IEEE COMMUNICATIONS LETTERS, VOL. 15, NO. 10, OCTOBER 2011
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On the Error Exponent of Amplify and Forward Relay Networks Bappi Barua, Student Member, IEEE, Mehran Abolhasan, Member, IEEE, Farzad Safaei, Member, IEEE, and Daniel R. Franklin, Member, IEEE
Abstract—In this letter we derive the exact random coding error exponent of a dual hop amplify and forward (AF) relay network with channel state information (CSI) assisted ideal relay gain. Numerical results have been presented, which provide insight about the performance tradeoff between the error exponent and the data rate of the network. Finally we present the capacity analysis of this relay network. Index Terms—Random coding error exponent, amplify and forward, ergodic capacity.
C
I. I NTRODUCTION
Ooperative relay communication has been proven to provide better reliability against the multipath fading process. Depending on the CSI and allowable complexity, relays retransmit the received signals utilizing different approaches. Amplify and forward is the commonly used relaying protocol due to its simplicity and ease of deployment. Significant research results on cooperative relay networks are already available [1]–[3]. The performance of a simple dual hop network has been studied in [3] for Rayleigh fading channels. In particular, performance analysis utilizing random coding error exponent (RCEE) has received considerable attention in recent years. The error exponent defined in [4], [5] is the exponent along with codeword length that imposes a tight upper bound on the probability of error. Error exponent measurement provides information about the design requirement of a codeword to achieve any target rate R below the capacity C of the channel. Particularly, we can derive different capacity terms such as, the ergodic capacity, cut-off rate and the critical rate of a network utilizing the error exponent expression [4]. Using Gallager’s exponent over Nakagami-𝑚 fading channels performance analysis of a dual hop relay network has been conducted in [6] assuming ideal inverted channel gain in AF relays. More recently [7] has analyzed the system performance using error exponent method in two way relay communication channels. In [6] the authors have derived the RCEE using the hypothetical relay gain which is, in fact, a simplified assumption of CSI assisted power constraint relay gain factor 𝐺2 = 𝑃𝑅 proposed by [2] when 𝑊1 is set to zero. 𝑃𝑆 𝑃𝑆 ∣ℎ1 ∣2 +𝑊1 and 𝑃𝑅 are the source and the relay power respectively, and Manuscript received March 28, 2011. The associate editor coordinating the review of this letter and approving it for publication was D. Michalopoulos. This research was supported by the Australian Research Council (ARC) discovery research grant No. DP0879507. B. Barua, M. Abolhasan, and D. R. Franklin are with the Faculty of Engineering and IT, University of Technology Sydney, 15 Broadway, Ultimo, NSW 2007, Australia (e-mail:
[email protected], {mehran.abolhasan, daniel.franklin}@uts.edu.au). F. Safaei is with the Faculty of Informatics, University of Wollongong, NSW 2522, Australia (e-mail:
[email protected]). Digital Object Identifier 10.1109/LCOMM.2011.081211.110668
