Zero-forcing based Decode-and-Forward Cooperative Relaying Scheme over Doubly Selective Fading Channels

Ho Van Khuong, Tho Le-Ngoc

Department of Electrical and Computer Engineering, McGill University, Montreal, QC, Canada, H3A 2A7 khuong.hovan@mail.mcgill.ca, tho.le-ngoc@mcgill.ca

Abstract—This paper evaluates the performance of a decode-and-forward scheme using orthogonal frequency-division multiplexing (OFDM) over both time-selective and frequency-selective fading channels. We consider a simple detection based on the zero-forcing (ZF) principle to eliminate inter-carrier interference (ICI) caused by high mobility and to combine signals received from the source and the relay for high spatial diversity at the destination. The symbol error rate (SER) of the ZF-based cooperative relaying scheme is derived and validated by computer simulations. Various results show the superiority of cooperative relaying to direct transmission under same transmission power and bandwidth efficiency for any relay position, signal-to-noise ratio (SNR), path-loss exponent, and normalized Doppler frequency.

Keywords-Decode-and-forward cooperative relaying scheme; doubly selective fading; OFDM; zero-forcing detection

I. INTRODUCTION OFDM modulation is a bandwidth-efficient technique to

obviate inter-symbol interference (ISI) caused by quasi-static1 frequency-selective fading channels, and is widely used in many applications and standards such as xDSL, Digital Audio Broadcasting (DAB), Terrestrial Digital Video Broadcasting (DVB-T), IEEE 802.11 and 802.16 standards [1]. However, as users move very fast (e.g., broadband mobiles supported by WiMAX with speeds beyond 120 km/h) the time variation of the channel over an OFDM symbol period results in a loss of subcarrier orthogonality, which leads to ICI [2]. Various detection techniques are proposed, including ZF, minimum mean-squared error (MMSE), decision-feedback (DF), maximum a posteriori probability (MAP) equalizers [2]-[4]. Naturally, there is a performance-complexity trade-off among these equalizers. The ZF equalizer is the least complicated but has the lowest performance while the MAP equalizer has the highest performance but highest complexity.

Recently, cooperative relaying (CR) in which users share their own antennas to form a virtual MIMO system [5] has been proposed to achieve the potentials of the space diversity in addition to conventional diversity techniques [6] for further system performance enhancement [7]. It is attractive in some scenarios where terminals cannot afford multiple transmit/receive antennas, e.g., due to size or cost.

Various cooperative relaying schemes, which are broadly classified as decode-and-forward (DF) and amplify-and-forward (AF), have been proposed, e.g., in [8]-[9] with performance analysis in terms of SER, outage probability, diversity-multiplexing trade-off, ... for a simple 3-terminal scenario including a source, a relay and a destination in quasi-

1 The channel is unchanged in one or more OFDM symbol periods.

static flat fading channels. In DF scheme, each relay decodes information from the source, re-encodes it, and then forwards it to the destination. In AF scheme, each relay simply amplifies the received signal and forwards it to the destination.

In CR, cooperating terminals are at different locations and time synchronization among them needs to be considered. In [10]-[11], this problem is solved when OFDM is used with inserted cyclic prefix (CP) longer than the maximum channel delay spread in quasi-static frequency-selective fading channels. A soft-decode-and-forward scheme in asynchronous OFDM systems over doubly selective fading channels (i.e., channel changed within an OFDM symbol period) has been considered in [13]. Cooperative relaying takes place in two phases. In the first phase, the source broadcasts information to both relay and destination. Then the relays can either recover the source information (for hard-decode-and-forward scheme) or generate a soft copy of the source information (for soft-decode-and-forward scheme) based on the signal at the output of the block decision-feedback equalizer (BDFE), and forwards the processed source information to the destination in the second phase. Based on signals received from the source and the relays in both phases, the destination performs the maximum likelihood (ML) detection to restore the source information. However, in order to design BDFE, both schemes assume no error propagation in the feedback process [13, (17)]. Furthermore, although the soft-decode-and-forward scheme outperforms the hard-decode-and-forward scheme, the destination requires from the relay a significant amount of overhead information for detection such as the matrices Q and Φ in [13, (21)], resulting in reduced bandwidth efficiency and high complexity for estimating these information. Moreover, ML detection makes the destination’s receiver sophisticated.

