Space-time block codes from designs for fast-fading channels

(1)

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 chan- nels by giving a matrix representation of the multi- antenna fast-fading channels.

I. EXTENDED SUMMARY

A linear STBC, S E C L X N , in K complex variables { z k = A ~ ~ + I x ~ Q where the 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. T h e received signal vtj at the antenna j at time t, is given by

X k I + ’ j X k Q } f z i iS a matrix such that

s

⁼ A Z k X k j

+

N - 1

Y - v ,

i=O

With perfect channel state information (CSI) at the receiver, the ML decision rule is min,

hijtstiI2. For simplicity we assume M = 1 a n d accordingly (1) can be written as

N - 1

E:;’

Jvtj -

Ci=o

V = S H + 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 L is the Extended codeword ma- trix (ExCM) (as opposed to codeword matrix

S)

given by

S = [

7 ^... ^...

⁰

_I,.=[

^Ho

^;

] & H t = [

‘:I,

(3) where St =

[

st0 stl

...

~ t ( N - 1 )

1 ,

^H E C N L x l denotes the equivalent channel matrix (EChM) formed by stacking the channel vectors for different t a n d W E C L x 1 is i.i.d. complex Gaussian with zero mean and unit vari- ance. We denote the codeword matrices by boldface letters and the ExCMs by normal letters. For example, t h e ExCM

...

HL-1

...

^{S L - 1}

S for the Alamouti code, S =

[ -:: :: 1,

is given by S =

[ 7 7

^-x7 ^x:

^.

ObservethatforalinearSTBC, its EXCM

s

is also linear such that

s

= X k r A 2 k

+

X k ~ & k + l , where Ak are referred t o as extended weight matrices.

With these notions of ExCM, and EChM we observe that, O O 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 t h e 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] ap- plies t o fast-fading channels also-following t h e observa- tion that t r ( S H S ) = 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 t h e characterization of single-symbol decodable designs given in [6] for quasi-static fading channels for the case of fast- fading 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

= X k I A Z k

+

^xkqA2k+l

,

the ML metric, M ( S ) decomposes as M ( S ) = Mk(2k)

+

^{M c}

where MC = - ( K - 1)tr ( V H V ) , iff

AFAl + A y A k = 0,O

5

^k

# 1 5

2K - 1. (4) Theorem 2: For fast-fading channel, the maximum rate pos- sible 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 t h a t achieves full diversity over both quasi-static and fast-fading channels and provides single-symbol decoding.

‘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.

154 REFERENCES

V.Tarokh, H.Jafarkhani and A.R.Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Informa- tion Theory, vo1.45, 1313.1456-1467, July 1999.

Zafar Ali Khan and B. Sundar Rajan, “STBC from Ceordinate Interleaved Orthogonal designs,” ISIT 2002, June 30-July 5, pp.

275. Also, submitted to IEEE Trans. on Information Theory.

0.Tirkkonen and A.Hottinen, “Square matrix embeddable STBC,” IEEE ” a c t i o n on Information Theory, vol. 48, No.2, pp. 384-395, Feb. 2002.

V.Tarokh, N.Seshadri and AXCalderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. Information Theory, Z. Chen, J. Yuan and B. Vucetic

,

“An improved Space-time trel- lis coded modulation scheme on slow Rayleigh fading channels,”

ICC 2001,Helsinki, Finland Jun. 11-15 2001, pp.1110-1116.

Zafar Ali Khan and B. Sundar Rajan, “On Single-Symbol and Double-Symbol Decodable STBCs,” Proc. of ISIT 2003, June 30-July 5, Yokohama, Japan.

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Space-time block codes from designs for fast-fading channels