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Differential Quasi-Orthogonal Space-Frequency Trellis Codes

Jorge Flores, Jaime Snchez, Member, IEEE,and Hamid Jafarkhani, Fellow, IEEE

AbstractTwo rate-one differential quasi-orthogonal space-

frequency trellis codes (DQOSFTCs) for multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing(OFDM) channels are proposed. The DQOSFTCs are systemati-cally constructed within an OFDM symbol period, by combiningunitary quasi-orthogonal trellis codes and differential modulationover the frequency domain (DF-QOSFTC) or time domain(DT-QOSFTCs). Besides multipath diversity, our DQOSFTCsachieve high-coding gain and simple decoding when the channelstate information is not available at both the transmitter andthe receiver. Simulation results show that our proposed codessignificantly outperform the existing differential space-frequencytrellis codes.

Index TermsMIMO-OFDM, differential modulation, quasi-orthogonal codes, space-frequency codes, trellis codes.

I. INTRODUCTION

SPACE-FREQUENCY coded multiple-input multiple-

output orthogonal frequency division multiplexing

(MIMO-OFDM) systems are capable of achieving maximum

diversity over a frequency selective channel (FSC) [1].However, acquiring knowledge of the channel state

information (CSI) for an FSC with many taps is prohibitively

complex. As a result, exploiting the available spatial and

frequency diversities, if neither the transmitter nor the receiver

knows the CSI, have been recently addressed in the literature.

In order to achieve the space-time (ST) and space-frequency(SF) diversity, differential and noncoherent schemes were

proposed in [2][12].

In many of these schemes, maximum diversity and good

performance can be achieved by dividing an OFDM symbol

into several subcarriers groups, and perform differential en-

coding and decoding between adjacent groups. However, the

existing differential techniques have several drawbacks. Forexample, they have a large code size, which exponentially

increases the coding and decoding complexity. Moreover, bytrying to increase the data rate, they have a low coding gain.

In addition, these implementations require constant channel

frequency response (CFR) from group to group, which leadto severe error floor in FSCs. Differential SF trellis codes

(DSFTCs) presented in [10], can obtain rate-one (1 symbol

per subcarrier), spatial diversity, and a simple decoding com-

Manuscript received March 23, 2010; revised July 20, 2010; acceptedSeptember 11, 2010. The associate editor coordinating the review of thisletter and approving it for publication was X. Wang.

J. Flores and J. Snchez are with the Department of Electronicand Telecommunications, CICESE Research Center, 3918 Carr. Tijuana-Ensenada Zona Playitas , Ensenada, B. C. 22860 Mxico (e-mail: {jtroncos,

jasan}@cicese.mx).H. Jafarkhani is with the Center for Pervasive Communications & Com-

puting, University of California at Irvine, Irvine, CA 92697 USA (e-mail:hamidj@uci.edu).

Digital Object Identifier 10.1109/TWC.2010.102210.100463

plexity. Even though the DSFTCs increase the coding gain,

the error floor is still significant.The differential codes proposed in this letter, called differ-

ential quasi-orthogonal SF trellis codes (DQOSFTCs), solve

these drawbacks. The proposed DQOSFTCs achieve high

coding gain, and are capable of exploiting both spatial andfrequency diversity over a MIMO-OFDM system with lack

of CSI at the transmitter and at the receiver. In order to

guarantee both rate-one and full-diversity, we use the structure

of the generalized quasi-orthogonal space-time block code

(QOSTBC) derived in [13]. We stress, however, that the

construction in [13] does not take into account the coding

gain, and it is not suitable for differential encoded process.