In this paper, we consider ZF equalization at both the relay and the destination for its simplicity. We derive an exact symbol error rate (SER) expression for the proposed cooperative relaying scheme and validate it by computer simulations. Illustrative results indicate that the proposed cooperative relaying schemes significantly outperform the direct transmission (DT) counterpart under same transmission power and bandwidth efficiency for any relay position, signal-to-noise ratio (SNR), path loss exponent, and normalized Doppler frequency.

II. PROPOSED ZERO-FORCING BASED DECODE-AND-FORWARD SCHEME

Consider information transmission from a source S to a destination D with the assistance of a relay R. We assume that all nodes share the same bandwidth, and are equipped with single-antenna transceivers using OFDM modulation with N sub-carriers and ZF equalization. Each node operates in a half-

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duplex mode, and the source and the relay use time division multiplexing for channel access.

Assume that the channel between transmitter u∈{S,R} and receiver v∈{R,D} experiences time-varying frequency-selective (i.e., doubly selective) fading. The channel can be represented by Luv resolvable paths and the sampled time-varying channel impulse response of its l-th resolvable path at the time sampling instant t = mTs can be modeled as zero-mean complex Gaussian random variables huv(m, l) where Ts is the sampling period. We assume a wide-sense stationary uncorrelated scattering (WSSUS) channel model so that the fading channel coefficients huv(m, l) are correlated within each individual path but uncorrelated in different paths and have a Jakes Doppler power spectral density with an autocorrelation function given by [2]

( ) ( ){ } ( )( ) ( ),

* 2 20E , , 2

uv luv uv uv h d Sh m l h n r J f T m n l rσ σ π δ= − − , (1)

where all the paths are subject to the same Doppler spectrum, E{.} denotes the expectation, J0(.) is the zeroth-order Bessel function of the first kind, the product , denotes the average power of the channel coefficients of the lth path, fd is the maximum Doppler frequency, and δ(.) is the Kronecker delta function. The term captures the path-loss and the term , depends on the channel power delay profile [1]. Using the S-D distance as a reference (i.e., to set 1), the relative path-loss can be represented by / where duv is the distance between transmitter u∈{S,R} and receiver v∈{R,D} and α is the path-loss exponent.

CR consists of two phases. In the first phase, S broadcasts its signal which is received and processed by R and temporarily stored by D. If R successfully recovers the source information, it will forward the processed signal to D in the second phase. Otherwise, it keeps silent. Finally, based on the signals received from S and R in both phases, D performs detection for restoring the original information.

A. Detection in the first phase Consider a point-to-point uncoded OFDM system with N

subcarriers as shown in Figure 1. Assuming time and frequency synchronization, and employing a cyclic prefix length greater than the maximum channel delay spread, the OFDM input-output relation (see Figure 1) for the i-th OFDM symbol can be expressed by [2]

[ ] [ ] [ ] [ ]uv u uv u uvi P i i iy G s + w= , (2)

where yuv = [yuv(0), yuv(1), ..., yuv(N −1)]T is the received signal at the receiver v, su = [su(0), su(1), ..., su(N −1)]T is transmitted signal from the transmitter u with the unit symbol energy E{|su(k)|2} = 1 and su(k) is taken from the Gray-coded constellation (e.g., M-QAM), wuv = [wuv(0), wuv(1), ..., wuv(N−1)]T denotes the noise vector at the receiver v whose covariance matrix is NvIN with IN being the N×N identity matrix, and Pu is the average symbol power of the transmitter u. Without confusion, we drop the OFDM symbol index i in the sequel for notational simplicity. The (k,p)th element of the

N×N matrix Guv, denoted as [Guv]kp, represents the time-varying channel and is given by

( ) ( )11

2 /

0 0

1 ,uvLN

j lp m k p Nuv uvkp

m lG h m l e

Nπ

−−⎡ ⎤− + −⎣ ⎦

= =

=⎡ ⎤⎣ ⎦ ∑ ∑ . (3)

Mapping(M-QAM)bits N-point

IFFTCP

insertion

channelhuv(m,l)

+AWGN

CPremoval

De-mapping

recoveredbits

N-pointFFT

ZFequalizer

Transmitter u

Receiver v

su

yuv

Figure 1. Point-to-point uncoded OFDM system.

