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Application of perfect space time codes: Optimality and Practical Limits

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Mroueh, Lina; Institut Supérieur d'Electronique de Paris ISEP, SITe Rouquette-Léveil, Stéphanie; Thales Communication Belfiore, Jean-Claude; ENST, COMELEC

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IEEE Transactions on Communications

Digital communication

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Re IEEE Transactions on Communications

Page 1 of 23

1

Application of perfect space time codes: Optimality and Practical Limits Lina Mroueh∗† , Stéphanie Rouquette-Léveil‡ , and Jean-Claude Belfiore∗ ∗ †

ISEP Paris, 75006 Paris, France

Télécom ParisTech, 75013 Paris, France ‡

Thales Communication, France

[email protected], [email protected], [email protected]

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Fo Abstract

The optimal design of perfect space time codes constructed from cyclic division algebra (CDA)

ee

on the quasi-static uncoded MIMO channel has received lots of attention in industry over the last few years. However, the recent standards that uses multiple antennas terminals such as IEEE 802.11n

rR

or IEEE 802.16e, are based on more realistic assumptions involving the use of outer codes, and multi-taps channels. In this paper, we study the performance of these perfect codes in a standard context. We show that although these codes are optimal, the asymptotical gain of these codes is

ev

only significant at a very high SNR range and cannot be observed for a moderate Packet Error Rate (PER) probability.

Index Terms

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Bit Interleaved Coded Modulation (BICM), Orthogonal Frequency Division Multiplexing (OFDM), Space Time Coding (STC), Pairwise Error Probability (PEP), Perfect codes, IEEE 802.11n.

I. I NTRODUCTION The IEEE 802.11n is one of the latest evolutions of the previous 802.11 standards for wireless LANs. The main objective of this technology is to provide to the end-user modes Part of this work was performed while the first author was with the department of communication and Electronics in Telecom ParisTech. Lina Mroueh is now at the department of signal, image and telecommunications in the Institut Supérieur d’Electronique de Paris ISEP, France. This work was financed by Motorola Labs Paris, the Association National de Recherche Technique (ANRT) and the Fond Social Européen.

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2

of operation that are capable of much higher throughput than 11a/b and g, with a maximum throughput of at least 100Mbps, as measured at the MAC data Service Access Point (SAP). The major novelty of this version is to use Multiple Input Multiple Output (MIMO) techniques so that devices embed multiple transmitter and receiver antennas to allow an increased data throughput through spatial multiplexing and an increased range by exploiting the spatial diversity. In order to exploit fully the available diversity of a block fading uncoded MIMO channel, Tarokh et al. derive two fundamental criterion that should be satisfied by a space time code to be optimal. These two criterion are as follows - The rank criteria, stating that the difference between two distinct codewords must be a

Fo

full rank matrix.

- The determinant criteria, stating that the minimal determinant of space time code is maximized.

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Recently, Oggier et al. in [1] proposed a family of optimal space time codes known as perfect space time codes that fulfill the design criteria of Tarokh [2]. Moreover, it has been

ee

shown that these codes are the optimal codes over quasi-static uncoded1 MIMO channel since they achieve full rate and full diversity, preserve the mutual information and achieve

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the Diversity Multiplexing Tradeoff (DMT) [3]: by construction, these codes have a non vanishing determinant [4].

ev

Unlike the simplified quasi-static uncoded MIMO channel, industrial transmission schemes are based on more realistic assumptions involving the use of outer codes such as the con-

iew

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Page 2 of 23

volutional code, and frequency selective channels. This motivates us to study the design of space time codes in a practical scenario which is the MIMO-BICM case over a flat and a frequency selective channel.

Contributions: In this paper, we focus on the nt × nr MIMO-BICM transmission scheme. The main purpose of this paper is to study the performance of the perfect space time codes using such transmission system. For this end, we derive the pairwise error probability for both cases of flat fading and frequency selective channels. From the PEP expression, we derive space time design criterion that minimizes this PEP. We show that in both cases, perfect 1

In this paper, the uncoded MIMO system refers to the case when no outer codes such as the convolutional codes are

used.

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Page 3 of 23

3

codes are optimal as they fulfill the space time design criterion in a BICM-MIMO system.

Outline: The rest of the paper is organized as follows. Section II defines the BICM-MIMO system model and provides some background materials on perfect space time codes design over a uncoded MIMO block fading channel and the general expression of the pairwise error probability (PEP) in a MIMO-BICM system. Then, we derive in section III and section IV the pairwise error probability in the BICM-MIMO context for a flat fading channel and a frequency selective channel respectively. Simulations results given in the IEEE 802.11n context. Finally, section V concludes the paper.

Fo

Notations: The notations used in this paper are as follows. Boldface lower case letters v denote vectors, boldface capital letters M denote matrices. M† and MT denotes respectively

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the matrix transposition and conjugated transposition operations. kvk stands for the Euclidean norm of vector v. kHk2F = Tr{F† F} is the Frobenius norm of matrix. Tr{A} refers to the

ee

trace of matrix A. IN stands for the N × N identity matrix. CN represents the complex Gaussian random variable. EX {.} is the mathematical expectation of random variable X.

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Finally, A ⊗ B denotes the kronecker product between matrices A and B . Channel

Convolutional Code C

BICM MIMO system

Modulation

ML soft Decoder

Deinterleaver

Viterbi Decoder

iew

Fig. 1.

