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Paulraj et al., 1997. Space - Time Processing
for Wireless Communications, IEEE Signal
Processing Magazine, 49-83, 1997.
• Frequencies for wireless communication a
scarce resource
• Conventional “point to point”
communication limited by Shannon’s
capacity
• Space-time processing “point to volume”
SIMO (Sinlge Input Multiple Output) and
“volume to point” MISO (Multiple Input
Sigle Output): Shannon capacity ∝ log
number of antennas. Paper treats these
cases.
• Space-time processing “volume to
volume” MIMO (Multiple Input Multiple
Output): Shannon capacity ∝ smallest
number of antennas.
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Channel modelling
• Path loss
— Pr ∝ Pt/Dν , ν = 2.5 − 5. Pr - transmitted
power, Pr - received power
• Fading - spatial variability in
electromagnetic field (Figure 2).
— α(t) = αs(t)αr (t)
∗ αs(t) - slow fading (log normal)
∗ αr (t) - short term fading (Rayleigh or
Rician)
— Doppler, delay spread and angle spread
(Fig. 3).
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Channel model - reverse link
(mobile-to-base), SU-SIMO, TDMA (Fig. 6)
xi(t) =
PL
R
l=1 ai (θl )αl (t)u(t
− τl ) + ni(t)
• xi - received (baseband) signal at i-th array
element
• L - number of multipaths
• ai(θl ) - response og i-th array element to
signal along path l arriving at angle θl
• αlR (complex) fading along path l
• u(t − τl ) - transmitted signal along path l
delayed by τl ⇒ ISI - Inter symbol
interference.
P
Here u(t) = i g(t − iT )s(i)
— g(·) - pulse shaping waveform
— s(k) - symbol (information bits)
— T - symbol length
• ni(t) - additive noise
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Consider all elements xi(t) and define the
array response vector
a(θl ) = [a1(θl )a2(θl ), · · · , am(θl )]T .
Channel model - reverse link (mobile-to-base),
SU-SIMO
x(t) =
L
X
a(θl )αlR(t)u(t − τl ) + n(t)
l=1
Channel model - reverse link
(mobiles-to-base), MU-SIMO (Q - users ⇒
CCI - Co Channel Interference)
x(t) =
Q X
L
X
R
a(θlq )αlq
(t)uq (t − τlq ) + n(t)
q=1 l=1
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Channel model - forward link
(base-to-mobile), SU-MISO
Space only: w beamforming weight vector
x(t) =
L
X
wH a(θl )αlF (t − ti)u(t − τl ) + n(t)
l=1
Space-time :
x(t) =
X
wiH
i
L
X
a(θl )αlF (t)u(t − τl − ti) + n(t)
l=1
Channel model - forward link
(bases-to-mobile), MU-MISO
Space only - signal received at user d of Q:
xd(t) =
Q
X
q=1
wqH
L
X
F
a(θld)αld
(t)uq (t−τld−ti)+nd(t)
l=1
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Reverse link (mobile-to-base), SU-SIMO
Converts to “symbol response” channel model.
Sample at symbol rate tk = kT .
P
We have u(tk ) = i g(tk − iT )s(i). Assume
min(τl ) < 1 symbol length and max(τl ) is the
channel length corresponding to N symbol
lengths.
x(k) =
L
X
a(θl )αlR(t)u(tk − τl ) + n(k)
l=1
= Hs(k) + n(k)
Since a(θl ) is (m × 1), H is (m × N ) and


s(k)


..


s(k − N + 1)
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Reverse link (mobiles-to-base), MU-SIMO.
Symbol response model with CCI
x(k) = Hsss(k) +
Q
X
Hq sq (k)n(k)
q=1
“Final and versatile block model”
Define
X(k) = [x(k), · · · , x(k + M − 1)] , (m × M )
X(k) = HS(k) + N(k)
S(k) = [s(k), · · · , s(k + M − 1)] , (N × M ) Toeplitz
N(k) = [n(k), · · · , n(k + M − 1)] , (m × M )
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• Given channel parameters - easy to
compute H
• The challenge: Compute H and S from X
• Ill-posed problem - requires regularization
offered through structure (Figure 8)
• Assume first reverse link with H known
— Time only equalizer - generally effective
against ISI but not CCI (Figure 9)
— Space only array processor - generally
effective against CCI but not ISI (Figure
10)
— Space-time offers better performance
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Reverse link (mobile-to-base) space-time ,
SU-SIMO, H known
• ST - MLSE (Maximun Likelihood Sequence
Estimation) N Gaussian white in space and
time: argminS ||X − HS||2F (Not a regression problem)
Use Viterbi algorithm to carry out a
“dynamic programming like” search
Does not generalize well to correlated
noise/interference. CCI problematic.
• ST - MMSE (Minimum Mean Square Error)
Handles CCI better, but not as good at
handling ISI.
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Some blind identification methods
• Iterated MLSE (Maximum Likelihood
Sequence estimation)
• DOA (Direction Of Arrival) techniques
from array processing
• CM (Constant Modulus) methods (special
case of Bussgang algorithms)
• Unmixing through HOS (Higher Order
Statistics)
• Polyphase methods
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