It is seen from (3) that the Doppler shift due to the fading time variation makes the non-diagonal elements of Guv nonzero, leading to ICI. As a result, the received signal of the point-to-point OFDM system with single-antenna transceivers and N sub-carriers must be represented as an equivalent N×N MIMO system as shown by (2). In other words, the received signals at R and D can be, respectively, expressed as

SR S SR S SRPy G s + w= . (4)

SD S SD S SDPy G s + w= . (5)

Using ZF equalization, the relay recovers the source information based on

ˆ SR SR SR S S SRPy G y s e⊥= = + , (6)

where ⊥ is the pseudo-inverse of GSR and e ⊥ n . By the ZF equalization, ICI is completely removed but the resulting noise, eSR, is enhanced, causing the irreducible error floor. From (6), the source information can be recovered by a simple symbol-by-symbol maximum-likelihood detection.

If R incorrectly detects the source signal (e.g., indicated by the cyclic redundancy check (CRC)), it keeps silent. Otherwise, R forwards the decoded signal in the second phase to D with the average symbol power PR. Using (2), the signal received at D from R can be expressed as

RD R RD S RDPy G s + w= , (7)

where , if decodes successfully0 , otherwise . B. Detection in the second phase

Based on the signals (ySD, yRD) received in both phases, the destination arranges them as follows

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S SDSD SDS

RD RDR RD

P

Py nG

Gy ws +

y wG

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

. (8)

We assume the destination also applies the ZF equalization. Therefore, the source information is recovered from the following signal

Sy G y s G n⊥ ⊥= = + . (9)

In (9), it is observed that no ICI exists in the equalized signal but the noise enhancement still occurs.

The ZF equalization is considered in this paper for its simplicity. The inherent complexity of the ZF equalizer due to computing the matrix pseudo-inverse can be remedied by an approximation approach in [16]. However, this equalizer suffers a high bit error rate (BER) and an irreducible error floor due to the ill-conditioned matrix [2], resulting in a low cooperation probability2 between S and R, which eventually reduces the spatial diversity at D. When other high-performance equalizers such as MMSE and MAP are applied, the cooperation probability can be increased and this can lead to higher spatial diversity at D, and hence more performance improvement by cooperative relaying at the cost of increased implementation complexity.

Cooperative Relaying with Source Retransmission: In the above described cooperative relaying scheme, whenever the relay incorrectly detects the source signal, it keeps silent and hence the second phase is waste and furthermore, D does not benefit diversity in this situation. This can be improved by allowing S to re-send its information in the second phase whenever R keeps silent. To implement this feature, a fast and simple feedback signaling with one bit to indicate the relay detection status from R to S can be used as follows. Consider the relay detection status bit being included in the preamble of the packet sent by R in the second phase. If R successfully detects the source information, it sets the detection status bit to 1 in the preamble and relays the detected source information in the second phase. Upon reception of this successful status, S understands that its information is being re-sent by R, and hence, keeps silent in the second phase. On the other hand, if R incorrectly detects the source signal, R sets the detection status bit to 0 in the preamble, and just sends a short preamble, and then keeps silent for the remainder of the second phase. Upon reception of the relay unsuccessful detection status bit, S will re-transmit its information in the remainder of the second phase. Due to the time variation of the considered doubly selective fading channels within each OFDM symbol, the matrix GSD in (5) in the first and second phases can be different, and as a result, D still reaps the time diversity instead of the spatial diversity. The signals received at D can be rewritten as

2 The probability that the relay precisely detects the source signal.

* **

S SDSD SDS

SD SDR SD

P

Py nG

Gy ws +

y wG

⎡ ⎤⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

, (10)

where * in ySD*, GSD*, wSD* denotes the second phase, i.e., ySD*, GSD*, wSD* are the received signal, channel matrix and noise, respectively, in the second phase. The source information is recovered in the same manner as (9) with y and G given in (10). It is noted that in the second phase, S resends the information with the average power PR which is the average power of the relay instead of PS to maintain the total transmission power of the CR system of PS+PR.