ℎ1 is the channel gain of Source-Relay hop. 𝑊1 is the onesided noise spectral density at the relay node. Avoiding the noise figure in the denominator of the relaying gain factor has allowed the authors to produce the probability density function (PDF) of the receiver SNR in more mathematically tractable form. However this assumption may not be viable in a power constraint AF relay system when the channel gain of the source-relay hop is very low. In this letter we derive the exact random coding error exponent with CSI assisted AF relay without avoiding the denominator noise figure. II. S YSTEM M ODEL Consider a single source-destination pair communicating via a single antenna relay without any direct link. A half duplex AF protocol has been considered over independent Rayleigh fading channels. We assume the receiver and the relay have full CSI while the transmitter has no CSI. Using the CSI assisted relay gain the end-to-end signal-to-noise ratio (SNR) at the receiver using maximal ratio combining (MRC) is given by, 𝛾1 𝛾2 (1) 𝛾𝑑 = 𝛾1 + 𝛾2 + 1 Due to the Rayleigh fading assumption, the first and second hop SNR 𝛾1 and 𝛾2 are exponentially distributed with parameter 𝜆1 and 𝜆2 respectively1. The instantaneous and the average ∣ℎ𝑖 ∣2 and SNR of 1st and 2nd hop are denoted as 𝛾𝑖 ≜ 𝑃 𝑊 𝑖 𝛾 𝑖 ≜ 𝑃𝑊Ω𝑖𝑖 respectively, where, 𝑃 ∈ {𝑃𝑆 , 𝑃𝑅 }, ℎ𝑖 and Ω𝑖 are the instantaneous and the average channel gain of 𝑖 ∈ {1, 2}th hop and 𝑊𝑖 is the variance of zero mean circularly symmetric complex Gaussian noise at the relay or at the receiver node. The PDF of the end-to-end SNR 𝛾𝑑 can be invoked from [8] as, [ ( √ ) 𝑓𝛾𝑑 (𝛾) = 2𝑒−(𝜆1 +𝜆2 )𝛾 𝜆1 𝜆2 (2𝛾 + 1) 𝐾0 2 𝜆1 𝜆2 𝛾 (𝛾 + 1) ( √ )] √ (2) +(𝜆1 + 𝜆2 ) 𝜆1 𝜆2 𝛾 (𝛾 + 1) 𝐾1 2 𝜆1 𝜆2 𝛾 (𝛾 + 1) where, 𝐾𝜈 (𝑧) is the 𝜈th order modified Bessel’s function of second kind. III. E RROR E XPONENT: D UAL H OP N ETWORK The random coding error exponent is defined as a function of input distribution function 𝑄 (𝑥), a factor 𝜌 ∈ [0, 1] and rate 𝑅 ≤ 𝐶 (for details please read ch. 5 of [4]), which is jointly optimized over 𝑄 (𝑥) and 𝜌 at a desired rate 𝑅. However the Gaussian input distribution has often been used in 1 where, 𝜆 = 1 ; 𝑖 ∈ {1, 2} is the inverse of the average SNR of the 𝑖 𝛾𝑖 corresponding (1st or 2nd) hop.
c 2011 IEEE 1089-7798/11$25.00 ⃝
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IEEE COMMUNICATIONS LETTERS, VOL. 15, NO. 10, OCTOBER 2011
𝐸𝑟 (𝑅) = max {𝐸0 (𝜌) − 2𝜌𝑅} 0≤𝜌≤1
with
{( 𝐸0 (𝜌) = − ln 𝔼𝛾𝑑
𝛾 1+ 1+𝜌
(3)
)−𝜌 } (4)
𝔼𝛾𝑑 (𝛾) denotes the statistical expectation operation over the receiver SNR 𝛾𝑑 . We modify (4) by using (2) as, ] [ 𝐸0 (𝜌) = − ln 𝒥1 + 𝒥2 (5)
0.8
Randon Coding Error Exponent Er(R)
many publications such as in [7], [9] to avoid the mathematical complexity involved in the joint optimization of the reliability function. This assumption provides near optimal result for the error exponent at a rate near the channel capacity. The error exponent of the dual hop AF network with Gaussian input distribution can be written as [4],
CSI assisted ideal relay gain Hien et. al. [6] in Rayleigh fading
0.7 0.6 15dB
0.5 0.4 0.3
10dB
0.2 5dB
0.1 0.0 0.0
0dB
0.1
0.2
0.3 0.4 0.5 Rate R (nats/s/Hz)
0.6
0.7
0.8
Fig. 1. Random coding error exponent vs rate 𝑅 in nats/s/Hz with various total signal to noise power allocation.
