High-Rate Distributed Space-Time-Frequency Coding for Wireless Cooperative Networks

614 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 2, FEBRUARY 2011

High-Rate Distributed Space-Time-FrequencyCoding for Wireless Cooperative Networks

Jinsong Wu, Member, IEEE, Honggang Hu, and Murat Uysal, Senior Member, IEEE

Abstract—In this paper, we propose high-rate distributedspace-time-frequency codes (DSTFCs) to exploit maximumachievable diversity gains over frequency-selective fading chan-nels. The proposed designs achieve full-rate for any number ofcooperative nodes, and allow channel variations over multipleOFDM blocks within one DSTFC codeword. We analyze diversitygains of DSTFCs through both conditional and average pairwiseerror probability (PEP), and we proposes better design criteriabased on one-side channel conditional PEP. We show thatthe difference between the frequency-selective channel ordersof source-to-relay and relay-to-destination links may provideextra diversity advantages, thus additional performance gains.Through Monte-Carlo simulations, we demonstrate that pro-posed high-rate DSTFCs provide notable diversity advantagesover existing designs.

Index Terms—Amplify-and-forward relaying, high-rate, dis-tributed space-time-frequency coding, one-side conditional pair-wise error probability, diversity.

I. INTRODUCTION

THE performance of wireless communications is highlydegraded in the presence of fading, shadowing, and mul-

tiuser interference. To overcome these limitations, cooperativediversity has been proposed [1], [2] which extracts spatial di-versity advantages in a distributed manner by creating a virtualantenna array. Relay-based cooperative communications cannotably increase capacity, and this capacity gain can also betranslated into reduced power for the users [3].

To extract the spatial diversity in distributed scenarios, anumber of distributed space-time codes (DSTCs) have beendeveloped for cooperative communications over frequency-flatchannels [2], [4], [5]. On the other hand, broadband channelsexhibit frequency-selectivity and require the deployment oftransceiver techniques which will handle the resulting inter-symbol interference (ISI). Orthogonal frequency division mul-tiplexing (OFDM) is a kind of multicarrier communicationsystem where the high-rate data stream is demultiplexed andtransmitted over a number of frequency subcarriers. If thesubcarrier width is sufficiently small compared to the channelcoherence bandwidth, a frequency-flat channel model can beassumed for each subcarrier and channel distortion can beeasily compensated at the receiver.

Manuscript received April 15, 2010; revised August 28, 2010; acceptedOctober 27, 2010. The associate editor coordinating the review of this paperand approving it for publication was B. Sundar Rajan.

J. Wu and H. Hu are with the Department of Electrical and ComputerEngineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1(e-mail: [email protected], [email protected]).

M. Uysal is with the Faculty of Engineering, Ozyegin University, 34662,Istanbul, Turkey (e-mail: [email protected]).

H. Hu is supported by the National Natural Science Foundation of Chinaunder Grant No. 60802029.

Digital Object Identifier 10.1109/TWC.2011.120810.100615

There has been a growing attention on the design ofcooperative OFDM systems [6]–[8]. Seddik and Liu have pro-posed distributed space-frequency codes (DSFCs) assumingboth decode-and-forward (DF) and amplify-and-forward (AF)relaying [7]. The proposed code structure of AF DSFC isonly shown to achieve full diversity for some special cases[7]. In [6], Zhang et al. have introduced some AF DSFCsand provided diversity analysis for arbitrary number of relaynodes and channel lengths based on some approximations. Themaximal rate of AF-DSFCs proposed in both [7] and [6] is oneOFDM block per two-phase cooperative OFDM transmission,which is termed as “full rate”. Wang, et al. have proposed twolower-rate (i.e., less than full rate) DSFCs to combat multiplecarrier frequency offsets (CFO) [8].

Space-time-frequency codes (STFCs) [9] have been fur-ther proposed for use in cooperative OFDM systems withmultiple relays over multiple OFDM blocks and multiplesubcarriers [10]–[12]. Oguz et al. have proposed distributedSTFC (DSTFC) for a two-user multiple-access-channel withcooperating transmitters [10]. The DSTFC design proposedby Tran et al. [11] is to improve the system performanceof multi-band OFDM-based ultra-wideband communicationsusing the cooperation of two nodes. Yang et al. have studiedspace-time-frequency block coding for DF relaying version[12]. All the existing DSTFC designs [10]–[12] are basedon orthogonal space-time block-codes [13], which introducesome limitations to applications. The current designs [10]–[12] only consider two cooperative nodes. In practice, usuallymore than two cooperative nodes are available. Therefore,extending those designs for more than two cooperative nodesis of practical concern. However, orthogonal designs [14]based approaches cannot achieve full-rate if the number ofcooperative nodes is more than two. It should be also notedthat orthogonal designs ensure simpler maximum-likelihoodreceiver; however, this requires that the channels are constantover multiple OFDM blocks, that is to say, this prevents fromexploiting time diversity over multiple OFDM blocks.

To overcome the design limitations mentioned above, inthis paper, we propose high-rate DSTFC for relay-based AFcooperative communications. The proposed designs achievefull-rate for any number of cooperative nodes, and allowchannel variations over multiple OFDM blocks within oneDSTFC codeword. Unlike the designs of AF DSFC whichwere based on diversity analyses using average pairwise errorprobability (PEP) [6], [7], we design AF DSFC using one-side-channel conditioned PEP based diversity analysis, whichenables to shed light on code construction. In addition, wealso provide some discussions on average PEP based diversityanalysis for our design creteria, and show some insights on

1536-1276/11$25.00 c⃝ 2011 IEEE

WU et al.: HIGH-RATE DISTRIBUTED SPACE-TIME-FREQUENCY CODING FOR WIRELESS COOPERATIVE NETWORKS 615

differences between conditional and average PEP-based baseddesigns.

Notations: 𝑗 =√−1, (⋅)𝒯 matrix transpose, (⋅)∗ conju-

gate, (⋅)ℋ matrix conjugate transpose, ∘ Hadamard productoperator, 𝜙 denotes empty set, ⊗ Kronecker product, 𝛿 (⋅)Kronecker delta, [A]𝑎,𝑏 the (𝑎, 𝑏) entry (element) of matrixA, tr (⋅) matrix trace operation, Re (⋅) real part of the object(matrix or variable), Im (⋅) imaginary part of the object (matrixor variable), [A]

.𝑎 element-wise power of 𝑎 for matrix A, and𝜁𝑎 = exp

(𝑗 2𝜋𝑎

), and E𝛼 (⋅) expectation over random variable

or random variable set 𝛼.

II. SYSTEM MODEL AND PROPOSED APPROACH

We consider a wireless cooperative communication systemwhich consists of a source node, 𝐾 relay nodes, and adestination node. The source node sends 𝑁𝐶 symbols over𝑁𝐶

OFDM subcarriers per transmission. Perfect synchronizationis assumed for all transmissions between different relay nodesand the destination node. One OFDM block transmission inthe half-duplex cooperative-communications consists of twophases. In the first phase, the source node adds cyclic-prefix(CP) and broadcasts the OFDM block to all 𝐾 relay nodes.Each relay node receives the channel symbols with additivenoise at that relay. During the second phase, the sourcenode stops the transmission. All relay nodes remove the CPand process the received block, i.e., energy normalization,precoding, power-amplifying, etc. Then, all relay nodes (𝑘 =1, ...,𝐾) simultaneously retransmit the processed signals tothe destination node .

The frequency-selective channels are modeled using widelyadopted discrete symbol-spaced tap-delay-line (SSTDL) [15].The channel between the source node and the 𝑘-th relaynode in the 𝑡-th OFDM block experiences frequency-selective,temporally flat Rayleigh fading with channel coefficients

h(𝑘,𝑡) =

[ℎ(𝑘,𝑡)(0) , ..., ℎ

(𝑘,𝑡)

𝐿(𝑘)ℎ

]𝒯, 𝑘 = 1, ...,𝐾 . Similarly, the

channel coefficient vector between the 𝑘-th relay node andthe destination node in the 𝑡-th OFDM block is represented

by g(𝑘,𝑡) =

[𝑔(𝑘,𝑡)(0) , ..., 𝑔

(𝑘,𝑡)

𝐿(𝑘)𝑔

]𝒯, 𝑘 = 1, ...,𝐾 . Set 𝐿 =

max{𝐿(𝑘)ℎ , 𝐿

(𝑘)𝑔 , 𝑘 = 1, ...,𝐾} where 𝐿

(𝑘)ℎ and 𝐿

(𝑘)𝑔 are

the frequency-selective channel orders for source-to-relay andrelay-to-destination links, respectively. The entries of h(𝑘,𝑡)

and g(𝑘,𝑡) are Rayleigh fading channel gains, and are mod-eled as complex Gaussian with zero mean, whose variancesdepend on power delay profile. We assume that the channelcoefficients remain constant within one OFDM block and varyindependently among different OFDM blocks. We further as-sume channel power constraints as 𝐸

((h(𝑘,𝑡)

)ℋh(𝑘,𝑡)

)= 1

and 𝐸((

g(𝑘,𝑡))ℋ

g(𝑘,𝑡))

= 1. Let 𝐻(𝑘,𝑡)𝑝 denote the 𝑝-th

subcarrier channel gain from the source node to the 𝑘-th relaynode during the 𝑡-th OFDM block. It is given by

𝐻(𝑘,𝑡)𝑝 =

𝐿(𝑘)ℎ∑

𝑙=𝑜

ℎ(𝑘,𝑡)(𝑙) 𝑒−𝑗(2𝜋/𝑁𝑐)𝑙(𝑝−1) = w

(𝐿(𝑘)ℎ )

