The Silver Code is a fast-decodable space-time
block code for 2 transmit and 2 receive antennas, for the coherent MIMO
channel. It has been found by [HT04] [HTWBook] [TH02] [TK02]. Recently it was
re-found by [PGA07][SF07], which pointed out its fast-decoding properties and
it was also summarized by [BHV09].
The channel model considered is Y = H X + N, where H
={hij} is the 2x2 channel matrix with
complex fading coefficients
and N the 2x2 complex Gaussian noise matrix. The codewords X of
the Golden Code are 2x2 complex matrices of the following form :
X = Xa(s1,s2)
+ TXb(z1,z2) |
|
where
Xa(s1,s2) = |
s1 |
-s2* |
Xb(z1,z2) = |
z1 |
-z2* |
[z1, z2]T = U* |
s3 |
s2 |
s1* |
z2 |
z1* |
s4 |
si , i=1,...,4, are the information QAM
symbols
U = 1/sqrt(7) * |
1 + j |
-1 + 2j |
1 + 2j |
1 - j |
T = |
1 |
0 |
0 |
-1 |
Porperties
• Consider a linear space-time block
coded MIMO, where STBC carries 4 independent QAM information symboles. In a complex
vector form, we rephrase the received signal equation as vec(Y) = Fs
+ vec(N), where s = {si}, i=1,...,4, and
F = diag(H,...,H) x G, where G is the
generator matrix of the silver code given in [PGA07,BHV09], vec(·)
operator stacks the m column vectors of a n x m complex
matrix into a mn complex column vector.
• Let F = [f1 | f2 | f3 | f4], where fi
is a 4 dimensional column vector. Sphere decoding (SD) can be used to conduct
the MLD based on QR decomposition to minimize ||Q† vec(Y)
– Rs||, where (×)† denotes matrix Hermitian, and F = QR, where
Q is a 4 x 4 unitary matrix, and R is a 4 x 4 upper triangular matrix
with the following special structure, where <a,b> denotes the inner product of a and b,
R =
|
||d1|| |
0 |
<f3, e1> |
<f4, e1> |
0 |
|| d2|| |
<f3, e2> |
<f4, e2> |
|
0 |
0 |
|| d3|| |
0 |
|
0 |
0 |
0 |
|| d4|| |
where di = fi –
sum( Projej fi, j
= 1, …, i-1), where Projuv = <v,u>/<u,u> and ei = di/||di||.
• Note that there are two zeros in the
matrix R which lead to a reduced-complexity MLD [PGA07][SF07][BHV09].
1) <f2, e1> = 0 provides a saving of 2-dimensional complex
SD tree search, i.e., we employ 2-dimensional complex SD tree search to find s3,
s4, with complexity of M2. For the
remaining pair (s1,s2), an Alamouti
decoding is used with decoding complexity 2M. In summary, the worst-case
decoding complexity is 2M3.
2) <f4, e3> = 0 leads to a faster metric
computation in the relevant SD computation.
[HT04] |
A. Hottinen and O. Tirkkonen, ``Precoder designs for high rate space-time block codes,'' in Proc. Conference on Information Sciences and Systems, Princeton, NJ, March 17--19, 2004. |
[HTWBook] |
A.
Hottinen, O. Tirkkonen and R. Wichman, ``Multi-antenna Transceiver Techniques
for 3G and Beyound,'' WILEY publisher, UK. |
[TH02] |
O. Tirkkonen and A. Hottinen, ``Square-matrix embeddable space-time block codes for complex signal constellations,'' in IEEE Trans. Inform. Theory, vol. 48, no. 2, , pp. 384-395, February 2002. |
[TK02] |
O.
Tirkkonen and R. Kashaev, ``Combined information and performance optimization
of linear MIMO modulations,'' in Proc IEEE Int. Symp. Inform. Theory
(ISIT 2002), |
[PGA07] |
J. Paredes,
A.B. Gershman, and M. G. Alkhanari, ``A2x2 space--time code with
non-vanishing determinants and fast maximum likelihood decoding,'' in Proc
IEEE International Conference on Acoustics, Speech, and Signal Processing
(ICASSP2007), Honolulu, Hawaii, USA, pp. 877-880, April 2007. |
[SF07] |
M. Samuel
and M. P. Fitz, ``Reducing the detection complexity by using 2x2 multi-strata
space--time codes,'' in Proc IEEE Int. Symp. Inform. Theory (ISIT
2007), pp. 1946-1950, Nice, |
[BHV09] |
E.
Biglieri, Y. Hong and E. Viterbo, "On fast-decodable space-time block
codes," |
[HLRVV08] |
C. Hollanti,
J. Lahtonen, K. Ranto, R. Vehkalahti, and E. Viterbo, ``On the Algebraic
Structure of the Silver Code,'' in IEEE Information Theory Workshop,
Porto, Portugal, May 2008. |