A Joint Symbol Detection Algorithm
Efficient at Low SNR for a Multi-Device
STBC MIMO System
STBC-MIMO
Muhammad Naeem and Daniel C. Lee
1
Motivation
¾ The Optimal detector (maximum likelihood (ML) detector) for
Multi device STBC
Multi-device
STBC-MIMO
MIMO is computationally expensive
expensive.
¾ The Sphere decoding algorithms operation at a low SNR
requires inordinately high computation.
¾ There is need of low complexity algorithms for Multi-device
STBC-MIMO systems.
2
Problem Formulation
¾
¾
¾
¾
¾
¾
¾
¾
¾
K = Number of Mobile devices/users
NB = Total number of transmitted symbols
NR = Number of receive antennas
NT = Number of receive antennas
Xk = Symbols transmitted from the kth device
Y = received signal
M = Size of symbol constellation
b = log2(M)
m = Problem size
3
Problem Formulation
¾Consider the case of single mobile device, K=1.
¾For T time slots in the space
space-time
time code block
block,the
the relation
between the input and output signal is
NT ×NR
T ×NT T × NR
Y = S ⋅ H + Z, H ∈^
, S ∈ ^ ,Y ∈ ^ ,
where
S = ∑ q =1 (α q + j β q ) Cq + (α q − j β q ) Dq 
Q
Q is the number of symbols in a space time code block.
Q is complex.
¾An alternative representation is
2Q×2TN
NR
Y = X Ω + Z , Ω ∈ \ 2Q×2T
4
Problem Formulation
¾For multiple mobile devices
K
Y = ∑ k =1 Sk ⋅ H k + Z , Sk ∈ ^T × NT , H k ∈ ^ NT × N R
¾An alternative representation is
Y = [ X1
X2 "
XK ]
 Ω1 
Ω 
 2+Z
 # 
 
Ω K 
5
Optimum Detector
Maximum Likelihood (ML) Detection
arg min Y − ∑ k =1 X k Ωk
K
ML Complexity
¾ Exponential Complexity (NP-Hard Problem i.e.
Can not solve in polynomial time )
¾ Exhaustive Search with Search Space exponent
M
NB
=2
bN B
6
1
2
Mobile
Device 1
1
STBC
encoder
NT
1
2
Mobile
Device 2
STBC
encoder
2
NT
3
Receiver
1
2
Mobile
Device K
STBC
encoder
NT
NR
7
Cross Entropy Optimization
¾ The Cross-Entropy optimization is a generalized Monte Carlo
technique to solve combinatorial optimization problems
¾ In general the CEO first associates the target combinatorial
optimization problem with a rare-event estimation problem,
and then solves the problem iteratively in two phases.
¾ IIn first
fi t phase
h
the
th CEO generates
t a sample
l off random
d
data
d t
according to a specified random mechanism.
¾ In the second phase it updates the parameters of the random
mechanism, typically parameters of probability distribution, on
the basis of the data, to produce a better sample in the next
iteration
iteration.
8
Cross Entropy Optimization
¾ Consider a maximization problem: maximize
x∈X
F ( x)
¾ Let us denote a maximum by x* and the maximal function
value by γ*.
¾ f ( x; u ), x ∈ X is an arbitrary probability mass function used in
at a stage in iterations.
¾ Hypothetically, if pmf
g ( x; γ , u ) =
I { F ( x ) ≥ γ } f ( x; u )
∑
x∈X
I { F ( x ) ≥ γ } f ( x; u )
≡
I { F ( x ) ≥ γ } f ( x; u )
l (u , γ )
is used as the pmf at the next iteration, then every sample
generated from this distribution will be a high-quality
candidate.
9
¾ The CEO algorithm
g
uses in p
place of g ( x; γ , u ) the p
pmf that is
closest to g ( x; γ , u ) in terms of Kullback-Leibler (KL) distance
(cross entropy) . That is, the pmf v that minimizes
g ( x; γ , u )
D ( g ( x; γ , u ) f ( x; v ) ) = ∑ x∈X g ( x; γ , u ) ln
f ( x; v )
= ∑ x∈X g ( x; γ , u ) ln g ( x; γ , u ) − ∑ x∈X g ( x; γ , u ) ln f ( x; v )
¾ Minimizing this KL-distance by choosing pmf v is equivalent to
maximizing
∑
x∈X
g ( x; γ , u ) ln f ( x; v ) ,
¾ This is also equivalent to maximizing
∑
I
,
=
I
f
x
;
u
ln
f
x
;
v
E
ln
f
X
;
v
(
)
(
)
(
)
u
≥
γ
≥
γ
F
x
F
X
x∈X { ( ) }
 { ( ) }