Interleaver

Space Time Block Coding STBC

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II. P RELIMINARIES ON THE BICM SYSTEM MODEL In this section, we start by defining in subsection II-A the BICM system model that is based on the IEEE 802.11n transmission scheme. The properties of the space time codes used throughout this paper are reviewed in subsection II-B. Then, the general pairwise error probability (PEP) expression for the MIMO BICM system is given in subsection II-C. A. System model The block diagram of the considered system based on the IEEE 802.11n transmission scheme is depicted in Figure 1. During the transmission, the binary information elements b October 5, 2010

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4

are first encoded by a binary code of rate Rc e.g. a convolutional code with free distance dfree , and then interleaved by a bit interleaver which will be denoted by π. The coded and interleaved sequence c is fed into the 2m -QAM gray mapper and is mapped onto the signal sequence x ∈ X . The resulting symbols are coded by a space time block code Xp with spreading factor2 s. The coded space-time codes are finally transmitted on a multiple antenna channel H with nt transmit antennas and nr receive antennas. Channel model: In this paper, we adress the two cases where the channel is either flat fading or selective in frequency. The flat fading channel models the case when the channel remains constant over all realisations during the duration of the transmission, i.e.,

Fo

H(k) = H,

∀k

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We assume in this case that the transmission is done during a period of N s time slots, and we refer the coded space-time code index by k, with k = 1 . . . N . The frequency selective fading models the case when the channel. For a frequency selec-

ee

tive channel with L taps, a MIMO-OFDM system is considered. The channel is therefore decomposed into N parallel channels,

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H(k) = H(ej2πfk ),

k = 1...N

that are statistically correlated where fk is the subcarrier and k refers in this case to the

ev

channel frequency index. We assume in this case that the sequence of symbol x is coded into N space-time block codeword, and each space-time codeword is transmitted over one subcarrier k, where k = 1 . . . N .

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Page 4 of 23

0

0

Bit interleaver: The bit interleaver can be modeled as π: k → (k, i), where k denotes the original ordering of the coded bits ck0 , k denotes the index of the space-time code matrix and i indicates the position of the bits ck0 in the codeword. At the receiver, the received coded space time codeword is given by Y(k) = H(k)C(k) + Z(k),

(1)

where Z(k) is the complex Gaussian noise Z ∼ CN (0, N0 Inr ). The maximum likelihood (ML) soft decoder generates for each coded bit ck,i two metrics: λick =0 and λick =1 . Theses metrics correspond to the log-MAP (Maximum A-Posteriori) 2

or equivalently the number of slots required to code the information.

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5

computed over one codeword (refer to [5] for more details on λ-metrics), and are given by : X λi (ck ) = log p(Y(k) H(k), C), C∈Xci

k

= log

X

2

exp−kY(k)−H(k)C(k)kF .

C∈Xci

k

These metrics can be approximated by λi (ck ) ≈ mini kY(k) − H(k)Ck2F ,

(2)

C∈Xc

k

where Xbi denotes the constellation subset Xbi = {C ∈ Xp : li (C) = b},

Fo

and li (C) is the ith bit of the space time codeword matrix C. For low complexity algorithms, these metrics can be computed using the list sphere decoder [6].

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Then, the metrics associated to the interleaved bits are deinterleaved. Finally, the λ metrics are used by the Viterbi decoder to decode the information bits by finding the shortest path

ee

in the trellis according to

ˆc = arg min

X

c

λ(cik ).

(3)

rR k

0

B. Overview on space time coding scheme

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The space time block codes used in this paper are the perfect codes and the spatial division division multiplexing (SDM) schemes. The properties of these two coding schemes are summarized in the following.

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1) Perfect space time codes: Perfect space time codes fulfill the design criteria of Tarokh et al. in [2] for a quasi static uncoded MIMO fading channel. For this family of code, the spreading factor s = nt , and the matrix codeword given in (4),  x1 x2 ... xnt   γσ(x ) σ(x1 ) . . . σ(xnt −1 ) nt  C= . ..  .. .  γσ nt −1 (x2 ) γσ nt −1 (x3 ) . . . σ nt −1 (x1 )

      

(4)

belongs to a cyclic division algebra. The fundamental properties of perfect space time code that will be used in the following are: - Perfect space time codeword has full rank of nt .

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6

- Perfect space time code has a non-vanishing determinant min det{CC† } ≥ δ dmin

2nt

C6=0

,

with δ < 1 being the inverse of the discriminant of Q(θ) (refer to [1] for more details), and is independent of the constellation size. The minimal Euclidian distance between two symbols is dmin =

√2 Es

where Es is the energy per symbol (Es = 2, 10, 42 for

QPSK, 16QAM and 64QAM respectively).

2) Spatial division multiplexing: The spatial division multiplexing corresponds to the case when no space time code is used. The spreading factor in this case is s = 1, and the codeword

Fo

C is a nt × 1 vector of symbols. For SDM schemes, properties used in the following are - The rank of CC† is equal to 1.