𝑝 h(𝑘,𝑡), (1)

where w(𝐿)𝑝 =

[1, 𝜔𝑝−1, 𝜔2(𝑝−1), ⋅ ⋅ ⋅ , 𝜔𝐿(𝑝−1)

]𝒯. Let 𝐺(𝑘,𝑡)

𝑝

denote the 𝑝-th subcarrier channel gain from the 𝑘-th relay

node to the destination node during the 𝑡-th OFDM block. Itis given by

𝐺(𝑘,𝑡)𝑝 =

𝐿(𝑘)𝑔∑

𝑙=𝑜

𝑔(𝑘,𝑡)(𝑙) 𝑒−𝑗(2𝜋/𝑁𝑐)𝑙(𝑝−1) = w

(𝐿(𝑘)𝑔 )

𝑝 g(𝑘,𝑡). (2)

One DSTFC codeword is of size 𝑁𝐹 ×𝐾 × 𝑇 , i.e., across𝐾 relays, 𝑇 OFDM blocks, and 𝑁𝐹 subcarriers. Thus, 𝑇OFDM blocks contain 𝑛 = 𝑁𝐶

𝑁𝐹DSTFC codewords, and the

rest of subcarriers may be padded by zero. The constructionand transmission procedure for the 𝑖-th DSTFC codeword,𝑖 = 1, ..., 𝑛, is described as follows: At the source node, adata symbol vector of size 𝑁𝐹𝑇 × 1, x𝑖, is encoded in sourcenode by

x𝑖 = 𝜃s𝑖, 𝑖 = 1, ..., 𝑛, (3)

where s𝑖 is the source data vector carved from the signal alpha-bet 𝒜 ∈ ℤ [𝑗]. The matrix 𝜃 is designed such that, for any two

different non-zero vector s(1)𝑖 and s

(2)𝑖 ,[x(1)𝑖 − x

(2)𝑖

]𝑎,1

∕= 0

always holds for all 𝑎 = 1, ..., 𝑁𝐹𝑇 , where x(1)𝑖 = 𝜃s

(1)𝑖

and x(2)𝑖 = 𝜃s

(2)𝑖 . The design of 𝜃 can be carried out using

signal-space-diversity concepts [16], [17]. For example, 𝜃 canbe constructed as 𝜃 = [F𝑁𝑠 ]

ℋ𝑑𝑖𝑎𝑔

(1, 𝛼, ..., 𝛼𝑁𝑠−1

), where

𝑁𝑠 should be chosen as a power of 2, 𝑁𝑠 = 𝑁𝐹𝑇 , 𝛼 =

exp(𝑗 2𝜋4𝑁𝑠

), where [F𝑁𝑠 ]𝑎1,𝑎2

= exp(𝑗 2𝜋(𝑎1−1)(𝑎2−1)

𝑁𝑠

)[16]–[19].

Denote x(𝑖,𝑡) = [x𝑖](𝑡−1)𝑁𝐹+1:𝑡𝑁𝐹 ,1. During the 𝑡-th

OFDM block, using subcarriers{𝑝(𝑖)1 , ..., 𝑝

(𝑖)𝑁𝐹

}, the source

node transmits the sequence x(𝑖,𝑡) of size 𝑁𝐹 × 1 to the relaynode. The received signal vector r(𝑖,𝑘,𝑡)𝑓 at the 𝑘-th relay nodein frequency domain is given by

r(𝑖,𝑘,𝑡)𝑓 =

√𝑃1x

(𝑖,𝑡) ∘ h(𝑖,𝑘,𝑡)𝑓 + v

(𝑖,𝑘,𝑡)𝑓 , (4)

where h(𝑖,𝑘,𝑡)𝑓 =

[𝐻

(𝑘,𝑡)

𝑝(𝑖)1

, ⋅ ⋅ ⋅ , 𝐻(𝑘,𝑡)

𝑝(𝑖)𝑁𝐹

], v(𝑖,𝑘,𝑡)

𝑓 is the relevant

complex Gaussian noise vector at the 𝑘-th relay node, and

𝐸

(v(𝑖,𝑘,𝑡)𝑓


)ℋ)= I𝑁𝐹 .

At the destination node, the received signal vector infrequency domain is

y(𝑖,𝑡)𝑓 = u

(𝑖,𝑡)𝑓

+√𝑃1

𝐾∑𝑘=1

(𝛽(𝑘)

√𝑃

(𝑘)2 a(𝑖,𝑘,𝑡) ∘ r(𝑖,𝑘,𝑡)𝑓 ∘ g(𝑖,𝑘,𝑡)

𝑓

),

(5)

where g(𝑖,𝑘,𝑡)𝑓 =

[𝐺

(𝑘,𝑡)

𝑝(𝑖)1

, ⋅ ⋅ ⋅ , 𝐺(𝑘,𝑡)

𝑝(𝑖)𝑁𝐹

], 𝛽(𝑘) = 1√

𝑃1+1, u(𝑖,𝑘,𝑡)

𝑓

is the relevant complex Gaussian noise vector at the destina-

tion node, and 𝐸

(u(𝑖,𝑡)𝑓

(u(𝑖,𝑡)𝑓

)ℋ)= I𝑁𝐹 . The design of

a(𝑘,𝑡) will be later discussed in Section III and AppendicesA-B.

Combining (4) with (5), the received symbol vector duringthe 𝑡-th OFDM block can be rewritten as

y(𝑖,𝑡)𝑓 = u

(𝑖,𝑡)𝑓

+√𝑃1diag

(x(𝑖,𝑡)

) 𝐾∑𝑘=1

(𝛽(𝑘)

√𝑃

(𝑘)2 h

(𝑖,𝑘,𝑡)

𝑓

),

(6)


where h(𝑖,𝑘,𝑡)

𝑓 = a(𝑖,𝑘,𝑡) ∘ h(𝑖,𝑘,𝑡)𝑓 ∘ g

(𝑖,𝑘,𝑡)𝑓 , and u

(𝑖,𝑡)𝑓 =

u(𝑖,𝑡)𝑓 +

𝐾∑𝑘=1

(√𝑃

(𝑘)2 𝛽(𝑘)a(𝑖,𝑘,𝑡) ∘ v(𝑖,𝑘,𝑡)

𝑓 ∘ g(𝑖,𝑘,𝑡)𝑓

). Thus, the

received symbol vector for the 𝑖-th DSTFC codeword is givenby

y(𝑖)𝑓 =

√𝑃1diag

(x(𝑖))h(𝑖)

𝑓 + u(𝑖)𝑓 , (7)

where

x(𝑖) =

[[x(𝑖,1)

]𝒯, ...,[x(𝑖,𝑇 )

]𝒯 ]𝒯,

u(𝑖)𝑓 =

[[u(𝑖,1)𝑓

]𝒯, ...,[u(𝑖,1)𝑓

]𝒯 ]𝒯,

y(𝑖)𝑓 =

[[y(𝑖,1)𝑓

]𝒯, ...,[y(𝑖,1)𝑓

]𝒯 ]𝒯,

h(𝑖)

𝑓 =

[[h(𝑖,1)]𝒯

, ...,[h(𝑖,𝑇 )]𝒯 ]𝒯

, and h(𝑖,𝑡)

=

𝐾∑𝑘=1

(𝛽(𝑘)

√𝑃

(𝑘)2 h

(𝑖,𝑘,𝑡)

𝑓

). The received vectors are fed

to maximum likelihood (ML) decoder given by

argmaxs𝑖

(y(𝑖)𝑓

∣∣∣ s𝑖) =

argmins𝑖

∥∥∥y(𝑖)𝑓 −√

𝑃1diag(x(𝑖))h(𝑖)

𝑓

∥∥∥2 . (8)

III. DIVERSITY ANALYSIS AND DESIGN CRITERIA

In this section, in order to analyze the diversity gains,we first derive conditional PEP of the 𝑖-th codeword of theDSTFC under consideration and discuss relevant code designparameters. Second, we present an average PEP based analysisand point out the differences between average and conditionalPEP based designs.

