Power Allocation for OFDM-Based Cooperative Relay Systems by Victor Wu

JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 10, NO. 2, JUNE 2008

Power Allocation for OFDM-Based Cooperative Relay Systems Victor K. Y. Wu, Ye (Geoffrey) Li, Marilynn P. Wylie-Green, Tony Reid, and Peter S. S. Wang Abstract: Cooperative relays can provide spatial diversity and improve performance of wireless communications. In this paper, we study subcarrier power allocation at the relays for orthogonal frequency division multiplexing (OFDM)-based wireless systems. For cooperative relay with amplify-and-forward (AF) and decodeand-forward (DF) algorithms, we investigate the impact of power allocation to the mutual information between the source and destination. From our simulation results on word-error-rate (WER) performance, we find that the DF algorithm with power allocation provides better performance than that of AF algorithm in a single path relay network because the former is able to eliminate channel noise at each relay. For the multiple path relay network, however, the network structure is already resistant to noise and channel distortion, and AF approach is a more attractive choice due to its lower complexity. Index Terms: Cooperative diversity, cooperative relay, power allocation.

I. INTRODUCTION Recently, relays are being exploited to improve performance in wireless communications systems. The relays are a network of transceiver nodes between the transmitter and receiver that facilitate the transfer of information. This type of scheme is known as cooperation or cooperative communications in the literature because the relay network is cooperating with the transmitter and receiver to improve performance. One application example of such technologies is the MIT-initiated One Laptop per Child (OLPC) project [1], which aims to provide affordable laptops equipped with meshed networking functionality to children in the developing world. Since cellular and Internet connectivity is sparse and sporadic in these regions, such laptops can cooperate to make the best use of available bandwidth. In this paper, we restrict ourselves to a single path relay network and a multiple path relay network in the context of orthogonal frequency division multiplexing (OFDM) systems with power allocation. The authors in [2]â&#x20AC;&#x201C;[4] have provided several physical layer relay algorithms. These include amplify-and-forward (AF) and decode-and-forward (DF). In AF, a node amplifies its received symbols, subject to a power constraint, before forwarding them to the next node. This algorithm is obviously with low complexity. In DF, a node fully decodes the received symbols, re-encodes them and then forwards them. In other words, this scheme attempts to eliminate channel distortion and noise at Manuscript received November 7, 2007. Wu and Li are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA, email: vwu3@illinois.edu, liye@ece.gatech.edu. Green, Reid, and Wang are with the Nokia Siemens Networks, Dallas, Texas, USA, email:{marilynn.green, tony.reid, peter.wang}@nsn.com.

each node by means of the redundancy of error-correct coding. The authors in [5] and [6] have investigated cooperation for a single path of relays connected in series. The motivation for this network structure is that broader wireless coverage can be achieved while still maintaining a low power constraint at the transmitter. Analog relaying and digital relaying are considered as two possible relay algorithms. These are equivalent to the AF and DF algorithms, respectively. Each node has a certain transmit power limit. The outage probability is then minimized by allocating power among the relay network under these power constraints. This power allocation accounts for the channel conditions in the network in order to achieve the optimal outage probability. Simulations indicate that 2 dB of total power can be saved for 5 relays by using optimal power allocation instead of uniform power allocation. This is for the DF case. However, at high signal-to-noise ratio (SNR) values, the DF and the AF cases are almost the same. The authors in [7] have investigated cooperation for multiple paths of relays connected in parallel. In the conventional scheme, all relays use AF scheme. This is called all-participate AF (AP-AF). The authors also consider an algorithm where only one relay is selected in the transmission to maximize the mutual information. This is called selection AF (S-AF). S-AF selects the relay which results in the maximum mutual information between transmitter and receiver. Simulations of outage probability indicate that 5 dB of SNR can be saved for 3 relays by using S-AF instead of AP-AF. The authors in [8] derive symbol error probabilities for multiple paths of relays. In this paper, we continue to investigate the series and parallel cooperative relay networks using OFDM signals. We consider a single path relay network and a multiple path relay network. Using the AF relay algorithm, we derive the input-output relations and the mutual informations for both networks. Using a power constraint at each relay, we consider two relay power allocation schemes: Constant gain allocation and equal power allocation. Using the DF relay algorithm, we derive input-output relations for both networks. We also compare word-error-rates (WERs) for the two networks using the AF and DF relay algorithms. The rest of this paper is organized as follows. We study power allocation for the single path relay network in [5], [6] and the multiple path relay network in [7] in Sections II and III, respectively. Finally, Section IV concludes the paper and provides future research directions. II. SINGLE PATH RELAY NETWORK In this section, we consider the single path relay network. We first study the impact of power allocation to the mutual information for the AF and the DF relay networks, respectively, and then present their WER performance from computer simulation.

