Effect of Feedback Delay on the Performance of ... - CiteSeerX

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 12, DECEMBER 2011

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Effect of Feedback Delay on the Performance of Cooperative Networks with Relay Selection Mehdi Seyfi, Student Member, IEEE, Sami Muhaidat, Senior Member, IEEE, Jie Liang, Member, IEEE, and Mehrdad Dianati, Member, IEEE

Abstract—In this paper, we analyze the effect of feedback delay and channel estimation errors on the performance of a decodeand-forward (DF) cooperative transmission scenario with relay selection. In our relay selection scheme, only one relay with the best relay-to-destination (𝑅 → 𝐷) channel quality is selected among the set of relays that decode the source information correctly. Specifically, the destination terminal first estimates the channel state information (CSI) of all active 𝑅 → 𝐷 links and then sends the index of the best relay to the relay terminals via a delayed feedback link. Due to the time varying nature of the fading channels, selection is performed based on the old version of the channel estimate. Closed-form expressions for the outage probability, average capacity and average symbol error rate (ASER) are derived. Through asymptotic diversity order analysis, we show that the presence of feedback delay reduces the asymptotic diversity order to one, while the effect of channel estimation errors reduces it to zero. Finally, simulation results are presented to corroborate the analytical results. Index Terms—Relay selection, decode-and-forward, feedback delay, channel estimation error.

I. I NTRODUCTION

I

T has been demonstrated that cooperative diversity provides an effective means of improving spectral and power efficiency of wireless networks as an alternative to MIMO systems [1], [2]. The main idea behind cooperative diversity is that in a wireless environment, the signal transmitted by the source (𝑆) is overheard by other nodes, which can be defined as “relays” [3]. The source and its partners can then jointly process and transmit their information, thereby creating a “virtual antenna array”, although each of them is equipped with only one antenna. It is shown in [4], [5] that cooperative diversity networks can achieve a diversity order equal to the number of paths between the source and the destination. However, the need of transmitting the symbols in a time division multiplexing (TDMA) fashion reduces the maximum achievable capacity improvements. Additionally, due to power Manuscript received October 26, 2010; revised April 12, 2011 and August 22, 2011; accepted August 28, 2011. The associate editor coordinating the review of this paper and approving it for publication was S. Affes. The work of S. Muhaidat was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under grant RGPIN372049. The work of Jie Liang was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under grants RGPIN312262-05, EQPEQ330976-2006, and STPGP350416-07. M. Seyfi, S. Muhaidat, and J. Liang are with the School of Engineering Science, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada (e-mail: mehdi [email protected], [email protected], [email protected]). M. Dianati is with the Faculty of Engineering and Physical Sciences, University of Surrey, UK, GU2 7XH (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2011.101711.100901

allocation constraints, using multiple relay cooperation is not economical. To overcome these problems, relay selection is proposed to alleviate the spectral efficiency reduction caused by multiple relay schemes and to moderate the power allocation constraints [1], [6], [7]. In [6], Bletsas et al. propose a fast selection algorithm that is below the coherence time of the channel. In their work, they also present an outage probability analysis for a relay selection amplify-and-forward (S-AF) scheme. In [7], the performance of selection amplify-and-forward and all participate amplify-and-forward (AP-AF) cooperation is investigated. Outage probability and average symbol error rate (ASER) expressions of selection decode-and-forward (SDF) networks are derived in [8] and [9], respectively. In [10], Beres and Adve analyze relay selection in multi-source networks. Further results on the performance of relay selection in cooperative wireless networks can be found in [11]–[14]. Most of the current works on relay selection in cooperative networks assume an ideal feedback link and assume the availability of perfect channel state information (CSI) at the receiver. However, in practical scenarios, the fading gains are unknown and need to be estimated before performing relay selection. The index of the selected relay is then fed back to all potentially available relays. Due to the time varying nature of the fading channels, which is a function of the Doppler shift of the moving terminals, the CSI corresponding to the selected relay is time varying. Consequently, relay selection is performed based on outdated CSI and hence the selection algorithm may not yield the best relay. Related work and contributions: The setups of [6]–[14] assume CSI knowledge. However, in practical scenarios, the fading gains of communication links are unknown and need to be estimated. Since channel estimation is required for the selection procedures, performance degradation due to channel estimation errors is inevitable. In [15], we analyze the effect of channel estimation errors on the outage probability of SAF. Capacity of relay selection with imperfect CSI is studied in [16], [17]. The impact of channel estimation errors on the ASER performance of distributed space time block coded (DSTBC) systems is investigated in [18], assuming the AF mode. Building upon a similar set-up, Gedik and Uysal [19] extend the work of [18] to a system with 𝑀 relays. In [20], the symbol error rate performance is investigated for the same scenario as in [19]. Although there have been research efforts on conventional MIMO antenna selection with feedback delay and channel

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estimation errors (see for example [21]), only a few isolated results have been reported in the context of cooperative communications. Suraweera et al. analyze the effect of feedback delay on the performance of a partial relay selection scheme with the AF protocol [22]. In [23], Vicario et al. analyze the outage probability and the achievable diversity order of an opportunistic relay selection scenario with feedback delay. In this paper, we investigate the impact of feedback delay and channel estimation errors on the performance of a cooperative diversity scheme with relay selection. Assuming imperfect channel estimation and feedback delay, our contributions are summarized as follows: ∙ We derive exact closed-form formulation for outage probability. ∙ We derive a lower bound on the average capacity. ∙ We derive exact ASER and outage probability expressions for relay selection decode-and-forward (S-DF) relaying ∙ We demonstrate that the asymptotic diversity order is reduced to one with feedback delay which is in line with the results introduced by [23] and reduces to zero in the presence of channel estimation error. ∙ We present a comprehensive Monte Carlo simulation study to confirm the analytical observations and give insight into system performance. The rest of the paper is organized as follows: In Sec. II, we introduce our system setup and the selection strategy. In this section the channel estimation error as well as the delayed feedback link models are illustrated. In Sec. III and Sec. IV we study the information theoretic performance of the system. In particular, in Sec. III, we investigate the outage probability of the system and develop a closed-form expression for outage probability and in Sec. IV we introduce a lower bound on the instantaneous capacity and derive a closed-form analytical expression for the average lower bound capacity. In Sec. V, we study the ASER performance of the system in detail and derive exact expression for the ASER. In Sec. VI, we analyze the asymptotic order of diversity in S-DF. In particular, we investigate the effect of channel estimation errors and feedback delay on the diversity of the system. Simulation results are presented in Sec. VII, and the paper is concluded in Sec. VIII. II. SYSTEM MODEL Fig. 1 shows the selection cooperation system model studied in this paper. We consider a multi-relay scenario with 𝑀 relays. We assume that the relay 𝑅𝑚 , 𝑚 = 1, ..., 𝑀 , the source 𝑆, and the destination 𝐷 are equipped with single transmit and receive antennas. In our system model, we ignore the direct transmission between the source and its destination, due to shadowing. ℎ𝑠𝑚 and ℎ𝑚𝑑 represent the channel fading gains between 𝑆 → 𝑅𝑚 and 𝑅𝑚 → 𝐷, respectively. Assuming a half duplex constraint, the data transmission is performed in two time slots. In the first time slot the source terminal transmits its data to all potentially available 𝑀 relays. After receiving the source signal via independent channels, all relays, i.e., 𝑅𝑚 , 𝑚 = 1, 2, . . . 𝑀 , decode their received signals and check whether the transmitted signal is decoded correctly or not. We define the decoding set 𝒟(𝑠) as the set of all

Destination

Source

Relay Switching Center (RSC) M Relays

Feedback Delay

Signal Detection Center (RDC)

Relay Index Feedback

Relay Selection

Channel Estimation

Fig. 1. System model for decode-and-forward relay selection with delayed feedback.

relays that decode the transmitted signal correctly1. Clearly, only those relay nodes with a good source to relay channel can be in the decoding set 𝒟(𝑠). In the second time slot, the best relay that satisfies an index of merit participates in the transmission and broadcasts its decoded symbol towards the destination. In this paper we assume that the source has a power constraint of 𝑃 Joules/symbol and similarly each relay node in 𝒟(𝑠) can potentially transmit its information with 𝑃 Joules/symbol, and the receiver noise power is normalized to 1. A. Model of channel estimation error All channels are assumed to be independent and identically distributed (i.i.d), with ℎ ∼ 𝒞𝒩 (0, 𝜎ℎ2 ). A block fading scheme is considered where the channel realizations are assumed to be constant over a block and correlated across blocks. The correlation coefficient between the (𝑛− 𝑖)𝑡ℎ and the 𝑛𝑡ℎ block is 𝜌𝑓 = 𝐽𝑜 (2𝜋𝑓𝑑 𝑇 𝑖), where 𝑓𝑑 is the Doppler frequency, 𝐽𝑜 (⋅) is the zeroth order modified Bessel function of first kind and 𝑇 is the block duration. Least mean squares estimator (LMSE) is employed so that the estimation error is orthogonal to the channel estimate. We assume that the number of training symbols in each frame is sufficiently less than the data symbols, so that increasing the training symbols’ power would not lead to an increase in the average power; hence, we assume that the training symbols’ power is independently adjustable and thus, the channel estimation error is independent of the received data SNR. Let the true channel be ℎ and the estimated ˆ then ℎ and ℎ ˆ are related by channel via LMSE be ℎ, ˆ + 𝑢, ℎ=ℎ

(1)

ˆ ∗ } = 0 due where 𝑢 is the channel estimation error and E{𝑢ℎ ˆ to principle of orthogonality for optimal LMSE estimators. ℎ and 𝑢 are zero mean Gaussian random variables with variances 1 and 𝜎𝑢2 . The correlation coefficient between the true channel and the estimated channel is given by 𝜌ℎℎˆ =

E{ℎℎˆ∗ } 1 = . 𝜎ℎ 𝜎ℎˆ 𝜎ℎ

(2)

1 We have to stress that we can either consider that the relay is in the decoding set only if it passes the cyclic redundancy code (CRC) check, or assume that the relay is in the decoding set when the mutual information between the source and the relay is above a threshold limit. In this paper we have adopted the former scheme.

