Constructions of Multiblock Space–Time Coding Schemes That Achieve the Diversity–Multiplexing Tradeoff
Hsiao-Feng (Francis) Lu, Member, IEEE
Abstract—Constructions of multiblock space–time coding schemes that are optimal with respect to diversity–multiplexing (D-M) tradeoff when coding is applied over any number of fading blocks are presented in this correspondence. The constructions are based on a left-regular represen-tation of elements in some cyclic division algebra. In particular, the main construction applies to the case when the quasi-static fading interval equals the number of transmit antennas, hence the resulting scheme is termed a minimal delay multiblock space–time coding scheme. Constructions corre-sponding to the cases of nonminimal delay are also provided. As the number of coded blocks approaches infinity, coding schemes derived from the pro-posed constructions can be used to provide a reliable multiple-input mul-tiple-output (MIMO) communication with vanishing error probability.
Index Terms—Cyclic-division algebras, diversity–multiplexing (D-M) tradeoff, fading channels, multiblock space–time codes, multiple–input multiple–output (MIMO) channels, number fields, space–time codes.
I. INTRODUCTION
By deploying multiple antennas at both transmitter and receiver ends, the multiple–input multiple–output (MIMO) technology can significantly increase the ergodic channel capacity as well as improve the link reliability. For example, in a MIMO communication system with nt transmit and nr receive antennas, under the quasi-static MIMO Rayleigh block fading channel model, it is known [1] that the ergodic MIMO channel capacityC equals
C = minfnt; nrg log2SNR+ O(1) bits/channel use (1)
at high signal-to-noise ratio (SNR) regime.
Coding schemes dedicated to the MIMO systems to achieve higher transmission rate and better link reliability are specifically coined
space–time codes [2], [3]. LetT denote the quasi-static interval of the
quasi-static MIMO Rayleigh block fading channel. An(nt 2 mT )
multiblock space–time codeX is a collection of (nt2 mT ) matrices and is used to code messages over m fading blocks, meaning mT channel uses in all. The codeX transmits on average
R := 1mT log2jX j (2)
bits per channel use. Letr denote the normalized rate of X , also known as the multiplexing gain [4], given by
r := logR
2SNR:
(3) From (1), it can be seen that to have a reliable MIMO communication, the maximum achievable multiplexing gain equalsminfnt; nrg. Given
Manuscript received December 22, 2005; revised October 20, 2007. This work was supported by Taiwan National Science Council under Grants NSC 96-2219-E-194-002 and NSC 96-2628-E-194-001-MY3. The material in this correspondence was presented in part at the IEEE International Symposium on Information Theory, Seattle, WA, July 2006.
The author is with the Department of Communications Engineering, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: francislu@ieee.org).
Communicated by H. Boche, Associate Editor for Communications. Digital Object Identifier 10.1109/TIT.2008.926447
the multiblock codeX with multiplexing gain r, we say X achieves
di-versity gaind(r) if at high SNR regime, the codeword error probability
ofX is on the order of
Pe(r) := SNR0d(r): (4)
By :=, we mean the exponential equality defined in [4]. We say the functionf(SNR) := SNRbif and only if
lim
SNR!1
log f(SNR)
log SNR = b: (5)
The notations of _ and _ are defined similarly.
In their ground-breaking paper, Zheng and Tse [4] showed that there exists a fundamental tradeoff between the multiplexing and the diver-sity gains, referred to as the diverdiver-sity–multiplexing (D-M) tradeoff. For the cases whenT nt + nr 0 1 and when the m
consecu-tive fading blocks are statistically independent, the D-M tradeoff as-serts that the maximum possible diversity gaind3(r) for any (nt2
mT ) multiblock space–time coding schemes with multiplexing gain r is a piecewise linear function connecting the points(k; d3(k)); k = 0; 1; 1 1 1 ; minfnt; nrg, and
d3(k) = m(n
t0 k)(nr0 k): (6)
On the other hand, ifT < nt+ nr0 1, only upper and lower bounds ond3(r) are available in [4].