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JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 10, NO. 2, JUNE 2008

(0)

(m)

(m−1)

(1) hk (0)

Now, consider the variance of wk . Using (3) and (4), we have

rm+1

(m)

(m−1) hk

Rwk wk = E [wk wk∗ ]

Receiver

  m m N0 B X  Y (i) (i)  = b p N j=0 i=j+1 k k

Transmitter Fig. 1. Single path relay network.

∗

A. Amplify-and-Forward Fig. 1 shows the single path relay network. In the figure, r0 is the transmitter, rm+1 is the receiver, and r1 , · · ·, rm are m relay nodes connected in series forming a single path link between the transmitter and receiver. The relays perform AF relaying. We assume that OFDM with N subcarriers is used in the (0) (m) system. hk , · · ·, hk are the complex subchannel gains at the (0) (m) kth subcarrier in the link, for k = 1, · · ·, N . nk , · · ·, nk are the corresponding noises, which are assumed to be mutually independent and circular symmetric complex Gaussians all with zero mean and variance N0 B/N , where N0 is the power spectral density of the underlying continuous time noise process and (0) B is the OFDM bandwidth of the system. Let pk = Ptot /N be the transmit power on q the kth subcarrier, where Ptot is the net (l)

transmitter power and pk be the amplifying gain used in the AF algorithm at the lth relay,q for l = 1, · · ·, m. The kth receive (l)

symbol at rl is amplified by

pk before it is forwarded to the

(0)

next node. Let xk be the kth transmit symbol with zero mean and unit variance, yk be the kth receive symbol at the receiver, (l) (l) and xk be the kth receive symbol at the lth relay. Note that xk is also the kth transmit symbol at the lth relay. Using Fig. 1, the input-output relation at the lth relay is   q q ! l−1 l−1 l−1 X Y Y (i) (i) (0) (i) (j) (i) (l)  pk xk + pk  nk . hk hk xk = j=0

i=0

i=j+1

(1)

The input-output relation at the receiver is   q q ! m m m X Y Y (i) (i) (0) (i) (j) (i)  pk xk + pk  nk . hk hk yk = j=0

i=0

i=j+1

(2)

Denote hk =

m Y

i=0

(i) hk

q (i) pk ,

(j) γk

m Y

i=j+1

(i) hk

q (i) pk ,

˜ k xk + w ˜k (7) y˜k = h p p ˜ k = hk / Rw w , and w ˜k = where y˜k = yk / Rwk wk , h k k p wk / Rwk wk . The variances of w ˜k and y˜k are E [w ˜k w ˜k∗ ] = 1

(8)

and E

[˜ yk y˜k∗ ]

1 Rwk wk

m Y

(i) (i) b k pk

i=0

+ 1,

(9)

respectively. The cross terms do not appear in (9) because ˜hk , w ˜k , and xk are mutually independent. Note that the normalized system has unit variance noise. A.1 Mutual Information To derive the mutual information, note that the differential entropy of a circular symmetric complex Gaussian vector, v, with covariance matrix, K, is h (v) = log2 det (πeK) [9]. When the circular symmetric complex Gaussian is a scalar, v, the differential entropy is h (v) = log2 πeσv2 , where σv2 is the variance of v. Let Ik be the mutual information between the transmitter and receiver on the kth subcarrier Ik = h (˜ yk ) − h (w ˜k ) ! # " m Y 1 (i) (i) b p +1 . = log2 Rwk wk i=0 k k

(10)

The total mutual information between the transmitter and receiver, I, is the sum of all Ik divided by N . That is, after substituting (6) into (10), we have    (0) Qm (i) (i) N b b p X k i=1 k k 1 Q  I= log2 1 + SNR  P (i) (i) m m N b p j=0

i=j+1 k

(11)

where SNR = Ptot /N0 B. A.2 Relay Power Allocation

(l) (l) γk nk .

(4)

yk = hk xk + wk .