SEYFI et al.: EFFECT OF FEEDBACK DELAY ON THE PERFORMANCE OF COOPERATIVE NETWORKS WITH RELAY SELECTION

B. Feedback delay model

S → Rm D(s)

There are two main selection strategies in the literature. In the first scenario, the relay nodes are in charge of monitoring of their individual source and destination instantaneous CSI [6]. In this strategy, CSI is estimated at the relay nodes for further decisions on which relay is the best node to cooperate. In particular, when the relay nodes overhear a ready-to-send (RTS) packet form the source terminal, they estimate the source link CSI, i.e., ℎ𝑠𝑚. Then, upon receiving a clear-tosend (CTS) packet from the destination, the corresponding destination link at the relays, i.e., ℎ𝑚𝑑 is estimated. Right after receiving the CTS packet, the relay nodes ignite a timer which is a function of the instantaneous relay-destination CSI2 . The timer of the best relay expires first, and a flag packet is sent to other relays, informing them to stop their timers as the best relay is selected. Then the relays that receive the flag packet, keep silent and the selected relay participates in communication [6]. In the second strategy, the relay nodes estimate their individual uplink CSI. The relays that correctly decode their received symbols from the source, send a flag packet to the destination, announcing that they are ready to participate in cooperation. The destination terminal, on the other hand, estimates the downlink CSI, orders the received SNRs from all relays in the decoding set, and feedbacks the index of the best relay that introduces the maximum received SNR via a log 𝑀 bit feedback link. The selected relay then operates with full power [26]. In both strategies, a very important issue that must be taken into consideration is the selection speed. Communication links among terminals are time varying with a macroscopic rate in the order of Doppler shift, which is inversely proportional to the channel coherence time [6]. Any relay selection scheme must be performed no slower than the channel coherence time, otherwise selection is performed based on the old CSI, while the channel conditions are altered at the time that selection is performed. This might lead to a wrong selection of relay and therefore affect the performance of the system. In this paper, to simplify the notation, we adopt the second strategy in relay selection. However, the first strategy obeys the same rules and formulations. In the second strategy, since the feedback link only transmits the index of the selected relay, a lower feedback bandwidth is required. ˆ and the previously (old) Let the estimated channel be ℎ estimated CSI based on which the selection is performed be ˆ and ℎ ˆ 𝑠𝑙 are both zero mean-unit variance and ˆ 𝑠𝑙 . Since ℎ ℎ jointly Gaussian they can be related as follows [21] √ ( ) ˆ= 𝜌ℎ ˆ 𝑠𝑙 + 1 − 𝜌 2 𝑣 ℎ 𝑓 𝑓

(3)

ˆ 𝑠𝑙 is the channel which where 𝑣 ∼ 𝒩 (0, 1). We stress that ℎ is used for relay selection (using the prior knowledge of the channel), whereas ˆℎ is the channel which is used for decoding (using the current knowledge of the channel). 2 The timer is a function of relay-destination CSI for the DF mode and source-relay-destination CSI for the AF protocol.

ChannelEstimation Best Relay Index for Rm Feedback Relay selection

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→D

time slot I

Fig. 2. Signal transmission procedure.

III. O UTAGE P ROBABILITY A NALYSIS In this section we study performance of the our system from the information theory perspective. In the first time slot, the source terminal communicates with the relay terminals by broadcasting the signal 𝑥. Let the received signal at each relay in the first time slot be 𝑦𝑠𝑚,𝑠𝑙 , 𝑚 = 1, . . . 𝑀 . To decode the received symbol, each relay estimates the corresponding 𝑆 → 𝑅𝑚 channel and then decodes the received signal using a maximum likelihood (ML) decoder. The received signal at the 𝑚𝑡ℎ relay in the first time slot can be written as √ ∗ 𝑥ˆ = 𝑃 ˆℎ ,𝑠𝑙 𝑦𝑠𝑚,𝑠𝑙 √ 𝑠𝑚 √ ˆ∗ ( 𝑃 ℎ = 𝑃ℎ 𝑥 + 𝑛𝑠𝑚 ) (4a) 𝑠𝑚,𝑠𝑙 𝑠𝑚,𝑠𝑙 ) ] √ ∗ [√ ( ˆ = 𝑃 ˆℎ𝑠𝑚,𝑠𝑙 𝑃 ℎ + 𝑢𝑠𝑚 𝑥 + 𝑛𝑠𝑚 𝑠𝑚,𝑠𝑙 √ ∗ ∗ ˆ ˆ = 𝑃 ∣ℎ ∣2 𝑥 + 𝑃 ˆℎ𝑠𝑚 𝑢 𝑥 + 𝑃ℎ 𝑛 ,(4b) 𝑠𝑚,𝑠𝑙 ,𝑠𝑙 𝑠𝑚 𝑠𝑚,𝑠𝑙 𝑠𝑚 

 

   message component error component noise component

where we have substituted (1) into (4a). Using (4b), the received effective SNR at each relay in the first time slot (when selection is performed) is given by 𝛾ˆ𝑠𝑚eff = ,𝑠𝑙 ˆ ∣ℎ

ˆ 𝑃 ∣ℎ ∣2 𝑠𝑚,𝑠𝑙 = 𝑃 𝛾ˆ𝑠𝑚,𝑠𝑙 , 1 + 𝑃 𝜎𝑢2𝑠𝑚

(5)

∣2

𝑠𝑚,𝑠𝑙 where 𝛾ˆ𝑠𝑚,𝑠𝑙 = 1+𝑃 . Similarly, still in the first time slot, 2 𝜎𝑢 𝑠𝑚 the destination estimates the 𝑅𝑚 → 𝐷 channels for 𝑚 ∈ 𝒟(𝑠) and calculates the effective SNR of each link which is given by ˆ 𝑃 ∣ℎ ∣2 𝑚𝑑,𝑠𝑙 = = 𝑃 𝛾ˆ𝑚𝑑,𝑠𝑙 , (6) 𝛾ˆ𝑚𝑑eff ,𝑠𝑙 2 1 + 𝑃 𝜎𝑢 𝑚𝑑

where 𝛾ˆ𝑚𝑑,𝑠𝑙 =

ˆ ∣ℎ ∣2 𝑚𝑑,𝑠𝑙 2 1+𝑃 𝜎𝑢

𝑚𝑑

. Here, 𝛾ˆ𝑠𝑚,𝑠𝑙 and 𝛾ˆ𝑚𝑑,𝑠𝑙 are exponen-

tially distributed with parameters 𝜆𝑠𝑚,𝑠𝑙 = (1 + 𝑃 𝜎𝑢2𝑠𝑚 ) and 𝜆𝑚𝑑,𝑠𝑙 = (1 + 𝑃 𝜎𝑢2 ). 𝑚𝑑 If the relay 𝑅𝑚 is in the decoding set, by sending a flag packet, it signals its capability of participating in cooperation. Then, based on the first time slot channel realizations, the destination selects the relay with the best 𝑅𝑚 → 𝐷 link (see Fig. 2) 𝛾𝑚𝑑,𝑠𝑙 }. (7) 𝑅𝑚★ = arg max {ˆ 𝑚∈𝒟(𝑠)

A. Outage probability with feedback delay Once the best relay is selected by destination, the index of 𝑅𝑚★ is fed back to all relays via a delayed feedback link. This means, at the time the relays receive the index (beginning of the second time slot), the system’s CSI has changed due to the time varying nature of communication links. Let the current 𝑅𝑚★ → 𝐷 channel realization( channel realization in the second time slot), the corresponding estimate and the ˆ ★ normalized received SNR from the selected relay be ℎ𝑚★𝑑 , ℎ 𝑚𝑑

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and 𝛾ˆ𝑚★𝑑 , respectively. Due to feedback delay, relay selection is done based on the channel coefficients in the first time slot instead of current coefficients, i.e., 𝛾ˆ𝑚★𝑑 = 𝛾ˆ𝑚𝑑,𝑠𝑙 where 𝑚 = arg max {ˆ 𝛾𝑖𝑑,𝑠𝑙 }.