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- The minimum determinant of the nt × nt matrix xx† is equal to d2min , where dmin =

√2 Es

is the distance between two constellation points.

ee

C. General pairwise error probability derivation The pairwise error probability (PEP) of BICM-MIMO-OFDM using an orthogonal space

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time code such as the Alamouti code was derived in [7]. In this subsection, we extend the pairwise error probability expression of [7] to a more general case using a space-time code

ev

with a spreading factor s. We assume that the energy per coded symbol at each antenna and on each time slot is equal to one and that SNR = 1/N0 . The main result of this subsection is summarized by the following theorem.

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Page 6 of 23

Lemma 1: The PEP over a general channel model is upper bounded by, o 1 n X P(c → ˆc) ≤ EH exp − kH(k)C(k)k2F , 4N0 k0 ,d

(5)

free

where dfree is the free distance of the convolutional code, C(k) is a non-zero space time codeword matrix. P The notation k0 ,dfree means that we only consider the dfree bits indexed by k 0 , interleaved such that π : k 0 → (k, i).

Proof: The proof of this lemma is detailed in Appendix A.

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7

III. BICM-MIMO WITH FLAT FADING CHANNEL In this section, we specify our results to the case where the channel is flat fading. The PEP expression when perfect space time code is used, is derived in subsection III-A. The comparison with the spatial division multiplexing (SDM) schemes follows in subsection III-B. A. PEP for perfect code with flat fading channel The main result of this section is summarized in the following theorem. Theorem 1: For a MIMO-BICM system with a flat fading channel, perfect space time code allows to extract the full diversity order of nt nr and the PEP is upper-bounded by,

Fo

PEP ≤ dfree δ

−nr

Es

nt nr

SNR−nt nr .

(6)

Proof: When perfect space time codes are used, the Frobenius norm in the general PEP

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expression in (5) can be simplified by, X kHC(k)k2F , k0 ,dfree

X

Tr HC(k)C(k)† H† ,

k0 ,dfree

ee

X = Tr H C(k)C(k)† H† . k0 ,dfree

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In the following, A denotes the nt × nt matrix such that, X C(k)C(k)† = CC† , A= k0 ,dfree

ev

where

C = [C(k1 ) . . . C(kdfree )].

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Let A = UΛU† be the eigen-value decomposition of A, then n o X kHC(k)k2F = Tr (HU)Λ(HU)† , k0 ,dfree

=

nr X nt X

λj (A)βi,j ,

i=1 j=1

where βi,j are i.i.d. random variable resulting from the multiplication of H by the unitary matrix U and λj (A) are the non zero eigen-values of A. By averaging over all the βi,j variables which are Gaussian distributed as shown in [2], the PEP is bounded as, P(c → ˆc) ≤

nt Y j=1

!−nr λj (A)

1 4N0

−nt nr

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.

(7)

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8

Let ∆min ,

nt Y

λj (A)

j=1

be the minimal determinant of A. As all matrices C(k)C(k)† are positive definite matrices, then ∆min can be bounded such that †

∆min = det{AA } ≥

dfree X

det{C(k)C(k)† }.

k0 =1

From Lemma 1, we know that C(ki ) matrices are non-zero matrices and therefore, ∆min ≥ dfree ∆cmin = dfree δ(dmin )2nt , where ∆cmin is the minimal determinant of the code and dmin is the minimal Euclidian distance

Fo

between two symbols (dmin =

√2 ). Es

It is clear that the convolutional code over a flat fading

channel improves the coding gain. The PEP is therefore bounded by

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PEP ≤ dfree δ

−nr

Es

nt nr

SNR−nt nr .

(8)

ee

Notice here that the main difference between [8], [9], that using a combination of convolutional code and perfect space time codewords guarantees that all C(ki ) are non zero

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and therefore the coding gain is enhanced automatically, which is not the case in [8] and [9]. Combining convolutional code with perfect codewords can be another alternative that

ev

could be investigated to enhance the coding gain instead of performing set partitioning at the mapper as shown in [8], [9].

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Page 8 of 23

B. Perfect space time codes versus SDM

When no space time code is used (SDM scheme), the PEP can be bounded following the same steps as for the perfect code case such that, PEP ≤ dfree

−nr

Es

nr

SNR−nr .

(9)

In Figure 2, the asymptotical behavior of the coded SDM versus the coded 2×2 perfect code scheme (known as Golden code (GC) in [4]) is depicted for the 2 × 2 MIMO configuration using QPSK constellation. As it can be shown, when no convolutional code is used, the coding gain of SDM is dominant at very low SNR. However, the diversity gain of the Golden code became quickly dominant, and can be easily observed at reasonable PER range. When a MIMO BICM is used, it can be observed that the coding gain of SDM is largely October 5, 2010

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9

enhanced compared to the Golden code coding gain especially when the free distance of the convolutional code increases. At low SNR, this coding gain dominates the high diversity gain that can be achieved by the GC. However, at high SNR, the diversity gain became dominant. As shown in Figure 3 for the 2 × 2 MIMO flat fading channel and a convolutional code [5 7], Rc = 1/2 with dfree = 5 and a packet of 1000-bits, the gain provided by the full diversity of the GC dominates the coding gain for low SNR as well as for the high SNR order. A gain of 1.9 dB is obtained for a PER of 10−2 . However, when using a convolutional code with a large free distance, e.g. [133 171] - Rc = 1/2, with dfree = 10, as shown in Figure 4 the coding gain dominates the diversity gain. The impact of the additional diversity cannot be observed at a moderate range of PER.