This remarkable result has spurred a considerable amount of research activities on constructing space–time coding schemes to achieve this optimal tradeoffd3(r). Much progress has been made in the case of m = 1. When nt = nr = T = 2 and m = 1, several D-M
op-timal(2 2 2) schemes with tilted-QAM constellations can be found in [5]–[8]. These are the first instances showing that the D-M tradeoff (6) holds even in the case ofT < nt+ nr 0 1. In particular, the
“Golden code” proposed by Belfiore et al. [8] is shown to have the best error performance among the aforementioned(2 2 2) codes. By generalizing the Golden code construction and working on latices with unitary generating matrices, Oggier et al. provided in [9] the construc-tions of(3 2 3), (4 2 4), and (5 2 5) perfect space–time codes. El Gamal, Caire, and Damen [10] proposed a construction of(nt2 T )
coding schemes, termed lattice space–time (LAST) codes that are ob-tained from a nested lattice randomly drawn from an ensemble of lat-tices having good covering properties. The LAST codes are shown to be D-M optimal for allT nr+ nr0 1 with m = 1. By extending an earlier work [11], Kiran and Rajan [12] proposed a construction of (nt2nt) space–time codes that is based on the left-regular
representa-tion of elements in a cyclic division algebra (CDA) as square matrices for the cases ofnt = 2n; 3 1 2n, and3nfor some positive integern. The CDA-based codes are known to have a linear dispersion form [13], hence can be decoded using the sphere decoding technique [14]. A suf-ficient condition for codes to be D-M optimal as well as a general con-struction of(nt2nr) CDA-based codes satisfying this condition were
discovered by Elia et al. [15], [16]. In addition, it was shown in [16] that the CDA-based codes are approximately universal, a criterion proposed by Tavildar and Viswanath [17]. In [18], Liao and Xia proposed a trans-formation technique to balance the mean powers at different transmit antennas and introduced a multilayer structure for CDA-based codes. It was shown that the resulting(3 2 3) code has better performance than the previous ones.
While all the aforementioned constructions are D-M optimal, we remark that none of them is capable of providing a reliable MIMO communication [19] due to the nonvanishing error probability. In other words, the error probabilityPe(r) achieved by the above schemes is 0018-9448/$25.00 © 2008 IEEE
bounded away from zero whenever SNR< 1. This is due to that these constructions address only the case ofm = 1, i.e., the coding is con-fined within one fading block, and independent fading blocks are coded independently. Therefore, from the D-M tradeoff (6), it follows that in order to achieve a reliable MIMO communication, coding must be ap-plied over multiple fading blocks. In other words, at finite SNR regime, the vanishing error probabilityPe(r) can only be approached through
lengthening the coding scheme so thatm 1. This coincides exactly with what we have learned from the conventional SISO communication [19]. Motivated by this, for any set of parameters,m; nt; nr, andT , in this correspondence, we will aim at providing explicit constructions of (nt2 mT ) multiblock space–time codes with multiplexing gain r that
are D-M optimal (6). Namely, these newly proposed codes will have Pe(r) := SNR0d (r)0! 0
asm approaches infinity at high SNR regime whenever the transmis-sion rateR is set below the ergodic channel capacity C, or equivalently, the multiplexing gainr minfnt; nrg.
This correspondence is organized as follows. We will begin with the construction of the(nt2 mnt) minimal delay multiblock space–time
coding schemes1for the caseT = ntand for allm 1 in Section II.
A design example will also be given for illustration. It will be proved in the Appendix that coding schemes derived from this construction are D-M optimal (6), hence are able to provide a reliable MIMO com-munication asm ! 1. However, it is generally true that the MIMO communication channels are slowly varying and the assumption of in-dependent fading blocks might not hold. In fact, the consecutive fading blocks are expected to be correlated in time, and the degrees of corre-lations strongly depend upon the conditions of communication envi-ronment, such as number of multipaths, Doppler spread, and carrier frequency. In view of this, we will provide in the Appendix a much stronger proof to show that this newly proposed construction is D-M optimal for all kinds of wireless communication channels, including the ones having time correlations, antenna correlations, and having dif-ferent fading statistics.