Defining

Assuming that the communication between source and destination targets an end-to-end data rate 𝑅, the system is in outage if the 𝑆 → 𝑅𝑚★ → 𝐷 link observes an instantaneous capacity per bandwidth3 𝐶 ★ = 12 log (1 + 𝑃 𝛾ˆ𝑚★𝑑 ) that is below the required rate 𝑅, i.e., [ ] 1 log (1 + 𝑃 𝛾ˆ𝑚★𝑑 ) ≤ 𝑅 𝑃𝑜 = 𝑃 𝑟 2 ∑ = 𝑃 𝑟[𝒟(𝑠)]

and noting that 𝜒𝑚 and 𝛾ˆ𝑚𝑑 are independent, the conditioned outage probability on the decoding set is ∫ ∞  [ ] 𝒟(𝑠) = 𝑃 𝑟 𝛾ˆ𝑚𝑑 ≤ 𝑅𝑜 𝒟(𝑠), 𝛾ˆ𝑚𝑑,𝑠𝑙 𝑃𝑜 0 [  ] 𝛾𝑚𝑑,𝑠𝑙 ) 𝑑𝛾ˆ , × 𝑃 𝑟 𝛾ˆ𝑚𝑑,𝑠𝑙 ≥ 𝜒𝑚 𝒟(𝑠), 𝛾ˆ𝑚𝑑,𝑠𝑙 𝑓𝛾ˆ (ˆ 𝑚𝑑,𝑠𝑙 𝑚𝑑,𝑠𝑙 ∫ ∞ = 𝐹𝛾ˆ (𝑅𝑜 )𝐹𝜒𝑚 (ˆ 𝛾𝑚𝑑,𝑠𝑙 )𝑓𝛾ˆ (ˆ 𝛾𝑚𝑑,𝑠𝑙 ) 𝑑𝛾ˆ . (14)

𝑖 ∈ 𝒟(𝑠)

𝒟(𝑠)

[

𝑃 𝑟 𝛾ˆ𝑚★𝑑 ≤ 𝑅𝑜 , 𝛾ˆ𝑚★𝑑,𝑠𝑙 ∑ = 𝑃 𝑟[𝒟(𝑠)]

×

𝒟(𝑠)

×

∑

𝑚∈𝒟(𝑠)

𝑃 𝑟 ⎣𝛾ˆ𝑚𝑑 ≤ 𝑅𝑜 , 𝛾ˆ𝑚𝑑,𝑠𝑙

 ≥ max {ˆ 𝛾𝑖𝑑,𝑠𝑙 }𝒟(𝑠)⎦ ,(9)

𝐹𝛾ˆ (𝑥) 𝑚𝑑

𝑖 ∈𝒟(𝑠) 𝑖 ∕= 𝑚

The relay 𝑅𝑚 is in the decoding set 𝒟(𝑠) if the 𝑆 → 𝑅𝑚 link observes an instantaneous capacity per bandwidth 𝐶𝑠𝑚,𝑠𝑙 that is above the required rate 𝑅 ( ) 1 = log2 1 + 𝑃 𝛾ˆ𝑠𝑚,𝑠𝑙 ≥ 𝑅. 2

(10)

Noting that 𝛾ˆ𝑠𝑚,𝑠𝑙 is exponentially distributed, relay 𝑅𝑚 is in the decoding set if [10] ] [ 𝑃 𝑟 [𝑅𝑚 ∈ 𝒟(𝑠)] = 𝑃 𝑟 𝛾ˆ𝑠𝑚,𝑠𝑙 ≥ 𝑅𝑜 . ( ) = exp −𝜆𝑠𝑚,𝑠𝑙 𝑅𝑜 . (11) Finally, the probability of selecting a specific decoding set is [10] ∏ ) ( 𝑃 𝑟[𝒟(𝑠)] = exp −𝜆𝑠𝑚,𝑠𝑙 𝑅𝑜 𝑚∈𝒟(𝑠)

×

( )] 1 − exp −𝜆𝑠𝑚,𝑠𝑙 𝑅𝑜 .

(12)

𝑚∈ / 𝒟(𝑠)

C. Outage probability conditioned on the decoding set 𝒟(𝑠) Conditioned on the decoding set 𝒟(𝑠) with the old channel realizations (channels in the first time slot), the outage probability with the new CSI (CSI in the second time slot) is ⎡ ⎤  𝑃𝑜𝒟(𝑠) = 𝑃 𝑟 ⎣𝛾ˆ𝑚𝑑 ≤ 𝑅𝑜 , 𝛾ˆ𝑚𝑑,𝑠𝑙 ≥ max {ˆ 𝛾𝑖𝑑,𝑠𝑙 }𝒟(𝑠)⎦ . (13) 𝑖 ∈𝒟(𝑠) 𝑖 ∕= 𝑚

to channel estimation errors, the instantaneous capacity is a lower bound on the true instantaneous capacity. Consequently, the derived outage probability expression is also a lower bound in the presence of channel estimation error. For further details, please refer to Sec. IV.

2𝜌2

(1−𝜌2 ) . 𝑓

Therefore,

𝑓

𝑚𝑑

∞ ∑

=

𝑒

𝑐𝑚 𝜆 𝛾 ˆ 𝑚𝑑,𝑠𝑙 𝑚𝑑,𝑠𝑙 2

−

𝑘=0

×

∏ [

𝑚𝑑,𝑠𝑙

we can write 𝐹𝛾ˆ (𝑥) as [27]

B. Probability of decoding set

3 Due

parameter 𝜂𝑚 = 𝑐𝑚 𝛾ˆ𝑚𝑑,𝑠𝑙 , where 𝑐𝑚 =

−1 . 𝑃

𝐶𝑠𝑚,𝑠𝑙

𝑚𝑑,𝑠𝑙

Conditioned on 𝛾ˆ𝑚𝑑,𝑠𝑙 and using (3), 𝛾ˆ𝑚𝑑 has a non-central Chi-square distribution with two degrees of freedom and

2𝑅

2

max {ˆ 𝛾𝑖𝑑,𝑠𝑙 },

𝑖 ∈𝒟(𝑠) 𝑖 ∕= 𝑚

𝑚𝑑

0

⎤

⎡

𝑚∈𝒟(𝑠)

where 𝑅𝑜 =

]  = max {ˆ 𝛾𝑚𝑑,𝑠𝑙 }𝒟(𝑠) , (8)

Δ

𝜒𝑚 =

𝜆

( ) 𝑐𝑚𝜆𝑚𝑑,𝑠𝑙 𝛾ˆ𝑚𝑑,𝑠𝑙 𝑘 2

𝑥

𝛾(𝑘 + 1, 𝑚𝑑 2 ) , (𝑘!)2

(15)

where 𝛾(⋅, ⋅) is the incomplete gamma function defined as ∫ 𝑥 𝛾(𝑠, 𝑥) = 𝑡𝑠−1 𝑒−𝑡 𝑑𝑡. 0

Furthermore, 𝐹𝜒𝑚 (𝑥) is given by ] ∏ [ 1 − 𝑒(−𝜆𝑖𝑑,𝑠𝑙 𝑥) 𝐹𝜒𝑚 (𝑥) = 𝑖 ∈ 𝒟(𝑠) 𝑖 ∕= 𝑚

= 1−

∑

𝑒−𝜆𝑖𝑑,𝑠𝑙 𝑥 +

𝑖 ∈ 𝒟(𝑠) 𝑖 ∕= 𝑚

+ (−1)∣𝒟(𝑠)−1∣

∑

∑

𝑒−(𝜆𝑖𝑑,𝑠𝑙 +𝜆𝑗𝑑,𝑠𝑙 )𝑥 − . . .

𝑖, 𝑗 ∈ 𝒟(𝑠) 𝑖 ∕= 𝑚, 𝑗

𝑒−(𝜆𝑖𝑑,𝑠𝑙 +𝜆𝑗𝑑,𝑠𝑙 +...𝜆𝑙𝑑,𝑠𝑙 )𝑥 .(16)

𝑖, 𝑗, . . . , 𝑙 ∈ 𝒟(𝑠) 𝑖 ∕= 𝑚, 𝑗, . . . , 𝑙

Substituting (15) and (16) into (14) and noting that 𝑓𝛾ˆ 𝜆𝑚𝑑,𝑠𝑙 𝑒

−𝜆𝑚𝑑,𝑠𝑙 𝑥

and also uisng the fact that [28] ∫ ∞ 𝑘! 𝑥𝑘 exp(−𝑎𝑥)𝑑𝑥 = 𝑘+1 , 𝑎 0

(𝑥) =

𝑚𝑑,𝑠𝑙

we can write the outage probability, conditioned on the decoding set, as in (17) which can be evaluated as (18). Finally, substituting (12) and (18) into (9), we obtain (19). For the special case where 𝜆 = 𝜆𝑠𝑚,𝑠𝑙 = 𝜆𝑚𝑑,𝑠𝑙 = 𝜆𝑖𝑑,𝑠𝑙 , and 𝑐1 = 𝑐2 = . . . 𝑐∣𝒟(𝑠)∣ = 𝑐, (19) reduces to (20). IV. AVERAGE CAPACITY In this section, we derive an analytical expression for average capacity in the presence of channel estimation error and feedback delay. At the destination in the second time slot and after matched filtering, the received signal over the

