Space-Time Differential Modulation using Linear Constellation Precoding Alfonso Cano
Xiaoli Ma
Georgios B. Giannakis
Dept. of Signal Theory and Comms. Universidad Rey Juan Carlos 28943 Fuenlabrada, Madrid, Spain E-mail:
[email protected]
Dept.of ECE Auburn University Auburn, AL 36849, USA E-mail:
[email protected]
Dept.of ECE University of Minnesota Minneapolis, MN 55455, USA E-mail:
[email protected]
Abstract— Differential encoding is known to simplify receiver implementation because it by-passes channel estimation. Relying on linear constellation precoders designed in closedform for coherent multi-antenna systems, we derive a nonconstant modulus space-time differential scheme that enables diversity, does not sacrifice rate, and more interestingly, it allows for constellation designs with many degrees of freedom. Performance merits of this scheme are analyzed and compared with existing differential designs. Simulations corroborate our theoretical findings.
I. I NTRODUCTION In wireless fading communications, the use of multiple antennas has been widely acknowledged as a means of enhancing error performance and transmission rate compared to traditional single-antenna links. Space-Time (ST) coding at the transmitter based either on orthogonal designs (STOD) [12], or, linear constellation precoding (LCP) [13] were developed to enable spatial diversity. When there is no channel knowledge available at the receiver, these strategies require insertion of pilot symbols to obtain channel estimates. An alternative to training is the use of differential and noncoherent coding strategies. In noncoherent schemes, assuming that the channel stays invariant over T channel uses (a.k.a. coherence time), one aims for transmissions that allow decoding without any channel knowledge at the receiver [3]. For differential schemes, channel estimation is by-passed by encoding the information in the difference between transmitted blocks. The advantage is that one can thus allow coherent-like block-by-block demodulation. Moreover, because only the first block does not carry information, the bandwidth loss can be arbitrarily low. Differential unitary space-time modulation (DUSTM) has been introduced for flat-fading channels using diagonal matrices with constant-modulus (CM) entries [2], [4], and shown capable of enabling full spatial diversity. Differential orthogonal designs for two antennas using the Alamouti scheme and later for any number of antennas were proposed in [11] and [6], respectively. However, similar to their coherent counterparts, orthogonal differential designs in [6] incur rate loss when real-valued constellations are employed. Furthermore, when channel impairments such as time and/or frequency selectivity are present, differential schemes are
still possible to design; see [1] and references therein. The aforementioned differential schemes rely on CM constellations which guarantee stability in the differential recursion. In this paper, we develop a new framework to include non-CM differential encoders and decoders. The use of non-CM constellations over [2] or [11] are motivated by the fact that, as the number of bits per channel use (pcu) increases, the minimum Euclidean distance between points in the constellation in M -ary CM constellations reduces drastically. To prevent this, amplitude phase shift keying (APSK) symbols can be differentially encoded not only in phase but also in amplitude [7]. Differential nonCM constellations, e.g., M -ary quadrature amplitude modulation (QAM), which cannot be utilized in [11] for two antennas using orthogonal ST codes, have been suggested in [5], but tend to be inefficient for a general number of antennas. This paper develops differential diversity-enabling techniques based on LCP which does not sacrifice rate and takes full advantage of higher constellation distances by using