High-Rate Full-Diversity STBC Design with Option to Provide Unequal Error Protection

Sushanta Das, Monisha GhoshPhilips Research North America

Department of Wireless Communications and Networking (WiCAN)Briarcliff Manor, NY, USA.

sushanta.das, monisha.ghosh@philips.com

Abstract— A new rate- 74

full-diversity Space Time BlockCode for 2 TX has been designed by enlarging the signallingset of quaternions[14]. Power-scaling and constellation-rotation of information symbols have been incorporated inorder to extend the signalling set. The proposed codingscheme achieves a 75% rate enhancement in comparisonto the traditional Alamouti [3] code for QPSK modulationwithout requiring any additional system resources, e.g. poweror bandwidth. The use of power-scaling and constellation-rotation also guarantee full-rank of codeword differencematrices in the extended set, while maximizing the codinggain and minimizing the transmitted signal peak-to-averagepower ratio. We propose a reduced-complexity efficient de-coding algorithm, which does not result in any loss of usefulinformation in comparison to the full-length ML decoding.The proposed high-rate code can also be utilized to provideunequal error protection to two different levels of sourcebits and hence, is useful in applications where the notion ofunequal error protection supports varying degree of fidelityto source data, for example multimedia communications.Extensions to general M-PSK constellations and code designfor more than 2 transmit antennas are straightforward.

Keywords: Space-Time Block Code (STBC), Quaternion, Un-equal Error Protection (UEP), Rate Compatible Punctured Con-volutional Code (RCPC).

I. INTRODUCTION

In recent years, there has been significant researchinterests in multiple-input multiple-output (MIMO) com-munications in which antenna arrays are used at the trans-mitter and the receiver to improve capacity, reliability andcoverage [1], [2]. However, there is an inherent tradeoffbetween reliability (diversity) and rate, which implies thathigh multiplexing gain comes at the cost of decreaseddiversity gain and is a manifestation of a correspondingtradeoff between error probability and rate [12]. Overprovisioning of diversity to one layer of data would resultin loss of rate to that layer and as well as other layers.This is more apparent in applications that need to supportvariable rate and reliability requirements, for example,multimedia communications (triple play: audio, video, anddata) where real time (voice) data may require more errorprotection than non-real time data. In recognition of thefact that in a stream of encoded source bits, some groupsof bits encounter more detrimental effects due to bit errorsthan that of other groups, Unequal Error Protection (UEP)is often applied as a mean of Forward Error Correction

(FEC). In this approach, the bits are grouped accordingto some criteria capable of determining their importanceand required fidelity to the source reconstruction, anddifferent levels of channel codes from a given familyare assigned to the different groups of source bits. Forexample, [8]-[9] proposes to partition a digital video streaminto multiple substreams and apply different levels of errorprotection to each substream. A different approach basedon non-uniform constellations and convolutional codesis discussed in [4]. Various joint source-channel codingtechniques are discussed in [10]-[11].

Our objective in this paper is to design a new classof rate- 7

4, full-diversity STBC for 2 transmit antennas.

The proposed high-rate code can be concatenated withRCPC codes to extract possible time domain diversityand to provide unequal error protection to two differ-ent levels of source bits. The intrinsic high rate of theproposed code compensates for the rate-loss, which oth-erwise is a natural consequence of FEC. The code hasbeen designed by exploiting the algebraic structure ofquaternions, which is also the building block of the well-known space-time code design for 2 transmit antennascommonly referred to as the Alamouti code [3]. Anexpansion of the signalling set of the Alamouti code isachieved by introducing power-scaling and constellationrotations of information symbols. The extended signalingset, in which each candidate codeword still conforms tothe quaternionic structure, accommodates three additionalsource bits to be transmitted and reconstructed at thereceiver without requiring any additional system resources,e.g. power or bandwidth. The inherent algebraic propertiesof quaternions have been further exploited to design anefficient decoder that significantly reduces the decodingcomplexity of the proposed code. Our proposed reduced-complexity decoding strategy does not suffer from any lossof useful information in comparison to the optimum full-length maximum-likelihood (ML) decoding.