SEYFI et al.: EFFECT OF FEEDBACK DELAY ON THE PERFORMANCE OF COOPERATIVE NETWORKS WITH RELAY SELECTION

𝑃𝑜𝒟(𝑠) =

∫ ∞ ∑ 𝜆𝑚𝑑𝑘+1 𝛾(𝑘 + 1, 𝜆𝑚𝑑 𝑅𝑜 /2) ( 𝑐 )𝑘 ,𝑠𝑙 𝑘=0

+

∑

𝑚

(𝑘!)2

2

𝑒−(𝜆𝑚𝑑,𝑠𝑙 +𝜆𝑖𝑑,𝑠𝑙 +𝜆𝑗𝑑,𝑠𝑙 +𝜆𝑚𝑑,𝑠𝑙

𝑐𝑚 2

)ˆ 𝛾𝑚𝑑,𝑠𝑙

∞

0

⎡ 𝑘 ⎣𝑒−(𝜆𝑚𝑑,𝑠𝑙 +𝜆𝑚𝑑,𝑠𝑙 𝛾ˆ𝑚𝑑 ,𝑠𝑙

𝑐𝑚 2

)ˆ 𝛾𝑚𝑑,𝑠𝑙

−

∑

4165

𝑒−(𝜆𝑚𝑑,𝑠𝑙 +𝜆𝑖𝑑,𝑠𝑙 +𝜆𝑚𝑑,𝑠𝑙

𝑐𝑚 2

)ˆ 𝛾𝑚𝑑,𝑠𝑙

𝑖 ∈𝒟(𝑠)

∑

− . . . + (−1)∣𝒟(𝑠)−1∣

𝑖, 𝑗 ∈𝒟(𝑠) 𝑖 ∕= 𝑗

⎤

𝑒−(𝜆𝑚𝑑,𝑠𝑙 +𝜆𝑖𝑑,𝑠𝑙 +𝜆𝑗𝑑,𝑠𝑙 +...𝜆𝑙𝑑,𝑠𝑙 +𝜆𝑚𝑑,𝑠𝑙

𝑐𝑚 2

)ˆ 𝛾𝑚𝑑,𝑠𝑙 ⎥

𝛾𝑚𝑑,𝑠𝑙 , ⎦ 𝑑ˆ

𝑖, 𝑗, . . . , 𝑙 ∈𝒟(𝑠) 𝑖 ∕= 𝑗, . . . , 𝑙

(17)

𝑃𝑜𝒟(𝑠)

=

[ ∑ 𝜆𝑚𝑑𝑘+1 𝜆𝑚𝑑𝑘+1 𝑅𝑜 /2) ( 𝑐𝑚 )𝑘 ,𝑠𝑙 ,𝑠𝑙 𝑐𝑚 𝑘+1 − (𝑘!) 2 (𝜆𝑚𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 2 ) (𝜆 + 𝜆 + 𝑚𝑑 ,𝑠𝑙 𝑖𝑑 ,𝑠𝑙 𝑖 ∈𝒟(𝑠)

∞ ∑ 𝛾(𝑘 + 1, 𝜆

𝑚𝑑

𝑘=0

+

∑

𝑐𝑚 𝑘+1 2 )

(𝜆𝑚𝑑,𝑠𝑙 + 𝜆𝑖𝑑,𝑠𝑙

𝑖, 𝑗 ∈𝒟(𝑠) 𝑖 ∕= 𝑗

⎤

∑ 𝜆𝑚𝑑,𝑠𝑙 𝜆𝑚𝑑,𝑠𝑙 ⎥ ∣𝒟(𝑠)−1∣ 𝑐𝑚 𝑘+1 − . . . +(−1) 𝑐𝑚 𝑘+1 ⎦ . + 𝜆𝑗𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 2 ) (𝜆 + 𝜆 + 𝜆 + . . . 𝜆 + 𝜆 ) 𝑚𝑑 ,𝑠𝑙 𝑖𝑑 ,𝑠𝑙 𝑗𝑑 ,𝑠𝑙 𝑙𝑑 ,𝑠𝑙 𝑚𝑑 ,𝑠𝑙 2 𝑖, 𝑗, . . . , 𝑙 ∈𝒟(𝑠) 𝑘+1

𝑘+1

𝑖 ∕= 𝑗, . . . , 𝑙

(18)

𝑃𝑜 =

( )∏[ ( )] ∑ exp −𝜆𝑠𝑚,𝑠𝑙 𝑅𝑜 1 − exp −𝜆𝑠𝑚,𝑠𝑙 𝑅𝑜

𝒟(𝑠) 𝑚∈𝒟(𝑠)

⎡ ×⎣

∑ ∏

(𝜆𝑚𝑑,𝑠𝑙

𝑚/ ∈𝒟(𝑠)

∑ 𝜆𝑚𝑑𝑘+1 𝜆𝑚𝑑𝑘+1 ,𝑠𝑙 ,𝑠𝑙 𝑐𝑚 𝑘+1 − + 𝜆𝑚𝑑,𝑠𝑙 2 ) (𝜆 + 𝜆 + 𝑚𝑑,𝑠𝑙 𝑖𝑑,𝑠𝑙 𝑖 ∈𝒟(𝑠)

𝑚∈𝒟(𝑠)

𝑐𝑚 𝑘+1 2 )

+(−1)∣𝒟(𝑠)−1∣

+

𝑅𝑜 /2) ( 𝑐𝑚 )𝑘 (𝑘!) 2

∞ ∑ 𝛾(𝑘 + 1, 𝜆

𝑚𝑑

𝑘=0

∑ (𝜆𝑚𝑑,𝑠𝑙 + 𝜆𝑖𝑑,𝑠𝑙

𝑖, 𝑗 ∈𝒟(𝑠) 𝑖 ∕= 𝑗

𝜆𝑚𝑑𝑘+1 ,𝑠𝑙 − ... 𝑐 + 𝜆𝑗𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 2𝑚 )𝑘+1 ⎤

∑

𝑘+1

(𝜆𝑚𝑑,𝑠𝑙 + 𝜆𝑖𝑑,𝑠𝑙

𝑖, 𝑗, . . . , 𝑙 ∈𝒟(𝑠) 𝑖 ∕= 𝑗, . . . , 𝑙

𝜆𝑚𝑑,𝑠𝑙 ⎥ ⎦. 𝑐 + 𝜆𝑗𝑑,𝑠𝑙 + . . . 𝜆𝑙𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 2𝑚 )𝑘+1 (

𝑃𝑜 =

𝑀 ∑

exp (−𝑙𝜆𝑅𝑜 ) [1 − exp (−𝜆𝑅𝑜 )]

𝑀−𝑙

×𝑙×

𝑙=1

𝑘=0

selected link is given as

= = =

𝑙 ∑

(20)

defined as

√ ∗ ˆ ★ 𝑃ℎ 𝑑 √ 𝑚 ˆ ∗★ 𝑃ℎ 𝑚𝑑 √ ∗ ˆ 𝑃 ℎ𝑚★𝑑

𝑥 ˆ =

∞ ∑

) 𝑙−1 𝑚−1 𝛾(𝑘 + 1, 𝜆𝑅𝑜 /2) ( 𝑐 )𝑘 × . (𝑘!) 2 (𝑚 + 2𝑐 )𝑘+1 𝑚=1

(19)

𝑦𝑚★𝑑 , √ ( 𝑃 ℎ𝑚★𝑑 𝑥 + 𝑛𝑚★𝑑 ), [√ ( ) ] ˆ ★ +𝑢 ★ 𝑥+𝑛 ★ , 𝑃 ℎ 𝑚𝑑 𝑚𝑑 𝑚𝑑 √ ∗ ˆ ★ ∣2 𝑥 + 𝑃 ˆ ˆ ★ 𝑛 ★ . (21) 𝑃 ∣ℎ ℎ𝑚∗★𝑑 𝑢𝑚★𝑑 𝑥 + 𝑃 ℎ 𝑚𝑑 𝑚𝑑 𝑚𝑑 

 

  

message component error component

noise component

Defining Δ ˆ ★ ∣2 , 𝑞 = 𝑃 ∣ℎ 𝑚𝑑 √ ∗ Δ ∗ ˆ ˆ★ 𝑛★, 𝑛 ˜ = 𝑃 ℎ𝑚★𝑑 𝑢𝑚★𝑑 𝑥 + 𝑃 ℎ 𝑚𝑑 𝑚𝑑

(22a) (22b)

𝐶 𝒟(𝑠)

= = =

max ℐ(ˆ 𝑥, 𝑥)

𝑓𝑥 (𝑥)

max {ℋ(ˆ 𝑥) − ℋ(ˆ 𝑥∣𝑥)}

𝑓𝑥 (𝑥)

max {ℋ(ˆ 𝑥) − ℋ(˜ 𝑛∣𝑥)},

𝑓𝑥 (𝑥)

(24)

where 𝑓𝑥 (𝑥) is the PDF of the input signal 𝑥 and ℋ(⋅) is the differential entropy function. It can be deduced easily from (23) that 𝑛 ˜ and 𝑥 are uncorrelated, i.e., E{˜ 𝑛𝑥∗ } = 0. However, 𝑛 ˜ and 𝑥 are not independent and therefore ℋ(˜ 𝑛∣𝑥) ∕= ℋ(˜ 𝑛). This means that the standard procedures for deriving capacity expressions are not applicable to (24). Alternatively, noting that conditioning does not increase the entropy, i.e., ℋ(˜ 𝑛∣𝑥) ≤ ℋ(˜ 𝑛),

(21) can be written as 𝑥 ˆ = 𝑞𝑥 + 𝑛 ˜.