Fo

IV. BICM-MIMO WITH FREQUENCY SELECTIVE CHANNELS

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In this section, we consider the frequency selective fading channel. The PEP expression when perfect space time code and SDM scheme is used, is derived in subsection IV-A and IV-B. The comparison with the spatial division multiplexing (SDM) schemes follows in

ee

subsection IV-C in IEEE 802.11n context.

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A. PEP of perfect codes with frequency selective channel The main result of this section is summarized in the following theorem.

ev

Theorem 2: For a MIMO-BICM system with frequency selective fading channel, perfect space time code used over subcarriers allows to extract the full diversity order of nt nr min(L, D) and the PEP is upper-bounded by,

iew

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PEP ≤ G SNR−nr nt min(L,D) , where, G= αδ

−nr dfree

(10)

(Es )nt nr min(L,D) ,

where D ≤ dfree denotes the number of different subcarriers on which erroneous bits are received. The interleaver design allows to maximize the parameter D. The parameter α is a constant that depends on the covariance matrix. Proof: The expression in (5) can be simplified as following, X †¯† ¯ kH(k)C(k)k2F = Tr HCC H

(11)

k0 ,dfree

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10

¯ is the equivalent channel matrix over dfree frequency slots. where H i h ¯ = H(f1 ) . . . H(fd ) H free

(12)

and odfree n . CC† = diag C(fi )C(fi )†

(13)

i=1

¯ can be expressed Lemma 2 (Equivalent channel matrix): The equivalent channel matrix H as, ¯ = Hw V ⊗ Int , H

(14)

where Hw is the nr × Lnt i.i.d CN (0, 1) such that, h i ˜0 ... H ˜ L−1 , Hw = H 2π

w = e−j N , and,

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Fo 

   V=  

σ0

...

σ0



σ1 wf1 .. .

...

σ1 wfD

   .  

ee

(15)

σL−1 w(L−1)f1 . . . σL−1 w(L−1)fD

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Proof: The proof of this Lemma is detailed in Appendix B. Let,

† Θ = VL×dfree ⊗ Int CC† VL×dfree ⊗ Int ,

ev

(16)

be the effective matrix codeword. In the following, D denotes the number of different

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Page 10 of 23

subcarriers on which erroneous bits are received, such that D ≤ dfree . The interleaver design allows to maximize the parameter D. The interleaver design is not addressed in the scope of this paper. The interested reader can refer to [10] for suboptimal interleaver design. Equation (11) can be simplified using the eigen-value decomposition of Θ = UΛU† , to n o X ˜ ω U)Λ(H ˜ ω U)† kH(k)C(k)k2F = Tr (H k0 ,dfree

=

nr X r X

λj (Θ)βi,j

i=1 j=1

where r is the rank of Θ, βi,j are i.i.d random variable resulting from the multiplication of H by the unitary matrix U and λj (Θ) are the non zeros eigen-values of Θ. The rank and the eigen-values of Θ can be deduced from the following Lemma 3. October 5, 2010

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Page 11 of 23

11

Lemma 3: Let A be the M × M Hermitian matrix given by, A = B(CC† )B† , where B is M × N matrix having D different columns such that rank r ≤ min(M, D) and C is full rank M × M matrix. Then, the matrix A has the following property. a) The rank of A is equal to r, the rank of B. b) Let λk (A) and λk (B) be respectively the decreasing ordered eigen-values of A and B. The non zero eigen-values of A can be lower bounded by, λk (A) ≥ λr (BB† )λk (CC† ),

if M ≥ N,

λk (A) ≥ λr (BB† )λN −M +k (CC† ),

Fo

if N ≥ M.

Proof: The proof of Lemma 3 is given in appendix C.

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By applying Lemma 3 to Θ, it follows that rank{Θ} = rank{V ⊗ Int },

ee

= rank{V}nt .

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The rank of V can be deduced by considering only the D different subcarriers in V. These columns form a linear independent family of rank min(L, D) as the equivalent L × D matrix can be assimilated to a submatrix of a Vamdermonde matrix, and therefore the rank is equal

ev

min(L, D). This implies that,

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r = rank{Θ} = nt min(L, D). Using Lemma 3, the eigen-values are bounded such that, λj (Θ) ≥ αλm (CC† ), where m=

  j

if L ≥ dfree

 dfree − L + j

if L ≤ dfree ,

(17)

and α = λr (VV† ⊗ Int ). By averaging over all the Gaussian variable βi,j as described in [2], the PEP over a frequency selective channel is bounded by  −nr −nr nt min(L,D) nt min(L,D) Y 1   λj (Θ) . P(c → ˆc) ≤ 4N0 j=1 October 5, 2010

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12

The product of the non zero-eigen-values can be lowerbounded such that, nt min(L,D)

Y

nt min(L,D) 2nt min(L,D)

λj (Θ) ≥ (dmin )

j=1

Y

αµ ¯m (CC† ),

j=1

where dmin is the minimal Euclidian distance between two symbols (dmin =

√2 ), Es

µ ¯m (Θ)

are the normalized eigen-values and m = f (j) as defined in (17). From the normalization power constraint with a total energy equals to 1 per antenna at each time slot, we know that µ ¯m (C(k)C† (k)) ≤ 1,

∀k = 1 . . . N, j = 1 . . . nt min(L, D),

where C(k) denotes the matrix codeword over a subcarrier k. As CC† is a block diagonal matrix containing C(k)C(k)† , k = f1 . . . fdfree , then the eigen-values of CC are trivially

Fo

equals to the eigen-value of C(k)C(k)† , for all k = f1 . . . fdfree , and then, µ ¯j (CC† ) ≤ 1,

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∀j = 1 . . . dfree nt .