The rest of the paper is organized as follows. In SectionII, we describe the signal set, code design and transmissionmodel. In Section III, we introduce the optimum values ofthe power scaling factor and the constellation rotation an-gle. Section IV illustrates the reduced-complexity coherentdecoding algorithm utilizing the quaternionic structure ofthe code. Simulation results and concluding remarks are

978-1-4244-1645-5/08/$25.00 ©2008 IEEE 1448

presented in Section V and Section VI, respectively.

II. ENCODING SCHEME OF HIGH-RATE

FULL-DIVERSITY STBC

The Alamouti code designed for two antennas at thetransmitter (Nt = 2) is the 2 × 2 matrix:

Q(x1, x2) →[

x1 x2

−x2 x1

](1)

where (.) denotes the complex-conjugate transpose. Thecolumns of Q represent different time slots, the rows repre-sent different antennas, and the entries are the two symbolsto be transmitted. The Alamouti code achieves full-rateand full transmit diversity. The decoding of the Alamouticode is remarkably simple with a single matched filteringoperation. In our proposed design, the signalling set of theAlamouti code has been extended by introducing constel-lation rotation and power scaling of information symbols.Therefore, each candidate codeword in the extended setstill conforms to quaternionic structure. In this section,we illustrate the design and transmission mechanism ofour proposed code by using QPSK modulated symbols,however the code applies to general PSK modulation.

Figure 1 represents the transmission strategy of theproposed high-rate full-diversity STBC for QPSK mod-ulated symbols. A stream of seven information bits aretransmitted over four bit durations and hence, the termrate- 7

4 code. At the source, the stream of seven bits isdivided into two groups of size four and three, respectively.Henceforth, the four information bits in the first group willbe termed as category 1 bits, whereas the three informationbits in the second group will be termed as category 2 bits.The justification of categorizing the source bits into twostreams would be apparent when we perform numericalanalyses in Section V. The category 1 bits are mapped ontotwo QPSK symbols and are arranged in a 2 × 2 matrixfollowing the quaternion structure in (1). The remainingthree category 2 bits determine the codeword structureas follows: one of the bits determines the radius of theconstellation circle, i.e., whether the modulated symbols inQ are selected from QPSK constellation with unit radiusor with radius scaled by a factor of 1

k . The other twoof the remaining category 2 bits decide the rotation ofmodulated symbols according to one of the following fourchoices: i) no symbol rotation, ii/iii) rotation of symbolsin positive/negative slope diagonal in Q, and iv) rotationof all symbols in Q. The selected codeword is transmittedfrom two transmit antennas over two consecutive symbolperiods. To illustrate with an example, consider a streamof seven binary information bits (b0, b1, b2, b3, b4, b5, b6).The four information bits of category 1, for example(b3, b4, b5, b6), are mapped to two QPSK symbols x1 andx2. The remaining three bits of category 2 determine thecodeword structure as follows:

(b1, b2) = {0, 0} ⇒ Q1 =

[x1 x2

−x2 x1

],

(b1, b2) = {0, 1} ⇒ Q2 =

[x1e

jθ x2

−x2 x1e−jθ

],

TX2

TX1

ConstellationMapper

Q1

Q2

Q3

Q4

b0 b1,b2 b5,b6b3,b4

r=1/k

X1

X2

b0=0

b0=1

r=1

[b1,b2]=[1 0]

[b1,b2]=[1 1]

[b1,b2]=[0 1]

[b1,b2]=[0 0]

Fig. 1. The Block Diagram of Rate- 74

STBC for QPSK Modulation

(b1, b2) = {1, 0} ⇒ Q3 =

[x1 x2e

jθ

−x2e−jθ x1

],

(b1, b2) = {1, 1} ⇒ Q4 =

[x1e

jθ x2ejθ

−x2e−jθ x1e

−jθ

].