(23)

The instantaneous capacity conditioned on the decoding set is

we can lower bound the capacity as 𝑥) − ℋ(˜ 𝑛)}. 𝐶 𝒟(𝑠) ≥ max {ℋ(ˆ 𝑓𝑥 (𝑥)

(25)

4166

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 12, DECEMBER 2011

It must be noted that the distribution of 𝑛 ˜ is not Gaussian. However, following [30], Theorem 1, we may assume that 𝑛 ˜ has zero-mean complex Gaussian distribution with the same variance, which is the worse case distribution. In this case, we can re-write (25) as 𝑥) − ℋ(˜ 𝑛)} ≥ max { 𝐶 𝒟(𝑠) ≥ max {ℋ(ˆ 𝑓𝑥 (𝑥)

where 𝛼 and 𝛽 depend on the modulation scheme, and 𝑓𝛾ˆ𝑚★𝑑 is the probability density function (PDF) of 𝛾ˆ𝑚★𝑑 , which is given by (see Appendix for details) 𝑓𝛾ˆ𝑚★𝑑 (𝑥) =

𝑓𝑥 (𝑥) †

ℋ(ˆ 𝑥) − ℋ(𝑛 )}, (26) ) ˆ ★ ∣2 𝜎 2 + 𝑃 ∣ ℎ ˆ ★ ∣2 . Maximizing where 𝑛† ∼ 𝒞𝒩 0, 𝑃 2 ∣ℎ 𝑢𝑚★𝑑 𝑚𝑑 𝑚𝑑 (26) with respect to 𝑓𝑥 (𝑥) and using (23), a lower bound on the average capacity, conditioned on the decoding set, can be written as

+

(

𝐶 𝒟(𝑠)

≥

𝒟(𝑠)

𝐶lb

ˆ ∣ℎ

∣2

where 𝛾ˆ𝑚★𝑑 = 1+𝑃𝑚𝜎★𝑑2 . Hence, a lower bound on the S-DF 𝑢𝑚★𝑑 capacity can be obtained as ∑ 𝒟(𝑠) 𝐶lb 𝑃 𝑟[𝒟(𝑠)]. (28) 𝐶lb = 𝒟(𝑠)

Therefore, the average capacity bound, i.e., 𝐶¯lb , can be written as 1∑ 𝐶¯lb = 𝑃 𝑟[𝒟(𝑠)] 2 𝒟(𝑠) ∫ ∞ log (1 + 𝑃 𝛾ˆ𝑚★𝑑 ) 𝑓𝛾ˆ𝑚★𝑑 ∣𝒟(𝑠) (ˆ 𝛾𝑚★𝑑 )𝑑ˆ 𝛾𝑚★𝑑 .(29) ×

where

Δ

E𝑖(𝛾) =

∫ 𝛾

∞

1 exp(−𝑥)𝑑𝑥 𝑥

is the exponential integral function. In the special case where 𝜆 = 𝜆𝑠𝑚,𝑠𝑙 = 𝜆𝑚𝑑,𝑠𝑙 = 𝜆𝑖𝑑,𝑠𝑙 and 𝑐1 = 𝑐2 = . . . 𝑐∣𝒟(𝑠)∣ = 𝑐, the average capacity bound can be obtained as (32). V. AVERAGE S YMBOL E RROR R ATE In this section, we look into the performance of our system from the communication theory perspective. In particular, we derive a closed form expression for the ASER of S-DF in the presence of channel estimation error and feedback delay. Let the received SNR via the selected path in the second time slot = 𝑃 𝛾ˆ𝑚★𝑑 . Then, the ASER can be written be denoted by 𝛾ˆ𝑚eff ★𝑑 as ∫ ∞ ) (√ 𝑃¯𝑒 = 𝛼𝑄 𝛽𝑃 𝛾ˆ𝑚★𝑑 𝑓𝛾ˆ𝑚★𝑑 (ˆ 𝛾𝑚★𝑑 ) 𝑑ˆ 𝛾𝑚★𝑑 (33) 0

⎡ ⎣

∏

(1 − 𝐵𝑖 )

𝑖∈ / 𝒟(𝑠)

∏

⎤ 𝐵𝑖 ⎦ 𝑓𝛾ˆ𝑚★𝑑 ∣𝒟(𝑠) (𝑥),

𝑖∈𝒟(𝑠)

where 𝛿(𝑥) is ) the delta function and ∫ ∞ (√ 𝛼 0 𝑄 𝛽𝑃 𝛾ˆ𝑠𝑖𝑜 𝑓𝛾ˆ𝑠𝑖𝑜 𝑑ˆ 𝛾𝑠𝑖𝑜 is given as 4 √ [ ] 𝛽𝑃 𝛼 1− . 𝐵𝑖 = 2 𝛽𝑃 + 2𝜆𝑠𝑖𝑜

𝐵𝑖

=

(35)

Here, 𝑓𝛾ˆ𝑚★𝑑 ∣𝒟(𝑠) (𝑥) is the conditional PDF of the received signal via the selected path, which is given by 𝑓𝛾ˆ𝑚★𝑑 ∣𝒟(𝑠) (𝑥) =

∂𝐹𝛾ˆ𝑚★𝑑 ∣𝒟(𝑠) (𝑥) ∂𝑥

.

(36) 𝒟(𝑠)

Following the same procedure for obtaining 𝑃𝑜 we reach to (37). Regarding (36) and (37), the conditional PDF of the received signal can be then written as by (38). Finding a closed form formula for the integral in (33) is not tractable. However, by substituting (34) into (33), we obtain (39). where we have used the approximation [29] 𝑄(𝑥) ≈ 𝑒

0

Substituting (38) into (29) yields (30). Note that in our derivation, we have used the integration formula [28] [ ∫ ∞ 𝑘 ∑ (−1)𝑘−𝜇−1 𝑒1/𝑎 𝑘! 𝑘 −𝑥 ln(1 + 𝑎𝑥)𝑥 𝑒 𝑑𝑥 = (𝑘 − 𝜇)! 𝑎𝑘−𝜇 0 𝜇=0 ( ( ) 𝑘−𝜇 )𝑘−𝜇−𝑡 ] ∑ 1 1 ×E𝑖 − (𝑡 − 1)! − , (31) + 𝑎 𝑎 𝑡=1

∑

(34)

𝑚𝑑

(27)

𝐵𝑖 𝛿(𝑥)

𝑖=1

𝒟(𝑠)

( ) 𝑞2 1 = log 1 + , ˆ ★ ∣2 𝜎 2 + 𝑃 ∣ ℎ ˆ ★ ∣2 2 𝑃 2 ∣ℎ 𝑢 ★ 𝑚𝑑 𝑚𝑑 1 = log (1 + 𝑃 𝛾ˆ𝑚★𝑑 ) , 2

𝑀 ∏

2

− 𝑥2

𝑛𝑎 ∑

𝑎𝑛 𝑥𝑛−1 ,

(40)

𝑛=1

where 𝑎𝑛 =

(−1)𝑛+1 (𝐴)𝑛 , √ √ 𝐵 𝜋( 2)𝑛+1 𝑛!

(41)

with 𝐴 = 1.98 and 𝐵 = 1.135. In the special case where 𝜆 = 𝜆𝑠𝑚,𝑠𝑙 = 𝜆𝑚𝑑,𝑠𝑙 = 𝜆𝑖𝑑,𝑠𝑙 and 𝑐1 = 𝑐2 = . . . 𝑐∣𝒟(𝑠)∣ = 𝑐, (39) reduces to ) ] 𝑀 [( 1 𝑀 ∑ 𝑀 𝑙 𝑀−𝑙 ¯ 𝑃𝑒 = 𝐵 + (1 − 𝐵) 𝐵 × 𝛼𝑙 𝑙 2 𝑙=1 ( ) ( )𝑘+1 𝑛𝑎 𝑎 Γ 𝑘 + (𝑛+1) /𝑘! ( ) ∞ ∑ ∑ 𝑛 2 𝜆 𝑐 𝑘 × × 𝜆 𝑘+ (𝑛+1) 2 2𝑃 2 𝑘=0 𝑛=1 (1 + 2𝑃 𝛽 ) ( ) 𝑙−1 𝑙 ∑ 𝑚−1 × , (42) (𝑚 + 2𝑐 )𝑘+1 𝑚=1 where 𝐵 = 𝐵1 = . . . = 𝐵𝑀 . 4 We may assume the relay 𝑅 is in the decoding set if the 𝑆 → 𝑅 𝑖 𝑖 instantaneous SNR in the first time slot is above a predefined threshold limit (−𝜆𝑠𝑖,𝑠𝑙 .𝛾𝑡 ) (𝛾𝑡 ) = 1 − 𝑒 . Since no CRC 𝛾𝑡 , then in (34) 𝐵𝑖 = 𝐹𝛾ˆ 𝑠𝑖,𝑠𝑙 check is performed in this scenario there is a probability that the relay in the decoding set does not decode the signal correctly and this leads to error propagation.