The product of eigen-values can be therefore bounded as, nt min(L,D) nY nt min(L,D) t dfree Y 4α λj (Θ) ≥ µ ¯j (CC† ), E s j=1 j=1 nt min(L,D) 4α δ dfree . ≥ Es

rR

ee

Finally, the PEP can be bounded such that,

ev

P(c → ˆc) ≤ G SNR−nr nt min(L,D) , where, G= αδ

−nr dfree

(18)

iew

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Page 12 of 23

(Es )nt nr min(L,D) .

It can be noticed here that the full diversity order for the frequency selective fading channel can be extracted here without requiring to code across all the frequency subcarriers as shown in [11] or in [12] for the case when no outer code is used. As we show here, by coding the symbols in a MIMO BICM system using a perfect code over each subcarrier, the maximal diversity order can be achieved.

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Page 13 of 23

13

B. Perfect space time code versus SDM Similar to the case with perfect space time coding, the PEP expression can be derived for the SDM case. When no space time coding is used, the minimal rank of the effective codeword Θ in (16) arises when all the erroneous codewords at the dfree subcarriers are received on the same antenna. This implies that, C in (13) is proportional to ID ⊗C(k), where C(k) is a non zero SDM codeword received on an arbitrary subcarrier k, with rank equals to 1 for the SDM case. Therefore, the rank of the effective codeword matrix is equal, to rank{Θ} = rank{(V ⊗ Int )(ID ⊗C(k))},

Fo

= rank{V ⊗ C(k)} = min(L, D).

The eigen-values of Θ can be computed in a similar way as in Lemma 3. We emphasize

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here that the rank of C in Lemma 3 does not impact the eigen-values bound, but impacts only the rank of A. The proof can be easily repeated for the case where C is not full rank. The PEP for the SDM is therefore bounded such that, P(c → ˆc) ≤ α

ee

−nr dfree

(Es )nr min(L,D) SNR−nr min(L,D) ,

(19)

rR

where α depends on the covariance matrix. C. Numerical results

ev

In Figure 5, the asymptotical behavior of SDM versus coded GC is depicted for the 2 × 2 MIMO configuration using QPSK constellation, and a convolutional code [5 7] − 1/2, with

iew

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dfree = 5, and a over a multi-tap channel with L = 18. It can be observed that at low SNR, the coding gain of SDM is largely enhanced compared to the Golden code coding gain. At low SNR, this coding gain dominates the high diversity gain that can be achieved by the GC. Both schemes gain in diversity compared to the non coded case. At high SNR, the diversity gain became dominant. The diversity gain can be observed at a very low PER rate (range of 10−7 ). The performance of the Golden code versus SDM has been evaluated in the IEEE 802.11n context in terms of packet error rate (PER) versus SNR, for a packet length of 1000-bits. In the following, SNR gain will be related to a PER of 10−2 . The packet error rates in Figure 6 are evaluated over channel D using QPSK and 16QAM constellation. The channel D is characterized by a 50ns rms delay spread and 18 taps, and then by significant frequency October 5, 2010

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14

diversity. In the IEEE 802.11n context, the convolutional code [133 171] with a coding rate of Rc = 1/2 is used with dfree = 10. No additional gain is observed at a PER = 10−2 . The channel B in the IEEE 802.11n standard can be assimilated to a flat fading channel, for which the additional gain using a convolutional code with high free distance (dfree = 10) cannot be observed at reasonable PER. D. Practical limits of space time codes use in a standard context Recent standards that use MIMO system such that IEEE 802.11n and IEEE 802.16e aim to increase the throughput and the reliability of the system. However, increasing the reliability comes often at the expense of increased complexity at both transmitter and receiver side.

Fo

Scarifying the complexity order can be done if promising gains at reasonable PER range are observed. Although, the theoretical optimality of these codes is studied in the high SNR

rP

regime, practical assumptions are more realistic, and address generally a moderate SNR regime and moderate range of PER.

For the flat fading case, when no outer code is used, the huge gain observed by the Golden

ee

code over all other known code make it promising to be used in such systems. However, when a complete chain as BICM-MIMO-OFDM system is considered, the situation become

rR

considerably different. As we show in a BICM-MIMO-OFDM system, the diversity of BICMOFDM system can be extracted when no space time code is used. Additional diversity can be

ev

provided by using perfect space time codes over each subcarriers. This additional diversity comes at the expense of an increased lattice decoder, i.e instead of using a 2nt × 2nt ML

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Page 14 of 23

soft decoder a 2n2t × 2n2t is required. Moreover, the impact of this additional diversity cannot be unfortunately observed at moderate range of PER. V. C ONCLUSION