In Section II-B, we will generalize the construction to provide non-minimal delay coding schemes when the quasi-static intervalT > nt. Furthermore, for the cases ofT mnt, it will be seen that codes derived from this generalization might not be efficient in terms of sig-naling complexity, in the sense that when representing the code in its linear dispersion form [13], each entry of code matrix resulting from this construction is a large linear combination of many signal points drawn from the underlying constellation set. In view of this, a more efficient construction targeting atT mntthat requires lesser linear combinations will be given in Section II-C. In Section IV, we conclude this correspondence.
II. CONSTRUCTIONS OFMULTIBLOCKSPACE-TIMECODES
Consider a quasi-static MIMO Rayleigh block fading channel with nttransmit andnrreceive antennas. Throughout this correspondence, we will assume for simplicity thatnr ntwhile later in the Appendix it will be straightforward to see that the D-M optimality of the pro-posed constructions remains to hold even when the number of receiver antennas is less than the number of transmit. LetT be the quasi-static interval and let X be an (nt 2 mT ) multiblock space–time coding
scheme that sends coded information overm consecutive, yet statisti-cally independent fading blocks. We will first focus on the case whenT equalsnt. The cases ofT > ntwill be dealt with later in Sections II-B and II-C.
1We were informed that this minimal delay construction was independently discovered by Yang and Belfiore [20], [21] for constructing distributed space–time codes in MIMO amplify-and-forward cooperative channels.
AssumingX = (X0; . . . ; Xm01) 2 X is the code matrix chosen for transmission, the transmitter actually sends the(nt2nt) submatrix
Xiat theith fading block, i = 0; 1; . . . ; m 0 1. Thus, the received
signal matrix corresponding toXiat the receiver end is modeled as
Yi= HiXi+ Wi (7)
fori = 0; 1; . . . ; m 0 1, where Hi and Wi are, respectively, the (nr2 nt) channel and (nr2 T ) noise matrices. Entries of HiandWi
are modeled as independent identically distributed (i.i.d.), zero mean, circularly symmetric, complex Gaussian random variables with unit variance N (0; 1). The parameter is chosen to satisfy the following power constraint:
m01 i=0
kXik2F = m 1 T 1 SNR (8)
where byk 1 kF we mean the Frobenius norm of a matrix.
A. Minimal Delay Construction
Given the desired multiplexing gainr, we first identify the following QAM base alphabet [16]2that is a subset of the Gaussian integer ring
and is given by
A(SNR) = a + b{ : 0M + 1 a; b M 0 1; a; b
odd integers; M = SNR (9) where{ =p01.
Next, the construction calls for two number fields [22]L and K that are field extensions of ({). The number field L is a cyclic Galois extension of ({) with degree
n := [L : ({)] = mnt (10)
and the number fieldK is a subfield of L with degree [K : ({)] = m. We refer the readers to [15] and [16] for a systematic construction of such number fieldL.
Bearing with the above in mind, let be the generator of the Ga-lois group Gal (L= ({)) and it is clear that has order n. Since Gal (L= ({)) is cyclic, the Galois group Gal (L=K) is also a cyclic group generated by = mwhose order equalsnt. Therefore,L is cyclic Galois overK as well and [L : K] = nt. The Galois group for K over ({) is the quotient group
Gal (K= ({)) = hi=hi = fi: i = 0; . . . ; m 0 1g: (11)
It follows that = (L=K; ; ) is a CDA3for some nonnorm
ele-ment 2 K3. Moreover, can be embedded in an(nt2 nt) matrix
algebra overL through the left-regular representations of elements in [12]. We have = D = x0 (xn 01) 1 1 1 n 01(x1) x1 (x0) 1 1 1 n 01(x2) .. . ... . .. ... xn 01 (xn 02) 1 1 1 n 01(x0) : xi2 L (12) and it is known [23] that every such(nt2nt) matrix D has determinant
inK.
2For brevity, here we only provide the constructions of codes with QAM base alphabet. The ones for the HEX base alphabet [16] can be obtained in a similar fashion.
3For readers not familiar with the subject of CDA, we refer them to [12] and [11] for a nice introduction.
For simplicity, here we restrict ourselves to the case of 2 [{] while, in general, the construction can be generalized to take unit mod-ulus 2 OK to yield the multiblock version of perfect space–time codes [24], whereOKis the ring of algebraic integers inK.