(5)

wk =

where E [·] is the expectation operator, (·) is the complex con

(i) 2 (i) jugate operator for a scalar, and bk = hk , for i = 0, · · ·, m. Rwk wk is positive for a nonzero N0 . We define a normalized version of the system in (5)

k=1

(3)

and m X

(6)

l=0

Then, (2) can be written as

We assume that the net transmit power at the transmitter and at each each relay is Ptot , that is, ( q

2 ) N X

(l) (l) = Ptot . (12) E

pk xk

k=1

WU et al.: POWER ALLOCATION FOR ODFM-BASED COOPERATIVE DIVERSITY RELAY SYSTEMS

k=1

(13)

WER

At the transmitter, we assume a uniform power distribution, that (0) is, pk = Ptot /N . To derive the power constraint at each relay, substitute (1) into (12) to arrive at   ! N l−1 Y l−1 l−1 (l) X X Y pk  (0) 1 (i) (i) (i) (i) bk pk  = 1. b k pk + bk N SNR j=0 i=j+1 i=1 Note that (13) is defined recursively. The power constraint for (l) (1) (l−1) (1) pk depends on pk , · · ·, pk . pk is the base case in the recursion, which follows from (13), when l = 1. (l) One power allocation at the lth relay is to set pk constant (l) for all subcarriers. This results in moving pk in (13) out of the summation because it is no longer a function of k (l)

−1

Hard, AF, CT Hard, AF, EQ Hard, DF Soft, AF, CT Soft, AF, EQ Soft, DF −2



SNRb(0) k

l−1 Y

N SNR !

(i) (i) bk pct

i=1

l−1 Y l−1 X

j=0 i=j+1

 . (14)

(i) (i) bk pct 

We call this constant gain allocation (CT). This is in fact the AF scheme as defined in [2], where a receive symbol is multiplied by a constant gain to satisfy a power constraint. In our context, this means that every subcarrier is multiplied by the same gain, (l) pct . Note that this power allocation does not require each relay to have any channel state information (CSI). That is, we do not actually need to use (14). We can use (12) directly to solve for (l) pct . (l) A second power allocation is to choose pk such that every subcarrier transmits the same power at the lth relay. That is, q (l) (l) 2

(12) becomes E p x = Ptot /N , for k = 1, · · ·, N . k,eq k

This is equivalent to setting every summand on the left hand side of (13) to 1/N . We then have SNR !

(l)

pk,eq = (0)

SNRbk

l−1 Y

(i) (i)

bk pk,eq

i=1

l−1 Y l−1 X

. (15) (i) (i)

bk pk,eq

j=0 i=j+1

We call this equal power allocation (EQ). Note that this power allocation does require each relay to have the CSI of its upstream channels. B. Decode-and-Forward In DF, each relay fully recovers the information bits (with possible errors) after receiving an OFDM symbol. It then converts the information bits back into an OFDM symbol and then transmits it. The transmitter and all the relays transmit with the same uniform power distribution. That is, (0)

(l)

pk = pk

Ptot = N

(16)

for k = 1, · · ·, N and for l = 1, · · ·, m. (0) Let xk be the kth transmit symbol from the transmitter and (l) xk be the kth transmit symbol from the lth relay, all with with

WER

k=1

(a)

(l)

pk,ct = pct = N X

SNR (dB)

−1

Hard, AF, CT Hard, AF, EQ Hard, DF Soft, AF, CT Soft, AF, EQ Soft, DF −2

SNR (dB) (b) Fig. 2. WERs in a single path relay network with TU channels using AF and DF for N = 128: (a) m = 2 and (b) m = 4. (m+1)

zero mean and unit variance. Let yk be the kth receive (l) symbol at the receiver and yk be the kth receive symbol at the lth relay. Using Fig. 1, the input-ouput relation at the lth relay is r Ptot (l−1) (l) (l−1) (l−1) y k = hk x + nk . (17) N k The input-output relation at the receiver is r Ptot (m) (m+1) (m) (m) yk = hk x + nk . N k

(18)

C. Simulations We simulate WERs versus SNR for both the AF and DF cases. At the transmitter (and at the transmitter structure of a relay using DF), each information word contains 83 bits. We use a rate 1/3 convolutional encoder with generator sequences [111], [111], and [110] to encode the information word into a 255-bit codeword. A zero bit is padded at the end to make 256 bits. The bits are then interleaved and modulated onto N = 128 QPSK

JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 10, NO. 2, JUNE 2008

subcarriers to form one OFDM symbol. At the receiver (and at the receiver structure of a relay using DF), the codeword is recovered (with possible errors) using a matched filter and deinterleaving. A Viterbi decoder is used to decode the codeword. Both hard decisions and soft decisions are used. We consider the m = 2 and 4 relays cases. We assume that all distances between any two adjacent transceiver nodes are the same. Therefore, all path loss effects are normalized to 0 dB. Shadowing is assumed to be log-normally distributed. That is, the received power gain due to shadowing in dB is a zero-mean Gaussian with variance of 8 dB, which is typical for cellular land mobile applications [10]. We model frequency selective fading as Typical Urban (TU) channels [10]. We use an OFDM bandwidth of 800 kHz divided into N = 128 equal blocks. Maintaining OFDM orthogonality, this translates into an OFDM symbol period of Ts = 160 µs. The simulation results are shown in Fig. 2. As shown in the plots, there are significant error rate performance gains when using DF instead of AF. The gains are even larger when we increase the distance between the transmitter and receiver (and thus, add more relays). The AF error rates suffer because more channel distortion and noise enter the system. The DF error rates suffer only slightly because noise and channel distortion are eliminated at each relay. This results in the large performance gains for m = 4. In terms of power allocation when using AF, CT is the preferable choice since EQ requires CSI and only results in small performance gains over CT. As expected, soft decisions give better performance than hard decisions in Viterbi decoding. III. MULTIPLE PATH RELAY NETWORK In this section, we consider a multiple path relay network, following the same structure as Section II.

(0 |1)

(l)

symbol at rl is amplified by pk before it is forwarded to the next node. Let xk be the kth transmit symbol with zero mean (l) and unit variance and yk be the kth receive symbol from the lth path at the receiver.

(1|m+1)

rm+1 (0|m)

Transmitter

(m| m+1) (m|m+1) nk

Receiver

(0| m)

Fig. 3. Multiple path relay network.

Using Fig. 3, the input-output relation for the lth path is q q (l) (0|l) (l|m+1) (0) (l) pk pk xk + y k = hk hk q (l) (0|l) (l|m+1) (l|m+1) p k nk + nk . (19) hk

Denote

yk = 

  hk =  

(1)

(m)

···

(0|1) (1|m+1) hk hk

q q (0) (1) pk pk

(20) 

. (m|m+1)

(0|1)

nk .. .

    (0|m)  nk nk =  (1|m+1)  nk   ..  .

(m|m+1)

(21)



and

  .. , . q  q (0) (m) (0|m) (m|m+1) pk pk hk hk

q  (1) (1|m+1) pk hk   Γk =  

A. Amplify-and-Forward

transmitter power and pk be the amplifying gain used in the AF algorithm at the lth relay,q for l = 1, · · ·, m. The kth receive



Fig. 3 shows the multiple path relay network. In the figure, r0 is the transmitter, rm+1 is the receiver, and r1 , · · ·, rm are m relay nodes connected in parallel forming a multiple path link between the transmitter and receiver. The relays perform AF relaying. We assume that OFDM with N subcarriers is used in (0|1) (0|m) (1|m+1) (m|m+1) the system. hk , · · ·, hk , hk , · · ·, hk are the complex subchannel gains at the kth subcarrier in the link, for (0|1) (0|m) (1|m+1) (m|m+1) k = 1, · · ·, N . nk , · · ·, nk , nk , · · ·, nk are the corresponding noises, which are assumed to be mutually independent, zero-mean, circular symmetric complex Gaussians all with variance N0 B/N , where N0 is the power spectral density of the underlying continuous time noise process and B is (0) the OFDM bandwidth of the system. Let pk = Ptot /N be the transmitter power on the q kth subcarrier, where Ptot is the net

(1| m+1)

r1 (0 |1)

q (m) pk

  , Im×m  

(22)



     ,    

wk = Γk nk .

(23)

(24)

Then, (19), for all l = 1, · · ·, m, can be written as yk = hk xk + wk .

