ISlT 2003, Yokohama, Japan, June 29 -July 4,2003
Space-Time Block Codes from Designs for Fast-Fading Channels
Md. Zafar Ali Khan and B.Sundar Rajan’
ECE Department, Indian Institute of Science
Bangalore, India 560 012
{zafarBprotocol. ,bsrajanB}ece.iisc.ernet . i n
Abstract - We study Space-Time Block Codes from
orthogonal designs [l, 21 for use in fast-fading channels by giving a matrix representation of the multiantenna fast-fading channels.
I. EXTENDED
SUMMARY
A linear STBC, S E C L X N ,in K complex variables { z k =
XkI +’jXkQ}fzi
iS a matrix such that
=
AZkXkj
A ~ ~ + Iwhere
x ~ Qthe 2K complex matrices {A2k, A 2 k + l } , k =
0 , 1 , . .. , K - 1 are called the weight matrices of S. The
rate an symbols/channel use is K I L . Let N, M be the number
of transmit, receive antennas respectively, hijt = a;jtdeijt
denote the path gain from the transmit antenna i t o the receive
antenna j at time t, where j = fl and sti is the signal
transmitted on antenna i a t time t. The received signal vtj at
the antenna j at time t, is given by
s
+
N-1
Y
i=O
-
v,
With perfect channel state information (CSI) at the receiver,
N-1
the ML decision rule is min,
E:;’ Jvtj - Ci=o
hijtstiI2. For simplicity we assume M = 1 and accordingly
(1)can be written as
V=SH+W
(2)
where V E C L x l (C denotes the complex field) is the received
signal vector, S E C L X N Lis the Extended codeword matrix (ExCM) (as opposed to codeword matrix S) given by
S=[
7 I, . = [
...
...
...
0
...
SL-1
Ho
;
]&Ht=[
‘:I,
The well known distance criterion on the difference of
two distinct codeword matrices for fast fading channels
[4] is equivalent t o the rank criterion for the difference
of two ExCM.
The product criterion on the difference of two distinct codeword matrices for fast fading channels [4] is
equivalent t o the determinant criterion for the difference of two ExCM.
The trace criterion on the difference of two distinct
codeword matrices derived for quasi-static fading [5] applies t o fast-fading channels also-following the observation that t r (SHS)= t r (SwS).
The ML metric can be written in terms of ExCM, S, as
M ( S ) = t r ((V - S H ) H ( V- SH)) . This matrix form
for the ML decoding metric makes applicable the characterization of single-symbol decodable designs given in
[6] for quasi-static fading channels for the case of fastfading channels also. Applying this characterization we
obtain the following results:
Theorem 1: For a linear STBC in K complex variables,
whose ExCM is given by, s =
XkIAZk
xkqA2k+l , the
ML metric, M ( S ) decomposes as M ( S ) =
Mk(2k)
Mc
where MC = -(K - 1)tr ( V H V ) iff
,
+
AFAl + A y A k = 0,O 5 k # 1 5 2K - 1.
REFERENCES
(3)
where St = [ st0 stl ... ~ t ( N - 1 )
H E C N L x l denotes the equivalent channel matrix (EChM) formed by
stacking the channel vectors for different t and W E C L x 1
is i.i.d. complex Gaussian with zero mean and unit variance. We denote the codeword matrices by boldface letters
and the ExCMs by normal letters. For example, the ExCM
1,
S=
[7 7
its EXCM
Xk~&k+l,
s
-x7
O
x:
O I
[ -:: : 1,
is given by
. ObservethatforalinearSTBC,
+
is also linear such that s =
XkrA2k
where Ak are referred t o as extended weight
matrices.
With these notions of ExCM, and EChM we observe that,
‘This work was partly funded by the DRDO-IISc Program on
Mathematical Engineering through a grant to B.S.Rajan.
0-7803-7728-1103/$17.00
02003 IEEE.
(4)
Theorem 2: For fast-fading channel, the maximum rate possible for a full-diversity single-symbol decodable STBC using
N transmit antennas is 2/L. Hence, a rate-one, full-diversity,
single-symbol decodable design for fast-fading channel exists
iff L = N = 2.
Theorem 3: The CIOD of size 2 [2] is the only STBC that
achieves full diversity over both quasi-static and fast-fading
channels and provides single-symbol decoding.
HL-1
S for the Alamouti code, S =
+
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