non-CM constellations. Notation: Upper (lower) bold face letters will be used for matrices (column vectors); (·)H and (·)T denote Hermitian and transpose; [·]k,l denotes the (k, l)th entry of a matrix, and [·]k the kth entry of a vector; IN denotes the N × N identity matrix; 1N is the N × 1 all-one vector; ⊗ denotes Kroneker product; diag(x) is a diagonal matrix with x on its diagonal; det(X) is the determinant of matrix X; · is the Frobenius norm; vec(X) is a vector resulting from stacking columns of X; FN stands for the normalized fast Fourier transform (FFT) matrix with entries [FN ]k,l := N −1/2 exp(−j2πkl/N ); and CN (µ, σ 2 ) denotes the complex Gaussian distribution with mean µ and variance σ2 . II. S YSTEM M ODEL We consider a single-user transmission from Nt transmit to Nr receive antennas. Arriving bits at rate R bits pcu are parsed into blocks, indexed by index k, and mapped to a square matrix Xk , of size Nt × Nt . The received Nr × Nt matrix Yk results after multiplying the matrix Xk with the Nr × Nt flat fading channel matrix H. The corresponding
multi-input multi-output (MIMO) relationship is: Yk = HXk + Wk ,
(1)
where the noise term Wk has size Nr × Nt and its (n, i)-th entry wkn,i = [Wk ]n,i is modelled as wkn,i ∼ CN (0, N0 ), with N0 denoting the noise variance. Similarly, the i.i.d. fading coefficients in H, and hn,m = [H]n,m are modelled as hn,m ∼ CN (0, 1). We suppose that the channel fading coefficients in H remain invariant over a sufficient number of blocks. Time variability of H has been treated in the context of differential transmissions by [10], [1], [9], and will not be dealt with here. Matrix Xk will be designed with the following structure: Xk = UDxk , where U is unitary and the diagonal matrix Dxk conveys the symbols to be transmitted. Matrix U is introduced to distribute power among transmit antennas as desired. For simplicity, and without loss of generality, we can assume U = INt , which implies that only one transmit antenna is on per time slot. A. Constant-Modulus Differential Recursion In order to put our contribution in context, we first briefly review the results in [2]. These results are presented here in vector form with the intention to offer a unified framework for performance comparison. In [2], the following CM recursion is proposed: Dxk−1 Dvk , k ≥ 1 Dxk = , (2) k=0 INt , where the diagonal matrix Dvk belongs to a set V with cardinality L = 2RNt that carries the information. Each element Dvl of the set V is constructed as Dvl = diag([exp(j2πu1 /L), . . . , exp(j2πuNt /L)])l , (3) and integers u1 , . . . , uNt are chosen to optimize diversity and coding advantages as detailed in [2]. At the receiver, we substitute (2) into (1) to obtain Yk = HDxk−1 Dvk + Wk .
(4)
Considering the previously received block Yk−1 = HDxk−1 + Wk−1 , the current one can be expressed as Yk = Yk−1 Dvk + Wk
where := Wk −Wk−1 Dvk . Because entries in Dvk are constant-modulus, entries in Wk−1 remain white but with larger variance 2N0 , common to any differential detection [2]. The ML detection rule for detecting Dvk from Yk is: Dv ∈V
˜x where matrix D is built by normalizing each entry in k−1 Dxk−1 by its norm: ¯ −1 , ˜x = Dxk−1 D D xk−1 k−1
(6)
which clearly does not require knowledge of the channel. B. Linearly-Precoded (LCP) differential recursion Constraints limiting the constellation design for Dvk in (3) yield symbols with reduced minimum distance as bit
(8)
1/2 ¯x := (Dxk−1 DH . Matrix Duk contains with D xk−1 ) k−1 the information-carrying symbols and has the form Duk = R t diag(Θsk ), with sk ∈ AN s and As = 2 conveys bits in the k-th block of standard M -ary QAM/PAM constellations and Θ is a unitary matrix that performs LCP. The performance advantage offered by this matrix as well as its design criterion will become clear in the diversity analysis we pursue in the next section.