(25)

The large boldface zeros in (22) represent zero values in the off diagonal entries in the left m × m submatrix of Γk . Now, consider the covariance of wk . Using (21) to (24), we have   (1|m+1) (1) bk pk + 1 0 N0 B   .. Rwk wk =   . N (m|m+1) (m) 0 bk pk + 1 (26)

WU et al.: POWER ALLOCATION FOR ODFM-BASED COOPERATIVE DIVERSITY RELAY SYSTEMS

(i|j) 2 (i|j) where bk = hk , for i = 0, · · ·, m, for j = 1, · · ·, m + 1, and i 6= j. Since the diagonal entries of Rwk wk are never zero, −1

−1

−1 Rw and Rwk2wk are well defined, where Rwk2wk Rwk2wk = k wk 1

−1 2 2 Rw . Also, if we define Rwk wk as Rw k wk Rwk wk = Rwk wk , k wk −1

q (l) multiply its entire OFDM receive symbol by a constant, pct , such that the total transmit power is Ptot , similar to constant gain allocation in Section II-A. Equal power allocation (EQ) in this case is

−1

2 2 2 then Rwk2wk Rw k wk = Rwk wk Rwk wk = I. We define a normalized version of the system in (25)

˜ k xk + w ˜k = h ˜k y

(27) 1

−2 − ˜ k = Rw ˜k = ˜ k = Rwk2wk yk , h where y k wk hk , and w 1 −2 ˜ k and y ˜k are Rwk wk wk . The covariance matrices of w ˜ kH = I ˜ kw (28) E w

and

˜ H + I, ˜kh ˜ kH = h ˜k y E y k

(29)

˜ k, respectively. The cross terms do not appear in (29) because h ˜ k and xk are mutually independent. Note that the normalized w system has identity covariance noise. A.1 Mutual Information Let Ik be the mutual information between the transmitter and receiver on the kth subcarrier ˜ k) Ik = h (˜ yk ) − h (w −1 = log2 1 + hH k Rwk wk hk .

(30)

The total mutual information between the transmitter and receiver, I, is the sum of all Ik divided by N . That is, after substituting (21) and (26) into (30), we have " !# N m (0|i) (i|m+1) (i) X bk bk pk 1 X I= . (31) log2 1 + SNR (i|m+1) (i) N b p +1 i=1 k=1

A.2 Relay Power Allocation We assume that the net transmit power at the transmitter and at each each relay is Ptot , as in (12). At the transmitter, we (0) assume a uniform power distribution, that is, pk = Ptot /N . To derive the power constraint at each relay and thus, possible power allocations, we use a derivation similar to the one in Section II-A to arrive at N (l) X pk 1 (0|l) bk + = 1. N SNR

(32)

k=1

(l) pct

k=1

(0) SNRbk

(0|l)

SNRbk

(34)

Note that this power allocation does require each relay to have the CSI of its upstream channel. B. Decode-and-Forward In DF, each relay fully recovers the information bits (with possible errors) after receiving an OFDM symbol. It then converts the information bits back into an OFDM symbol and then transmits it. The transmitter and all the relays transmit with the same uniform power distribution. That is, (0)

(l)

pk = pk =

Ptot N

(35)

for k = 1, · · ·, N and for l = 1, · · ·, m. (0) Let xk be the kth transmit symbol from the transmitter and (l) xk be the kth transmit symbol from the lth relay, all with with (m+1) zero mean and unit variance. Let yk be the kth receive sym(l) bol at the receiver and yk be the kth receive symbol at the lth relay. Using Fig. 3, the input-ouput relation at the lth relay is r Ptot (0) (l) (0|l) (0|l) x + nk . (36) y k = hk N k The input-output relation at the receiver is ! r m X Ptot (i) (i|m+1) (m+1) (i|m+1) hk yk = x + nk . N k i=1

(37)

Note that in this situation, there are m (modified in gen(0) eral) copies of the original kth transmit symbol xk , namely, (1) (m) xk , · · ·, xk . Therefore, the receiver has to assume (incorrectly in general) that all m relays perform perfect recovery of (l) (0) the information bits so that xk = xk . That is, the receiver assumes the kth receive symbol is ! r m m X X Ptot (i|m+1) (0) (i|m+1) (m+1) nk . (38) xk + hk yk = N i=1 i=1 This allows the receiver to use a filter that is matched to m X (i|m+1) hk . C. Simulations

N SNR N X

SNR

(l)

pk,eq =

i=1

CT in this case is (l) pk,ct

. +1

(33)