III. PERFORMANCE ANALYSIS The received signals in (4), (5), (7), (8), (10) have the same

general form of an Mt×Mr MIMO system

Py Gs + w= , (11)

where y = [y(0), y(1), ..., y(Mr −1)]T is the received signal vector, s = [s(0), s(1), ..., s(Mt −1)]T is transmitted signal vector whose elements s(k) are taken from the Gray-coded constellation (e.g., M-QAM) with the unit symbol energy E{|s(k)|2} = 1, w = [w(0),w(1), ...,w(Mr − 1)]T denotes noise at the receiver whose covariance matrix is N0IMr, P is the average symbol power of the transmitter, G is the Mr×Mt channel matrix with the covariance matrix R = E{GHG}.

The transmitted symbols are recovered using the ZF equalizer as

1/ 2ˆ Py G y s e− ⊥= = + , (12)

where e = P-1/2G⊥n. Based on (12), the SNR γm on s(m), m∈{0, 1, …, Mt-1} is given by E | | / E e with [.]H being the Hermitian transpose operator and has the probability density function (pdf) given by [18]

( )( ) ( )

1

expm

d d

dd dfdγ

γ γγ

ηΓ η

− ⎛ ⎞= −⎜ ⎟

⎝ ⎠, (13)

where η / |λ | , λ R , d = Mr - Mt + 1, Γ(⋅) is the Gamma function.

A. SER of a MIMO system with the ZF equalizer We will derive the SER expression of the MIMO system

described in (11) for M-PSK and square M-QAM with M = 2k (k even) modulation schemes. Using / and / , where Q(.) is the Gaussian Q-function, the conditional SER for a given SNR γ can be expressed as [19]

PSK

QAM QAM

( ,[ 1] / ) , for -PSK 4 [ ( , / 2) ( , / 4)] , for -QAM

g M M Mp

K g K g Mγ

ψ πψ π ψ π

−⎧= ⎨ −⎩

, (14)

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where ψ g,Ω gΩ , gPSK sin , K 1 √ , gQAM .

By averaging (14) with respect to the random variable γm with the pdf given by (13), the SER in detecting s(m) is

{ } ( ); ,emp p F g dγ γ ηE=

PSK

QAM QAM

( ,[ 1] / ) , for -PSK4 [ ( , / 2) ( , / 4)] , for -QAM

g M M MK g K g M

Φ πΦ π Φ π

−⎧= ⎨ −⎩

, (15)

where using [22, (5A.35)]

( )( ) ( )

( ) ( )

1

0

1 1

0 0

2 1( , )1 4 1

1 sin 2 2221 4 1 2 2

d

kk

j kd k

kk j

kT cgkc c

k j Tkcjc c k j

ΩΦ Ωπ π

π

−

=

+− −

= =

⎛ ⎞= − ⎜ ⎟+ ⎡ ⎤⎝ ⎠ +⎣ ⎦

⎡ ⎤− −⎛ ⎞ ⎣ ⎦− ⎜ ⎟+ ⎡ ⎤⎝ ⎠ + −⎣ ⎦

∑

∑∑

with tan 1 ,2 1 sin 2Ω , 1 2 cos 2Ω 1 , ! ! !, η.

Then the average SER is computed by averaging the SERs of all symbols

1

0

1 tM

e emmt

p pM

−

== ∑ . (16)

From the average SER, the average BER can be approximated [20] as /log .