where,
( )−𝜌 ∫ ∞ 𝛾 𝒥1 ≜ 2𝜆1 𝜆2 1 + 𝑒−(𝜆1 +𝜆2 )𝛾 (2𝛾 + 1) 1+𝜌 0 ) ( √ × 𝐾0 2 𝜆1 𝜆2 𝛾 (𝛾 + 1) 𝑑𝛾
(6)
)−𝜌 ∫ ∞( 𝛾 𝑒−(𝜆1 +𝜆2 )𝛾 𝒥2 ≜ 2 (𝜆1 + 𝜆2 ) 1+ 1+𝜌 0 ( √ ) √ × 𝜆1 𝜆2 𝛾 (𝛾 + 1) 𝐾1 2 𝜆1 𝜆2 𝛾 (𝛾 + 1) 𝑑𝛾
(7)
Using series expansion of Bessel’s function given by [10, eq. (8.447.3)] and [10, eq. (8.446)], and, invoking the transformation of binomial power function to 𝐻-function by [11, eq. (8.4.2.5)] and [11, eq. (8.3.2.21)], we can modify (6) and (7) as, ∞ ∑ 𝑘 ( ) 𝑘+1 ∑ 𝑘 2(𝜆1 𝜆2 ) 𝐶 (𝑘, 𝜆) 𝒥1 = 2 𝑙 (𝑘!) Γ (𝜌) 𝑘=0 𝑙=0 [∫ [ ] ∞ 𝛾 (𝑙 − 𝜌, 1) 1,1 𝛾 2𝑘−𝑙 𝑒−(𝜆1 +𝜆2 )𝛾 𝐻1,1 × 1 + 𝜌 (0, 1) ] [ {0 (𝛾 − 1) 2,2 (0, 1) , (0, 1) 𝐻2,2 𝛾 × 1− (0, 1) , (0, 1) 2𝐶 (𝑘, 𝜆) [ ]} ] 1 (1, 1) , (1, 1) 1,2 − 𝐻 𝛾 (2𝛾 + 1) 𝑑𝛾 (8) (1, 1) , (0, 1) 2𝐶 (𝑘, 𝜆) 2,2 ] 𝛾 (𝑙 − 𝜌, 1) (𝜆1 + 𝜆2 ) 1,1 𝑒−(𝜆1 +𝜆2 )𝛾 𝐻1,1 𝑑𝛾 𝒥2 = Γ (𝜌) 1 + 𝜌 (0, 1) 0 ( ) ∞ ∑ 𝑣+1 𝑣+1 ∑ (𝜆1 + 𝜆2 ) (𝜆1 𝜆2 ) 𝐶(𝑣, 𝜆) 𝑣 + 1 + 𝑣! (𝑣 + 1)!Γ (𝜌) 𝑢 𝑣=0 𝑢=0 ] [ [∫ ∞ 𝛾 (𝑙 − 𝜌, 1) 1,1 𝛾 2𝑣+2−𝑢 𝑒−(𝜆1 +𝜆2 )𝛾 𝐻1,1 × 1 + 𝜌 (0, 1) {0 [ ] (𝛾 − 1) 2,2 (0, 1) , (0, 1) × 1+ 𝐻2,2 𝛾 (0, 1) , (0, 1) 𝐶 (𝑣, 𝜆) [ ]} ] 1 (1, 1) , (1, 1) 1,2 𝐻 𝛾 𝑑𝛾 (9) + (1, 1) , (0, 1) 𝐶 (𝑣, 𝜆) 2,2 ∫
∞
[
where, 𝐶 (𝑘, 𝜆) = 𝜓 (𝑘 + 1) − 12 ln (𝜆1 𝜆2 ) 𝐶 (𝑣, 𝜆) = ln (𝜆1 𝜆2 ) − 𝜓 (𝑣 + 1) − 𝜓 (𝑣 + 2)
and,
To arrive at (8) and (9) we represent the natural log functions in terms of 𝐻-functions by using [11, eq. 8.4.6.11] and [11, eq. 8.4.6.5]. Now using [12, eq. 2.19] and [13, eq. 2.6.2] and after some manipulations we can represent the solution of eq. (8) and (9), as eq. (10) and (11), shown at the top of the last page, where, 𝑠 = 𝜆1 + 𝜆2 . Finally replacing 𝒥1 and 𝒥2 from (10) and (11) in (5) we will have the desired error exponent expression. IV. E RGODIC C APACITY The ergodic capacity of this dual hop network is given by, [ ] 1 ∂𝐸0 (𝜌) 𝐶= 2 ∂𝜌 𝜌=0 ∫ ∞ 1 = ln (1 + 𝛾) 𝑓𝛾𝑑 (𝛾) 𝑑𝛾 (12) 2 0 