It is worth stressing that unlike Dvk in (3), Duk in (7) has non-CM entries and so does Dxk . As a result, additional degrees of freedom become available in designing vector Duk , with the final objective of improving performance. At the receiver, we substitute (7) into (1) and obtain ˜ x Du + Wk . Yk = HD k−1 k
(9)
˜ k−1 := HD ˜ x , we can relate Defining now the term Y k−1 ˜ k−1 with the previously received matrix Yk−1 and the Y ˆ x , both of which are previously estimated matrix D k−1 available at the receiver ˜ k−1 = Yk−1 D ¯ −1 . Y xk−1
(10)
Finally, we arrive at [c.f. (5)] ˜ k−1 Du + W , Yk = Y k k
(11)
˜ k−1 and where now Wk := Wk − W ¯ −1 Du . ˜ k−1 = Wk−1 D W xk−1 k
(12)
¯ −1 Du affects the noise distribution, let Since the term D xk−1 k us take a close look at its mean and variance: E{Wk } = 0
(5)
Wk
ˆ v = arg min Yk − Yk−1 Dv 2 , D k
rates of the number of transmit antennas increase. Motivated by this limitation, we propose here an alternative differential recursion using non-CM constellations, as follows: ˜ x Du , k ≥ 1 D k−1 k , (7) Dxk = k=0 INt ,
with
E{Wk (Wk )H } = N0 Σw ,
(13)
¯ −2 Du DH . Σw = INt + D xk−1 uk−1 k−1
(14)
Since Σw is clearly a diagonal matrix, we deduce that the ¯x and Duk−1 ensure that the noise non-CM entries in D k−1 matrix Wk remains uncorrelated but with unequal variance entries. Considering (10) and (14) and substituting Duk = diag(Θsk ), we obtain the ML detector for decoding sk from Yk in (11) as: ˜ k−1 diag(Θs))Σ−1/2 2 , ˆsk = arg min (Yk − Y w N
s∈As t
(15)
˜ k−1 depends not only Compared to (6), we notice that Y on Yk−1 but also on the estimation of the previouslyˆ x . This dependence can cause error received symbol, D k−1 propagation, and its effect on error performance will be assessed through simulations in Section III. If needed, there are means of mitigating such error propagation at the price of higher complexity using, e.g., Multiple-Symbol-Detection (MSD) [10], or the Viterbi Algorithm (VA) [9]. C. Performance Comparison We now compare the relative performance of the systems introduced in subsections II-A and II-B. Our figure of merit will be the pairwise error probability (PEP), which defines the probability that the ML detector incorrectly decodes an information block sk as sk = sk . We first derive bounds on the PEP of the LCP-based differential transmission. As usual, this PEP can be upperbounded using the Chernoff bound [1], [2], [9], [13] 2 d (s, s )Es ˆx ≤ exp − Pr s → s |Yk−1 , D . (16) k−1 4N0 ˆx We will assume that D = Dxk−1 ; that is, previous k−1 symbols are perfectly estimated and thus there is no error propagation. Defining the error vector e := s − s , we can find from (15) the distance ˜ x diag(Θe)Σ−1/2 2 . d2 (s, s ) = Es−1 HD w k−1
(17)
Using the fact that for any matrix M, we have M = vec(M), it is possible to conveniently reshape the righthand-side of (17) as ˜ x diag(Θe)Σ−1/2 )]h2 , d2 (s, s ) = Es−1 [INr ⊗ (D w k−1 (18) where the vector h := vec(HT ) collects all Nt Nr fading coefficients. Let us also define the Nt Nr × Nt Nr matrix 1/2 1/2 Ae := (Rh )H Be (Rh ), with ˜ x diag(Θe))H Σ−1 (D ˜ x diag(Θe))] Be := INr ⊗ [(D w k−1 k−1 (19) ¯h ¯ H } has rank rh := rank(Rh ). The where Rh := E{h average PEP can then be expressed as [13] −Ge,d Es , (20) Pr [s → s ] ≤ Ge,c 4N0 where Ge,d := rAe with rAe := rank(Ae ), and Ge,c :=
(1/rAe ) rAe −1 λr is the coding gain, with λr denoting r=0 the non-zero eigenvalues of Ae . The diversity order is defined as Gd := mins=s Ge,d . Supposing henceforth that Rh = INr Nt , and thus rh = Nt Nr , the maximum diversity order the system can collect is := Nt Nr . Based on these definitions, we can establish Gmax d the following result on the diversity performance of LCPbased differential modulations:
Proposition 1 If Θ is constructed such that |θ Tm (s − s )| = 0,
t ∀m ∈ [1, Nt ], ∀s, s ∈ AN s ,
(21)
θ Tm
with denoting the m-th row of matrix Θ, the differential recursion and detector in (7) and (15) achieve the full diversity Ge,d = Nt Nr . Proof. We need to prove that matrix Be in (19) has full rank. First, we notice that rank(Be )