Let ~X be an (nt2 nt) space–time coding scheme
~ X := x0 (xn 01) 1 1 1 n 01(x1) x1 (x0) 1 1 1 n 01(x2) .. . ... . .. ... xn 01 (xn 02) 1 1 1 n 01(x0) : xi= n j=1
ai;jej; ai;j2 A(SNR) (13)
whereB := fe1; e2; . . . ; eng is an integral basis for L over ({). Then,
the proposed(nt2 mnt) multiblock space–time coding scheme X is given by
X := X = ~X; ( ~X); . . . ; m01( ~X) : ~X 2 ~X : (14)
In other words, if ~X was the code matrix chosen from ~X for transmis-sion, then the transmitter actually sendsi( ~X) during the ith fading block,i = 0; 1; . . . ; (m 0 1). One direct consequence of the above construction is the following.
Proposition 1: Let ~X and X be defined as above; then for every
nonzero codeword(X0; . . . ; Xm01) 2 X , we have
m01 i=0
det(Xi) 1: (15)
Proof: First, note thatXi = i( ~X) for some nonzero ~X 2 ~X .
As the nonnorm element lies in [{], the entries of ~X are in OL, the ring of algebraic integers inL, i.e., the integral closure of in L. It then follows from [16] and [23] that0 6= det( ~X) 2 OK, whereOK
is the ring of algebraic integers inK. Now the proof is complete after noting m01 i=0 det(Xi) = m01 i=0 det(i( ~X)) = m01 i=0 i(det( ~X)) = NK= ({) det( ~X) 2 [{]
whereNK= ({)(a) denotes the algebraic norm of a from K to ({).
The above property should be regarded as the generalized
nonvan-ishing determinant property. To see this, settingm = 1 in Proposition
1 yieldsj det(X)j 1 for every nonzero code matrix X 2 X , and we recover the nonvanishing determinant criterion stated in [8], [12], and [16]. Furthermore, as
jX j = j ~X j = jA(SNR)jn n= SNRrn
the (nt 2 mnt) multiblock space–time coding scheme X achieves
transmission rateR = r log2SNR bits per channel use and has full rate in terms of the size of A(SNR). To ensure that X satisfies the power constraint (8), the parameter should be set at
2 := SNR10 (16)
due to m01i=0 kXik2F := 2SNR .
All in all, the proposed coding schemeX is full-rate, has a signal constellation that is a linear combination of points inA(SNR), and satisfies the “generalized” nonvanishing determinant property. The fol-lowing theorem shows thatX is in fact optimal with respect to the D-M tradeoff. The proof to this theorem is relegated to the Appendix. Fur-thermore, by adopting techniques from [20], it will be shown that the proposed construction satisfies the approximately universal property, meaning that this code is D-M optimal for any kinds of fading distri-butions, including the time-correlated channels.
Theorem 2: LetX be the (nt2mnt) multiblock space–time coding
scheme defined as in (14); then,X is optimal with respect to the D-M tradeoff (6), i.e., it achieves simultaneously multiplexing gainr and diversity gaind3(r).
In the following, we give an example construction of multiblock (2 2 2m) space–time codes for better understanding of this construction.
