Stratified Diagonal Layered SpaceTime
Architecture
Mathini Sellathurai* and Gerard J. Foschini**
*Communications Research Centre, Canada
(This work has been carried out at Bell Labs during an internship)
**Bell Laboratories, Crawford-Hill, NJ
DMACS Workshop on Signal Processing for Wireless Transmission
(M,N) communication systems
• A communication structure uses multiple transmit
(M) and multiple receive (N) antennas and
provides enormous capacity in rich scattering
environments.
Heavy Scattering
M transmit antennas
N receive antennas
Known as BLAST systems
LAYERED SPACETIME CODES
*SPACETIME CODES DESIGNED USING 1-D CODER(S)
Primitive
bit stream
1-Dim Complex no.
stream
encoder
**HOW ARE 1-D CODED STREAM(S) ARRAYED IN SPACETIME TO
ENABLE 1-D RECEIVER SIGNAL PROCESSING?
SPACE
Transmit
anennas
(1
(2
(3
(4
(5
...
  symbol duration
TIME
*** CAN WE ACHIEVE NEAR CAPACITY IN A QUASISTATIC FADING ENVIRONMENT?
BLAST: Prior Work
*Introduction by Foschini (’96)
– Diagonal-Layered Spacetime : Foschini (’96)
• Wrapped spacetime coding: Caire and Colavolpe (‘2001)
• Trellis coding for D-BLAST: Wesel and Matache (‘2002)
– Horizontal Layered Spacetime (also known as V-BLAST):
Foschini and colleagues (’98) Codes designed for AWGN
channels
– Iterative detection for BLAST
• Hammons & El Gamal (’99), Sellathurai & Haykin (’99),
Ariyavastikul (’00)
**Stratified Diagonal Layered SpaceTime (SD-BLAST)
(M,N) CHANNEL ENVIRONMENT
• NARROWBAND, QUASI-STATIC
• LONG BURST PERSPECTIVE
• CHANNEL MATRIX, G, UNKNOWN TO TRANS, KNOWN TO REC
• EQUATION FOR N-D RECEIVE VECTOR, r
r (t )  Gs(t )   (t )
s: M-D trans vector, power P/M per transmit antenna
: N-D AWGN noise vector,  2var /component
G: M x N matrix Gaussian channel. Entries are independent and
identically distributed complex N(0,1) random variables
2
E Gij  1
2
Gij ~  22
(M,N) Generalization of Shannon’s (1,1)
2


C

log
1


g
Capacity Formula
2
• LogDet Formula



C  log 2 det  I N  GG H 
M


• Capacity Lower Bound
 2


C   log 2 1 
 2( N M  k ) 
 M

k 1
M
•Formula used in inventing BLAST
DIAGONAL BLAST
• Capacity Lower-bound
•
 2


C   log 2 1 
 2( N M  k ) 
 M

k 1
M
(5,5) system with interference nulling and cancellation
Space
Code/mod
Primitive
5:1
data
Demux
stream
Code/mod
Code/mod
Code/mod
Code/mod
(1
(2
(3
(4
(5
 2

log 2 1 
10 
M


 2

log 2 1 
6 
M


 2

 2

log 2 1 
2 
log 2 1 
8 
M


M


...
Time
 2

log 2 1 
2 
 M

THE RECEIVER BASED ON MAXIMIZING THE SIGNAL TO
INTERFERENCE AND NOISE RATIO
OF THIS ARCHITECTURE “ATTAINS” LogDet
FOR ALL M AND N, AND ALL  AND ALL CHANNELS.
DIAGONAL BLAST (D-BLAST)
ADVANTAGES - THEORETICAL STANDPOINT
• LogDet capacity “attained” for all (M,N) and
all channels all SNRs with 1-D codecs
DISADVANTAGES
• Payload may not be adequate
Wasted spacetime at start and at end of a burst can result
in not enough layers (coded blocks).
• Somewhat problematic to code for periodic SINR
BURST
Five
point
space
...
..
THIN:
(one symbol)
Poor code but
little waste
Burst duration
Wasted
Wasted
Symbol time ||
...
...
Time
HORIZONTAL
Capacity lower-bound:
Primitive
5:1
data
stream Demux
Code/mod
Code/mod
Code/mod
Code/mod
Code/mod
BLAST
 2