˜ x diag(Θe)]H Σ−1 [D ˜ x diag(Θe)]). (22) = Nr rank([D w k−1 k−1 But recalling condition (21), diag(Θe) has all its diagonal entries to be non-zero and likewise for the previously esti˜x and Σw . Consequently, for any s, s ∈ mated symbol D k−1 t ˆ AN s , and for any previously-estimated symbol Dxk−1 , the overall product is a diagonal matrix with non-zero diagonal entries and thus with ˜ x diag(Θe)]) = Nt . ˜ x diag(Θe)]H Σ−1 [D rank([D w k−1 k−1 (23) Plugging the latter into (22) completes the proof. Eq. (21) is what we term symbol detectability criterion, which means that block s can be detected even if fading nullifies all-except-one entries of Θs [13]. This same intuition applies also to CM-based constellations. In fact, the mapping in [2] turns out to have the following property: entries in the diagonal of matrix Dv are unique to that matrix in that specific position; and thus, if it happens that only one entry of Dv is available at the receiver, we know which matrix it belongs to and the Nt − 1 remaining entries that were nullified by fading can be estimated. Coming back to the LCP-based design, [13] provides guidelines to design matrix Θ using either parametrization techniques or algebraic tools. The diversity maximizing matrix Θ is not unique, but is available for any Nt . Among all designs available in [13], we rely on the Vandermonde structure given by Nt −1 ), Θ = FH Nt diag(1, α, . . . , α
(24)
where α := exp(j2π/P ) and the choice of P follows from number theoretic arguments provided in [13]. Specifically, for Nt = 2i , we have i ∈ N and P = 4Nt ; for Nt = 3, P = 18; and for Nt = 5, P = 35. We further recall that one can also rely on ad-hoc numerically-optimized constructions of Θ. Once proven that both strategies achieve the maximum diversity order, our interest is to show that the use of LCP can be advantageous in terms of coding gain compared to CM-differential alternatives. The coding gain is defined as Gc = mins=s Ge,c and can be written as: 1
1
1
Gc = min [det(Ae )] rh = [det(Rh )] rh min [det(Be )] rh . s=s
s=s
(25)
TABLE I P RODUCT DISTANCES FOR Nr = 1, Nt = 2, 3, 4 AND R = 2.
Nt δ CM / κCM GCM tot δ LCP / κLCP GLCP tot
3 3.6e-3 / 128 0.030 2.7e-3 / 32 0.087
With error propagation No error propagation
4 9.05e-5 / 1024 0.017 0.062 / 6272 0.029
−1
10
BER
2 0.086 / 32 0.051 0.29 / 96 0.054
0
10
Let us now define
−2
10
δ
LCP
:= min
s=s
[det(Be )]
(26)
as the minimum product distance when using LCP and δ CM = (1/2)Nt minDv =Dv [det(INr ⊗(Dv −Dv )2 )] (27) as its CM counterpart. Notice that the term (1/2) in (27) comes from the fact that in CM-based transmissions Σw = 2INt . Because each system uses a different bit-mapping, a fair comparison needs to consider also the kissing number κ [8], which is the number of (s, s ) pairs with the same minimum product distance. To this end, we can introduce the following performance indicator which captures both parameters:
−3
10
Nt
2
4
6
8
10 SNR
12
14
16
18
20
Fig. 1. Differential-LCP with and without error propagation effects (2 bits pcu, Nt = 2, Nr = 1) 0
10
Diff−LCP Diff−CM [2] Coherent LCP in [13]
−1
10
(28)
Given that the constellation design in [2] is only available through numerical search, we can not rely on any analytical performance comparison. Instead, we will resort to numerical tests. In Table I we compare δ, κ and Gtot for R = 2 bits pcu and different values of Nt . We observe that for a sufficiently large constellation size or number of antennas, the coding gain advantage in differential-LCP outperforms that of the differential-CM modulation in Section II-A. This is intuitive because, on the one hand, CM-based constellations in [2] always lie ont the unit circle. Consequently, if one increases information rate, the minimum distance decreases exponentially. On the other hand, non-CM designs allow blocks to be designed with any complex-valued constellation, and thus minimum distances between points decrease less. III. S IMULATED P ERFORMANCE In this section, we compare the average bit-error-rate (BER) of the proposed differential-LCP system with differential designs such as the DUSTM in [2] and the orthogonal designs in [11] and [5]. Also, we test the effect that error propagation has on the system performance. Vector s will be drawn from M -ary QAM constellations at a fixed transmission rate expressed in bits per channel use (pcu) common to all systems. Here, signal-to-noise ratio (SNR) is defined as SNR:= Dx 2 /E{W2 }, and the channel matrix H has i.i.d. entries. Coherence time is set to T = 20Nt . 1) Error-propagation effect: In Section II-B, we have claimed that the proposed differential detection causes unavoidable error propagation. To delineate the effects of error
BER
Gtot = κ1/Nr Nt Gc
0
−2
10
−3
10
−4
10
0
Fig. 2.