Again, this power allocation does not require each relay to have any channel state information (CSI). The lth relay only has to

We simulate WERs versus SNR for both the AF and DF cases. We use exactly the same configuration in Section II-C. The simulation results are shown in Fig. 4. The performance gains resulting from using DF instead of AF diminish as we add more paths between the transmitter and receiver (and thus, add more relays). This is because for m = 4

JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 10, NO. 2, JUNE 2008 0

rithms, such as hybrid schemes. For example, depending on the channel conditions, a relay can AF, DF, or even just discard subcarrier symbols. This in turn leads to more possibilities for relay power allocation. In this paper, we only investigate the single path relay network and the multiple path relay network. Other general relay networks need to be considered in the context of OFDM as well. This will lend more insight into developing a general theoretical framework for OFDM in cooperative relay networks.

WER

−1

Hard, AF, CT Hard, AF, EQ Hard, DF Soft, AF, CT Soft, AF, EQ Soft, DF

REFERENCES [1]

−2

SNR (dB) (a) 0

WER

−1

Hard, AF, hard, AF, CT CT hard, Hard, AF, AF, EQ EQ hard, Hard, DF DF soft, Soft, AF, AF, CT CT soft, Soft, AF, AF, EQ soft, Soft, DF −2

(2005) One laptop per child. [Online]. Available: http://laptop.media.mit. edu/ [2] J. N. Laneman, “Cooperative diversity in wireless networks: algorithms and architectures,” Ph.D. dissertation, MIT, Cambridge, MA, Sept. 2002. [3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, pp. 3062–3080, Dec. 2004. [4] J. N. Laneman, G. Wornell, and D. N. C. Tse, “An efficient protocol for realizing cooperative diversity in wireless networks,” in Proc. IEEE ISIT 2001, June 2001, p. 294. [5] M. O. Hasna and M. Alouini, “Performance analysis of two-hop relayed transmission over rayleigh-fading channels,” in Proc. IEEE VTC 2002-fall, Sept. 2002, pp. 1992–1996. [6] ——–, “Optimal power allocation for relayed transmissions over rayleighfading channels,” IEEE Trans. Wireless Commun., vol. 3, pp. 1999–2004, Nov. 2004. [7] Y. Zhao, R. Adve, and T. J. Lim, “Improving amplify-and-forward relay networks: Optimal power allocation versus selection,” in Proc. IEEE ISIT 2006, July 2006, pp. 1234–1238. [8] A. Ribeiro, X. Cai, and G. B. Giannakis, “Symbol error probabilities for general cooperative links,” IEEE Trans. Wireless Commun., vol. 4, pp. 1264–1273, May 2005. [9] I. E. Telatar, “Capacity of multi-antenna gaussian channels,” Eur. Trans. Telecom., vol. 10, pp. 585–596, Nov. 1999. [10] G. L. Stüber, Principles of Mobile Communications, 2nd ed. Norwell, MA: Kluwer Academic Publishers, 2001.

SNR (dB) (b) Fig. 4. WERs in a multiple path relay network with TU channels using AF and DF for N = 128: (a) m = 2 and (b) m = 4.

relays and thus, for 4 paths, the system is already resistant to noise and channel distortion. Therefore, DF cannot provide any more significant improvement over AF. In such a situation, AF might be a more attractive choice due to its lower complexity. In terms of power allocation when using AF, CT is the preferable choice because it gives better performance than EQ and does not require CSI. As expected, soft decisions give better performance than hard decisions in Viterbi decoding. IV. CONLCUSIONS In this paper, we exploit cooperative relays by studying two subcarrier power allocation schemes as well as the system mutual information in OFDM-based wireless networks. WER simulations indicate that the DF relay algorithm provides better performance than that of AF in a single path relay network because the former is able to eliminate channel noise at each relay. For the multiple path relay network, however, the network structure is already resistant to noise and channel distortion, and AF is more attractive due to its lower complexity. Future research includes investigating additional relay algo-

Victor K. Y. Wu received his B.A.Sc. in 2005, from the Division of Engineering Science at the University of Toronto, Toronto and his M.S. in 2006, from the School of Electrical and Computer Engineering at the Georgia Institute of Technology, Atlanta. He is currently pursuing a Ph.D. from the Department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign. His research interests are wireless networks and RFID systems.