B. SER of the direct transmission In DT, S sends its information directly to D without the

help of R. As a result, the received signal at D is expressed in (5) which is of the same form as (11) with Mt = Mr = N. Therefore, the average SER of DT is given as in (16)

1

0

1 N

eDT emDTm

p pN

−

=

= ∑ , (17)

where g; 1, η given in (15) with η/ |λ | , and λ . Based on the WSSUS channel model shown in (1), the (k,p)th element of the N×N matrix R E is given by

1

0E

N

DT SD SDkp kt tpt

R G G−

∗

=

⎧ ⎫= ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎨ ⎬⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎩ ⎭∑ ( ), , , , ,d s SDk p N f TΘ C

( )( ) ( ) ( ),

1 11 1 12 2

020 0 0 0 0

1 2SD SD

SD l

L LN N Nlt m k t rp n t p

SD h d St m l n r

J f T n m zN

σ σ π− −− − −

+ − − − −

= = = = =

= −∑∑ ∑ ∑ ∑ ,

(18)

where z = ej2π/N, and Cuv denotes the channel power delay profile of the u-v link with parameters: Luv, , , .

C. SER of the proposed cooperative relaying scheme The transmission model from S to R is given in (4). Let pc,R

be the probability that R successfully detects the source signal. Since the vector sS consists of independent N symbols, this probability is given by

( )1

,0

1N

c R emRm

p p−

=

= −∏ , (19)

where g; 1, η is the SER of the symbol sS(m) at R with η / |λ | , and λ . The (k,p)th element of the N×N matrix R E is given by

Θ , , , , , .

In the proposed CR scheme, R keeps silent in the second phase if it fails to detect the source signal with the probability of 1-pc,R. In this case, D only bases on the signal from S for detection and the average SER at D is denoted as pe1. Otherwise, R forwards the source signal with the probability of pc,R and D combines both signals from S and R for detection. We denote pe2 as the average SER at D in this case. As a result, the average SER at D can be computed as follows

( )1 , 2 ,1eCR e c R e c Rp p p p p= − + , (20)

where ∑ / , ∑ / with g; 1, η , η / |λ | , and λ in which R1 has the same form as RDT given in (18). Furthermore, it is worth noting that for the case of Cooperative Relaying with Source Retransmission, we have g; 1, η , η 1 / |λ | , and E Θ , , , , , .

To compute pem2 which is the SER of the symbol sS(m) at D

with the transmission model given in (8), we note that (8) is equivalent to the MIMO system with Mr = 2N and Mt = N. Therefore, it is expressed as

( )2 2; 1,emp F g N η= + , (21)

where η 1 / |λ | and λ with R2 = E{GH

G} in which G is given in (8). The (k,p)th element of the N×N matrix R2 is given by

Θ , , , , , Θ , , , , , .

IV. ILLUSTRATIVE RESULTS Since CR completes in two phases, its required bandwidth

efficiency in each phase must be double that of DT to achieve the same overall bandwidth efficiency. For example, to achieve3 1 b/s/Hz, DT uses BPSK while CR must use QPSK.

Let PDT denote the average symbol power in DT. For the same energy consumption between CR and DT, it is required that 2PDT =PS +PR. Finding the optimal power allocation for the source power PS and the relay power PR satisfying this constraint is so involved. As an illustrative example, we

3 Due to space limitation, the results for 2 b/s/Hz are omitted.

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assume a simple power allocation scheme [17] where the channels from S and R to D result in the same average received SNR, i.e., / / . Combining this with 2PDT =PS +PR results in 2 / and 2 / .

In simulations, we assume that all channels have the same power delay profile with LSD = LRD = LRD = L. The channel coefficients huv(m,l) are uncorrelated zero-mean Gaussian random variables with the exponential power delay profile , . in (1), where C is a scalar factor that ensures that ∑ , is normalized to unity. Channel state information is assumed to be available at receivers4, not at transmitters. The normalized Doppler frequency is defined as fd/Δf where Δf = TsN is the subcarrier spacing [3].

In all the figures, SNR is defined as PDT/N0 where the noise variances at receivers are normalized so that ND = NR = N0.

Figure 2. Validation of closed-form SER formulas.

A. Validation of closed-form SER formulas Figure 2 compares the simulated and numerical results for

the uncoded OFDM system with N = 8, fd/Δf = 0.15, L = 2, 1. The exact SER formulas in (17) and (20) are used to obtain the analytical results for DT and CR, respectively. Figure 2 shows that the analytical and simulation results are in a good agreement and CR achieves higher spatial diversity than DT.