C  M  log 2 1 
 2( N  M 1) 
 M

(1
(2
(3
(4
(5
• Performs well and is easy to implement (Extremely Practical)
•Encoders can be 1-D codes designed for AWGN
•Receiver uses interference avoidance and cancellation
• Achieves near capacity for N>>M
Maximum capacity can be achieved by optimizing
*the number of antennas used
* detection order.
...
Time
CAPACITY CONTRASTS
16 RECEIVE ANTENNAS,  = 18 dB, OPEN LOOP
- COMPLEMENTARY DISTRIBUTION FUNCTIONS Probability {Capacity > Rate=1-e}
1
(1,16)
(1,8)
(1,4)
0.9
(1,2)
0.8
(1,1)
Log Det(16,16)
V-BLAST(16,16)
0.7
0.6
0.5
0.4
0.3
Receive
diversity
0.2
0.1
0
0
10
20
30
40
50
60
70
80
90
Rate(bps/Hz)
Outage capacity
e  PrC , G  R
(M,N) BURST PERFECTLY PACKED
WITH M “CONTINUOUS DIAGONALS”
EX: M = 5 WITH GREEN AND BLUE “CONTINUOUS DIAGONALS”, THERE ARE TWO OTHERS.
Space
five
transmit
antennas
………..
Time
Burst duration
-ISSUE: POSSIBILITY OF EFFICIENT COMMUNICATION OVER
THE M PARALLEL 1-D “HELICAL LAYERS”
- Used with iterative interference cancellation detection
We will introduce Stratified Diagonal Layers
(SD-BLAST).
Stratified Diagonal BLAST
• Fundamentals of stratification
• SD-BLAST transmitter-receiver structure
• Mutual information of SD-BLAST: Highlights of
the derivation
• Outage capacity evaluation of SD-BLAST:
Monte-Carlo method
STRATIFICATION AND CAPACITY
C = log2[1+(P/2)] bps/Hz
• Assume infinite signaling intervals
SIGNAL OF
POWER P
AWGN (2)
REPLACE
+
BY K
SIGNALS
OF TOTAL
POWER P
WITH SUITABLE ENCODING THE
CAPACITY DOESN’T CHANGE.
• Exploited in multi-level coding schemes to
achieve high band efficiency using binary codes
• Equal powers and various rates
• Various powers
WHY STRATIFY ALL M = 5 TRANSMISSIONS?
No stratification.
Stratification is harmless but pointless.
Only
purple
layer is
stratified
as shown
to left.
Stratification effective when all five are stratified.
PLY
For example, now
suffers less interference.
SD-BLAST: Transmitter
DEMUX
Stratification
1:4
CODE/MOD
CODE/MOD
CODE/MOD
CODE/MOD
S
1:4
DEMUX
1:5
1:4
CODE/MOD
CODE/MOD
CODE/MOD
S
S
CODE/MOD
CODE/MOD
DEMUX
STREAM
PRIMITIVE
CODE/MOD
CODE/MOD
CODE/MOD
CODE/MOD
CODE/MOD
S
CODE/MOD
1:4
CODE/MOD
DEMUX
DATA
DEMUX
1:4
DEMUX
CODE/MOD
CODE/MOD
CODE/MOD
CODE/MOD
S
CYCLE
EVERY
SYMBOL
STRATIFIED DIAGONALS: KEY FEATURES
*DIAGONALIZATION  TRANSMIT ANTENNAS FARE DIFFERENTLY
•STRATIFICATION  SELF INTERFERING VECTOR SIGNALS
•ONION PEELING RECEIVER EACH STRATUM FACES DIFFERENT
MUTUAL INFORMATION
...
Symbol duration ||
Time
- CONTINUOUS DIAGONALS: Good code, spacetime perfectly packed no waste, binary alphabet acceptable.
Detection
• Peeling away of successive plies (five strata each)
from outside-in.
White Noise
r1
r2
r3
r4
• Each onion ring (ply) is coded using 1-D Codes
(for example Turbo or LDPC codes )
•Equal powers and various rates
•Various powers and various rates
Mutual Information of SDBLAST: Highlights
• Power into a transmit antenna in an (M,N) system
P
M
• Power into a stratum in an (M,N) system with
n stratification
P
M n
PEELING OFF ONE PLY  PEELING
OFF ITS M HELICAL CONSTITUENTS
Each helix winding on the ith strata has constituents:
smi(t), signal from mth ant. on ith strata with interf. + noise
rm =  g1 , g 2 ,
, gM
 0