Consider that the first subset S1 = {Q1 ∪Q2 ∪Q3 ∪Q4}corresponds to (b0 = 0) and the second subset S2 =

{Q5∪Q6∪Q7∪Q8} = {Q1k

∪ Q2k

∪ Q3k

∪ Q4k} corresponds to

(b0 = 1). The extended signalling set of the proposed codeS = {S1 ∪S2}. It is easy to show that for selected valuesof k and θ, the codeword difference matrix B = (Si−Sj)for any Si,Sj ∈ S, i �= j, is always full rank. Hence, thecategory 1 bits, which are modulated as space-time codedsymbols x1 and x2, and also the category 2 bits whichdetermine the structure of the space-time code, achievefull transmit diversity of two. The above encoding scheme,therefore, enhances the rate of the traditional Alamouticode from rate-1 to rate- 7

4 for QPSK modulation.

III. OPTIMUM SCALING FACTOR AND CONSTELLATION

ROTATION ANGLE

The introduction of scaling factor k and the constellationrotation angle θ ensure that the codeword difference matrixbetween any two distinct codewords in the set S is alwaysfull rank. Therefore, the values of k and θ are so chosenthat the codeword difference matrix B for any choices ofSi,Sj ∈ S, i �= j has rank two. For example, k = 1 andθ = nπ

2 for integer n, are not permissible values of k and θ,since these values result in loss of rank of B. However, dueto the plurality of the set of permissible values, we need todiscern other criteria to select unique (if exists) optimumk and θ. Here, we have identified two important criteria: i)Maximizing the coding gain (CG) of the codewords in S,and ii) Minimizing the peak-to-average power ratio (PAPR)resulting from the power scaling of information symbolsin S, to identify the optimum values of k and θ.

We consider QPSK modulated symbols with unit energyand with energy 1

k2 in case of power-scaling. Therefore,

1449

the peak-to-average power ratio (PAPR) equals 2k2

1+k2 . Inaddition, we consider k > 1 so that the average transmittedpower is even reduced compared to that of no scaling. Thecoding gain (CG) is defined as the minimum product of thenonzero singular values of codeword difference matrix B

over all distinct pairs of codewords. The selection criterionfor k and θ follows the optimization function:

arg maxCG

PAPR= arg max

k>1θ �= nπ

2

minSi,Sj∈S,i�=j det(BB)2k2

1+k2

Numerical analysis yields the optimum value of k =1.6. The optimum rotation angle for QPSK constellationis π

4 radians [13] for which the minimum distance betweenany two distinct codewords is maximized. Figure 2 showsthe optimum value of the power-scaling factor k when theconstellation rotation angle θ = π

4 radians.

IV. EFFICIENT REDUCED-COMPLEXITY DECODING

The orthogonal structure of quaternions results in re-markably simple decoding of the Alamouti [3] code usinga single matched filtering operation. With the introductionof power-scaling and constellation rotation of informationsymbols in our proposed coding scheme, a single matchedfiltering operation will not be able to regenerate the sym-bols at the receiver. On the contrary, for QPSK modulationthe optimum full-length ML search size is 256 over theextended constellation set,1 and is not an attractive optiondue to the added complexity in comparison to the simpledecoding of the Alamouti code.

In this context, we introduce an efficient alternativeto the full-length ML decoding, which significantly re-duces the decoding complexity of the proposed code.Our proposed reduced-complexity decoding algorithm doesnot result in any loss of useful information or in anydegradation of the error-rate performance in comparisonto full-length ML decoding. The proposed scheme re-quires a simple matched filtering operation followed bya maximum-likelihood search of size eight. We illustratethe decoding algorithm in the following:

We consider the quasi static flat fading channel wherespace-time coded information is transmitted over Nt =2 antennas and received using Nr = 1 antenna. It isassumed that the receiver has perfect channel knowledge.The channel is constant over a coherence interval of T = 2symbols and changes independently from one coherenceinterval to the next. After demodulation and sampling, thereceived signal can be written as