A. Conditional PEP based analysis and code design

Given the channel coefficients h(𝑖,𝑘,𝑡)𝑓 , g

(𝑖,𝑘,𝑡)𝑓 ,

𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇 , the conditional PEP can bebounded by

Pr

(x(𝑖) → x(𝑖)

∣∣∣∣∣ h(𝑖,𝑘,𝑡)𝑓 ,g

(𝑖,𝑘,𝑡)𝑓 ,

𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

)

≤ exp

⎛⎜⎜⎝−∥∥∥√𝑃1Δ

(𝑖)h(𝑖)

𝑓

∥∥∥22𝐸{

v(𝑖,𝑘,𝑡)𝑓 ,u

(𝑖,𝑡)𝑓

}(∥∥∥u(𝑖)

𝑓

∥∥∥2)⎞⎟⎟⎠,

(9)

where Δ(𝑖) = diag(x(𝑖) − x(𝑖)

). The expectation of u

(𝑖,𝑡)𝑓

over{v(𝑖,𝑘,𝑡)𝑓 ,u

(𝑖,𝑡)𝑓

}can be performed as

𝐸{v(𝑖,𝑘,𝑡)𝑓 ,u

(𝑖,𝑡)𝑓

}(∥∥∥u(𝑖)

𝑓

∥∥∥2) =

𝑇∑𝑡=1

(𝐸{

v(𝑖,𝑘,𝑡)𝑓 ,u

(𝑖,𝑡)𝑓

}(∥∥∥u(𝑖,𝑡)

𝑓

∥∥∥2)), (10)

where


(𝑖,𝑡)𝑓

}(∥∥∥u(𝑖,𝑡)

𝑓

∥∥∥2)= tr

(𝐸{

u(𝑖,𝑡)𝑓

}(u(𝑖,𝑡)𝑓

(u(𝑖,𝑡)𝑓

)𝐻))+ 11 + 𝑃1

𝐾∑𝑘=1

tr

(𝑃

(𝑘)2 a

(𝑖,𝑡)g R

(𝑖,𝑘,𝑡)v𝑓

(a(𝑖,𝑡)g

)𝐻)= 𝑁𝐹 + 1

1 + 𝑃1

𝐾∑𝑘=1

(𝑃

(𝑘)2

(g(𝑖,𝑘,𝑡)𝑓

)𝐻g(𝑖,𝑘,𝑡)𝑓

),

(11)

Here

R(𝑖,𝑘,𝑡)v𝑓

= 𝐸{v(𝑖,𝑘,𝑡)𝑓

}(v(𝑖,𝑘,𝑡)𝑓


)𝐻),

a(𝑖,𝑘,𝑡)g = diag(a(𝑖,𝑘,𝑡) ∘ g(𝑖,𝑘,𝑡)

𝑓

),

and

diag

(a(𝑖,𝑘,𝑡) ∘

(a(𝑖,𝑘,𝑡)

)ℋ)= I𝑁𝐹 .

The equality in (10) is based on the assumption that theequivalent noise vectors, u(𝑖,𝑡)

𝑓 , 𝑡 = 1, .., 𝑇, are independent.Now we return our attention to the expression∥∥∥√𝑃1Δ

(𝑖)h(𝑖)

𝑓

∥∥∥2 in (9). It can be easily shown that∥∥∥√𝑃1Δ(𝑖)h

(𝑖)

𝑓

∥∥∥2 = 𝑃1tr(Δ(𝑖)A

(𝑖)ℎ

(Δ(𝑖)

)ℋ) (12)

where

A(𝑖)ℎ = A(𝑖)

(h(𝑖)𝑓 ∘ g(𝑖)

𝑓

)(h(𝑖)𝑓 ∘ g(𝑖)

𝑓

)ℋ (A(𝑖)

)ℋ,

h(𝑖)𝑓 =

[[h(𝑖,1)𝑓

]𝒯, ...,[h(𝑖,𝑇 )𝑓

]𝒯 ]𝒯,

g(𝑖)𝑓 =

[[g(𝑖,1)𝑓

]𝒯, ...,[g(𝑖,𝑇 )𝑓

]𝒯 ]𝒯,

A(𝑖) = diag(A(𝑖,1), ...,A(𝑖,𝑇 )

),

h(𝑖,𝑡)𝑓 =

[[h(𝑖,1,𝑡)𝑓

]𝒯, ...,[h(𝑖,𝐾,𝑡)𝑓

]𝒯 ]𝒯,

g(𝑖,𝑡)𝑓 =

[[g(𝑖,1,𝑡)𝑓

]𝒯, ...,[g(𝑖,𝐾,𝑡)𝑓

]𝒯 ]𝒯,

A(𝑖,𝑡) =

[𝛽(1)

√𝑃

(1)2 A(𝑖,1,1), ..., 𝛽(𝐾)

√𝑃

(𝐾)2 A(𝑖,𝐾,𝑡)

],

and A(𝑖,𝑘,𝑡) = diag(a(𝑖,𝑘,𝑡)

).

Using the results in Appendix A, (12) can be rewritten as∥∥∥√𝑃1Δ(𝑖)h

(𝑖)

𝑓

∥∥∥2 =

𝑃1tr(Δ(𝑖)M(𝑖)𝜓gh (h)𝐻 (𝜓g)

𝐻 (M(𝑖))𝐻 (

Δ(𝑖))𝐻)

= 𝑃1 (h)𝐻M

(𝑖)Δ h,

(13)

where M(𝑖)Δ = (𝜓g)

ℋ (M(𝑖)

)ℋ (Δ(𝑖)

)ℋΔ(𝑖)M(𝑖)𝜓g. Through

averaging over{h(𝑖,𝑘,𝑡)𝑓 , 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

}, the PEP

conditioned on channel{g(𝑖,𝑘,𝑡)𝑓 , 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

}can be bounded by (14), where the probability density function𝒫 (h) of h is defined in (15). Denote R

(𝑖)Δ in (16). Note that


Pr(x(𝑖) → x(𝑖)

∣∣∣ {g(𝑖,𝑘,𝑡)𝑓

}, 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

)

⩽ 𝐸{h} exp

⎛⎜⎜⎜⎝− 𝑃1 (h)ℋ M

(𝑖)Δ h

2

(𝑇∑𝑡=1

(𝑁𝐹 + 1

1 + 𝑃1

𝐾∑𝑘=1

(𝑃

(𝑘)2


)ℋg(𝑖,𝑘,𝑡)𝑓

)))⎞⎟⎟⎟⎠

=

∫ ⎧⎨⎩exp

⎛⎜⎜⎜⎝− 𝑃1 (h)ℋM

(𝑖)Δ h

2

(𝑇∑𝑡=1

(𝑁𝐹 + 1

1 + 𝑃1

𝐾∑𝑘=1

(𝑃

(𝑘)2



)))⎞⎟⎟⎟⎠𝒫 (h)

⎫⎬⎭𝑑h

𝜋𝑇

𝐾∑𝑘=1

(𝐿

(𝑘)ℎ +1

)

det (h)

.

(14)

𝒫 (h) =1

𝜋𝑇

𝐾∑𝑘=1

(𝐿

(𝑘)ℎ +1

)

det (Rh)

exp{− (h)ℋ (Rh)

−1 h}. (15)

R(𝑖)Δ =

⎛⎜⎜⎜⎝ 𝑃1M(𝑖)Δ

2

(𝑇∑𝑡=1

(𝑁𝐹 + 1

1 + 𝑃1

𝐾∑𝑘=1

(𝑃

(𝑘)2



))) + (Rh)−1

⎞⎟⎟⎟⎠−1

. (16)

1

𝜋𝑇

𝐾∑𝑘=1

(𝐿

(𝑘)ℎ +1

)

det(R

(𝑖)Δ

)∫ {

exp

(− (h)

ℋ (R

(𝑖)Δ

)−1

h

)}𝑑h = 1. (17)


∣∣ {g(𝑖,𝑘,𝑡)𝑓

}, 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

)≤

det(R

(𝑖)Δ

)det (Rh)

= 1

det

((R

(𝑖)Δ

)−1

Rh

)= 1

det

⎛⎜⎜⎜⎝ 𝑃1M(𝑖)Δ Rh

2

(𝑇∑𝑡=1

(𝑁𝐹 + 1

1+𝑃1

𝐾∑𝑘=1

(𝑃

(𝑘)2



))) + I𝑇

𝐾∑𝑘=1

(𝐿

(𝑘)ℎ +1

)

⎞⎟⎟⎟⎠.

(18)

the relation (17) always holds, and thus (14) can be simplifiedas (18), where the second equality in (18) is obtained throughinserting (16) within.

The corresponding rank and product criteria based onconditional PEP bound in (18) can be expressed as

1) Rank criterion: The minimum rank of M(𝑖)Δ Rh over

all pairs of different x(𝑖) and x(𝑖) should be as large aspossible.

2) Product criterion: The minimum value of the productof all non-zero eigenvalues of M(𝑖)

Δ Rh over all pairs ofdifferent x(𝑖) and x(𝑖) should be maximized.

To investigate the maximal achievable conditional diversityorder, i.e. the rank of M(𝑖)

Δ Rh, we further impose the following

assumptions:Assumption 1: Channels for DSTFC are assumed uncorre-

lated over different relays and OFDM blocks, although themulti-path channels to or from each relay may be corre-lated, and Rh is assumed to be full rank, i.e, rank (Rh) =

𝑇𝐾∑𝑘=1

(𝐿(𝑘)ℎ + 1

).

Assumption 2: Assume that, for all 1 ≤ 𝑘 ≤ 𝐾 and 1 ≤𝑡 ≤ 𝑇 , there exists at least one entry in g(𝑘,𝑡),

[g(𝑘,𝑡)

]𝑐(𝑘,𝑡),1

,

such that[g(𝑘,𝑡)

]𝑐(𝑘,𝑡),1

∕= 0, 0 ≤ 𝑐(𝑘,𝑡) ≤ 𝐿(𝑘)𝑔 , for each (𝑘, 𝑡)

pair.The achievable conditional diversity order for conditional

PEP in (18) is analyzed in Appendix B along with the design


of a(𝑖,𝑘,𝑡), where 𝑘 = 1, ...,𝐾 and 𝑡 = 1, ..., 𝑇 . Those resultsare summarized in the following theorem.