In this paper, we studied the performance of non-vanishing determinant code constructed from cyclic division algebra in a standard context. The PEP is derived for BICM-MIMO for the cases of flat fading and frequency selective channels when respectively perfect codes and spatial division multiplexing schemes are used at each subcarrier. When the channel is flat, we show that the diversity order remains the same as the non-coded case. However, the coding gain is improved compared to the non coded case. We noticed that the coding gain of the SDM is largely enhanced compared to the coding gain of perfect code especially for a convolutional with a large free distance. For the frequency selective channel case, we show October 5, 2010

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Page 15 of 23

15

that using perfect codes at each subcarrier allow to extract the transmit diversity. We show then that the diversity order of BICM-OFDM system can be also extracted when no space time code is used. In a practical context, such as in IEEE 802.11n, the numerical results we provide show that this gain in diversity cannot be observed at a moderate range of PER. VI. ACKNOWLEDGMENTS The authors would like to thank Olivier Rioul for his valuable comments on the PEP derivation and to thank Mohamed Kamoun and Ghaya Rekaya-Ben Othman for helpful discussions in the early stages of this work.

Fo

A PPENDIX A P ROOF OF L EMMA 1

rP

Assuming that the code sequence c is transmitted and ˆc is detected, the PEP given the channel knowledge can be written as

nX

P(c → ˆc|H) = Prob

k0

X k

0

C∈Xcˆ

k

min kY(k) −

H(k)C(k)k2F

o

rR

≤

mini kY(k) − H(k)C(k)k2F

ee C∈Xci

.

(20)

k

Let dfree be the minimum Hamming distance of the convolutional code. Error occurs when the distance between the incorrect path associated to cˆ and the correct one associated to c

ev

0

is equal to dfree . In this case, Xcik and Xciˆk in (20) are equal for all k except for the dfree 0

distinct values of k . Therefore, only dfree terms are different in the inequality (20). ˜ ˆ Let us denote in the following C(k) and C(k) as

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˜ C(k) = arg mini kY(k) − H(k)C(k)k2F , C∈Xc

k

ˆ C(k) = arg min kY(k) − H(k)C(k)k2F , C∈Xci

k

where Xcik is the complementary set of Xcik . The PEP can be written as P(c → ˆc|H) = Prob

n X

2 ˆ kY(k) − H(k)C(k)k F

k0 ,dfree

≤

X

o 2 ˜ kY(k) − H(k)C(k)k F ,

k0 ,df ree

October 5, 2010

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16

where

P

k0 ,dfree

˜ (k) means that we only consider the dfree terms of the inequality for which x

ˆ (k) are different. Consequently, and x P(c → ˆc|H) = Prob

n X

o ωk ≤ 0 ,

k0 ,df ree

where ω =

X

ωk ,

0

k ,df ree

ωk

2

ˆ ˜ = H(k) C(k) − C(k) +Z(k) − kZ(k)k2F , F | {z } D(k)

Fo

where D(k) is also a space time codeword as the considered space time block code is linear. Due to the bit interleaver, bits are uncorrelated, and then ω is the sum of dfree independent

2

2 Gaussian variable ωk with mean He (k)D(k) and variance 4N0 He (k)D(k) . Conse-

rP

F

F

quently, ω is a Gaussian variable, such that X X kH(k)D(k)k2F N kH(k)D(k)k2F , 4N0 k0 ,dfree

ee

k0 ,dfree

For a Gaussian variable ω ∼ CN (µ, σ 2 ), it is well known that µ µ2 Prob{ω < 0} = erf ≤ exp − 2 , σ 2σ

rR

where erf(.) is the error function. It follows that kH(k)D(k)k2 Y F P (c → ˆc|H) ≤ . exp − 8N0 /2 0

iew

k ,df ree

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Page 16 of 23

By averaging over all the channel, the PEP over a general channel model can be written such that

h oi 1 n X P(c → ˆc) ≤ EH exp − kH(k)D(k)k2F . 4N0 k0 ,d free

From the definition of D(ki ), we can note that all matrices D(ki ) (i = 1 . . . dfree ) are non zero matrix for all c 6= cˆ. This remark follows from the definition of D(ki ). Note that in this ˆ i ) and D(k ˜ i ) belongs to two case this matrix contains at least one non zero symbol, as D(k complementary set.

October 5, 2010

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Page 17 of 23

17

A PPENDIX B P ROOF OF L EMMA 2 In a MIMO OFDM system, each hi,j (k) term ((i, j) = {1 . . . nt } × {j = 1 . . . nr }) can ˜ i,j , where σl is the be written as the fast fourier transform of the channel taps hi,j [l] = σl h l

variance of the channel tap and

˜ i,j h l

hi,j (k) =

∼ CN (0, 1) such that, L−1 X

˜ i,j , wk,l σl h l

k = 1 . . . N.

l=0 2π

where wk,l = wkl and w = e−j N . The corresponding channel matrix over a subcarrier k, is therefore

Fo

H(k) =

L−1 X

˜ l. wk,l σl H

(21)

l=0

Using (21), it can be easily checked that the equivalent channel matrix in (12) can be written i i h h ˜0 ... H ˜ L−1 V ⊗ Int H(f1 ) . . . H(fdfree ) = H A PPENDIX C

ee

rP

P ROOF OF L EMMA 3

˜ the eigen-values ordered in To simplify the notations, we denote in the following by λ(.)

rR

increasing order, and λ(.) the eigen-values ordered in a decreasing order. As A is an Hermitian matrix, its rank is equal to the rank of BC. It can be easily checked

ev

from the product matrix rank property in (22) that for D ∈ Ca×b , E ∈ Cb×c , rank{D} + rank{E} − b ≤ rank{DE} ≤ min rank{D}, rank{E} , and the fact that C is a full rank matrix, that,

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(22)

rank{B} + p − p ≤ rank{A} ≤ rank{B}, which implies that, rank{A} = rank{B}. A. Case 1 : M = N Using the fact that for a square matrix M ∈ Ca×a , λ(MM† ) = λ(M† M), implies that λk (A) = λk (C† B† BC).