R = HS + Z, (2)

where R ∈ CNr×T is the received sequence, H ∈ CNr×Nt

is the quasi static channel fading matrix, S ∈ CNt×T is the

1Lets denote the QPSK alphabet set by a and the cardinality of this set‖a‖ = 4. The extended alphabet set A comprises of {a, ak, aθ, akθ},where ak, aθ and akθ denotes the scaled, rotated and scaled-rotatedversions of a. The cardinality of the extended set ‖A‖ = 16.

space-time code matrix with transmit power constraint P ,and Z ∈ CNr×T is assumed to be additive white Gaussiannoise with variance N0. The above expression can berepresented in an orthogonal 2 × 2 matrix form:

[r1 r2

−r2 r1

]︸ ︷︷ ︸

R

=

[h1 h2

−h2 h1

]︸ ︷︷ ︸

H

[x1 x2

−x2 x1

]︸ ︷︷ ︸

S

+

[z1 z2

−z2 z1

]︸ ︷︷ ︸

Z

,

The transmitted codeword S ∈ Qi, ∀i = 1 . . . 8. Theorthogonal property of the channel matrix H yields: HH =(|h1|2 + |h2|2)I2×2,, where I2×2 is 2× 2 Identity matrix. Asimple matched filtering operation is applied as follows:

HR = (|h1|2 + |h2|2)S + HZ

Observe that post-processing noise samples still remainindependent of each other, have zero mean and variance(|h1|2 + |h2|2)N0. From the output of the matched filter,eight candidate codewords Si, ∀i = 1 . . . 8, are generatedwhere each Si relates to the corresponding Qi. Thesecandidate codewords are then compared using the met-ric: arg min ‖[r1 r2] − [h1 h2] Si‖2. The decoding of(b0, b1, b2) follows directly once the decision on Si ismade. The proposed efficient decoding, therefore, reducesthe decoding-complexity from a formidable 256 length MLsearch to only a simple matched filtering operation plus alength 8 ML search.

V. NUMERICAL ANALYSIS

The simulation results presented in this section areintended to portray the main theme of our work that thenew coding scheme provides higher rate than the traditionalAlamouti code, and also provides an additional degree offlexibility in the selection of FEC rate according to thesystem requirements. For simulation purposes, we considera frame to be in error if any information bit in the frame isdecoded incorrectly, and a frame in error is not consideredfor retransmission. The channel is assumed to remainstatic over the duration of one codeword and changesindependently from one codeword to another over theframe. All symbols are normalized such that the averagepower of the Alamouti code matches with that of theproposed coding scheme.

Figure 3 shows that at high SNR (FER ≈ 0), our pro-posed code achieves a higher throughput level of 3.5 bitsper channel use (PCU), whereas the achievable throughputfor the Alamouti code is only 2 bits PCU assuming QPSKmodulation for both schemes. We observe a cross-overpoint at an input SNR of 17 dB. Since it is reasonableto assume that we know the operating input SNR level atthe transmitter, we can switch between the Alamouti codeand our proposed code to maximize throughput at all SNRlevels. Figure 3 also shows that the achievable throughputof the Alamouti code with 8PSK modulation is 3 bits PCUand the cross-over point is at 19 dB.

1450

In addition to the inherent full-diversity provided to allsymbols, the proposed high-rate code can also achievea well-balanced trade-off between rate and reliability byjudiciously selecting different FEC code rate for differentlayers of source bits. To illustrate, consider two infor-mation bit streams of length K1 and K2 representingcategory 1 and Category 2 bits, respectively. Puncturedconvolutional coding and bit interleaving are independentlyapplied on each stream to generate coded bit streams oflength N1 and N2, respectively. The bit error rate andeffective throughput2 of the proposed code have beencompared with those of the Alamouti code. The proposedrate- 7

4 code translates to a spectral efficiency of 3.5 bitsPCU and any modulation scheme alone cannot match thisspectral efficiency for the Alamouti code. Therefore, wehave chosen equal number of information bits supportedby different rates of convolutional code while comparingthese two schemes. The number of information bits andrespective convolutional code rates for Category 1 andCategory 2 layers of the proposed scheme are as follows:(3, 3

4 ), (2, 23 ). The Alamouti code supports 5 information

bits with rate- 56 convolutional code. Figure 4 shows that

the BER of category 1 bits performs 2 dB better than thatof the Alamouti code and the BER of Category 2 bits alsooutperforms the Alamouti code at SNR 15.5 dB and above.