Example 1: We wish to construct a multiblock(222m) space–time
code withnt = T = 2 to code information over m independently faded blocks. Our construction calls for the number fieldL that is a degree-2-m cyclic Galois extension over ({) and the field K that is a subfield ofL with degree m over ({). For instance, say m = 2. We use methods described in [15] and [16] to construct such fields, as shown in the following diagram, where!20 = exp({2=20) and !5 = exp({2=5):
The Galois groupGal (L= ({)) is generated by : !5 7! !52, hence the Galois groupGal (L=K) is
Gal (L=K) = h = 2i = f
1; 4g (17) where byiwe mean the automorphismi: !57! !i5. Furthermore, it
can be verified that = { is a valid nonnorm element for the cyclic divi-sion algebra = (L=K; ; = {). Noting that B = f1; !5; !25; !35g
is an integral basis forL over ({), the resulting (nt2 nt) and (nt2
2nt) space–time coding schemes are given, respectively, by
~
X = X =~ 3i=0ai!5i { 3i=0bi!54i 3
i=0bi!i5 3i=0ai!54i : ai; bi2 A(SNR)
(18) X = X0= 3 i=0ai!5i { 3i=0bi!54i 3 i=0bi!i5 3i=0ai!4i5 ; X1= 3 i=0ai!52i { 3i=0bi!3i5 3 i=0bi!2i5 3i=0ai!3i5 : ai; bi2 A(SNR) (19)
for the QAM base alphabetA(SNR) [{] of size SNR with 0 r 2 given in (9). It can be easily verified that
1 j=0
det 3i=0ai!52 1i { 3i=0bi!52 1i 3 i=0bi!52 1i 3i=0ai!25 1i = NK= ({) det 3 i=0ai!5i { 3i=0bi!4i5 3 i=0bi!5i 3i=0ai!54i
lies in [{] for all ai; bi2 [{], as claimed in Proposition 1. Theorem
2 then asserts that the codeword error performance of ~X at high SNR regime is on the order of
Pe(r) := SNR0d (r) (20)
where the optimal tradeoffd3(r) is given by the piecewise-linear func-tion connecting the points(k; d3(k)), and d3(k) = 2(nr0 k)(2 0 k) fork = 0; 1; 2.
Here we remark that the codeX in the above example can also be derived from the constructions provided in [21], and was used for the purpose of distributed space–time coding.
B. Nonminimal Delay Multiblock Construction forT > Nt.
In this section, we will extend the minimal delay multiblock con-struction in Theorem 2 to the case whenT > nt. To this end, we set the base alphabetA(SNR) as
A(SNR) = a + b{ : 0M + 1 a; b M 0 1; a; b
odd integers; M = SNR : (21) Comparing to the earlier construction (9), this time we only need a smaller QAM constellationA(SNR) to begin with. Next, let L be a number field that is cyclic Galois over ({) with [L : ({)] = mT and letK be a subfield of L with [K : ({)] = m. Let be the generator of the Galois groupGal (L= ({)); then, we have Gal (L=K) = h = mi, which is a cyclic group generated by with order T . Again, the
Galois group ofK over ({) is the quotient group hi=hi.
Now let ~X be a (T 2 T ) space–time coding scheme obtained by the left-regular representation of elements of the cyclic division algebra = (L=K; ; ) for some nonnorm element 2 [{], and by re-strictingai;jto be in the setA(SNR) [cf. (13)]. Removing any, but in a fixed fashion,(T 0 nt) rows of the matrices in ~X gives the resulting
(nt2 T ) space–time coding scheme ^X . Thus, we have the following
construction.
Theorem 3: Let ~X and ^X be defined as above; then, the (nt2 mT )
multiblock space–time coding schemeX
X = X = ^X; ( ^X); . . . ; m01( ^X) : ^X 2 ^X (22)
is optimal with respect to the D-M tradeoff.
C. Stacking Construction of Nonminimal Multiblock Codes When
T mnt
In the previous sections, we have provided constructions of (nt 2 mT ) multiblock space–time codes that are D-M optimal for
any number of transmit antennasnt, any number of blocks m, and forT nt. Here we wish to give an alternative construction when the quasi-static intervalT mnt, meaning the channel is extremely slowly varying. It will be seen that this alternative construction requires much lesser signaling complexity than that resulting from Theorem 3.
LetA(SNR) be a QAM base alphabet given by A(SNR) = a + b { : 0M + 1 a; b M 0 1; a; b
odd integers; M = SNR (23) and letE be a number field that is cyclic Galois over ({) with de-greeT ; then, = (E= ({); ; ) is a cyclic division algebra, where is the generator of the Galois group Gal (E= ({)) and 2 [{] is some nonnorm element. Let ~X be the (T 2 T ) space–time coding scheme obtained by left-regular representation of elements in and by restrictingai;j to the setA(SNR) [cf. (13)]. Remove any fixed (T 0 mnt) rows from code matrices in ~X , and let ^X denote the
re-sulting(mnt2T ) space–time coding scheme. By rearranging the rows
of the(mnt2T ) code matrices in ^X , we offer the following construc-tion, termed stacking construction. Specifically, given any(mnt2 T )
code matrix ^X 2 ^X , the construction first vertically partitions ^X into m submatrices, each of size (nt2 T ). Say X0; Xi; . . . ; Xm01are the resultingm submatrices; then, the stacking construction will put these m submatrices side-by-side to yield the desired code matrix of size (nt2 mT ). It turns out that such (nt2 mT ) code is D-M optimal.