10
¾ IIn order
d to
t avoid
id computational
t ti
l complexity,
l it a CE algorithm
l ith
finds in the family of pmfs, a pmf v that results in the largest
1 Γ
I{F ( X )≥γ } ln f ( X i ; v ) ,
∑
i
Γ i =1
11
¾ In
I general,
l CE algorithm
l ith proceeds
d as:
1. Define v0=u,
u, Set l=1(Iteration
l 1(Iteration counter)
2. Generate samples X1,…,XΓ from the density f(:,vl-1)
3. Evaluate the objective function and order them from the
smallest to largest: F1 ≤ F2 ≤ " ≤ FΓ Then set the (1-ρ)-quantile
γl as γl = F((1−ρ)Γ)
4 Use the same samples X1,…, XΓ to obtain a new pmf that
4.
results in largest (10). Denote this pmf by index vl.
5. If stopping criterion satisfied then terminate otherwise set
l=l+1 and reiterate from step 2.
12
F1 ≤ F2 ≤ " ≤ FΓ
F((1−ρ )Γ )

13

Cross Entropy Optimization For STBC-MIMO
STBC MIMO
We define the following parameters
1 F denotes
1.
d
the
h fitness
fi
or objective
bj
i
function,
f
i
which
hi h is
i the
h
target function to be optimized.
2. Γ is the sample size, or population.
3 Parameter m is the dimension of each sample or
3.
individual.
4. P is the parameter vector, which indicates the pmf used to
generate the random samples.
samples
5. ρ is the rarity parameter. In each iteration, a threshold γ
is set to select ρΓ best samples among Γ samples to
update
p
P.
6. α is the smoothing parameter.
7. ε is the tolerance. The iteration is stopped when F is
within ±ε of the target value.
8. I is the maximum number of iterations. The iteration is
stopped when I is reached.
14
Pseudo code of CEO For Sensor Selection
15
Simulation Results
16
Bit Error Rate (BER) Performance
Parameters
MMSE
V-BLAST
SDR
SD
CEO
-1
10
NT=4
K=3
4-QAM
4
QAM
-2
BER
10
NR=4
Γ = 100
α = 0.7
-3
10
ρ = 0.3
-4
10
m = 24
0
2
4
6
SNR
8
10
12
17
Bit Error Rate (BER) Performance
0
10
Parameters
MMSE
V-BLAST
SDR
SD
CEO
-1
10
NT=2
K=5
BER
4-QAM
4
QAM
NR=5
-2
10
Γ = 200
α = 0.7
ρ = 0.3
-3
10
m = 20
-4
10
0
2
4
6
SNR
8
10
12
18
Bit Error Rate (BER) Performance
MMSE
V-BLAST
SDR
SD
CEO
-1
10
Parameters
NT=2
K=4
4-QAM
4
QAM
-2
BER
10
NR=6
Γ = 80
α = 0.7
-3
10
ρ = 0.3
-4
10
m = 16
0
2
4
6
SNR
8
10
12
19
Complexity Comparison
Problem Dimension m ≡ NB log2 M
¾ ML
O (2
¾ MMSE
O ( m3 )
¾ V-BLAST
O ( m3 )
¾ Sphere Decoder
¾ CEO
m
)
O ( m6 )
O ( ΓI
)
20
Conclusions
¾ The Optimal detector (maximum likelihood (ML) detector) for
Multi device STBC
Multi-device
STBC-MIMO
MIMO is computationally expensive
expensive.
¾ The performance of CEO algorithm is close to the Sphere
decoder and better than SDR.
¾ Complexity of CEO is less than sphere decoder and SDR.
21
Questions ?
22