Theorem 1: Based on discrete frequency-selective chan-nel model for the OFDM-based relay network as de-scribed in Section II and under Assumptions 1 and 2,the achievable conditional diversity order of the 𝑖-thcodeword of the proposed DSTFC for PEP conditionedon channel

{g(𝑖,𝑘,𝑡)𝑓 , 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

}in (18) is

𝑇𝐾∑𝑘=1

(𝐿(𝑘)ℎ + 1

). This can be achieved through using

a(𝑖,𝑘,𝑡) =[f (𝑖)].((𝑘−1)𝐿𝑀)

(19)

and

𝑁𝐹 ≥𝐾∑𝑘=1

(𝐿(𝑘)𝑔 + 𝐿

(𝑘)ℎ + 1

), (20)

where 𝐿𝑀 > max

{ (𝐿(𝑘)𝑔 + 𝐿

(𝑘)ℎ + 1

),

1 ≤ 𝑘 ≤ 𝐾

}, 𝑘 = 1, ...,𝐾 ,

and 𝑡 = 1, ..., 𝑇 . These results hold for all 𝐿(𝑘)𝑔 < 𝐿

(𝑘)ℎ ,

𝐿(𝑘)ℎ = 𝐿

(𝑘)𝑔 , and 𝐿

(𝑘)𝑔 > 𝐿

(𝑘)ℎ cases.

Example: Based on Theorem 1, one DSTFC design exam-ple is provided here. Suppose 𝑁𝐶 = 32, 𝐿(𝑘)

ℎ = 𝐿(𝑘)𝑔 = 1,

𝑘 = 1, 2, and 𝐾 = 𝑇 = 2, thus 𝑁𝐹 ≥ 6. As mentionedearlier in Section II, 𝑁𝐹 has to be a power of 2, thus𝑁𝐹 = 8, 𝛼 = exp(𝑗 𝜋

32 ). We therefore have a(𝑖,𝑘,𝑡) =[𝜔128(𝑖−1)(𝑘−1), ..., 𝜔16(8𝑖−1)(𝑘−1)

]𝒯, where 𝑖 = 1, 2, 𝑘 =

1, 2, and 𝑡 = 1, 2.Now we return our attention to find PEP conditioned on

channel{h(𝑖,𝑘,𝑡)𝑓 , 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

}. Through aver-

aging (9) over{g(𝑖,𝑘,𝑡)𝑓 , 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

}, the PEP

conditioned on channel{h(𝑖,𝑘,𝑡)𝑓 , 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

}can be bounded by (21). Using the approximations (22)and (23), (21) can be approximately written as (24).Based on (24), similar to our previous discussions, itcan be shown that the achievable conditional diversityorder of the 𝑖-th codeword of the proposed DSTFC

is 𝑇𝐾∑𝑘=1

(𝐿(𝑘)𝑔 + 1

). However, unlike the result condi-

tioned on channel{h(𝑖,𝑘,𝑡)𝑓 , 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

}, this

conditional diversity order bound conditioned on channel{g(𝑖,𝑘,𝑡)𝑓 , 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

}for (24) may be notably

larger than the actual achievable conditional diversity orderbound due to the use of the aforementioned approximations.

B. Discussions on differences between average and condi-tional PEP -based designs

In Theorem 1 of [6], it is claimed that, in the case of 𝐿(1)ℎ =

... = 𝐿(𝐾)ℎ = 𝐿ℎ and 𝐿

(1)𝑔 = ... = 𝐿

(𝐾)𝑔 = 𝐿𝑔, the achievable

diversity for DSFCs is 𝐾min {𝐿ℎ, 𝐿𝑔}, which implies that,when 𝐿ℎ ∕= 𝐿𝑔, 𝐾 (max {𝐿ℎ, 𝐿𝑔} −min {𝐿ℎ, 𝐿𝑔}) extrafading paths cannot provide diversity benefits for systemperformance. However, according to our present diversityanalysis based on conditional PEP, even in the case of 𝐿 =

𝐿(1)ℎ = ... = 𝐿

(𝐾)ℎ = 𝐿ℎ > 𝐿𝑔 ≥

{𝐿(1)𝑔 , ..., 𝐿

(𝐾)𝑔

}, the

achievable diversity order of DSFC based on conditional PEPcan be 𝐾(𝐿ℎ + 1). Unlike our design approach, the DSFCswere designed under average PEP based diversity analysis, theresulting 𝑁𝐹 was set as 𝐾min (𝐿ℎ + 1, 𝐿𝑔 + 1) to achievetheir claimed diversity 𝐾min {𝐿ℎ, 𝐿𝑔}, which actually maynot fully exploit available diversity in the relay frequency-selective channels.

To explain the reasons for these somehow conflicting con-clusions, we provide the following remarks:

1) The full frequency diversity properties of DSTFC cannotbe determined only by the numbers of paths

2𝐾∑𝑘=1

(min

(𝐿(𝑘)ℎ + 1, 𝐿(𝑘)

𝑔 + 1))

,

where the number 2 is used for counting both sides ofmultipath channels. The extra number of paths

𝐾∑𝑘=1

⎛⎝ max(𝐿(𝑘)ℎ + 1, 𝐿

(𝑘)𝑔 + 1

)−

min(𝐿(𝑘)ℎ + 1, 𝐿

(𝑘)𝑔 + 1

) ⎞⎠may contribute to the system diversity of DSTFC.

2) The diversity analyses using one-side-channel condi-tional PEP leads to different DSTFC design parametersfrom those using average PEP. This difference intro-duces performance gains, which will be verified throughsimulations in Section IV.

3) There are two kinds of different one-side-channel con-ditional PEP ( source-to-relay and relay destination ).However, both one-side-channel conditional PEP baseddiversity analyses will lead to the same design parame-ters of DSTFC.

4) Note that, in [6], it was claimed that the diver-sity order be limited by 𝐾min (𝐿ℎ + 1, 𝐿𝑔 + 1).However, we have found that the diversity limi-tation of 𝐾min (𝐿ℎ + 1, 𝐿𝑔 + 1) is only valid infrequency flat fading relay channels. In the caseof{𝐿(𝑘)𝑔 = 𝐿

(𝑘)ℎ = 0, 𝑘 = 1, ...𝑘

}, i.e. in flat (non-

frequency-selective) fading channels, the numbers ofpaths

2

𝐾∑𝑘=1

(min

(𝐿(𝑘)ℎ + 1, 𝐿(𝑘)

𝑔 + 1))

does fully determine conditional diversity properties ofDSTFC for each OFDM block, which is coincidentalwith average PEP based diversity analysis of DSTC in[20].

To provide some further insights on differences betweenaverage and conditional PEP based designs, it is beneficialto also investigate diversity gains directly through derivingaverage PEP. Here, as an example, we only consider the caseof 𝐿(𝑘)

ℎ > 𝐿(𝑘)𝑔 > 0, 𝑘 = 1, ..,𝐾 .

Denote g =[[g(1)]𝒯

, ...,[g(𝑇 )

]𝒯 ]𝒯with g(𝑡) =[[

g(1,𝑡)]𝒯

, ...,[g(𝐾,𝑡)

]𝒯 ]𝒯. We use set partitioning to cat-

egorize whether or not there exists at least one non-zero entryin g(𝑘,𝑡). Adopting similar proof steps of Appendices A andB and using

{a(𝑖,𝑘,𝑡)

}specified in Theorem 1, the following

relations can be proved:



∣∣ {h(𝑖,𝑘,𝑡)𝑓

}, 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

)

≤ 𝐸{g(𝑖,𝑘,𝑡)𝑓

} exp

⎛⎜⎜⎜⎜⎜⎝−

∥∥∥∥∥√𝑃1diag(Δ(𝑖)

) 𝐾∑𝑘=1

(𝛽(𝑘)

√𝑃

(𝑘)2 h

(𝑖,𝑘)

𝑓

)∥∥∥∥∥2

2𝐸{v(𝑖,𝑘,𝑡)𝑓 ,u

(𝑖,𝑡)𝑓

}(∥∥∥u(𝑖)

𝑓

∥∥∥2)⎞⎟⎟⎟⎟⎟⎠ .

(21)

𝐸{g(𝑖,𝑘,𝑡)𝑓

} exp

⎛⎜⎜⎜⎜⎜⎝−


) 𝐾∑𝑘=1

(𝛽(𝑘)

√𝑃

(𝑘)2 h

(𝑖,𝑘)

𝑓

)∥∥∥∥∥2


(𝑖,𝑡)𝑓

}(∥∥∥u(𝑖)

𝑓

∥∥∥2)⎞⎟⎟⎟⎟⎟⎠

≈ 𝐸{g(𝑖,𝑘,𝑡)𝑓

} exp

⎛⎜⎜⎜⎜⎜⎝−


) 𝐾∑𝑘=1

(𝛽(𝑘)

√𝑃

(𝑘)2 h

(𝑖,𝑘)

𝑓

)∥∥∥∥∥2


(𝑖,𝑡)𝑓 ,g

(𝑖,𝑘,𝑡)𝑓

}(∥∥∥u(𝑖)

𝑓

∥∥∥2)⎞⎟⎟⎟⎟⎟⎠

(22)


(𝑖,𝑡)𝑓

}(

𝑇∑𝑡=1

(𝐾∑𝑘=1

(𝑃

(𝑘)2



)))≈

𝑇∑𝑡=1

(𝐾∑𝑘=1

(𝑃

(𝑘)2 𝑁𝐹

)),

(23)


∣∣ {h(𝑖,𝑘,𝑡)𝑓

}, 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

)

<∼𝐸{g(𝑖,𝑘,𝑡)𝑓

} exp

⎛⎜⎜⎜⎜⎜⎝−


) 𝐾∑𝑘=1

(𝛽(𝑘)

√𝑃

(𝑘)2 h

(𝑖,𝑘)

𝑓

)∥∥∥∥∥2

2

𝑇∑𝑡=1

(𝐾∑𝑘=1

(𝑃

(𝑘)2 𝑁𝐹

))⎞⎟⎟⎟⎟⎟⎠ .