 0

  smi
 0


 0






n


l i





 s1l

 s2 l
( mi )



s
 Ml





  .







N-D vector signal gmsmi, plus spatially colored “noise”
Whitening Matched Filter Receiver
Identify “additive noise”, xmi, and convert impaired signal
to form for Max Ratio Combining (MRC)
Additive Noise:
 mi
 s1l 
 
n , mi 
 s2 l 
G   

l i
 
s 
 Ml 
Whitened
Noise
Whitening and MRC:
g m2 rm   g m2 g m smi   m 




H
m
1
Noise covariance
H
m
1
Mutual Information
• Signal-to-Interference plus noise ratio over a
symbol-stratum interval
1




 P 
 P 
mi  g mH  I N   2 GG H (1  1 / n)  g m  2
  o(1 / n)



M


  Mn 















m


• Mutual information of a stratum is
1
Ci 
M
M
 log 1   
m 1
2
mi
Mutual Information (MI)
of a Ply
PLY
• MI added by the ith strata
PLY
1
Ci  M
M
1



P


 P 
H
H
log 2 1  g m  I N   2 GG (1  i / n)  g m  2


 M 
  Mn 
m 1



M
• Asymptotic MI as n  
• Small e approximation log 1  e   e
• Sum goes over to an integral
Asymptotic sum capacity
As n  , asymptotic sum capacity
n
lim
n 
 Ci G  
i 1
1
P
1
P
H
H 
G
I

GG
  Gd
 N
2
2

 M n ln 2 0
M


1

P2j
  log 2 1  2
 M
j 1

MIN




P

H
 log 2 det  I N 
GG 
2
M


• Achieves LogDet Capacity
Outage capacity
• Unlike traditional outage capacity definition, in SDBLAST we need to define outage capacities for
.
– n-strata capacities Ci, i=1,2,….,n.
– For comparison purposes, we also define outage capacity
based on sum capacity
n
C
n
SD
  Ci
i 1
• How much outage capacity can we achieve using
finite strata (n)?
Outage Capacity: Upper-bound
• Say, the sum capacity demand at outage level e is
n
C
n
SD
e ,     Ci e ,  
i 1
• The traditional outage capacity is based on the sum capacity
e  Pr C
n
SD
 , G   C e ,  
n
SD
• However, we have additional demands to satisfy:
• if any one of the strata fails to satisfy its corresponding
capacity demand Ci e ,   , then the message will fail.
• Thus, we need to relax the capacity demand.
Outage capacity-lower bound
• We relax (reduce) the demand on the outage capacity
n
C
n
SD
e  e ' ,     Ci e  e ' ,  
• Then minimize the relaxation
i 1
e ' 0
e  PrCi  , G   Ci e  e ' ,  , any i 
• A tighter capacity bound can be found by minimizing
n slack var iables e i'  0, i  1,2,...., n.
SD-BLAST and Eigenvalue Hardening
• The capacity accumulated through each stratum is a
function of eigenvalues of the channel matrix
Ci  f  j , j  1,2,...n , i  1,2,..., n
• SD-BLAST achieves capacity
– All (M,1) and (N,1) case (single eigenvalue)
– As set of eigenvalues of (M,N) systems harden,
M  N and M  
N  M and N  
min( M , N )  
DEALING WITH UNCERTAINTY IN BITS/SEC/PLY
- Monte-Carlo Method Q%: OUTAGE REQUIRED WITH K PLIES
• COMPUTE WEAKER TOTAL
CAPACITY DEMANDS FOR OUTAGE
q% < Q%
Outage (%)
Required Q 
Q
• CALCULATE Q% OF CHANNELS
NOT MEETING WEAKER
DEMANDS FOR STRATA
CAPACITIES (CLEARLY Q% > q%)