In summary, the contributions of this letter are the follow-ing: 1) In order to get rate-one and high-coding gain, we obtain

a sufficient number of full-diversity unitary quasi-orthogonal

codes, and following a similar procedure as in [14], we

perform set partitioning. Then, we systematically design the

unitary quasi-orthogonal SF trellis codes (QOSFTCs). 2) Wepropose to perform the differential encoding in 2 ways: a) over

the frequency domain, b) over the time domain. 3) Accordingto the number of orthogonal matrices (subblocks) within a

QOSFTC, we divide the OFDM subcarriers into equidistant

groups. In addition, we take advantage of orthogonality of

the inner subblocks of the QOSFTCs, by using a simple

Maximum-Likelihood (ML) decoder without requiring CSI atthe receiver, which is formed by a differential decoder and

a Viterbi decoder. Therefore, we obtain a simple differential

encoding process, and the assumption of a constant CFR from

group to group can be relaxed. Besides being computationally

efficient and easy to implement, we show through numerical

simulations that in the presence of FSCs, our proposed codessignificantly outperform the existing DSFTCs.

II. SYSTEMMODEL

Consider a system implemented with transmit and receive antennas. We assume no spatial fading correlation

exists between antennas. Each transmit antenna employs an-subcarrier OFDM modulator. We denote the transmitted space-

frequency code at the th OFDM symbol period by C =c1

c C , where c = [(1)(2) ()]

is transmitted from the th antenna ( = 1 ), and() is the complex data transmitted at the th subcarrier(= 1 ); superscript () denotes the vector transposeand C represents the complex field of dimension .Moreover, C satisfies the power constraint C2 = ,where2is the Frobenius norm. In order to avoid the InterSymbol Interference (ISI) which is produced by the multipath

delay of the channel, a cyclic prefix with the proper length is

added to each OFDM symbol.

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We assume that the channel impulse response (CIR) be-

tween the transmit antenna and receive antenna has independent delay paths on each OFDM symbol and anarbitrary power delay profile, which is denoted as () =1

=0 ()( ), where represents the th path delay,and each fading coefficient () at delay is a complexGaussian random variable with zero mean and variance 2.

For normalization purposes 1=0 2 = 1. The CFR at theth subcarrier between transmit antennaand receive antenna is expressed as

() =1=0

()exp

(2

(1)

where =is the th path delay normalized and isthe sampling rate of the OFDM system.

Assuming perfect sampling time and carrier synchronization

at the receiver, after matched filtering, removing the cyclic

prefix and FFT processing, the signal at th receive antennaand th subcarrier is given by

() =

=1

()() + () (2)

where () is a circularly symmetric zero-mean and unit-variance Gaussian noise term at the th symbol duration.

III. DIFFERENTIALSF TRELLISCODEDMODULATION

In this section, we derive the rate-one unitary QOSTBC

construction, and describe two differential encoding designs.

We also show the systematic construction for the rate-one

DQOSFTCs and the transmitted codeword C.

A. Unitary quasi-orthogonal block codes

The first stage of information processing in the transmitter

is the trellis encoder. Similar to the method in [14], depending

on the input information bits and the current state in the

trellis encoder, a codeword is selected from a constellation

of possible unitary-QOSTBCs. Then, there is a transition

towards the next state in the trellis, and the encoding processis performed again to obtain a frame with codewords. Letus consider the general class of QOSTBC defined in [13] as

= 12

A(1 2) 0

... . . .

...

0 A(21 2)

(3)where space goes horizontally, C22, and A( )denotes the Alamouti code [15] for any indeterminate symbols

, . In order to support a data rate of, defined as thenumber of bits per subcarrier use (b/s/Hz), it is necessary for

a sequence of information bits to pick a codewordas (3) at each state of the trellis encoder. Then, we denoteas the set of all possible information codewords, whichmust have a cardinality of = 2. Furthermore, inthe interest of performing differential encoding after trellis

encoding, the information codewords to be transmitted must be

unitary matrices, such that = I2, where Iis the

identity matrix and superscript () denotes the conjugatetranspose.

Let us denote by O the set of all unitary codewords. For the sake of providing the required cardinality

= 2, we introduce unitary rotation matrices U =diag(1 2 2), = 0 1 and diag()denotesa diagonal matrix, such that =OU0 OU. Thus,we express an information codeword from the trellis encoder

output as = U. Since Uis a unitary matrix, then is also a unitary matrix. Due to the diagonal structure of (3),

we can write the inner orthogonal matrices as

B(21 2) = 1

2A(21 2)diag(21 2)

(4)

where = 1 . Then, we define theth unitary codewordsent at period from a trellis encoder as

=

B(1 2) 0

... . . .