(24)

1) For 𝑁𝐹 ≥𝐾∑𝑘=1


(𝑘)ℎ + 1

), we have

𝑁 (𝑖,g,1) = rank(M

(𝑖)Δ Rℎ

)∣∣∣g

= rank(M

(𝑖)Δ B)∣∣∣

g

=𝑇∑𝑡=1

(𝐾∑𝑘=1

((𝐿(𝑘)ℎ + 1

)𝛿(g(𝑘,𝑡) ∕= 0

))),

(25)

2) For 𝑁𝐹 =𝐾∑𝑘=1

(min

(𝐿(𝑘)ℎ + 1, 𝐿

(𝑘)𝑔 + 1

)), we have

(26)

Denote the PEP conditioned on channel{g(𝑖,𝑘,𝑡)𝑓 , 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

}as 𝛼(𝑖∣g), i.e.,

𝛼(𝑖∣g) = Pr

⎛⎜⎝x(𝑖) → x(𝑖)

∣∣∣∣∣∣∣{g(𝑖,𝑘,𝑡)𝑓

},

𝑘 = 1, ...,𝐾,𝑡 = 1, ..., 𝑇

⎞⎟⎠ . (27)

To remove the conditions of channel

{g(𝑖,𝑘,𝑡)𝑓 , 𝑘 = 1, ...,𝐾, 𝑡 = 1, ..., 𝑇

}, we need to calculate

𝛼(𝑖,𝑏) =

∫∫∫⊙𝛼(𝑖∣g)𝑚g𝑑g, (28)

where 𝑑g =𝐾∏𝑘=1

𝑇∏𝑡=1

𝐿(𝑘)𝑔∏

𝑙𝑔=0

𝑑𝑔(𝑘,𝑡)𝑙𝑔

, and 𝑚g is the probability

density function of g. (28) can be rewritten as the form ofdiscrete summations

𝛼(𝑖,𝑏) =∑

...∑

𝛼(𝑖∣g)𝑚gΔg (29)

where Δg is the discrete multiple integration unit area.It is useful to derive the inequality (30), where {𝜆𝑚}

are non-zero eigenvalues of (Rh)1/2 M

(𝑖)Δ (Rh)

1/2, 𝑁 (𝑖) =

rank((Rh)

1/2M

(𝑖)Δ (Rh)

1/2)

= rank(M

(𝑖)Δ Rh

), 𝑔(𝑖,Δ) is

defined in (31), 𝜌g is symbol signal-to-noise ratio (SNR)conditioned on g, and 𝜇 is some positive constant dependingon the system structure.

Denote 𝛼(𝑖∣g) in (27) for the case of 𝑁𝐹 ≥


𝑁 (𝑖,g,2)= rank(M

(𝑖)Δ Rℎ

)∣∣∣g= rank

(M

(𝑖)Δ Rℎ

)=

𝑇∑𝑡=1

(𝐾∑𝑘=1

(min

(𝐿(𝑘)ℎ + 1, 𝐿(𝑘)

𝑔 + 1)𝛿(g(𝑘,𝑡) ∕= 0

))). (26)

(det

((R

(𝑖)Δ

)−1

Rh

))−1

=

(det

((R

(𝑖)Δ

)−1

Rh

))−1

=

⎛⎝det

⎛⎝𝑔(𝑖,Δ)𝑃1 (Rh)1/2

M(𝑖)Δ (Rh)

1/2+ I

𝑇𝐾∑

𝑘=1

(𝐿

(𝑘)ℎ +1

)

⎞⎠⎞⎠−1

=𝑁(𝑖)∏𝑚=1

(1 + 𝜇𝜌g𝜆𝑚)−1 ≤ (𝜇min {𝜆𝑚} 𝜌g)−𝑁(𝑖)

.

(30)

𝑔(𝑖,Δ) =

(2

(𝑇∑𝑡=1

(𝑁𝐹 +

1

1 + 𝑃1

𝐾∑𝑘=1

(𝑃

(𝑘)2



))))−1

. (31)

𝐾∑𝑘=1


(𝑘)ℎ + 1

)as 𝛼

(𝑖∣g)1 . Denote 𝛼(𝑖∣g) for the case

of 𝑁𝐹 =𝐾∑𝑘=1

(min

(𝐿(𝑘)ℎ + 1, 𝐿

(𝑘)𝑔 + 1

))as 𝛼

(𝑖∣g)2 . Using

(25), (26), and (30), (27) can be bounded as

𝛼(𝑖∣g)1 ⩽ 𝑑(𝑖,g,1) (𝜌g)

−𝑁(𝑖,g,1)

= t(𝑖,𝜌g,1) (32)

for 𝑁𝐹 ≥𝐾∑𝑘=1


(𝑘)ℎ + 1

)and

𝛼(𝑖∣g)2 ⩽ 𝑑(𝑖,g,2) (𝜌)−𝑁(𝑖,g,2)

= t(𝑖,𝜌g,2), (33)

for 𝑁𝐹 =𝐾∑𝑘=1

(min

(𝐿(𝑘)ℎ + 1, 𝐿

(𝑘)𝑔 + 1

)), where 𝑑(𝑖,g,1)

and 𝑑(𝑖,g,2) are two positive values conditioned on g. In high

SNR, (𝜌g)−𝑁(𝑖,g,1)

and (𝜌g)−𝑁(𝑖,g,1)

dominate t(𝑖,𝜌g,1) andt(𝑖,𝜌g,2) instead of 𝑑(𝑖,g,1) and 𝑑(𝑖,g,2), respectively. Using(25), (26), (32), and (33), we have the following observations:If g ∕= 0, we have 𝑁 (𝑖,g,1) > 𝑁 (𝑖,g,2) and 𝛼(𝑖∣g)

1 < 𝛼(𝑖∣g)2 , and

the slope for log(t(𝑖,𝜌g ,1)

)versus SNR conditioned on g is

larger than that for log(t(𝑖,𝜌g,2)

)versus SNR conditioned on

g. If g = 0, 𝑁 (𝑖,g,1) = 𝑁 (𝑖,g,2), it means that communicationlinks from relay nodes to the destination node are completelydisconnected (which usually happens at very low probability),and DSTFC cannot provide any help in this case.

Using the above mentioned results, we conclude that thefreedom order for average PEP (28) or (29) under 𝑁𝐹 ≥𝐾∑𝑘=1


(𝑘)ℎ + 1

)is larger than that under 𝑁𝐹 =

𝐾∑𝑘=1

(min

(𝐿(𝑘)ℎ + 1, 𝐿

(𝑘)𝑔 + 1

)).

IV. SIMULATION RESULTS AND DISCUSSIONS

In this section, we present Monte-Carlo simulation resultsto demonstrate the error-rate performance of the proposedschemes. We assume that the underlying frequency-selectivechannels follow uniform power delay profile, and remainconstant within a fixed integer number of OFDM blocks,denoted as the channel change interval (𝐶𝐶𝐼), and change

independently from one block to another block. The followingare further assumed in all simulations:

∙ 𝐿(1)ℎ = ... = 𝐿

(𝐾)ℎ = 𝐿ℎ and 𝐿

(1)𝑔 = ... = 𝐿

(𝐾)𝑔 = 𝐿𝑔.

∙ 𝑃1 = 𝑃 and 𝑃(1)2 = ... = 𝑃

(𝐾)2 = 𝑃

𝐾 .∙ CCIs for source-to-relay and relay-to-destination chan-

nels are equal to 1.∙ Before DSTFC precoding, data symbols are modulated

using 4-QAM modulation.The horizontal axes of performance figures is 10 log10 𝑃 . Notethe average symbol SNR is proportional to 𝑃 .

In Fig. 1, for 𝑇 = 1 and 𝐾 = 2, we compare the biterror rate (BER) performance of our proposed DSFC withthat proposed in [6]1. From Figure 1, it is observed that,

∙ The performance of DSFC for the case of 𝐿ℎ = 𝐿𝑔 = 2is worse than that for the cases of (𝐿ℎ = 5) > (𝐿𝑔 = 2)and (𝐿ℎ = 2) < (𝐿𝑔 = 5),

∙ The performance of DSFC for the case of (𝐿ℎ = 5) >(𝐿𝑔 = 2) is better than that for the case of (𝐿ℎ = 2) <(𝐿𝑔 = 5),

∙ The performance of DSFC for the case that 𝑁𝐹 is set toat least 𝐾(𝐿ℎ + 𝐿𝑔 + 1) is better than for the case that𝑁𝐹 is set to at least 𝐾min {𝐿ℎ + 1, 𝐿𝑔 + 1}.

The second observation implies that the extra frequency-selective order for source-to-relay paths provided strongerdiversity advantages over that for relay-to-destination paths.This phenomenon incurs due to the existing relay noises, andthus we would provide the following conjecture:

Conjecture 1: Due to the deterioration incurred by relaynoises, the degradation of the conditional diversity orderof DSFC and DSTFC in frequency-selective fading relaychannels is much larger than that of DSTC in frequency-flatfading relay channels. This effect may be more significant inlow and medium SNR regions.

1We remark that the discussions in [6] are based on general discretefrequency-selective channel models (channel taps are not necessarily uni-formly spaced), while our discussions are based on SSTDL as described inSection II. In the comparisons of Figure 1, the BER performance resultsfor DSFC approaches of Zhang, et al. are also based on the same SSTDLfrequency-selective channel models.