SEYFI et al.: EFFECT OF FEEDBACK DELAY ON THE PERFORMANCE OF COOPERATIVE NETWORKS WITH RELAY SELECTION

⎤ ⎡ ∑ ∏ ) ∏ [ ( )] ( 1 ¯lb = ⎣ 1 − exp −𝜆𝑠𝑚,𝑠𝑙 𝑅𝑜 ⎦ exp −𝜆𝑠𝑚,𝑠𝑙 𝑅𝑜 𝐶 2 ln 2 𝒟(𝑠) 𝑚∈𝒟(𝑠) 𝑚/ ∈𝒟(𝑠) ⎡ ( ) ( 𝑐𝑚 ) 𝑘 ( ( ) 𝑘−𝜇 )𝑘−𝜇−𝑡 ] 𝜆 ∞ ∑ 𝑘 𝑘−𝜇−1 𝑚𝑑 ∑ ∑ ∑ 𝜆𝑚𝑑 𝜆𝑚𝑑 ⎢ (−1) 2𝑃 2 × ×E𝑖 − (𝑡 − 1)! − + ⎣( )𝑘−𝜇 𝑒 (𝑘 − 𝜇)! 2𝑃 2𝑃 2𝑃 𝜇=0 𝑡=1 𝑚∈𝒟(𝑠)

⎡ ×⎣

(𝜆𝑚𝑑,𝑠𝑙

𝑘=0

𝜆𝑚𝑑

𝑘+1 𝑘+1 𝑘+1 ∑ ∑ 𝜆𝑚𝑑,𝑠𝑙 𝜆𝑚𝑑,𝑠𝑙 𝜆𝑚𝑑,𝑠𝑙 𝑐𝑚 𝑘+1 − 𝑐𝑚 𝑘+1 + 𝑐 + 𝜆𝑚𝑑,𝑠𝑙 2 ) (𝜆𝑚𝑑,𝑠𝑙 + 𝜆𝑖𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 2 ) (𝜆 + 𝜆𝑖𝑑,𝑠𝑙 + 𝜆𝑗𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 𝑚 )𝑘+1 2 𝑖 ∈𝒟(𝑠) 𝑖, 𝑗 ∈𝒟(𝑠) 𝑚𝑑,𝑠𝑙 𝑖 ∕= 𝑗

∑

− . . . + (−1)∣𝒟(𝑠)−1∣

¯lb = 𝐶

⎤ 𝑘+1

(𝜆𝑚𝑑,𝑠𝑙 + 𝜆𝑖𝑑,𝑠𝑙

𝑖, 𝑗, . . . , 𝑙 ∈𝒟(𝑠) 𝑖 ∕= 𝑗, . . . , 𝑙

⎥ 𝜆𝑚𝑑,𝑠𝑙 ⎥ 𝑐𝑚 𝑘+1 ⎦ . + 𝜆𝑗𝑑,𝑠𝑙 + . . . 𝜆𝑙𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 2 )

(30)

[ ( 𝑐 )𝑘 ) ( 𝑘 𝑀 ∞ ∑ ∑ 𝜆 1 ∑ (−1)𝑘−𝜇−1 ( 2𝑃 2 ) × E𝑖 − 𝜆 exp (−𝑙𝜆𝑅𝑜 ) [1 − exp (−𝜆𝑅𝑜 )]𝑀 −𝑙 × 𝑙 × 𝑒 × ( 2𝑃 )𝑘−𝜇 2 ln 2 𝑙=1 𝜆𝑘+1 (𝑘 − 𝜇)! 2𝑃 𝑘=0 𝜇=0 𝜆 ( ) 𝑙−1 ] ( ) 𝑘−𝜇 𝑙 𝑘−𝜇−𝑡 ( )𝑘 ∑ ∑ 𝑚−1 𝑐 𝜆 + (𝑡 − 1)! − × . 2𝑃 2 (𝑚 + 2𝑐 )𝑘+1 𝑡=1 𝑚=1

[ ∞ 𝑘+1 ∑ ∑ 𝜆𝑚𝑑𝑘+1 𝜆𝑚𝑑,𝑠𝑙 𝛾(𝑘 + 1, 𝜆𝑚𝑑 𝑥/2) ( 𝑐𝑚 )𝑘 ,𝑠𝑙 𝐹𝛾ˆ ★ ∣𝒟(𝑠) (𝑥) = − 𝑐 𝑚𝑑 (𝑘!) 2 (𝜆𝑚𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 𝑚 )𝑘+1 𝑖 ∈𝒟(𝑠)(𝜆𝑚𝑑,𝑠𝑙 + 𝜆𝑖𝑑,𝑠𝑙 + 2 𝑘=0 +

4167

∑ (𝜆𝑚𝑑,𝑠𝑙 + 𝜆𝑖𝑑,𝑠𝑙

⎤

∑ ⎥ 𝜆𝑚𝑑,𝑠𝑙 𝜆𝑚𝑑,𝑠𝑙 ∣𝒟(𝑠)−1∣ ⎥ 𝑐𝑚 𝑘+1 − . . . +(−1) 𝑐𝑚 𝑘+1 ⎦ . + 𝜆𝑗𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 2 ) (𝜆 + 𝜆𝑖𝑑,𝑠𝑙 + 𝜆𝑗𝑑,𝑠𝑙 + . . . 𝜆𝑙𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 2 ) 𝑖, 𝑗, . . . , 𝑙 ∈𝒟(𝑠) 𝑚𝑑,𝑠𝑙 𝑘+1

𝑖, 𝑗 ∈𝒟(𝑠) 𝑖 ∕= 𝑗

𝑐𝑚 𝑘+1 ) 2

(32)

𝑘+1

𝑖 ∕= 𝑗, . . . , 𝑙

(37)

𝑓𝛾ˆ

𝑚★𝑑 ∣𝒟(𝑠)

+

∑ (𝜆𝑚𝑑,𝑠𝑙 + 𝜆𝑖𝑑,𝑠𝑙

𝑖, 𝑗 ∈𝒟(𝑠) 𝑖 ∕= 𝑗

(𝑥) =

∞ ∑ 𝜆𝑚𝑑 𝑒(− 𝑘=0

𝜆

𝑚𝑑 2

𝑥

2𝑘!

) 𝜆𝑚𝑑 𝑥 𝑘 ( 2 )

( 𝑐 )𝑘 𝑚

2

⎡ ⎣

(𝜆𝑚𝑑,𝑠𝑙

∑ 𝜆𝑚𝑑𝑘+1 𝜆𝑚𝑑𝑘+1 ,𝑠𝑙 ,𝑠𝑙 𝑐𝑚 𝑘+1 − 𝑐 + 𝜆𝑚𝑑,𝑠𝑙 2 ) (𝜆 + 𝜆 + 𝜆𝑚𝑑,𝑠𝑙 𝑚 )𝑘+1 𝑚𝑑,𝑠𝑙 𝑖𝑑,𝑠𝑙 2 𝑖 ∈𝒟(𝑠) ⎤

𝑘+1 ∑ ⎥ 𝜆𝑚𝑑𝑘+1 𝜆𝑚𝑑,𝑠𝑙 ,𝑠𝑙 ∣𝒟(𝑠)−1∣ ⎥ 𝑐𝑚 𝑘+1 − . . . + (−1) 𝑐𝑚 𝑘+1 ⎦ . + 𝜆𝑗𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 2 ) (𝜆 + 𝜆 + 𝜆 + . . . 𝜆 + 𝜆 ) 𝑖𝑑,𝑠𝑙 𝑗𝑑,𝑠𝑙 𝑙𝑑,𝑠𝑙 𝑚𝑑,𝑠𝑙 2 𝑖, 𝑗, . . . , 𝑙 ∈𝒟(𝑠) 𝑚𝑑,𝑠𝑙 𝑖 ∕= 𝑗, . . . , 𝑙

(38)

) ( ⎡ ⎤ ( )(𝑘+1) ∞ 𝑎𝑛 Γ 𝑘 + (𝑛+1) /𝑘! ( 𝑀 ∞ ∑ ∏ ∑ ∑ ∏ ∏ ∑ 2 𝜆𝑚𝑑 𝑐𝑚 )𝑘 1 ⎣ 𝐵𝑖 + (1 − 𝐵𝑖 ) 𝐵𝑖 ⎦ × 𝛼 × 𝑃¯𝑒 = 𝜆𝑚𝑑 𝑘+ (𝑛+1) 2 𝑖=1 2 2𝑃 2 𝒟(𝑠) 𝑖 ∈𝒟(𝑠) / 𝑖 ∈𝒟(𝑠) 𝑚 ∈𝒟(𝑠) 𝑘=0 𝑛=1 (1 + 2𝑃 𝛽 )

[ (𝜆𝑚𝑑,𝑠𝑙

𝑘+1 𝑘+1 ∑ ∑ 𝜆𝑚𝑑𝑘+1 𝜆𝑚𝑑,𝑠𝑙 𝜆𝑚𝑑,𝑠𝑙 ,𝑠𝑙 −... 𝑐𝑚 𝑘+1 − 𝑐𝑚 𝑘+1 + 𝑐 + 𝜆𝑚𝑑,𝑠𝑙 2 ) (𝜆𝑚𝑑,𝑠𝑙 + 𝜆𝑖𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 2 ) (𝜆 + 𝜆𝑖𝑑,𝑠𝑙 + 𝜆𝑗𝑑,𝑠𝑙 + 𝜆𝑚𝑑,𝑠𝑙 𝑚 )𝑘+1 2 𝑖 ∈𝒟(𝑠) 𝑖, 𝑗 ∈𝒟(𝑠) 𝑚𝑑,𝑠𝑙 𝑖 ∕= 𝑗