Next, we consider the number of information bits andrespective convolutional code rates for two layers of theproposed scheme and the Alamouti scheme as follows:(2, 1

2 ), (2, 23 ) and (4, 2

3 ). In Figure 5, we observe that theBER of Category 1 bits is 4 dB better than that of theAlamouti code. The Alamouti code, however, outperformsthe BER of Category 2 bits by 3 dB. In Figure 6, it isshown that the effective throughout of the rate- 7

4 code andthe Alamouti code has a cross-over point at SNR 8.2 dB.Each coding scheme achieves a maximum throughput of 2information bits PCU.

Finally, we consider one 16QAM and one 8PSK mod-ulated symbols, which are supported by rate- 1

2 and rate-23 convolutional code respectively, for the Alamouti codeto achieve a spectral efficiency of 3.5 bits PCU. Thenumber of information bits and respective convolutionalcode rates for Category 1 and Category 2 layers of theproposed scheme are as follows:(2, 1

2 ),(2, 23 ). In Figure 7,

we observe that the BER of Category 1 bits outperformsboth layers of the Alamouti code, however BER of eachlayer of the Alamouti code performs better than that ofthe Category 2 bits. We also observe that unlike theproposed rate- 7

4 code, the Alamouti code with two differentmodulated symbols cannot provide distinct UEP to twolayers of source bits since both layers demonstrate closelycomparable performances.

2Effective Throughput η = (1−FER)∗R∗ log2(M)∗ KN

, whereR is the rate of STBC, M is the constellation size, and FER denotesthe frame error rate and K

Nis the error correction (FEC) code rate.

1 1.5 2 2.5 3 3.5 4 4.5 50

0.01

0.02

0.03

0.04

0.05

0.06

Scaling Factor, K

CG

/PA

R

CodingGain/PARCoding Gain

Fig. 2. Coding Gain to PAPR Ratio for Different K Values

5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

SNR in dB

Effe

ctiv

e T

hrou

ghpu

t

Effec. Thruput of the Rate 7by4 CodeEffec. Thruput of the Alamouti w/ QPSKEffec. Thruput of the Alamouti w/ 8PSK

Fig. 3. Effective Throughput Comparison between the Proposed Rate- 74

Code and the Alamouti Code with QPSK and 8PSK

8 9 10 11 12 13 14 15 16 1710

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR in dB

Bit

Err

or R

ate

Category 1: Rate−3/4 CC, QPSKCategory 2 : Rate−2/3 CCAvergae BERAlamouti, 8PSK, Rate−5/6 CC

Fig. 4. BER Comparison between the Proposed Code and the AlamoutiCode with 5 Information Bits for Each Code

1451

2 4 6 8 10 12 14 1610

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR in dB

Bit

Err

or R

ate

Category 1:Rate−1/2 CC,QPSKCategory 2:Rate−2/3 CCAvergae BERAlamouti, 8PSK, Rate−2/3 CC

Fig. 5. BER Comparison between the Proposed Code and the AlamoutiCode with 4 Information Bits for Each Code

2 4 6 8 10 12 14 160

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

SNR in dB

Effe

ctiv

e T

hrou

ghpu

t

Proposed Rate−7/4 Code, QPSKAlamouti Code, 8PSK, Rate−2/3 CC

Category 1: Rate 1/2 CCCategory 2: Rate 2/3 CC

Fig. 6. Effective Throughput Comparison between the Proposed Codeand the Alamouti Code

2 4 6 8 10 12 14 1610

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR in dB

Bit

Err

or R

ate

Prop.Code:Ctg. 1,Rate−1/2 CCProp.Code:Ctg. 2,Rate−2/3 CCAlamouti:16QAM,Rate−1/2 CCAlamouti:8PSK,Rate−2/3CC

Fig. 7. BER Comparison between the Proposed Code and the AlamoutiCode with Equal Spectral Efficiency of 3.5 Bits PCU.

