MIMO SYSTEMS
3.3 Space-Frequency Coding
normalized Doppler frequency of PfDNcT. Increasing the number of subcarriers Nc
increases the OFDM period PNCT but unfortunately results in more correlations due to relatively high mobility of the wireless communication channel. It can be seen from (3.16) that even in time selective fading OFDM systems, the effect of multipaths is multiplicative and can lead to performance improvement.
where c;(w) denotes the channel symbol transmitted over the wth subcarrier by the rth transmit antenna. The SF code is assumed to satisfy the energy constraint
E | | c | > = NCM,. The Nc point IFFT is then applied to each column of matrix C and after appending the cyclic prefix, each OFDM symbol is simultaneously transmitted from its corresponding transmit antenna. We define the SF code rate as the symbol rate per channel use and is given by
XSF'K'K- (3.18)
For proper OFDM operation, we assume that the path gains between each pair of transmit and receive antennas remains constant over one OFDM symbol period but is frequency selective with L independent paths. Hence after removing the cyclic prefix and performing FFT, the received signal at the yth receive antenna and wth subcarrier can be expressed as
Mi
yt (") = Z C' (*H J («) + Zj (") ' (3.19) with a frequency selective fading gain whose complex frequency channel response is defined as in (3.13). We rewrite the received signal in vector form as
Y = DH + Z . (3.20)
where D is an NcMr x MtLMr matrix derived from the SF codeword and is given by
D = /„
a
D-D, M,
(3.21)
such that each D,=\diag(*l)WHdiag(cl)WTi---diag(cl)Wt"~\ for i = \,2,---Mt,
Wt, -^y,e-i*",M. ...e-j2MrliN.-w».J a n d C( i s t h e 0 F D M s y m b o l o n t h e / t h transmit antenna. The received vector Y and the noise vector Z are of the same size NcMr x 1 and are given by
Y=[ymy^)---yM-^y2^>'-y2(Nc-i)yK(oy-'yMr(Nc-i)J, (3.22) and
Z = [zl(0)zx(\)-z](Nc-\)z2(0)-z2(Nc-\)zM(0)-zMr(Nc-\)J, (3.23) respectively. The channel vector H is of the size MtLMr xl and is defined as
where h , ./= [ a °/a ' . - - - a ^ ' ]T.
3.3.2 Performance Criteria
Let D and D be two different matrices derived from two different SF codewords C and C respectively. Assuming that the receiver has perfect knowledge of the channel, the PEP between Dand D is upper bounded as [104], [160]
P(D,D)< ( 2 r - l Y ' ^
r PI4
v r A *-i J 'SNR^
KM,J
(3.25)
where r is the rank of matrix ( D - D j r ( D - D j , Al,A2,---Ar are nonzero eigenvalues of ( D - D ) r ( D - D )W and r = £{HHK} is an MrM,LxMrM,L correlation matrix of H . Based on the upper bound in (3.25), the design criteria for SF codes was proposed to be diversity and product criterion. The product criteria maximizes the coding advantage.
The primary interest of SF code design is the diversity gain. The diversity criteria state that to maximize the diversity advantage, the minimum rank of ( D - D ) T ( D - D ) over all distinct codewords C and C should be as large as possible. Hence
r = rank
= M.rank
( D - D ) T ( D - D )
(T0diag IC - CI axdiag (C - C) • • • crL^diag (C - CI
(3.26)
where crnl = 0,1,•••1,-1 is the power profile on the /th path. It can be easily seen that the rank of SF codes in MIMO frequency selective fading channels is given by
r < mm{MrNc,LMrMt}, (3.27) In most scenarios, Nc >M,L, hence the maximum achievable diversity gain is at most r < MtMrL. However, for full diversity advantage, r = M,MrL i.e. the maximum diversity gain for SF is the product of spatial diversity gain MtMr and frequency diversity due to the L multipaths.
Note that from (3.27), the maximum achievable diversity gain is the same as that of ST coded OFDM [147], [153]. In SF the fading channel is assumed constant over one OFDM symbol in contrast to an entire STBC-OFDM codeword. Hence in the presence of time correlated fading, space-frequency block codes (SFBC) would perform better than STBC-OFDM which has a longer symbol period.
3.3.3 Example of SFBC from STBC
x[n] SFBC Encoder
x,[n]
x2[n]
7x \ 7
IFFT
\\.M
Tx.
\7^
IFFT -1
Ki M
~--A.4V
£[«! ML
Decoder y[«] *— FFT
Rx
Figure 3.2: Space-frequency block coded OFDM.
Figure 3.2 shows an example of a two-branch SFBC scheme signalling through a frequency selective fading channel. The two-branch SFBC is an extension of a simple orthogonal transmitter diversity scheme first shown in [88]. The data symbol vector on the nth block x[n] is encoded by space-frequency block code (SFBC) to two vectors as
x[n] = [x, (n)x2(«)••• xNc (n)]
^[n] = [x}(n)-x'2(n)---xK^{n)-x'K^ . (3.28)
^2[n] = \_x2(n)^(.n)---xNe(n)x'N,lj
During the nth OFDM block instance, x,[«] is transmitted from the first transmit antenna while x2[n] is simultaneously transmitted from the second transmit antenna. SF encoding and decoding processes can be described in terms of odd x0[n] and even
xe[n] component vectors of x[n] such that
X2 , o Xe X2 , « Xo
(3.29)
where xl 0,x] c and x2u,x2e are odd and even vectors of x,[«] and x2[n] respectively.