October 5, 2010

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18

˜ 0p−s ]. Then, Let B† B = UΛU† be the eigenvalue decomposition of B† B, with Λ = [Λ λk (A) = λk (C† UΛU† C), = λk (Λ1/2 U† CC† UΛ1/2 ). ˜ be the s × s principal submatrix of Ω. Then, Let Ω = U† (CC† )U and Ω λk (A) = λk (Λ1/2 ΩΛ1/2 ),

(23a)

˜ 1/2 Ω ˜Λ ˜ 1/2 ), = λk (Λ

(23b)

˜ 1/2 in (23b) is non singular matrix and Ω ˜ is Hermitian, The Ostrowski theorem in [13] As Λ can be applied,

Fo

˜ k (Ω), ˜ λk (A) ≥ λ1 (Λ)λ

(23c)

≥ λ1 (BB† )λk (Ω),

(23d)

= λ1 (BB† )λk (CC† ).

(23e)

rP

˜ is a s × s submatrix of the Hermitian matrix Ω, (23d) follows from the application As Ω

ee

of theorem 4.3.15 in [13]. Finally, (23e) follows from the fact that U is unitary matrix, and therefore λk (Ω) = λk (CC† ). B. Case 2: M ≥ N

ev

rR

Let AN be the N × N principal submatrix of the M × M matrix A, such that AN = BN CC† B†N

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Page 18 of 23

where BN = B([1 : N ], [1 : N ]) is N × N minor of matrix B with rank r ≤ N . Assuming that, the increasing eigen-values of AN and A are such that,

˜ 1 (AN ) = . . . = λ ˜ N −r (AN ) = 0, λ ˜ N −r+1 (AN ) . . . ≤ λ ˜ N (AN ) 0≤λ and ˜ 1 (A) = . . . = λ ˜ M −N (A) = . . . = λ ˜ M −r (A) = 0 λ ˜ M −r+1 (A) ≤ . . . ≤ λ ˜ M (A) 0≤λ Then, using theorem theorem 4.3.15 in [13], ˜ k (AN ) ≤ λ ˜ k+M −N (A), λ

k = 1...N

October 5, 2010

(24) DRAFT


Page 19 of 23

19

The non-zero eigen-values of A can be bounded by taking k = N − r + 1 . . . N in the inequality (24), or equivalently ˜ M −r+i (A) ≥ λ ˜ N −r+i (AN ), λ

i = 1 . . . r,

and in term of the decreasing order eigen-values, λk (A) ≥ λk (AN ),

k = 1 . . . r.

As AN is a square matrix, the eigen-values of AN can be lower bounded using Lemma 3, and therefore, λk (A) ≥ λk (AN ) ≥ λr (BN B†N )λk (CC† ),

Fo

k = 1 . . . r.

The matrix BN B†N is a N × N submatrix of the Hermitian matrix BB† , and therefore,

and therefore,

λr (BN B†N ) ≥ λr (BB† ),

rP

λk (A) ≥ λr (BB† )λk (CC† ),

rR

C. Case 3 : M ≤ N

k = 1 . . . r.

ee

In this case, A is a M × M principal submatrix of the N × N matrix AN with rank r, such that

ev

AN = BN CC† B†N

and BN is a N × N matrix constructed such that B is its M × N minor and BN has rank

iew

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N − M + r. Assuming that, the increasing eigen-values of AN and A are such that, ˜ 1 (AN ) = . . . = λ ˜ M −r (AN ) = 0 λ

˜ M −r+1 (AN ) . . . ≤ λ ˜ N (AN ), 0≤λ and ˜ 1 (A) = . . . = λ ˜ M −r (A) = 0, λ ˜ M −r+1 (A) . . . ≤ λ ˜ M (A). 0≤λ Then, using Theorem 4.3.15 in [13], ˜ k (A) ≥ λ ˜ k (AN ), λ

k = 1 . . . r,

October 5, 2010

(25)

DRAFT



20

and in term of the decreasing order eigen-values, λk (A) ≥ λN −M +k (AN ),

k = 1 . . . r.

As AN is a square matrix, the eigen-values of AN can be lower bounded using Lemma 3, and therefore, λk (A) ≥ λN −M +k (AN )

Fo

≥ λr (BN B†N )λN −M +k (CC† ),

k = 1 . . . r.

The matrix BN B†N is a N × N submatrix of the Hermitian matrix BB† , and therefore,

rP

λr (BN B†N ) ≥ λr (BB† ),

and therefore,

ee

λk (A) ≥ λr (BB† )λN −M +k (CC† ),

10

-4

10

-6

10

-8

10

-10

CC-SDM gain Intersection point CC-GC gain

10

15

20

iew

10

Non coded GC Non coded SDM Coded GC - dfree = 10 Coded SDM - dfree = 10 Coded GC - dfree = 5 Coded GC - dfree = 5

ev

PEP

Intersection point -2

k = 1 . . . r.