Strata
capacity
iterate

q
• ITERATE ON (q,Q) UNTIL Q%  Q%
0
Total
capacity
Numerical Results
Finite M, N
(M,1) case
(8,1) SD-BLAST, SUM CAPACITY vs
AVERAGE SNR at 10% outage
M=8, N=1
8
Capacity (bits/sec/Hz)
7
D-BLAST
6
64
32
5
16
4
8
3
4
V-BLAST
2
2
n=1
1
SD-BLAST
0
-5
0
5
10
15
Average SNR (dB)
20
25
Power
optimization
M > N case
(8,3) SD-BLAST, SUM CAPACITY vs
AVERAGE SNR at 10% outage
M=8, N=3
SD-BLAST (Upper Bound)
SD-BLAST (1-64 Strata True Capacity)
Capacity (bits/sec/Hz)
20
64
32
D-BLAST
16
15
8
4
10
2
n=1
5
SD-BLAST
V-BLAST
0
-5
0
5
10
15
Average SNR ( dB)
20
25
M > N case
SUM CAPACITY CONTRASTS
(4,2), (8,3) and (16,5) systems  = 10 dB, OPEN LOOP
- COMPLEMENTARY DISTRIBUTION FUNCTIONS 1
Probability {Capacity > Rate}=
1 e
0.9
0.8
0.7
(8,3)
0.6
(16,5)
0.5
(4,2)
0.4
0.3
0.2
SD-BLAST, n=64
SD-BLAST, upper-bound
D-BLAST
V-BLAST
0.1
0
0
2
4
6
8
10
12
Rate (bps/Hz/dim)
14
16
18
20
M < N case
SUM CAPACITY CONTRASTS
(2,4) systems  = 10, 18 dB, OPEN LOOP
- COMPLEMENTARY DISTRIBUTION FUNCTIONS 1
CCDF for (2,4) system
0.9
Probability {Capacity > Rate }=
1  e 0.8
SD-BLAST, n=64
SD-BLAST, upper-bound
D-BLAST
0.7
D-BLAST
0.6
0.5
V-BLAST
V-BLAST
0.4
10dB
0.3
18dB
0.2
0.1
0
2
4
6
8
10
12
Rate (bps/Hz/dim)
14
16
18
Conclusions
• SD-BLAST ARCHITECTRE OFFERS ENORMOUS
CAPACITY (NEAR-CAPACITY)
-AVOIDS WASTE OF SPACETIME RESOURCE
-HIGH BAND WIDTH EFFICIENCY
-IMPLEMENTATION WITH 1-D CODECS
-BINARY CODES
- CAPACITY CAN BE APPROACHED FOR-
–(M,1) systems
–As set of eigenvalues harden
• Acknowledgement
– Dr. Reinaldo Valenzuela Director Wireless
Communications Research Department, Bell
Labs, Lucent Technologies, Holmdel, NJ, for
financial assistance
– Members of wireless research group, Holmdel,
NJ, for many interesting discussions.
Thank you