Theorem 4: AssumingT mnt, let ^X be defined as above; then, the(nt2 mT ) multiblock space–time coding scheme X
X = (X0; X1; . . . ; Xm01) : ^X = X0 .. . Xm01 2 ^X (24) is D-M optimal, where the submatricesXiare of size(nt2 T ).
To see the advantage of the above construction, recall that in The-orem 3 the construction of(nt2mT ) multiblock codes with T mnt calls for a number fieldL with [L : ({)] = mT . It in fact means that entries of the resulting code matrix are linear combinations ofmT points drawn fromA(SNR). However, in the alternative construction of Theorem 4, onlyT linear combinations are required, hence it has much lower signaling complexity.
III. SIMULATIONRESULTS
In this section, we will provide some simulation results of the multi-block code given in Example 1 for a MIMO system withnt = 2 transmit andnr= 2 receive antennas at the transmission rate of 4 bits
per channel use. The results are shown in Fig. 1 and all the codes used in simulation are normalized to satisfy the power constraint (8). First, we consider the case when the channel quasi-static intervalT equals 2 and the channel varies independently for every consecutive fading block. Using sphere decoding [14], the performance result of the(224) multiblock codeX given in (19) is shown in solid line in Fig. 1, and it is seen thatX achieves the codeword error probability of 1004at SNR= 17 dB. Comparing withX , the Golden code [8] that is the known best code for the(2 2 2) MIMO system requires SNR = 23 dB to achieve the same codeword error probability. The gain of 6 dB in SNR for the multiblock codeX is due to the fact that the Golden code was origi-nally designed to code information within one fading block only, not across consecutive fading blocks. On the other hand, it is true that for the very slowly varying fading channels, the consecutive fading blocks could be almost the same. Thus, in Fig. 1, we have also considered the case when the two consecutive fading blocks are identical. For such channel, the performance result ofX is shown in dash line in Fig. 1, and it can be seen that the codeX achieves 1004at SNR= 20.2 dB. Clearly, the degradation in performance is due to the lesser degrees of freedom in channel variation. However, even in this case, the multi-block codeX still shows an excellence performance and gains in SNR for about 2.8 dB, compared to that of the Golden code.
Fig. 1. Performance simulations of the multiblock codeX given in Example 1 and the Golden code for the (2 2 2) MIMO system.
IV. CONCLUSION
In this correspondence, we had presented systematic constructions of multiblock space–time codes that can be used to encode and transmit coded information over quasi-static MIMO fading channels. The con-structions are based on a left-regular representation of elements in a cyclic division algebra whose center is a field extension of the quadratic number field of finite degree. Constructions of codes with minimal or nonminimal delays were both given. In particular, when the MIMO channel is extremely slowly varying, an alternative construction was also given to reduce the number of linear combinations required for encoding, and to yield codes with lower signaling complexity. We had proved that all the constructions proposed in this correspondence are optimal in terms of D-M tradeoff, and can be used to provide a re-liable MIMO communication with vanishing error probability when the number of coded blocksm is sufficiently large. Furthermore, we had given a stronger proof showing that codes resulting from the pro-posed constructions are approximately universal and can cope with sit-uations when the fading coefficients are correlated in time, are corre-lated among different antenna, and/or are of different kinds of statistics.