∑

⎤ 𝑘+1

⎥ 𝜆𝑚𝑑,𝑠𝑙 ⎥ 𝑐𝑚 𝑘+1 ⎦ , (𝜆 + 𝜆 + 𝜆 + . . . 𝜆 + 𝜆 ) 𝑖𝑑,𝑠𝑙 𝑗𝑑,𝑠𝑙 𝑙𝑑,𝑠𝑙 𝑚𝑑,𝑠𝑙 2 𝑖, 𝑗, . . . , 𝑙 ∈𝒟(𝑠) 𝑚𝑑,𝑠𝑙

+(−1)∣𝒟(𝑠)−1∣

𝑖 ∕= 𝑗, . . . , 𝑙

(39)

4168

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 12, DECEMBER 2011

VI. A SYMPTOTIC D IVERSITY O RDER

0

10

−2

10

−3

10

ideal feedback link −4

10

−5

10

S−DF with FD, M=4, simulation S−DF with FD, M=4, analytical

−6

10

0

5

10

15 SNR(dB)

20

25

30

Fig. 3. Outage Probability for 𝑀 = 4 and perfect CSI in the presence of delay in the feedback link. In this figure 𝑃𝑒 = 1 and 𝜌𝑓 = 0.6, 0.7, . . . 1. 0

10

S−DF with FD M=2,3,4 −1

10

−2

10

Pout

where 𝑔(𝑛, 𝑘) is a constant depending on 𝑘 and 𝑛. In the following, we discuss two different scenarios:

ρf = 0.6, ...0.9

−1

10

Pout

In the previous section, we have derived an exact ASER expression with feedback delay and channel estimation errors, which is valid for the entire SNR range. To gain further insights into the system’s performance, we focus here on the high SNR regime and analyze the asymptotic ASER for S-DF conditioned on the decoding set, i.e., 𝑃¯𝑒 ∣𝒟(𝑠)5 . Without loss of generality, we assume that 𝜆 = 𝜆𝑠𝑚,𝑠𝑙 = 𝜆𝑚𝑑,𝑠𝑙 = 𝜆𝑖𝑑,𝑠𝑙 and 𝑐1 = 𝑐2 = . . . 𝑐∣𝒟(𝑠)∣ = 𝑐. Using (42), we can write the ASER, conditioned on the decoding set, as ) ( 𝑛𝑎 𝑎 Γ 𝑘 + (𝑛+1) /𝑘! ( ) ∞ ∑ ∑ 𝑛 2 𝑐 𝑘 𝑃¯𝑒∣𝒟(𝑠) = 𝛼 (𝑛+1) 𝜆 𝑘+ 2 2 𝑘=0 𝑛=1 (1 + 2𝑃 𝛽𝜎2 ) ) ( 𝑙−1 ( )𝑘+1 ∑ 𝑙 𝑚−1 𝜆 × , 2𝜎 2 𝑃 𝛽 (𝑚 + 2𝑐 )𝑘+1 𝑚=1 ( )𝑘+1 𝑛𝑎 ∞ ∑ ∑ 𝜆 𝑔(𝑛, 𝑘) = , 𝜆 𝑘+ (𝑛+1) 2𝜎 2 𝑃 𝛽 2 𝑘=0 𝑛=1 (1 + 2𝑃 𝜎2 𝛽 ) (43)

−3

10

−4

10

ideal feedback Link

A. Diversity order with feedback delay and perfect CSI

10

With perfect CSI, we have 𝜆 = 1. Therefore, as 𝑃 → ∞, the dominant term in (43) would be the term corresponding to 𝑘 = 0 and 𝑛 = 1, i.e. 𝑃¯𝑒∣𝒟(𝑠)

1 1 × , 1 + 𝑘0 /𝑃 𝑃 1 ∝ , 𝑃 + 𝑘0 = 𝑂(𝑃 −1 )

M=3

M=4

−5

−6

10

0

S−DF, with FD, simulation. S−DF, with FD, analytical 5

10

15 SNR(dB)

20

25

30

Fig. 4. Outage Probability for 𝑀 = 2, 3, 4 and perfect CSI in the presence of delay in the feedback link. In this figure 𝜌𝑒 = 1 and 𝜌𝑓 = 0.9.

∝

VII. S IMULATION R ESULTS (44)

where 𝑘0 is a constant. Thus, in this case, the achievable diversity order in the presence of delayed feedback is 1. B. Diversity order in the presence of feedback delay and imperfect CSI In the presence of channel estimation error, since 𝜆 is proportional to 𝑃 , it should be taken into consideration. From (43), we have [∞ 𝑛 ] ( 𝜆 )𝑘+1 𝑎 ∑∑ 𝑃 lim 𝑃¯𝑒∣𝒟(𝑠) ∝ lim , 𝑘1 𝜆 𝑘+ (𝑛+1) 𝑝→∞ 𝑝→∞ 2 (1 + 𝑘=0 𝑛=1 𝑃 ) = 𝐾 (45) and 𝑘1 and 𝐾 are constants. Hence, the asymptotic diversity order in the presence of feedback delay and channel estimation errors is 0. This means that the outage probability or the ASER curves are expected to be saturated at high SNR with imperfect CSI.

In the section, we investigate the performance of selection cooperation in the presence of channel estimation errors and feedback delay through Monte-Carlo simulation. The transmitted symbols are drawn from an antipodal BPSK constellation, which means, 𝛼 = 1 and 𝛽 = 2. The node-to-node channels are assumed to be zero mean independent Gaussian processes, ˆ and ℎ ˆ 𝑠𝑙 with variance 𝜎ℎ2 = 𝜎ℎ2 𝑠𝑙 = 1 + 𝜎𝑢2 . In this case, ℎ are zero mean Gaussian processes with unit variance. We also define 𝜌𝑒 = 𝜎ℎˆ2 /(𝜎ℎˆ2 + 𝜎𝑢2 ) = 1/(1 + 𝜎𝑢2 ) as in [25]. The variance of noise components is set to 𝑁0 = 1 and 𝑅 = 1 bps/Hz. Outage Probability: Fig. 3 shows the performance of the system with 𝑀 = 4 relays with feedback delay (FD) for 𝜌𝑓 = 0.6, 0.7, 0.8, 0.9 and 1. In Fig. 3, we assume perfect CSI, i.e., 𝜌𝑒 = 1. We observe a perfect match between the analytical and simulation results. It can also be deduced from the slope of the curves that, for 𝑀 = 4 and 0 < 𝜌𝑓 < 1, the diversity order 𝑀 (42) the order associated ) with 𝐵 is 𝑀 ]when 𝜆 = 1, and the order [( ∑𝑀 𝑀 (1 − 𝐵)𝑙 𝐵 𝑀 −𝑙 is zero. Therefore, in the associated with 𝑙=1 𝑙 case that 𝜆 = 1 the diversity order is min(𝑀, 𝒪(𝑃𝑒∣𝒟(𝑠) )). 5 In

SEYFI et al.: EFFECT OF FEEDBACK DELAY ON THE PERFORMANCE OF COOPERATIVE NETWORKS WITH RELAY SELECTION

0

4169

0

10

10

FD and CE error ρe = 0.9

S−DF, with FD, simulation, M=3 S−DF, with FD, analytical, M=3 S−DF, ideal feedback link, M=3

−1

10

−1

10

ρe = 0.95

delayed feedback link

−2

10

−2

Pout

10

ASER

ρe = 1 −3

−3

10

10

−4

10

ideal feedback link with perfect CSI −4

10

−5

10

S−DF, with FD and CE error, simulation, M=2 S−DF, with FD and CE error, analytical, M=2 −5

0

5

10

15 SNR(dB)

20

25

Fig. 5.Outage probability for 𝑀 = 2 in the presence of delay in the feedback link and channel estimation error. In this figure 𝜌𝑓 = 0.9. 6

0

5

10

15 SNR(dB)

20

25

30

Fig. 7. ASER for 𝑀 = 3 and perfect CSI in the presence of delay in the feedback link. In this figure 𝜌𝑒 = 1, 𝜌𝑓 = 0.6, 0.7, . . . , 1. 0

10

S−DF, average capacity, simulation, M=2 S−DF, average capacity, analytical, M=2

5

C¯lb /W (b/s/Hz)

−6

10

30

S−DF, with DF and CE error, simulation, M=2 S−DF, with DF and CE error, analytical, M=2 S−DF, perfect CSI and ideal feedback link, M=2

−1

10

ρe = 1 ρf = 0.7, 0.8, 0.9, 1

4

FD and CE Error ρe = 0.9

−2

10

ρf = 0.9, 1 ρe = 0.85, 0.9, 0.95

3

2

ASER

10

ideal feedback link

ρe = 0.95

−3

10

ρe = 1 −4

10

1

−5

10

ideal feedback link and perfect CSI

Dashed line: average capacity with ideal feedback link 0 0

5

10

15 SNR(dB)

20

25

30

Fig. 6. Lower bound average capacity versus SNR for different values of 𝜌𝑓 and 𝜌𝑒 .

of the selection scheme for is 1. In the case of ideal feedback link, i.e., 𝜌𝑓 = 1, the diversity order of 4 is observed. Fig. 4 illustrates the performance of the system for 𝑀 = 2, 3, 4, with 𝜌𝑓 = 0.9, 1. In Fig. 4, the same slope for different number of relays is noticed, confirming our analytical observations. For the case of 𝜌𝑓 = 1 and choosing 𝑀 = 3, 4, the diversity orders of 3 and 4 are observed, respectively. In Fig. 5, we study the effect of both feedback delay and channel estimation errors. An error floor, in the presence of channel estimation error, is noticed, as predicted by (36). Average Capacity: Fig. 6 shows the lower bound average capacity in bits per second per Hz per bandwidth versus SNR. It is observed that the presence of channel estimation errors result in capacity ceilings in the average capacity curves. It can be also seen that feedback delay aggravates the average capacity performance of the system. However, channel estimation errors have a greater effect in worsening the average capacity performance of the system. ASER performance: Fig. 7 shows the ASER in the presence of feedback delay for 𝑀 = 3. We assume perfect CSI knowledge. Assuming an ideal feedback link, the full diversity order of 3 is achieved. However, with feedback delay, the diversity order is reduced to 1, as predicted earlier.