...

0 B(21 2)

(5)

where

C22, = 0 1 1, and =2

.

For convenience, we assume that = 2for some integer. Note that the essential structure of the QOSTBC derived in[13] is preserved by the unitary matrix defined in (5), therefore

the full-diversity ofis still maintained.

B. Differential modulation

For simplicity, in the sequel, we consider a scenario with

= 2 transmit antennas. In order to perform differentialencoding keeping multipath diversity, we exploit the indepen-

dence of the orthogonal matrices in (5) by partitioning the totalsubcarriers into groups of subcarriers. Then, we proposeencoding differentially over the frequency domain (DF), and

over the time domain (DT) as follows.

1) DF: We consider that the channel remains constantduring one OFDM symbol period and changes independently

from symbol to symbol. We also consider that the correlation

between adjacent subcarriers is high, which is a reasonable as-

sumption ifis large enough. In the differential encoder, therecursive construction of the unitary codeword

C22is given by

C

(S) = I2 = 0B(S)C1(S) 1

(6)

where for the sake of brevity, we use the symbol Sto denotethe pairwise (21 2), = 1 2 .

2) DT: In this case we assume that the channel is quasi-

static over one OFDM symbol period and slowly varies

between adjacent OFDM symbols. Let

C22 be thedifferential encoded matrix to be transmitted at the th OFDMsymbol period. At the beginning of the transmission (= 0),the differential encoder sends C0(21 2) = I2 for all = 0 1 1, where = 1 2 . The th transmitcodeword is generated as

C(S) =

I2 = 0B(S)C

1 (S) 1

(7)

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Afterwards, the differentially encoded matrices are mapped

to the same subblock of two adjacent time domain OFDM

symbols.

C. Subcarrier assignment

After picking the unitary codeword (5) from the trellisencoder output and the differential encoding process, we map

the codeword to the OFDM-subcarrier groups. Then, weexpress a SF construction for two transmit antennas and subcarrier groups as

C =

C0(1 2)

C1(1 2)

C0(3 4)

C1(3 4)

C0(21 2)

C1(21 2)

(8)

where C 2.

D. Differential QOSFTCs

In what follows, we propose rate-one differential QOSFTC

designs for two transmit antennas, assuming a two-path fading

channel model, i.e. = 2. However, it is noteworthy that themethod described here can be extended in a straightforward

way to >2.

1) Codeword sets: Let 1 = (1+ 3), 3 = (13),2 = (2 + 4) and 4 = (2 4) be the symbolstransmitted in (5) by setting = 2, where 1, 2 belongto a M-PSK constellation, and 3, 4belong to the rotatedconstellation . Also, let = be the optimal rotationthat provides the maximum coding gain for the code in (5)[13]. In the sequel, we asume that the system employs a

QPSK constellation. Therefore, = 2 denotes a QPSK

symbol, where = 0 1 2 3 represents the index ofthe corresponding QPSK symbol. Thus, we obtain only 128

unitary codewords inO. Therefore, in order to get a data rateof= 2b/s/Hz, we need two unitary matrices U0U1 suchthat=OU0 OU1and = 256.

Due to the symmetry in (5), if (1 2) and(3 4) , we get another set of 128 different unitarycodewords called P. Let 1 and 2 be the rotation anglesfor the symbols (1 2) and (3 4), respectively. Then,we assign 0 =

1 = 0 2 = 4

for the 128 unitary

codewords belonging to set O, and1=1= 4 2 = 0for the 128 unitary codewords that belong to set P. Thus, weobtain a new set=PU0 PU1such that =256.

2) Design criteria: Here we borrow the design criteria

for differential space-time-frequency codes of [7], which are

identical to those proposed for differential super-orthogonal

space-time trellis codes over flat channels in [11]. Let 1and2 be two codewords as defined in (5) so that:

i) Full diversity is achieved if the difference matrix D =1 2 has full rank over all possible pairs of distinctcodewords1and 2.

ii) The minimum of the determinantdet DD 1=2corresponds to the coding gain distance (CGD) andmust be maximized.