APPENDIX A. Proof of Theorem 2
Recall that givenX = (X0; X1; . . . ; Xm01) 2 X , the code matrix transmitted overm fading blocks, the (nr2nt) received signal matrix
at theith fading block is
Yi= HiXi+ Wi
where2= SNR10 is given in (16), and where we have setT = nt. The goal here is to show the codeword error probability ofX is on the order ofPe(r) := SNR0d (r). We will adopt some techniques from [16] and [20]. For any distinct pair of code matricesX 6= X0 2 X withX = (X0; . . . ; Xm01) and X0 = (X00; . . . ; Xm010 ), let i;1
i;2 1 1 1 i;n and`i;1 `i;2 1 1 1 `i;n be, respectively,
the ordered eigenvalues of the matricesHiyHiand1Xiy1Xi, where 1Xi= Xi0Xi0. Then, by using the mismatch bound [25], [16], it can
be shown that the squared Euclidean distance between the noise-free received signal matrices corresponding, respectively, toX and X0is
d2 E(X; X0) := m01 i=0 kHiXi0 HiXi0kF2 2m01 i=0 n j=1 i;j`i;j: (25)
Moreover, we may reorder and reindex the values
(0;1; . . . ; 0;n ; . . . ; m01;n )
as a nondecreasing sequence
1 2 1 1 1 n
and similarly, (`0;1; . . . ; `0;n; . . . ; `m01;n) as a nonincreasing
sequence
`1 `2 1 1 1 `n
wheren = mnt. Now with the above reordering,d2E(X; X0) can be further lower bounded by
d2 E(X; X0) 2 n i=1 i`i 2 n i=n0k+1 i`i _2 n i=n0k+1 i`i (26)
for k = 1; 2; . . . ; n, where the last inequality follows from the arithmetic–geometric mean inequality. In particular, by making use of
Proposition 1 and again by the arithmetic–geometric mean inequality, the serial product of`ican be lower bounded by
n i=n0k+1 `i= m01 i=0 det(4Xi4Xiy) n0k i=1 `i n0k1 i=1 `i n0ki=1 `i n 0 k 0(n0k) _k4Xk02(n0k) F := SNR0 : (27) As1 > 0 with Probability 1, define
i:= 0 logSNRi and := [ 1; . . . ; n]t: (28)
Then, substituting (27) into (26) yields d2 E(X; X0) _ 2 n i=n0k+1 i k4Xk02F = SNR ( ) := d2 E;k( ) (29)
which is independent of the choices ofX and X0, and k( ) := 1 0 rn t 0 1k n i=n0k+1 i0 r(n 0 k)n tk = 1 0 rmk 0 1k n i=n0k+1 i (30)
fork = 1; 2; . . . ; n. Thus, the codeword error probability given can be upper bounded by Pe(rj ) Pr m01 i=0 kWik2F d 2 E;k( ) 4 = exp 0d2E;k4( ) n mT 01 t=0 d2 E;k( )=4 t t! := Pk( ) (31)
and it should be noted thatPk( ) := 0 if k( ) > 0. Since Pk( ) 1,
it follows that
Pe(r) min
k f Pk( )g
Pr f : k( ) 0; k = 1; . . . ; ng:
Define
i;j := 0 logSNRi;j: (32) Now bearing in mind that i;1 i;21 1 1 i;n and 1 21 1 1
n, by arguing similarly as [20] and [26], it can be shown that
f : k( ) 0; k = 1; . . . ; ng = : m01 i=0 n j=1 (1 0 i;j)+ rm (33) where(x)+ := maxf0; xg. A proof similar to (33) can also be found in [27]. Note that the right-hand side of (33) equals the channel outage probability, that is, we have
Pe(r) Pr : m01 i=0 n j=1 (1 0 i;j)+ mr := Pr 1mm01 i=0 log In + SNRHiyHi SNRr
whereIn is the identity matrix of sizent. With the outage bound from [4], we have proved that the codeX is approximately universal and is D-M optimal for any kinds of fading distributions, including the time-correlated channels. In particular, for quasi-static Rayleigh fading
channel with independent fading blocks, the probability density func-tion of can be derived from Wishart distribution (see [4] and [26]), and it can be shown without using (33) that the diversity gaind(r) achieved byX equals d3(r) defined in (6). The missing details can be found in the conference version [26] of this correspondence.
B. Proof of Theorem 3
By construction, the size ofX equals
jX j = j ~X j = jA(SNR)jmT 1T := SNRrmT
since there is a one-to-one correspondence between matrices in ^X and ~
X . This follows from the fact that the difference between every distinct pair of matrices in ~X has full rank T . Thus, X achieves multiplexing gain at valuer. To ensure the power constraint (8), this time we will set the parameter at
2 := SNR10 : (34)
Furthermore, we may assume without loss of generality that ^X is ob-tained by removing the last(T 0 nt) rows of code matrices in ~X .