10 15 20 25 30 3510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

10log10

P (dB)

BER

DSFC (Zhang et. al. settings),NF=8,L

h=2,L

g=2


h=2,L

g=5


h=5,L

g=2

DSFC (Our settings),NF=16,L

h=2,L

g=2


h=2,L

g=5


h=5,L

g=2

Fig. 1. Performance comparison of proposed DSFCs with those in [6] (𝑁𝐶 =64).

27 28 29 30 31 32 3310

−4

10−3

10log10

P (dB)

BE

R

DFC (Our settings),K=1,T=1,NF=16,L

h=6,L

g=6


h=6,L

g=9


h=9,L

g=6

Fig. 2. BER Performance of DFCs (𝑁𝐶 = 32).

As mentioned in Section III, the authors of [6] have statedthat the achievable diversity order based on average PEPis 𝐾min {𝐿ℎ + 1, 𝐿𝑔 + 1}, and the corresponding minimum𝑁𝐹 is set to be at least 𝐾min {𝐿ℎ + 1, 𝐿𝑔 + 1}. How-ever, according to our results based on the conditional PEPbound in Section III, we have stated that the achievablediversity order based one-side-channel conditional PEP is𝐾max {𝐿ℎ + 1, 𝐿𝑔 + 1}, and the corresponding minimum𝑁𝐹 is set to be at least 𝐾(𝐿ℎ + 𝐿𝑔 + 1). Observations fromFig. 1 support our statements.

For 𝐾 = 1 and 𝑇 = 1, DSTFC becomes frequency-onlycoding, which can be termed as distributed frequency code(DFC). In Fig. 2, we investigate the BER performance of DFCunder different frequency-selective channel conditions. It isobserved that,

∙ The performance of DFC for the case of 𝐿ℎ = 𝐿𝑔 = 6

10 15 20 25 30 3510

−5

10−4

10−3

10−2

10−1

10log10

P (dB)

BER

DSFC (Zhang et. al. settings),K=2,T=1,NF=4,L

h=1,L

g=1

DSFC (Our settings),K=2,T=1,NF=8,L

h=1,L

g=1

DSTFC (Our settings),K=2,T=2,NF=8,L

h=1,L

g=1

Fig. 3. Performance comparison of DSTFC and DSFCs (𝑁𝐶 = 32).

12 14 16 18 20 22 24 2610

−6

10−5

10−4

10−3

10−2

10−1

10log10

P (dB)

BE

R

DSTFC (Our settings),K=2,T=2,N

F=8,L

h=1,L

g=1

DSTFC (Our settings),K=2,T=2,NF=8,L

h=2,L

g=1

Fig. 4. Performance of DSTFC under different channel conditions (𝑁𝐶 =32).

is worse than that for the cases of (𝐿ℎ = 6) < (𝐿𝑔 = 9)and (𝐿ℎ = 9) > (𝐿𝑔 = 6),

∙ The performance of DFC for the case of (𝐿ℎ = 9) >(𝐿𝑔 = 6) is better than that for the case of (𝐿ℎ = 6) <(𝐿𝑔 = 9),

The above results for single relay-case further confirm ourdiversity analysis in Section III.

Figure 3 illustrates performance comparisons betweenDSTFC and DSFC approaches. In time-varying channels (i.e,channel changes from one OFDM block to another), theproposed DSTFC demonstrates significant performance gainsover DSFC at the price of longer decoding delay and highercomputational complexity. In Fig. 4, the BER performancesof DSTFC under different channel conditions are compared,which further confirms that, the channel order differences be-tween source-to-relay and relay-to-destination may contribute


to extra gains in diversity performance as discussed in SectionIII.

V. CONCLUSION

We have proposed high-rate distributed space-time-frequency coding for AF OFDM-based cooperative relay net-works over frequency-selective fading channels. Using bothconditional and average pairwise error probability, we haveanalyzed DSTFC diversity performance in block time-varyingfrequency-selective fading channels and pointed out the signif-icant differences between diversity performance in frequency-selective channels and that in frequency-flat fading channels.Such observations help obtain better design criteria for DST-FCs. We have shown that high-rate DSTFC may significantlyoutperform high-rate DSFC due to efficient exploitation oftime diversity over multiple OFDM blocks.

APPENDIX ADERIVATION OF A

(𝑖)ℎ

Denote f (𝑖) =

[𝜔𝑝

(𝑖)1 −1, ..., 𝜔

𝑝(𝑖)𝑁𝐹

−1

]𝒯︸︷︷︸

𝑁𝐹

.

The term 𝐻(𝑘,𝑡)𝑝 𝐺

(𝑘,𝑡)𝑝 is derived in (34),

where[z(𝑘)𝑝

]𝑙2+

(𝐿

(𝑘)𝑔 +1

)𝑙1+1,1

= 𝜔−(𝑙1+𝑙2)(𝑝−1),[b(𝑘,𝑡)

]𝑙2+

(𝐿

(𝑘)𝑔 +1

)𝑙1+1,1

= ℎ(𝑘,𝑡)(𝑙1)

𝑔(𝑘,𝑡)(𝑙2)

, 𝑙1 =

0, ..., 𝐿(𝑘)ℎ , 𝑙2 = 0, ..., 𝐿

(𝑘)𝑔 , b(𝑘,𝑡) = 𝜓(𝑘,𝑡)h(𝑘,𝑡), and

𝜓(𝑘,𝑡) = diag(I𝐿

(𝑘)ℎ

⊗ g(𝑘,𝑡)). Therefore, we have (35),

where[Ω(𝑖,𝑘)

]:,𝑙2+

(𝐿

(𝑘)𝑔 +1

)𝑙1+1

=[f (𝑖)].(𝑙1+𝑙2).

Note that

h(𝑖)

𝑓 =

𝐾∑𝑘=1

(𝛽(𝑘)

√𝑃

(𝑘)2 h

(𝑖,𝑘)

𝑓

)=[Ξ(𝑖,1), ...,Ξ(𝑖,𝑇 )

]= M(𝑖)b.

(36)

In (36), each element can be written as

Ξ(𝑖,𝑡) =

𝐾∑𝑘=1

(𝛽(𝑘)

√𝑃

(𝑘)2

[a(𝑖,𝑘,𝑡) ∘ h(𝑖,𝑘,𝑡)

𝑓 ∘ g(𝑖,𝑘,𝑡)𝑓

]𝒯 )= M(𝑖,𝑡)b(𝑡),

(37)

where

h(𝑡) =

[[h(1,𝑡)

]𝒯, ...,[h(𝐾,𝑡)

]𝒯 ]𝒯,

h =

[[h(1)]𝒯

, ...,[h(𝑇 )

]𝒯 ]𝒯,

M(𝑖) = diag(M(𝑖,1), ...,M(𝑖,𝑇 )

),

M(𝑖,𝑡) =[M(𝑖,1,𝑡), ...,M(𝑖,𝐾,𝑡)

],

M(𝑖,𝑘,𝑡) = 𝛽(𝑘)

√𝑃

(𝑘)2 diag

(a(𝑖,𝑘,𝑡)

)Ω(𝑖,𝑘),

b(𝑡) =

[[b(1,𝑡)

]𝒯, ...,[b(𝐾,𝑡)

]𝒯 ]𝒯,

b =

[[b(1)]𝒯

, ...,[b(𝑇 )

]𝒯 ]𝒯,

b(𝑡) = 𝜓(𝑡)g h(𝑡), 𝜓(𝑡)

g = diag(𝜓(1,𝑡)g , ..., 𝜓

(𝐾,𝑡)g

), b = 𝜓gh,

and 𝜓g = diag(𝜓(1)g , ..., 𝜓

(𝑇 )g

). Therefore, A(𝑖)

ℎ is calculatedas

A(𝑖)ℎ = M(𝑖)b (b)

ℋ (M(𝑖)

)ℋ= M(𝑖)𝜓gh (h)ℋ (𝜓g)

ℋ (M(𝑖))ℋ

.(38)

APPENDIX BACHIEVABLE RANK OF M

(𝑖)Δ Rh

In this section, we investigate the rank of M(𝑖)Δ Rh. Under

Assumption 1, we have Rh = diag (Rh(1) , ...,Rh(𝑇)), where

Rh(𝑡) = 𝐸h(𝑡)

(h(𝑡)

(h(𝑡))ℋ)

= diag (Rh(1,𝑡) , ...,Rh(𝐾,𝑡)),

Rh(𝑘,𝑡) = 𝐸h(𝑘,𝑡)

(h(𝑘,𝑡)

(h(𝑘,𝑡)

)ℋ). Further, denote Φh =

(Rh)12 = diag (Φh(1) , ...,Φh(𝑇)), where Φh(𝑡) = (Rh(𝑡))

12 =

diag (Φh(1,𝑡) , ...,Φh(𝐾,𝑡)), Φh(𝑘,𝑡) = (Rh(𝑘,𝑡))12 , 𝑘 = 1, ...,𝐾 ,

𝑡 = 1, ..., 𝑇 .Recall that M

(𝑖)Δ = (𝜓g)

ℋ (M(𝑖)

)ℋ (Δ(𝑖)

)ℋΔ(𝑖)M(𝑖)𝜓g,

where𝜓g = diag

(𝜓(1)g , ..., 𝜓(𝑇 )

g

),

𝜓(𝑡)g = diag

(𝜓(1,𝑡)g , ..., 𝜓(𝐾,𝑡)

g

),

𝜓(𝑘,𝑡)g = diag

(I𝐿

(𝑘)ℎ

⊗ g(𝑘,𝑡)).