2
4
6
8
10 SNR
12
14
16
18
20
Differential-LCP versus DUSTM (2 bits pcu, Nt = 2, Nr = 1)
propagation, we compare two cases here: one is with error propagation, i.e., the estimate of the previous symbol is used in (15); and the other is without error propagation, which means that the true value is assumed in (15). The BER performance using QPSK, Nr = 1 and Nt = 2 is depicted in Fig. 1. We notice that error-propagation affects performance by about 0.15dB, which is considerably small; if BPSK is employed, it would be negligible; and with 16QAM it would be about 0.5dB. This should not be surprising because the initial block is known, and the decision of the current block only depends on a single previous block. 2) Comparisons with [2]: Fig. 2 compares LCP-based differential modulation with the DUSTM in [2] at 2 bits pcu, Nr = 1, and Nt = 2 antennas. As a performance reference, coherent LCP detection is also included in the same figure. As expected, both schemes achieve the maximum diversity
0
0
10
10 Diff−CM in [2] Diff−LCP
Diff−LCP Diff−STOD in [5] ([11])
−1
BER
BER
10
−1
10
−2
10
−3
10
−2
10
−4
0
5
10
15
20
25
SNR
Fig. 3.
Differential-LCP versus DUSTM (4 bits pcu, Nt = 2, Nr = 1)
gain of 2, but LCP-based schemes exhibit better coding gains. In test case 1), we identified that error propagation increases with the constellation size. Our intention here is to illustrate that this degradation is less than that of the decreasing product-distances in CM-based transmissions. For this reason, we fix Nt and Nr to be the same, we increase the transmission rate to R = 4 pcu, and compare again both systems. For [2], we use [u1 , u2 ] = [1, 181] and s is now drawn from a 16-QAM constellation. Fig. 3 shows that in this case the performance gap increases considerably and reflects the fact that decreasing distances of CM-based transmissions are more harmful than errorpropagation effects. Furthermore, se infer from Table I that as the number of antennas increases, the performance gap increases accordingly. 3) Comparisons with [11] and [5]: Differential STODs in [5], [6], [11] have well-known merits in terms of decoding complexity as well as diversity performance, specially when Nt = 2. Designed using non-CM constellations, codes in [5] can be understood as the STOD counterparts of our codes. In Fig. 4, we compare performance of both systems. At 2 bits pcu, Nr = 1 and Nt = 2, codes [5] and [11] are equivalent. As with coherent codes [13], differential STODs outperform LCP-based codes by 1-2dBs. However, when the number of antennas increases, the only available differential designs are those built over generalized orthogonal designs in [6], known to incur 50% rate loss. On the other hand, LCPbased transmissions are still rate-1 and their performance advantage markedly improves. Simulation results in this case are analogous to the coherent ones in [13]. ACKNOWLEDGMENTS Work in this paper was prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U. S. Army Research Laboratory under the Collaborative Technology Alliance
10
0
5
10
15
20
25
SNR
Fig. 4. Differential-LCP versus differential-STOD (2 bits pcu, Nt = 2, Nr = 1
Program, Cooperative Agreement DAAD19-01-2-0011. The U. S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The work in this paper was also supported by the Electronics & Telecommunications Research Institute. The work of the first author was supported by the Spanish government TEC2005-06766-C03-01/P-TIC-000223-0505. The work of the second author was supported by ARO Grant No. W911NF-04-1-0338.
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