−6

10

0

5

10

15 SNR(dB)

20

25

30

Fig. 8. ASER for 𝑀 = 2 in the presence of delay in the feedback link and channel estimation error. In this figure 𝜌𝑓 = 0.9.

Fig. 8 illustrates the performance of the ASER in the presence of channel estimation error and feedback delay for 𝑀 = 2. It is clear from Fig. 8 that channel estimation errors reduce the diversity order of the system to zero, confirming our earlier analysis. Diversity order: Noting that the asymptotical diversity order 𝑑 is given by the magnitude of the slope of ASER against average SNR in a log-log scale [2]: ) (  log 𝑃¯𝑒 𝒟(𝑠) . (46) 𝑑 = lim − SNR→∞ log SNR Fig. 9 shows the diversity performance of the system in the presence of feedback delay link for different values of 𝜌𝑓 . We assume perfect CSI knowledge. It is observed that the asymptotical diversity order of the system tends to 1 for all 𝜌𝑓 values. This illustrates the destructive effect of feedback delay and demonstrates that relay selection in this case is annihilated. Fig. 10 illustrates the asymptotical diversity for different number of relays i.e., 𝑀 = 2, 3, 4. It is obvious that at high SNR the diversity order is independent of the number of relays in the system.

4170

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 12, DECEMBER 2011

1.6

3

diversity order with FD and perfect CSI

1.4

ρf = ρe = 1

2.5

2

1

Diversity Order

Diversity Order

1.2

0.8 0.6

1.5

1

0.4

ρf = 0.9 0.5

0.2 0 0

100

200

SNR(dB)

300

400

Fig. 9.Diversity order versus SNR. 𝜌𝑒 = 1 and 𝑀 = 4, The figure is plotted for different values of 𝜌𝑓 and is obtained from (46). 1.6

100

150

200 SNR(dB)

M=3

1

300

350

400

Fig. 11. Diversity order versus SNR. 𝜌𝑓 = 0.9 and 𝑀 = 3, The figure is plotted for different values of 𝜌𝑒 and is obtained from (46).

𝛾𝑚★𝑑 (relays in 𝒟(𝑠)are on)

𝒟(𝑠)

M=2

The 𝑖𝑡ℎ relay decodes its received signal erroneously with probability 𝐵𝑖 which is given in (35). Since all 𝑆 → 𝑅𝑖 links are statistically, we have

0.8 0.6 0.4

𝑃 𝑟(all relays are off) =

0.2 0 0

250

 𝑓𝛾𝑚★𝑑 (𝑥) = 𝑃 𝑟(all relays are off)𝑓 (𝑥) 𝛾𝑚★𝑑 (all relays are off) ∑  + 𝑃 𝑟(relays in 𝒟(𝑠) are on)𝑓 (𝑥). (47) 

M=4

Diversity Order

50

of 𝛾𝑚★𝑑 is written as

1.4 1.2

0 0

500

𝑀 ∏

𝐵𝑖 .

(48)

𝑖=1 50

100

150

200

250 300 SNR(dB)

350

400

450

500

Fig. 10. Diversity order versus SNR. 𝜌𝑒 = 1, 𝜌𝑓 = 0.9 and 𝑀 = 2, 3, 4. The figure is obtained from (46).

Fig. 11 depicts the asymptotical diversity order in the presence of channel estimation errors. It is noticed that the asymptotical diversity order in this case is reduced to zero, confirming our earlier observations in Figs. 5 and 8. VIII. C ONCLUSION In this paper, we discuss relay selection for DF protocol in cooperative networks. We show that the presence of channel estimation errors and feedback delay degrades the performance and also reduces the diversity order of S-DF. In particular in the presence of feedback delay the diversity order reduces to one where in the presence of channel estimation error the diversity order becomes zeros and an error floor is visible in the ASER and outage probability versus SNR curves. We derive exact analytical expressions for the outage probability, average symbol error rate and average capacity bound. A PPENDIX D ERIVING THE P ROBABILITY DENSITY FUNCTION OF 𝛾𝑚★𝑑 Let 𝛾𝑚★𝑑 denote the normalized received SNR at the destination terminal over the 𝑆 → 𝑅𝑖★ → 𝐷 link. Then, the PDF

On the other hand, if all the relays are off, then no communication would occur between source and destination terminal. The received SNR at the destination terminal would be zero. Therefore, the conditional PDF can be written as [31] 𝑓𝛾

𝑚★𝑑

 (all relays are off) (𝑥) = 𝛿(𝑥).

(49)

The probability of decoding set, given that there is at least one relay in 𝒟(𝑠), is given by ⎤ ⎡ ∏ ∏ 𝑃 𝑟(relays in 𝒟(𝑠) are on) = ⎣ (1 − 𝐵𝑖 ) 𝐵𝑖 ⎦ . (50) 𝑖∈ / 𝒟(𝑠)

𝑖∈𝒟(𝑠)

Inserting (48), (49) and (50) in (47) yields (34).

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SEYFI et al.: EFFECT OF FEEDBACK DELAY ON THE PERFORMANCE OF COOPERATIVE NETWORKS WITH RELAY SELECTION

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for the Gaussian Q-function,” IEEE Commun. Lett., vol. 12, no. 9, pp. 669–671, Sep. 2008. [30] B. Hassibi and B. M. Hochwald, “How much training is needed in multiple-antenna wireless links?” IEEE Trans. Inf. Theory, no. 4, pp. 951–963, Apr. 2008. [31] N. C. Beaulieu and J. Hu, “A closed-form expression for the outage probability of decode-and-forward relaying in dissimilar Rayleigh fading channels,” IEEE Commun. Lett., vol. 10, no. 12, pp. 813–815, Dec. 2006. Mehdi Seyfi (S’06) received his B.Sc and M.Sc both in Electrical Engineering from Shiraz University in 2005 and 2008, respectively. He is currently pursuing towards the PhD degree in Electrical Engineering at Simon Fraser University (SFU), Burnaby, BC, Canada. Since May 2009 he has been with the communications research laboratory at SFU. His research interest spans special topics in communications theory and signal processing, including cooperative communications, and statistical/adaptive signal processing. He has served as the reviewer for IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS, IEEE T RANSACTIONS ON S IGNAL P ROCESSING, and IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY. Sami Muhaidat (S’01-M’07-SM’11) received the M.Sc. in Electrical Engineering from University of Wisconsin, Milwaukee, USA in 1999, and the Ph.D. degree in Electrical Engineering from University of Waterloo, Waterloo, Ontario, in 2006. From 1997 to 1999, he worked as a Research and Teaching Assistant in the Signal Processing Group at the University of Wisconsin. From 2006 to 2008, he was a postdoctoral fellow in the Department of Electrical and Computer Engineering, University of Toronto, Canada. He is currently an Assistant Professor with the School of Engineering Science at Simon Fraser University, Burnaby, Canada. Dr. Muhaidat is an Associate Editor for IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY. Jie Liang (S’99-M’04) received the B.E. and M.E. degrees from Xi’an Jiaotong University, China, the M.E. degree from National University of Singapore, and the Ph.D. degree from the Johns Hopkins University, Baltimore, MD, in 1992, 1995, 1998, and 2003, respectively, all in Electrical Engineering. In May 2004, he joined the School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada, where he is currently an Associate Professor. From 2003 to 2004, he was with the Video Codec Group of Microsoft Digital Media Division, Redmond, WA. His research interests include Signal Processing, Image/Video Coding and Processing, Information Theory, and Digital Communications. He is currently serving on the Editorial Boards of Circuits, Systems, and Signal Processing (CSSP), EURASIP Journal on Image and Video Processing, and Signal Processing: Image Communication. Mehrdad Dianati (M’06) received the BSc. and MSc. in Electrical Engineering from Sharif University of Technology (Tehran, Iran) in 1992 and K.N.Toosi University of Technology (Tehran Iran) in 1995, respectively. He worked as a hardware/software developer and technical manager from 1992 to 2002. From 2002 to 2006, he completed his Ph.D. in Electrical and Computer Engineering at the University of Waterloo, Ontario, Canada. Dr. Dianati is currently a Lecturer (Assistant Professor) in the Centre for Communication Systems Research (CCSR) at the department of Electronic Engineering of the University of Surrey in United Kingdom.