Provided that Δ(𝑖) is to be full rank, i.e., rank(Δ(𝑖)

)= 𝑁𝐹𝑇 ,

the following holds

rank(M

(𝑖)Δ Rh

)= rank

(Δ(𝑖)M(𝑖)𝜓gΦh

)= rank

(M(𝑖)B

) (39)

where B = 𝜓gΦh.

A. Investigation of B

Note that matrix B can be calculated as B =diag

(B(1), ...,B(𝑇 )

), where B(𝑡) = 𝜓

(𝑡)g Φh(𝑡) , B(𝑡) =

diag(B(1,𝑡), ...,B(𝐾,𝑡)

), B(𝑘,𝑡) = 𝜓

(𝑘,𝑡)g Φh(𝑘,𝑡) . We know that

Rh(𝑘,𝑡) , 𝑘 = 1, ...,𝐾 , 𝑡 = 1, ..., 𝑇 , are positive definiteHermitian matrices, thus all diagonal entries of Rh(𝑘,𝑡) arenon-zero, so are Φh(1,𝑡) , 𝑘 = 1, ...,𝐾 , 𝑡 = 1, ..., 𝑇 .

B. Investigation of M(𝑖)

Recall that M(𝑖) = diag(M(𝑖,1), ...,M(𝑖,𝑇 )

),

M(𝑖,𝑡) =[M(𝑖,1,𝑡), ...,M(𝑖,𝐾,𝑡)

], M(𝑖,𝑘,𝑡) =

𝛽(𝑘)

√𝑃

(𝑘)2 diag

(a(𝑖,𝑘,𝑡)

)Ω(𝑖,𝑘). We thus know that

rank(M(𝑖)

)=

𝑇∑𝑡=1

rank(M(𝑖,𝑡)

). To maximize the rank

of M(𝑖), it is necessary to maximize the rank of eachM(𝑖,𝑡), respectively. This requires to properly design a(𝑘,𝑡).Recall that

[Ω(𝑖,𝑘)

]:,𝑙2+

(𝐿

(𝑘)𝑔 +1

)𝑙1+1

=[f (𝑖)].(𝑙1+𝑙2), where


𝐻(𝑘,𝑡)𝑝 𝐺(𝑘,𝑡)

𝑝 =

⎛⎝𝐿(𝑘)ℎ∑

𝑙=𝑜

(ℎ(𝑘,𝑡)(𝑙) 𝑒−𝑗(2𝜋/𝑁𝑐)𝑙(𝑝−1)

)⎞⎠⎛⎝𝐿(𝑘)𝑔∑

𝑙=𝑜

(𝑔(𝑘,𝑡)(𝑙) 𝑒−𝑗(2𝜋/𝑁𝑐)𝑙(𝑝−1)

)⎞⎠=

𝐿(𝑘)ℎ∑

𝑙1=𝑜

⎛⎝𝐿(𝑘)𝑔∑

𝑙2=𝑜

(ℎ(𝑘,𝑡)(𝑙1)

𝑔(𝑘,𝑡)(𝑙2)

𝑒−𝑗(2𝜋/𝑁𝑐)(𝑝−1)[𝑙1+𝑙2])⎞⎠ =

[z(𝑘)𝑝

]𝒯b(𝑘,𝑡).

(34)

h(𝑖,𝑘,𝑡)𝑓 ∘ g(𝑖,𝑘,𝑡)

𝑓 =

⎛⎝𝐿(𝑘)ℎ∑

𝑙1=𝑜

(ℎ(𝑘,𝑡)(𝑙1)

[f (𝑖)].(𝑙1))⎞⎠⎛⎝𝐿(𝑘)

𝑔∑𝑙2=𝑜

(𝑔(𝑘,𝑡)(𝑙2)

[f (𝑖)].(𝑙2))⎞⎠ = Ω(𝑖,𝑘)b(𝑘,𝑡). (35)

f (𝑖) =

[𝜔−(𝑝(𝑖)1 −1

), ..., 𝜔

−(𝑝(𝑖)𝑁𝐹

−1)]𝒯

︸︷︷︸𝑁𝐹

, 𝑙1 = 0, ..., 𝐿(𝑘)ℎ , and

𝑙2 = 0, ..., 𝐿(𝑘)𝑔 .

Assume that our design ensures[f (𝑖)].(𝑎1) and

[f (𝑖)].(𝑎2)

are linear-independent for all 𝑎1 ∕= 𝑎2, which can be achievedby choosing subcarrier indices

{𝑝(𝑖)𝑎

}such that 𝑝(𝑖)𝑎 + 𝑏 =

𝑝(𝑖)𝑎+1, where 𝑎 = 1, ..., 𝑁𝐹 − 1. If 𝑏 = 1, adjacent subcarrier

indices are chosen. Observing that[Ω(𝑖,𝑘)

]:,𝑙2+

(𝐿

(𝑘)𝑔 +1

)𝑙1+1

=[f (𝑖)].(𝑙1+𝑙2), where 𝑙2 = 0, 1, ..., 𝐿

(𝑘)𝑔 , 𝑙1 = 0, 1, ..., 𝐿

(𝑘)ℎ , al-

though Ω(𝑖,𝑘) have(𝐿(𝑘)ℎ + 1

)(𝐿(𝑘)𝑔 + 1

)columns, there are

only(𝐿(𝑘)𝑔 + 𝐿

(𝑘)ℎ + 1

)different columns regardless of differ-

ent coefficients, since there are totally only(𝐿(𝑘)𝑔 + 𝐿

(𝑘)ℎ + 1

)different element-wise power 𝑎 for

[f (𝑖)].(𝑎)

. This leads

to rank(Ω(𝑖,𝑘)

)=


(𝑘)ℎ + 1

). Note that, in

the case of diag(a(𝑘,𝑡)

)= I𝑁𝐹 , the maximal rank

of M(𝑖,𝑡) =

[𝛽(1)

√𝑃

(1)2 Ω(𝑖,1), ..., 𝛽(𝐾)

√𝑃

(𝐾)2 Ω(𝑖,𝐾)

]is

max{𝐿(𝑘)𝑔 + 𝐿

(𝑘)ℎ + 1, 𝑘 = 1, ...,𝐾

}. Now we set

a(𝑖,𝑘,𝑡) =[f (𝑖)].((𝑘−1)𝐿𝑀 )

, (40)

where 𝐿𝑀 > max{𝐿(𝑘)𝑔 + 𝐿

(𝑘)ℎ + 1, 𝑘 = 1, ...,𝐾

}. Using

a(𝑖,𝑘,𝑡) as specified in (40), M(𝑖,𝑡) has𝐾∑𝑘=1


(𝑘)ℎ + 1

)linear independent columns, in other words, rank

(M(𝑖,𝑡)

)=

𝐾∑𝑘=1


(𝑘)ℎ + 1

), for all 𝑡 = 1, ..., 𝑇 .

C. Rank of M(𝑖)B

Now we are ready to investigate the rank of M(𝑖)B. We canwrite M(𝑖)B as

M(𝑖)B = diag(M

(𝑖,1)B , ...,M

(𝑖,𝑇 )B

), (41)

where M(𝑖,𝑡)B = M(𝑖,𝑡)B(𝑡) =

[M

(𝑖,1,𝑡)B , ...,M

(𝑖,𝐾,𝑡)B

], and

M(𝑖,𝑘,𝑡)B = M(𝑖,𝑘,𝑡)B(𝑘,𝑡)

= 𝛽(𝑘)

√𝑃

(𝑘)2 diag

(a(𝑖,𝑘,𝑡)

)Ω(𝑖,𝑘)B(𝑘,𝑡)

= 𝛽(𝑘)

√𝑃

(𝑘)2 Ω

(𝑖,𝑘,𝑡)a B(𝑘,𝑡)

= 𝛽(𝑘)

√𝑃

(𝑘)2 Ω

(𝑖,𝑘,𝑡)B ,

(42)

where[Ω

(𝑖,𝑘,𝑡)a

]:,𝑙2+

(𝐿

(𝑘)𝑔 +1

)𝑙1+1

=[f (𝑖)].(𝑙1+𝑙2+(𝑘−1)𝐿𝑀)

,

Ω(𝑖,𝑘,𝑡)B = Ω

(𝑖,𝑘,𝑡)a B(𝑘,𝑡).

Denote Ω(𝑖,𝑘,𝑡)a(𝑙1)

=[Ω

(𝑖,𝑘,𝑡)a

]:,(𝐿

(𝑘)𝑔 +1

)𝑙1+1:𝐿

(𝑘)𝑔 +

(𝐿

(𝑘)𝑔 +1

)𝑙1+1

,

and thus Ω(𝑖,𝑘,𝑡)a =

[Ω

(𝑖,𝑘,𝑡)a(0) , ...,Ω

(𝑖,𝑘,𝑡)

a(𝐿(𝑘)ℎ )

]. Recall

B(𝑘,𝑡) = 𝜓(𝑘,𝑡)g Φh(1,𝑡) and 𝜓

(𝑘,𝑡)g = diag

(I𝐿

(𝑘)ℎ

⊗ g(𝑘,𝑡)).