Fig. 1. Final set partitioning of sets (a) with 0, and (b) with 1.

Fig. 2. Set partitioning for sets Oand Pusing QPSK.

3) Set partitioning:Note that the codeword structure in (5)

guarantees the full diversity criterion. Moreover, the minimum

CGD between codewords at each level of an optimal set

partitioning must be maximized. In order to avoid expanding

the original QPSK constellation, we need to search for the

optimal values of U ( = 0 1). In addition, the designcriteria for the codewords in set or must be satis-fied. With numerical search, we obtain the optimal matrices

U0 = diag(1 1 1), U1 = diag(1 1 1). As aconsequence, each set andcontains 256 rate-one full-diversity unitary quasi-orthogonal codes using QPSK.In order to design a trellis code, we partition the sets following

a procedure similar to the procedure in [14]. Due to the lack

of space, we only show the final set partitioning. Fig. 1 shows

the set partitioning for setsandwith QPSK, while Fig.2 shows a partial set partitioning for O and P.

4) Differential trellis encoding: We now show how to use

the proposed set partitioning schemes to systematically design

rate-one full-diversity differential QOSFTCs. A key concept

is that those codewords that do not belong to the same set

are assigned to different states. Moreover, we must assign

codewords diverging (or merging) into a state such that both

difference matrices have full rank, and the CGD between allpairs of codewords must be the largest CGD.

In Fig. 3 we propose 4-state and 8-state trellises containingbranches with 64 and 32 parallel transitions, respectively.

According to the current state and the 8 input bits, the trellis

encoder builds theth codeword

based on (5).Full-diversity and high-coding gain QOSFTCs with different

number of states can be systematically designed using both the

corresponding set partitioning and the design rules described

above.

After constructing the codeword

, in order to build a

DF-QOSFTC or a DT-QOSFTC, we first perform the differ-

ential encoding using (6) or (7), respectively. Then, at the

th OFDM symbol period we form a SF code as in (8) for= 2, such that C =

C0(1 2)

C1(1 2)

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Fig. 3. QOSTFTCs using QPSK (a) 4-state, (b) 8-state.

C0(3 4)

C1(3 4)

.

E. ML Decoder

The successful differential decoding of the proposed DF-

QOSFTCs and DT-QOSFTCs depends on the assumption that

the fading channel remains constant over at least four adjacent

subcarriers or two successive OFDM symbols, respectively. It

is worth mentioning that both, orthogonality and independence

of the inner building matrices in the proposed codes, make it

possible to simplify the complexity of the decoding process

by separating the decoding of these subblocks. In order toestimate the information subblock B(21 2), no CSIis required at the receiver. The proposed differential codes

are decoded using Maximum-Likelihood (ML) decoding, com-

prised by the differential decoder derived in [3] and a Viterbi

decoder. Note that to perform differential decoding for theDT-QOSFTCs, two consecutive received OFDM symbols are

required.

IV. SIMULATIONRESULTS

In this section, we provide simulation results for the perfor-

mance of the differential QOSFTCs proposed. We assume a

128-tone OFDM system with two transmit antennas, a single

receive antenna, and a total bandwidth of 1 MHz. We add

a cyclic prefix of 20 s to avoid the ISI. The performancecurves are described by means of OFDM symbol error rate

(SER) versus the receive SNR. We simulate the system over

the following channel scenarios: 1) A quasi-static channel

with 2-path uniform power delay profi

le, which changesindependently for each OFDM symbol. The delay betweenthese 2 paths is one OFDM sample duration. 2) The channel

is quasi-static over one OFDM symbol period and slowly

changes between adjacent OFDM symbols by varying the

normalized Doppler frequencies , with a) =0.0025and b) = 0.0125, which correspond to mobile speedsof 6 and 30 m/s, respectively. To properly model an FSC,

we adopt a typical urban (TU) six-path power delay profile.

