − τ
l
), i = 1 ...,m ; j = 1,...,N
(68.6)
where M
d
is the maximum path delay and 2T is the duration of the pulse shaping waveform g(t).
68.2.5 Co-Channel Interference
In wireless networks a cellular layout with frequency reuse is exploited to support a large number of
geographicallydispersed users. InTDMAandFDMAnetworks, whenaco-channelmobile operatesin
a neighboring cell, co-channel interference (CCI) will be present. The average signal-to-interference
power ratio (SIR), also called the protection ratio [24], depends on the reuse factor (K). It is 18.7 dB
for reuse K = 7 (IS-54), and 13.8 dB for reuse K = 4 (GSM). In sectored cells, CCI is significant
mainly from cells that lie within the sector beam. The received signal at a base station will therefore
be a sum of the desired signal and co-channel interference.
68.2.6 Signal-Plus-Interference Model
The overall signal-plus-interference-and-noise model at the base station antenna array can now be
rewritten as

of x(·) corresponding to time instants k,...,k+ M − 1, (and dropping subscripts for a moment),
we get
X(k) = HS(k) + N(k)
(68.8)
where X(k), S(k), and N(k) are defined as
X(k) =[x(k) ···x(k + M − 1)] (m × M)
S(k) =[s(k) ···s(k + M − 1)] (N × M)
N(k) =[n(k) ···n(k + M − 1)] (m × M)
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1999 by CRC Press LLC
Note that S(k) by definition is constant along the diagonals and is therefore Toeplitz.
68.2.8 Spatial and Temporal Structure
Given the signal model at Eq. (68.8), an important question is whether the unknown channel, H,
and data, s, can be determined from the observations X. This leads us to examine the underlying
constraints on H and S(·) which we call structure.
Spatial Structure
From Eq. (68.6), the jth column of H is given by
H
1:m,j
=
L

l=1
a(θ
l
)α
R
l
g((M

map of an underlying finite alphabet. For example, the IS-54 signal is a π/4 shifted DQPSK signal
given by
u(t) =

p
A
p
g(t − pT ) + j

p
B
p
g(t − pT )
A
p
= cos(φ
p
), B
p
= sin(φ
p
), φ
p
= φ
p−1
+ φ
p
(68.11)
where g(·) is the pulse shaping function (which is a square root raised cosine function in the case of
IS-54), and φ