We then obtain

Ω(𝑖,𝑘,𝑡)B =

[g(𝑖,𝑘,𝑡)(0) , ...,g

(𝑖,𝑘,𝑡)

(𝐿(𝑘)ℎ )

]Φh(𝑘,𝑡) , (43)

where g(𝑖,𝑘,𝑡)(𝑙1)

= Ω(𝑖,𝑘,𝑡)a(𝑙1)

g(𝑘,𝑡), 𝑙1 = 0, ..., 𝐿(𝑘)ℎ .

g(𝑖,𝑘,𝑡)(𝑙1)

= Ω(𝑖,𝑘,𝑡)a(𝑙1)

g(𝑘,𝑡)

=𝐿(𝑘)

𝑔∑𝑙2=0

(𝑔(𝑘,𝑡)𝑙2

[f (𝑖)].(𝑙1+𝑙2+(𝑘−1)𝐿𝑀)

)=

𝐿𝑁(𝑘,𝑡)𝑔∑𝑚2=1

(𝑔(𝑘,𝑡)𝜏(𝑚2)

[f (𝑖)].(𝑙1+𝜏(𝑚2)+(𝑘−1)𝐿𝑀)

),

(44)

where 𝐿𝑁(𝑘,𝑡)𝑔 is the total number of non-zero entries in

vector g(𝑘,𝑡), the non-zero entries of g(𝑘,𝑡) are denotedas 𝑔

(𝑘,𝑡)𝜏(1) , ..., 𝑔

(𝑘,𝑡)

𝜏(𝐿𝑁(𝑘,𝑡)𝑔 )

, 0 ≤ 𝜏 (1) < 𝜏 (2) < ... <

𝜏(𝐿𝑁(𝑘,𝑡)𝑔

)≤ 𝐿

(𝑘)𝑔 . Based on Assumption 2, 𝐿𝑁(𝑘,𝑡)

𝑔 ∕= 0

hold for all 1 ≤ 𝑘 ≤ 𝐾 and 1 ≤ 𝑡 ≤ 𝑇 . Since g(𝑖,𝑘,𝑡)(𝑙1)

is

the linear combination of 𝐿𝑁(𝑘,𝑡)𝑔 linear independent columns{[

f (𝑖)].(𝑙1+𝜏(𝑚2)+(𝑘−1)𝐿𝑀)

∣∣∣ 1 ≤ 𝑚2 ≤ 𝐿𝑁(𝑘,𝑡)𝑔

}, g

(𝑖,𝑘,𝑡)(𝑙1)

∕=0 holds for all 0 ≤ 𝑙1 ≤ 𝐿

(𝑘)ℎ , 1 ≤ 𝑘 ≤ 𝐾 , 1 ≤ 𝑡 ≤ 𝑇 , and

0 ≤ 𝑙1 ≤ 𝐿(𝑘)ℎ . Note that

[f (𝑖)].(𝑙1+𝜏(𝐿𝑁(𝑘,𝑡)

𝑔 )+(𝑘−1)𝐿𝑀)does

not exist in the linear combination within g(𝑖,𝑘,𝑡)(0) , ...,g

(𝑖,𝑘,𝑡)(𝑙1−1),

respectively, where 0 ≤ 𝑙1 ≤ 𝐿(𝑘)ℎ . Thus, g

(𝑖,𝑘,𝑡)(𝑙1)

is linear

independent to all g(𝑖,𝑘,𝑡)((𝑙2)

for 𝐿(𝑘)ℎ ≥ 𝑙1 > 𝑙2 ≥ 0. Using this


recursive way, it can be proven that any two vectors within

the set

{g(𝑖,𝑘,𝑡)(0) , ...,g

(𝑖,𝑘,𝑡)

(𝐿(𝑘)ℎ )

}are pairwise independent.

Now we can calculate the rank of Ω(𝑖,𝑘,𝑡)B . Using

rank (Φh(𝑘,𝑡)) = 𝐿(𝑘)ℎ , the following holds

rank(Ω

(𝑖,𝑘,𝑡)B

)= rank

([g(𝑖,𝑘,𝑡)(0) , ...,g

(𝑖,𝑘,𝑡)

(𝐿(𝑘)ℎ )

])= 𝐿

(𝑘)ℎ + 1.

(45)

Further, note that the set{ [f (𝑖)].(𝑙1+𝜏(𝑚2)+(𝑘1−1)𝐿𝑀 )

,

1 ≤ 𝑚2 ≤ 𝐿𝑁(𝑘1,𝑡)𝑔 , 0 ≤ 𝑙1 ≤ 𝐿

(𝑘1)ℎ

}and

the set

{ [f (𝑖)].(𝑙1+𝜏(𝑚2)+(𝑘2−1)𝐿𝑀 )

,

1 ≤ 𝑚2 ≤ 𝐿𝑁(𝑘2,𝑡)𝑔 , 0 ≤ 𝑙1 ≤ 𝐿

(𝑘2)ℎ

}do not have

any common element vector, where 1 ≤ 𝑘1 ∕= 𝑘2 ≤ 𝐾 ,

𝐿𝑀 > max

{ (𝐿(𝑘)𝑔 + 𝐿

(𝑘)ℎ + 1

),

1 ≤ 𝑘 ≤ 𝑀

}, thus any element

vector in

{g(𝑖,𝑘1,𝑡)(0) , ...,g

(𝑖,𝑘1,𝑡)

(𝐿(𝑘)ℎ )

}is linear-independent to any

element vector in

{g(𝑖,𝑘2,𝑡)(0) , ...,g

(𝑖,𝑘2,𝑡)

(𝐿(𝑘)ℎ )

}. Therefore,

rank(M

(𝑖,𝑡)B

)=

𝐾∑𝑘=1

rank(Ω

(𝑖,𝑘,𝑡)B

)(46)

and

rank(M

(𝑖)Δ Rh

)= rank

(M(𝑖)B

)=

𝑇∑𝑡=1

rank(M

(𝑖,𝑡)B

)= 𝑇

𝐾∑𝑘=1

(𝐿(𝑘)ℎ + 1

) (47)

To achieve the achievable rank provided in (47), there

are𝐾∑𝑘=1


(𝑘)ℎ + 1

)available element vectors in set{[

f (𝑖)].(𝑙1+𝑙2+(𝑘−1)𝐿𝑀 )

}, and note that those element vectors

have to be linear-independent. Thus we have to choose

𝑁𝐹 ≥𝐾∑𝑘=1


(𝑘)ℎ + 1

). (48)

ACKNOWLEDGEMENTS

The authors would like to thank the editor and anonymousreviewers for invaluable comments and suggestions.

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Jinsong Wu received the B.Eng. degree fromChangsha Railway University (now part of CentralSouth University), Changsha, China, in 1990, theM.A.Sc. degree from Dalhousie University, Halifax,NS, Canada, in 2001, the Ph.D. degree from Queen’sUniversity, Kingston, ON, Canada, in 2006, all inelectrical engineering.

He experienced research and development posi-tions of information technology or telecommunica-tions industry with STONE in China, Great-WallComputer in China, Nortel Networks in Canada,

Philips Research North America in USA, and Sprint-Nextel in USA. He iscurrently working as a research fellow in University of Waterloo, Canada.His recent research interests lie in general communications theory and signalprocessing, space-time-frequency processing and coding, cooperative com-munications, iterative decoding or processing, communication optimization,and network protocols with applications. He has served as technical programcommittee members in IEEE Global Communications Conference 2008 and2010 and IEEE International Conference on Communications 2009 and 2011.


Honggang Hu received a B.S. degree in Mathemat-ics in June 2000, and a B.E. degree in ElectricalEngineering in June 2001 from the University ofScience and Technology of China (USTC), Hefei,China, and the Ph.D. degree in Electrical Engineer-ing from the Graduate School of Chinese Academyof Sciences, Beijing, China, in July 2005. From July2005 to April 2007, he was a Postdoctoral Fellowat the Institute of Software, Chinese Academy ofSciences, Beijing, China. From August 2007 toJuly 2009, he was a Postdoctoral Fellow at the

University of Waterloo, Canada. Since August 2009, he has been a ResearchAssociate at the University of Waterloo, Canada. His research interests includepseudorandom sequences, cryptography, and coding theory.

Murat Uysal was born in Istanbul, Turkey in 1973.He received the B.Sc. and the M.Sc. degree in elec-tronics and communication engineering from Istan-bul Technical University, Istanbul, Turkey, in 1995and 1998, respectively, and the Ph.D. degree in elec-trical engineering from Texas A&M University, Col-lege Station, Texas, in 2001. Since 2002, he has beenwith the Department of Electrical and ComputerEngineering, University of Waterloo, Canada, wherehe is now an Associate Professor. He is currently onleave at Ozyegin University, Istanbul, Turkey. His

general research interests lie in communications theory and signal processingfor communications with special emphasis on wireless applications. Specificresearch areas include MIMO communication techniques, space-time coding,diversity techniques and coding for fading channels, cooperative commu-nication, and free-space (wireless) optical communication. Dr. Uysal is anAssociate Editor for IEEE TRANSACTIONS ON WIRELESS COMMUNICA-TIONS, IEEE COMMUNICATIONS LETTERS, and Wireless Communicationsand Mobile Computing (WCMC) journal. In the past he served as a GuestCo-Editor for WCMC Special Issue on “MIMO Communications” (October2004) and IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS

Special Issue on “Optical Wireless Communications” (December 2009). Overthe years, he has served on the technical program committee of more than70 international conferences and workshops in the communications area. Herecently co-chaired IEEE ICC 2007 Communication Theory Symposium andCCECE’08 Communications and Networking Symposium. He is currentlyserving as the Tutorial Chair of IEEE WCNC 2011. Dr. Uysal is a SeniorIEEE member.

High-Rate Distributed Space-Time-Frequency Coding for Wireless Cooperative Networks

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