B. Performance comparison between DT and CR For simplicity in illustration, we assume the relay to be

located on a straight line between S and D. The direct path length S-D is normalized to be 1. We also denote d as the distance between S and R. In all figures below, we consider N = 128 and L = 8.

Figure 3 compares the BER performance of CR with that of DT with fd/Δf = 0.15, α = 3 and three typical positions of the relay: d = 0.1 (R is close to S), d = 0.5 (R is in the middle of S and D), and d = 0.9 (R is near D). fd/Δf = 0.15 generally represents a high Doppler spread condition. For example, it corresponds to a mobile speed of 324km/h, a carrier frequency of 10 GHz, and a subcarrier spacing of 20 kHz [3]. It is seen from Figure 3 that CR brings a significant performance improvement over DT for any relay position. In particular, the SNR gain of CR over DT is more than 10 dB at the target BER

4 Channel estimation is outside the scope of this paper.

of 10-2 for any relay position and this gain increases for lower target BERs since the slope of the BER curve in CR is steeper than that in DT. Additionally, the irreducible error floor of DT is much higher than that of CR. As a result, the performance of DT is almost unacceptable in real systems where the required BER is usually lower than 10-3.

Figure 3. BER versus SNR for fd/Δf = 0.15 and α = 3.

Figure 4. BER versus fd/Δf for α = 3, 1 bit/s/Hz, SNR = 20 dB.

Figure 4 investigates the impact of fd/Δf on the BER performance of two considered schemes under SNR = 20dB and α = 3. It is observed that the BER performance is dramatically degraded as fd/Δf increases. CR still outperforms DT for any value of fd/Δf but the performance gain of CR over DT becomes negligible at high fd/Δf.

Figure 5. BER versus the path loss exponent for fd/Δf = 0.3 and SNR = 20 dB.

The effect of α on the BER performance is shown in Figure 5 for the typical range of α∈[2, 5], SNR = 20 dB, and fd/Δf = 0.3. We see that the performance of DT is independent

0 2 4 6 8 10 12 14 1610

-3

10-2

10-1

100

SNR (dB)

SE

R

1 bit/s/Hz

DT-Simulation

DT-TheoryCR-Simulation

CR-Theory

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR (dB)

BE

R

1 bit/s/Hz

DT

CR-d=0.1CR-d=0.5

CR-d=0.9

0 0.2 0.4 0.6 0.8 110

-5

10-4

10-3

10-2

10-1

100

Normalized Doppler frequency

BE

R

1 bit/s/Hz

DT

CR-d=0.1CR-d=0.5

CR-d=0.9

2 3 4 510

-2

10-1

100

path loss exponent

BE

R

1 bit/s/Hz

DT

CR-d=0.1CR-d=0.5CR-d=0.9

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of α since the distance between S and D is normalized to unity (i.e., 1) and CR is better than DT for any value of α and d. In addition, the performance gain increases with the increased α. This is natural since besides the spatial diversity benefit, CR takes advantage of the lower path loss of shorter links such as the S-R and R-D links.

A common remark withdrawn from all results in Figures 3-5 is that the nearer to the source the relay is, the higher the cooperation probability between S and R is, and the better performance CR obtains for any simulation conditions.

C. Cooperative Relaying with Source Retransmission (CR with SR)

Figure 6. BER versus SNR for fd/Δf = 0.1 and α = 3.

Figure 6 illustrates the BER performance of three schemes: DT, CR, and CR with SR for fd/Δf = 0.1 and α = 3. It is observed that CR with SR significantly outperforms the others with the SNR gain of over 5dB at the target BER of 10-3 for any relay position. Additionally, CR with SR offers better diversity than CR.

V. CONCLUSIONS A low-complexity high-performance zero-forcing based

decode-and-forward cooperative relaying scheme is proposed and theoretically analyzed in this paper. Various results considering multiple factors impacting on the system performance such as the relay position, SNR, the path loss exponent, the bandwidth efficiency, the severity of the doubly selective fading channel (reflected through the normalized Doppler frequency) demonstrate that cooperative relaying performs considerably better than direct transmission.

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978-1-4244-5638-3/10/$26.00 ©2010 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.