Ye (Geoffrey) Li received his B.S.E. and M.S.E. degrees in 1983 and 1986, respectively, from the Department of Wireless Engineering, Nanjing Institute of Technology, Nanjing, China, and his Ph.D. degree in 1994 from the Department of Electrical Engineering, Auburn University, Alabama. He was a teaching assistant and then a lecturer with Southeast University, Nanjing, China from 1986 to 1991, a research and teaching assistant with Auburn University, Alabama from 1991 to 1994, and a post-doctoral research associate with the University of Maryland at College Park, Maryland from 1994 to 1996. He was with AT&T Labs—Research at Red Bank, New Jersey as a senior and then a principal technical staff member from 1996 to 2000. Since 2000, he has been with the School of Electrical and Computer Engineering at Georgia Institute of Technology as an associate and then a full professor. He is also holding a visiting chair professor position at the University of Electronic Science and Technology of China since March 2006. His general research interests include statistical signal processing and telecommunications, with emphasis on OFDM and MIMO techniques, cross-layer optimization, and signal processing issues in cognitive radios. In these areas, he has published over

WU et al.: POWER ALLOCATION FOR ODFM-BASED COOPERATIVE DIVERSITY RELAY SYSTEMS

100 papers in refereed journals or conferences and filed about 20 patents. He also has two books, entitled, Blind Equalization and Identification (co-authored with Z. Ding, published by Mercel Dekker, Inc. in 2000) and OFDM for Wireless Communications (co-authored with G. St端ber, published by Springer in 2006). He is active in professional societies. He once served or is currently serving as an editor, a member of editorial board, and a guest editor for 6 technical journals. He organized and chaired many international conferences, including technical program vice-chair of the IEEE 2003 International Conference on Communications. He has been awarded an IEEE fellow for his contributions to signal processing for wireless communications in 2005.

Marilynn P. Wylie-Green received the B.S. and M.S.E. degrees in electrical engineering from Temple University in 1989 and 1991, respectively and the M.S.E.E. and Ph.D. degrees in electrical engineering from the University of Pennsylvania in 1993 and 1995, respectively. In 1995, she worked at the Wireless Information Laboratory (WINLAB) of Rutgers University on the research topic of mobile location estimation. In 1997, she joined the Nokia Research Center where she became involved with various aspects of advanced physical layer design for several wireless systems, including GSM, W-CDMA, UWB, and WiMAX. In 2007, she transitioned into the Research and Technology arm of Nokia Siemens Networks. She is the author of approximately 45 international publications and 26 granted/pending patents. She is a senior member of the IEEE. She is a member of several national honor societies including Eta Kappa Nu National Engineering Honor Society, Golden Key National Honor Society, Alpha Lambda Delta National Honor Society, and she is a former recipient of the National Science Foundation Fellowship for graduate study. Her current research interests include statistical signal processing, optimized transceiver design and cross-layer optimization.

Anthony Reid received his B.S.E.E. (Cum Laude) from Rensselaer Polytechnic Institute, M.S.E.E. from Stanford University and Ph.D. from Southern Methodist University (SMU). He has over 3 decades of industry experience in the telecommunications and military defense arenas. He has worked in the telecomm industry at Bell Labs, Nortel Networks and Nokia Research Center (NRC) and in the defense industry at Sandia Laboratories, TI, and Raytheon Company. Most recently he is an Engineering Fellow at Raytheon working in the areas of algorithms and systems for Space and Airborne Systems. As a Principal Scientist at NRC his work covered design, research and standardization of physical layer for wireless mobile systems (e.g., WiMAX, WLAN, 4G, and GSM/EDGE). He is senior member of IEEE. He has over 20 publications and 25 granted/pending patents. His research interests are in mobile wireless communications and also applied optimal estimation theory and statistical signal processing.

Peter S. S. Wang received his M.S.E. and Ph.D. degrees in 1986 and 1991, respectively, from the Department of Electrical Engineering Department at the University of Texas at Arlington, Texas. He was a visiting assistant professor with University of Texas at Arlington from 1991 to 1994. He was a staff engineer with Motorola Research Lab. in Texas from 1994 to 1998. He was a senior research scientist at Nokia Research Center in Texas from 1998 to 2006. Since 2006, he has been work with Nokia Siemens Networks in Texas. His work covers wireless mobile location services, RF channel modeling, and wireless communication systems. His research interests are mobile wireless communications and diffractive optics.