VI. CONCLUSION AND FUTURE WORK

In this paper, we have presented an encoding methodfor a new class of high-rate, full-diversity STBC for 2transmit antennas. We have used the Alamouti code asthe building block and then extended the signalling setwith the incorporation of power-scaling and constellationrotation. The power-scaling factor and rotation angle areoptimized to provide full-diversity, maximum coding gainand minimum peak-to-average power ratio. An efficientdecoding method is proposed to reduce the complexity bymany folds without sacrificing performance in comparisonto the optimum ML decoding.

As an extension to this work, we will utilize similarencoding procedure as presented in this paper to enhancethe rate and the reliability levels of the linear additiveembedded diversity codes [6], [7]. The efficient decodingmethod will also be applicable to reduce the decoding com-plexity of resulting high-rate embedded diversity codes.

REFERENCES

[1] E. TELATAR. Capacity of Multi-Antenna Gaussian Channels.European Transactions on Telecommunications, 10(6):585–596,November-December 1999.

[2] G.J. FOSCHINI. Layered Space-Time Architecture for Wire-less Communication in a Fading Environment when using Multi-Element Antennas. Bell Labs Technical Journal, 1(2):41–59, Sep-tember 1996.

[3] S.M. Alamouti. A Simple Transmit Diversity Technique forWireless Communications. IEEE Journal on Selected Areas inCommunications, 16(8):1451–1458, October 1998.

[4] A.R. Calderbank, N. Seshadri. Multilevel Codes for Unequal ErrorProtection. IEEE Transactions on Information Theory, Vol.39,pp.1234-1248, July 1993.

[5] S.Das, N. Al-Dhahir, A.R. Calderbank, J.Chui. Novel Full-DiversityHigh-Rate STBC for 2 and 4 Transmit Antennas. IEEE Communi-cation Letter, March 2006.

[6] S. N. Diggavi, N. Al-Dhahir, and A. R. Calderbank. Diversity em-bedding in multiple antenna communications, advances in networkinformation theory. DIMACS Series in Discrete Mathematics andTheoretical Computer Science, pages 285-301, 2004.

[7] S. Das, N. Al-Dhahir, S. Diggavi, R. Calderbank. OpportunisticSpace-Time Block Codes. IEEE Vehicular Technology Conference,Fall-2005, Vol. 3, Sept.2005, pp.2025-2029.

[8] H. Gharavi, S.M. Alamouti. Multipriority Video Transmission forThird-Generation Wireless Communication Systems. Proceedingsof the IEEE, Vol.87, pp.1751-1763, October 1999.

[9] H. Gharavi, C.I. Richards. Partitioning of MPEG Coded Video BitStremas for Wireless Transmission. IEEE Signal Processing Letters,Vol.4, pp.153-155, June 1997.

[10] H.X. Tie, A. Goldsmith and M. Effros. Joint Design of Fixed-Rate Source Codes and UEP Channel Codes for Fading Channels.Conference Record of the Thirty-Second Asilomar Conference onSignals, Systems and Computers, Vol.1, pp.92-96, 1998.

[11] S.Z. Manji, N. Mandayam. Variable Rate Channel Coding andEnhanced Interleaving for Image Transmission Using an OutageCriterion. IEEE Wireless Communications and Networking Confer-ence, Vol.1, pp.344-348, 1999.

[12] L. Zheng, D. N. Tse. Diversity and Multiplexing: A FundamentalTradeoff in Multiple-Antenna Channels. IEEE Transactions onInformation Theory, Vol. 49, no. 5, pp. 1073:1096, May 2003.

[13] W. Su and X.-G. Xia. Signal Constellations for Quasi-OrthogonalSpace-Time Block Codes with Full Diversity IEEE Transactions onInformation Theory, pages 2331–2347, Vol.50,No.10,October 2004.

[14] J.H.CONWAY AND D.H.SMITH, Quaternions and Octonions.Cambridge University Press, 2003.

1452