Asymptotical PEP behavior

0

10

rR

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 23

25

30

SNR

Fig. 2.

Asymptotical behavior of the PEP over a flat fading channel

October 5, 2010

DRAFT


Page 21 of 23

21

QPSK - CC = [5 7] , 1/2 (dfree = 5)

0

10

2x2 MIMO - SDM 2X2 MIMO - GC

-1

PER

10

-2

10

Fo

-3

10

2

4

6

8

10

12

14

16

SNR (dB)

rP

Fig. 3. Simulation results for a packet of 1 ko: SDM vs the Golden code in a 2 × 2 BICM system for a QPSK modulation

ee

with [5 7] encoder (dfree = 5)

rR

QPSK - CC = [133 171] , 1/2 (dfree = 10)

0

10

-1

10

iew

PER

2x2 MIMO - SDM 2x2 MIMO - GC

ev

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-2

10

-3

10

-4

10

2

4

6

8

10

12

14

16

SNR (dB)

Fig. 4. Simulation results for a packet of 1 ko: SDM vs the Golden code in a 2 × 2 BICM system for a QPSK modulation with [133 171] encoder (dfree = 10)

October 5, 2010

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22

Asymptotical PEP behavior

0

10

-5

10

-10

10

-15

10

-20

10

-25

PEP

10

Intersection point

Non coded SDM Non coded GC coded OFDM-GC coded OFDM-SDM

Fo 6

10

12

14

16

18

20

SNR

Asymptotical behavior of the PEP over a frequency selective channel

rR

ee

Fig. 5.

8

rP

Error Probability of GC vs SDM in IEEE 802.11n 0

10

ev

10-1

PER

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Page 22 of 23

10-2

10-3

10-4

Fig. 6.

Channel D - QPSK 1/2 - SDM Channel D - QPSK 1/2 - GC Channel D - 16QAM 3/4 - SDM Channel D - 16 QAM 3/4 - GC Channel B - 16QAM 3/4 - SDM Channel B - 16QAM 3/4 - GC 2

4

6

8

10

12 14 Eb/N0(dB)

16

18

20

22

24

Golden Code vs SDM in IEEE 802.11n context for a QPSK - 1/2 Modulation with spectral efficiency = 2bpcu -

Channel model D

October 5, 2010

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Page 23 of 23

23

R EFERENCES [1] F. Oggier, G. Rekaya, J.-C. Belfiore, and E. Viterbo, “Perfect space time block codes,” IEEE transactions on information theory, vol. 52, no. 9, Septembre 2006. [2] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space time codes for high data rate wireless communication : performance criterion and code construction,” IEEE transactions on information theory, vol. 44, no. 2, pp. 744–765, March 1998. [3] L. Zheng and D. Tse, “Diversity and multiplexing : A fundamental tradeoff in multiple antenna channels,” IEEE Transactions on Information Theory, vol. 49, no. 5, pp. 1073 – 1096, May 2003. [4] J. C. Belfiore, G. Rekaya, and E. Viterbo, “The Golden Code: A 2 × 2 full rate space time code with non vanishing determinants,” IEEE Transactions on information theory, vol. 51, no. 2, pp. 1432 – 1436, April 2005. [5] G. Caire, G. Taricco, and E. Biglieri, “Bit-Interleaved Coded Modulation,” IEEE transactions on information theory, vol. 44, no. 3, May 1998.

Fo

[6] J. Boutros, N. Gresset, L. Brunel, and M. Fossorier, “Soft-input soft-output lattice sphere decoder for linear channels,” IEEE Globecom, Dec 2003.

[7] E. Akay and E. Ayanoglu, “Achieving Full Frequency and Space Diversity in Wireless Systems via BICM, OFDM,

rP

STBC, and Viterbi Decoding,” IEEE transactions on communications, vol. 54, no. 12, Dec 2006. [8] L. Luzzi, G. Rekaya, J. C. Belfiore, and E. Viterbo, “The Golden Code: A 2 × 2 full rate space time code with non vanishing determinants,” submitted to IEEE Transactions on information theory, 2008, available on arxiv http://arxiv.org/abs/0805.4502.

ee

[9] Y. Hong, E. Viterbo, and J. C. Belfiore, “Golden space-time trellis coded modulation,” IEEE Transactions on Information Theory, 2006.

rR

[10] N. Gresset, L. Brunel, and J. Boutros, “Space-time coding techniques with bit interleaved coded modulation over block-fading MIMO channels,” IEEE Transactions on Information Theory, vol. 54, no. 5, pp. 2156–2178, May 2008. [11] L. Mroueh and J.-C. Belfiore, “How to achieve the optimal dmt of selective fading channels?” To appear in Information Theory Workshop, Dublin, 2010.

ev

[12] P. Coronel and H. Bölcskei, “Diversity multiplexing tradeoff in selective fading MIMO channels,” in IEEE International Symposium on Information Theory, Nice, France, June 2007, pp. 2841 – 2845. [13] R. Horn and C. Johnson, Matrix Analysis.

Cambridge University Press, 1999.

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