In order to validate the work we have carried out, we then

compare the performance of our proposed DF-QOSFTCs and

DT-QOSFTCs, to that of the 8-state DSFTC in [10], which is

based on group codes with the same rate =2 b/s/Hz. Wehave added a comparison with coherent QOSFTCs and the

result is a 3 dB difference, as happens with most well-designed

10 12 14 16 18 20 22 24 26 28 3010

5

104

10

3

102

101

100

SNR [dB]

OFDMSym

bolErrorRate(SER)

8state Coherent 8state DSFTC [10] 4state DFQOSFTC 8state DFQOSFTC 4state DTQOSFTC 8state DTQOSFTC

2ray uniform

TU fDn=0.0025

Fig. 4. SER performance for =2 b/s/Hz under the 2-ray and TU powerdelay profiles.

10 12 14 16 18 20 22 24 26 28 3010

4

103

102

101

100

SNR [dB]

OFDMSymbolErrorRate(SER)

8state Coherent 8state DSFTC [10] 8state DFQOSFTC 8state DTQOSFTC

TU fDn

=0.0025

TU fDn

=0.0125

Fig. 5. SER performance for =2 b/s/Hz under the TU power delay profile.

differential schemes. As can be seen from Figs. 4 and 5,

our proposed codes all considerably outperform the existing

DSFTC. Fig. 4 provides SER performance for the channel

scenarios 1) (showed in dashed lines), and 2a) (showed in solid

lines). Fig. 4 shows that the DF-QOSFTCs and DT-QOSFTCs

over channel scenarios 1) and 2a), respectively, achieve thefull-diversity order of 4. Note that we have designed codes

for = 2 and therefore the maximum achievable diversityis four, independent of the actual number of taps in the

system. However, although DF-QOSFTCs also reach coding

gain under the channel scenario 2a), at high SNRs, they

suffer a diversity loss. This is because the success of the DF-

QOSFTCs depends on a high correlation between adjacent

subcarriers, which is not preserved in hostile FSCs. This effect

can be mitigated either by increasing the number of subcarriers

or implementing codes for > 2. Thus, the number ofsubcarrier groups is increased leading to a better multipath-

diversity gain.

Next, we compare the performance of the 8-state proposedcodes versus existing 8-state DSFTC by varying . We cansee in Fig. 5 that both DF-QOSFTC and DSFTC have a robust

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performance in the case of time-varying channels. However,

time variations of the channel affect the diversity of the DT-

QOSFTC, because of the differences between the channel gainof subcarrier groups for adjacent OFDM symbols. Neverthe-

less, the DT-QOSFTC notably outperforms all other codes.

Moreover, the transmission efficiency for the DT-QOSFTCs

is greater than that achieved in the frequency domain by

for example DF-QOSFTCs and DSFTC. In addition, under

practical fading channels, significant gains can be achievedby increasing the number of states in a DT-QOSFTC and the

simple decoding complexity is hold.

V. CONCLUSIONS

We have proposed rate-one differential space-frequency

trellis codes for MIMO-OFDM systems, where CSI is not

available at the transmitter and at the receiver. The proposed

differential codes have been designed based on a generalizedclass of unitary quasi-orthogonal space-time block codes. Also

by trellis encoding and grouping of OFDM-subcarriers, our

proposed codes achieve full-diversity and high-coding gain

over practical FSCs. Simulation results show that, comparedto the existing DSFTC, the proposed codes provide better

performance in simulated channel scenarios. Moreover, the

independence of inner orthogonal matrices in the proposed

codes, allows a reduced encoding and decoding complexity.

Furthermore, because the differentially encoded codewords are

transmitted within one OFDM symbol period, we obtain a re-

duced decoding delay. The provided examples have only beenable to achieve a maximum diversity order of four with two

transmit antennas, but it is straightforward to design similar

codes for more than two taps, and two transmit antennas. This,

of course, is at the expense of increased decoding complexity.

In addition, unlike differential and noncoherent SF schemesin the literature, our proposed codes achieve rate one and can

be designed in a simple systematic way.

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