Sup-pose thatX = ( ^X; . . . ; m01( ^X)) 2 X was transmitted, and that ~
X 2 ~X is the corresponding (T 2T ) code matrix. The received signal matrix at theith fading block can be written as
Yi= Hi i( ^X) + Wi= ~Hi i( ~X) + Wi
fori = 0; . . . ; m 0 1, where ~Hiis the equivalent(nr2 T ) ith fading
channel matrix given by ~
Hi:= Hi 0n 2(T 0n ) :
0n 2(T 0n ) denotes the (nr 2 (T 0 nt)) all-zero matrix.
Now for any X 6= X0 2 X ; X = ( ^X; . . . ; m01( ^X)) and X0 = ( ^X0; . . . ; m01( ^X0)), let ~X (resp., ~X0) be the code matrix
in ~X that is associated with ^X (resp., ^X0). Arguing similarly as in part A of the Appendix, the squared Euclidean distanced2E(X; X0) is lower bounded by d2 E(X; X0) = m01 i=0 k ~Hii( ~X 0 ~X0)k2F _2 mn i=mn 0k+1 i~`m(T 0n )+i (35) fork = 1; 2; . . . ; mnt.1 2 1 1 1 mn and ~`1 ~`2 1 1 1 ~`mT are, respectively, the ordered nonzero eigenvalues of the
set of matricesfHiyHig and f[i(4 ~X)][i(4 ~X)]yg, where 4 ~X =
~
X 0 ~X0. Furthermore, a similar argument as in part A of the Appendix
shows that mn i=mn 0k+1 ~`m(T 0n )+i= mT i=mT 0k+1 ~`i_SNR0 (mT 0k): (36)
Substituting (36) into (35) yields thek( ) defined as in (29)
k( ) = 1 0 rT 0 1k mn i=mn 0k+1 i0 r(mT 0 k)kT = 1 0 rmk 0 1k mn i=mn 0k+1 i
fork = 1; . . . ; mnt, where is defined as in (28). By arguing in exactly the same way as in part A of the Appendix, we see that the code X is approximately universal and that the diversity gain achieved by X equalsd3(r) given in (6) for the case of independent block fading. The proof is now complete.
C. Proof of Theorem 4
For simplicity, here we only prove the case of T = mnt, and the case of T > mnt can be done by extending the arguments in part B of the Appendix. First, to satisfy the power constraint (8), the parameter is set at
2 := SNR10 : (37)
Next, given the transmitted code matrixX = (X0; X1; . . . ; Xm01) 2 X , we rearrange the received signal matrices Y0; . . . ; Ym01as the fol-lowing(mnr2 T ) matrix: ~ Y := Y0 .. . Ym01 = H0 . .. Hm01 X0 .. . Xm01 + W0 .. . Wm01 = ~H ~X + ~W (38)
where ~H is the (mnr 2 mnt) block-diagonal channel matrix and
where ~W is an (mnr 2 T ) noise matrix. It should be noted that the matrix ~X 2 ~X by construction. Thus, a similar argument from part A of the Appendix shows that for everyX 6= X02 X ; d2E(X; X0) can be lower bounded by
d2E(X; X0) _2 T i=T 0k+1
i~`i 1=k
fork = 1; 2; . . . ; T , where 1 2 1 1 1 T and ~`1 ~`2
1 1 1 ~`T are, respectively, the ordered nonzero eigenvalues of the
ma-trices ~HyH and 4 ~~ X4 ~Xy1 4 ~X = ~X 0 ~X0and ~X0is the code matrix obtained by rearrangingX0as (38). In particular, we have
T i=T 0k+1
~`i_k4 ~Xk02(T 0k)F := SNR0 :
Moreover, the exponentk( ) defined in (29) now equals k( ) = 1 0 rn t 0 1k T i=T 0k+1 i0 r(T 0 k)kn t = 1 0 rmk 0 1k T i=T 0k+1 i:
Now by the same arguments as in part A of the Appendix, it can be shown thatX is approximately universal and achieves diversity gain d3(r) defined in (6) for the case of independent block fading.
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