It can be seen from (3.29) that the Alamouti's equivalent SFBC transmission matrix is given by
G[n]
-x* x* (3.30)
At the receiver assuming perfect knowledge of the channel, the receive signal vector after removing the cyclic prefix can be expressed in terms of odd and even components as
y „ ["] = x0 (") A, „ («) + xe (n) A2 0 (n) + z0 («)
(3.31) y > ] = -<(n)A,.(n) + x* («)A2 e(n) + ze(n)
where z0(n),ze(n) and A/0(n),A(e(») are odd and even components of the AWGN and the complex fading channel coefficients respectively. The channel coefficients are defined as in (3.13). To maintain SFBC orthogonality, the complex channel gains
between at least two adjacent subcarriers are assumed constant, i.e. Al 0(«)« \e(n) and A2o(«) w A2 e(«), hence A^T OFDM symbol period. In the presence of time selective fading, SFBC-OFDM would perform better than STBC-OFDM which has a longer symbol period. The transmitted symbols are recovered from the received signal in a similar manner to the Alamouti scheme [88]. Since the above example uses the Alamouti structure, it thus reasonable to expect SFBC from STBC to have the same diversity performance as the STBC-OFDM in Figure 3.1. Hence, although the above example can achieve spatial diversity, it still fails to achieve frequency diversity offered by channel selectivity.
3.3.4 Full Diversity Space-Frequency Codes
The design parameters for good SFC are vastly different from those of STC in narrowband fading channels. Employing known STC as SFC by coding across space and frequency (rather than space and time) in general provides spatial diversity but fails to exploit the available frequency diversity [153], [151]. SF codes offering full diversity by coding across the multipaths, hence maximizing the frequency diversity have been proposed. These codes use techniques such as linear precoders or constellation rotation [161], [162], [163], linear constellation decimation [164], or simply repetition codes to code across multipaths in addition to space coding [160]. This subsection presents the signal design and criteria for SF codes derived from STC using repetition coding across multipath.
3.3.4.1 Signal Design
In order to achieve full diversity M,MrL, the matrix ( D - D ) r ( D - D ) in (3.26) has to be full rank for every distinct pair of the SF codewords C and C . This subsection presents a derivation of how full diversity SF codes can be constructed from STBC designed for a narrowband fading channel using repetition coding similar to [160]. Note that constellation rotation, cyclic shift, or permutation techniques used to generate full diversity SF codes are a form of repetition coding with varying coding gains and code rates. Similar techniques have been used in the design of STBC to achieve full diversity or full rate [101], [102].
Suppose a STBC codeword V in (3.1) which is a M,xM, square matrix, i.e. P = Mt, an Nc x M, SF codeword is formed by concatenating V STBC matrices L times.
Hence a SF codeword is formed by repeating each row of the V matrix L times and adding some zero rows where MtL is less than the number of subcarriers Nc, i.e.
C = GL(y)
O (Nc-LM,)xM, (3.32)
The mapping GL is defined as
G
i(V) = [l
M,®l
M]
TV
: (3.33) where lM is an Mt x M, identity matrix, lix] is an all one matrix with size L x 1 and ® denotes the tensor product [177]. It can be seen from (3.32) that the symbol rate per channel use 9?SF = LM, I Nc is less than one if LMt < Nc.3.3.4.2 Diversity Criteria
From the performance criteria, the rank/diversity criteria states that the minimum rank of the matrix over all distinct codewords C and C should be as large as possible,
= ra«*;j(D-D)r(D-D)W
=rank< I
(c-c)(c-c)'
(3.34)
where o is the Hadamard product [177],
(c-c)(c-c)
? ^ ( V - V ^ G ^ V - V )o 'NC~LM, 0 NC-LM,
(3.35)
and since for quasi-static fading the channel characteristics for one OFDM smbol are assumed constant an JV x N correlation matrix T is defined as
r = i
M, (3.36)given htJ = [afjal • • • aft ]T for i = 1,2,• • ;M, •
The aim of this subsection is to prove that the matrix in (3.34) has a rank of at least LMtMr. Since ST codes achieve full diversity for quasi-static flat fading channels, ( V - V J has Mt rank for any two distinct codewords. Note from (3.36) that
r = E{hijhZ} = diag{cr„au-• •,C7£_,}, (3.37) is an L x L diagonal matrix whose power profile on the /th path is given by a,. Hence for any number of transmit antennas Mt the correlation T has a minimum rank of L.
Therefore the rank of the matrix in (3.34) is
r = rank {1M ®
-> \H
(v-v)(v-v)
= rank (lM) rank j(v ~ v ) ( v -\f%ank (r) (3.38)
= MrM,L, hence, achieving full diversity.