Equation first by] replacing (12) can] easily be [solved [ (1, 1) , (1, 1) 1,2 𝛾 (𝑙 − 𝜌, 1) with 𝐻2,2 𝛾 and some 1+𝜌 (0, 1) (1, 1) , (0, 1) related scalar manipulations in eq. (8) and (9) and then utilizing [12, eq. 2.19] and [13, eq. 2.6.2] we will arrive to a similar form of equations as eq. (10) and (11). Detail of the capacity expression is avoided here due to space constraint. 1,1 𝐻1,1
V. N UMERICAL R ESULTS AND D ISCUSSION For numerical evaluation we assume that the Gaussian noise at the relay node and at the receiver have same variance. Since the transmitter does not have any knowledge of CSI the total system power has been allocated equally between the source and the relay node. To assist with comparison the numerical results have been plotted with the result of [6], which has studied the error exponent with AF relays avoiding the noise figure at the denominator of the relay gain factor. Fig. 1 plots the error exponent as a function of rate R for total average SNR 0, 5, 10 and 15dB. The result of [6] sets a clear upper bound at these low SNR regions on our results. For example, at a fixed rate 0.1 nats/s/Hz and at total SNR of 10dB, the error exponent differs by 0.04 due to neglecting the noise figure in the denominator of the amplification factor. However, we observed that at SNR values above 25dB the
BARUA et al.: ON THE ERROR EXPONENT OF AMPLIFY AND FORWARD RELAY NETWORKS
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] [ [ 𝑘 ∞ ∑ 𝑘+1 ( ) ∑ (𝑙 − 2𝑘 − 1, 1), (𝑙 − 𝜌, 1) 1 𝑘 𝑙−2𝑘−2 2 (𝜆1 𝜆2 ) 1,2 𝒥1 = 2𝐶(𝑘, 𝜆)𝐻2,1 𝑠 (0, 1) 𝑠(1 + 𝜌) (𝑘!)2 Γ (𝜌) 𝑙 𝑘=0 𝑙=0 ⎤ ⎡ (2𝑘 − 𝑙 + 3, 1) 1 ] [ (𝑙 − 2𝑘, 1), (𝑙 − 𝜌, 1) ⎢ 𝑠 (0, 1) , (0, 1) ; (1 − 𝜌, 1) ⎥ 1 1,2,1,2,1 1 1,2 ⎥ ⎢ − 𝐻1,[2:1],0,[2:1] + 𝑠𝐶(𝑘, 𝜆)𝐻2,1 ⎦ ⎣ 1 −−− (0, 1) 𝑠(1 + 𝜌) 𝑠 𝑠(1+𝜌) (0, 1) , (0, 1) ; (0, 1) ⎡ ⎡ ⎤ ⎤ (2𝑘 − 𝑙 + 2, 1) (2𝑘 − 𝑙 + 2, 1) 1 1 ⎢ 𝑠 (0, 1) , (0, 1) ; (1 − 𝜌, 1) ⎥ ⎢ 𝑠 (1, 1) , (1, 1) ; (1 − 𝜌, 1) ⎥ 1 1,2,1,2,1 ⎢ ⎢ ⎥ − 𝐻 1,2,1,1,1 ⎥ + 𝐻1,[2:1],0,[2:1] 1,[2:1],0,[2:1] ⎣ ⎣ 1 ⎦ ⎦ −−− −−− 2 1 𝑠(1+𝜌) (0, 1) , (0, 1) ; (0, 1) 𝑠(1+𝜌) (1, 1) , (0, 1) ; (0, 1) ⎧ ⎡ ⎡ ⎤ ⎤⎫⎤ (2𝑘 − 𝑙 + 1, 1) (2𝑘 − 𝑙 + 1, 1) 1 1 ⎬⎥ 𝑠 ⎢ 𝑠 (0, 1) , (0, 1) ; (1 − 𝜌, 1) ⎥ ⎢ ⎥ 𝑠 ⎨ 1,2,1,2,1 (1, 1) , (1, 1) ; (1 − 𝜌, 1) 1,2,1,1,1 ⎥ ⎢ ⎢ ⎥ ⎥ 𝐻1,[2:1],0,[2:1] ⎣ − 𝐻 + 1,[2:1],0,[2:1] ⎣ ⎦ ⎦⎦ −−− −−− 2 1 1 ⎩ ⎭ 𝑠(1+𝜌) (0, 1) , (0, 1) ; (0, 1) 𝑠(1+𝜌) (1, 1) , (0, 1) ; (0, 1) [ ] ∑ ( ) ∞ ∑ 𝑣+1 𝑣+1 (0, 1), (𝑙 − 𝜌, 1) 1 𝑣 + 1 𝑢−2𝑣−2 (𝜆1 𝜆2 ) 1 1,2 𝒥2 = 𝐻 + 𝑠 Γ (𝜌) 2,1 𝑠(1 + 𝜌) (0, 1) 𝑣!(𝑣 + 1)!Γ (𝜌) 𝑢 𝑣=0 𝑢=0 ⎤ ⎡ (2𝑣 − 𝑢 + 4, 1) 1 ] [ [ (𝑢 − 2𝑣 − 2, 1), (𝑙 − 𝜌, 1) ⎢ 𝑠 (0, 1) , (0, 1) ; (1 − 𝜌, 1) ⎥ 1 1,2,1,2,1 1 1,2 ⎥ ⎢ + 𝐻1,[2:1],0,[2:1] × 𝐶 (𝑣, 𝜆)𝐻2,1 ⎦ ⎣ 1 −−− (0, 1) 𝑠(1 + 𝜌) 𝑠 𝑠(1+𝜌) (0, 1) , (0, 1) ; (0, 1) ⎡ ⎡ ⎤ ⎤⎤ (2𝑣 − 𝑢 + 3, 1) (2𝑣 − 𝑢 + 3, 1) 1 1 ⎢ 𝑠 (0, 1) , (0, 1) ; (1 − 𝜌, 1) ⎥ ⎢ 𝑠 (1, 1) , (1, 1) ; (1 − 𝜌, 1) ⎥⎥ 1,2,1,2,1 ⎢ ⎢ ⎥ + 𝐻 1,2,1,1,1 ⎥⎥ − 𝐻1,[2:1],0,[2:1] 1,[2:1],0,[2:1] ⎣ 1 ⎣ 1 ⎦ ⎦⎦ −−− −−− 𝑠(1+𝜌) (0, 1) , (0, 1) ; (0, 1) 𝑠(1+𝜌) (1, 1) , (0, 1) ; (0, 1)
1.5
Capacity C (nats/s/Hz)
1.0
0.5
Fig. 2.
0
5 10 15 Total signal to noise power ratio (dB)
(11)
R EFERENCES
CSI assisted ideal relay gain Hien et. al. [6] in Rayleigh fading
0.0
(10)
20
Capacity (nats/s/Hz) vs total signal power to noise ratio in dB.
difference between the two models becomes negligible. Fig. 2 shows the capacity of the channel behavior with total SNR in Rayleigh fading channels. We can see that the channel capacity increases with the total SNR in the system and about 10dB increase in total SNR can double the channel capacity. In both figures the result of [6] is an upper bound on our result, which become progressively loose at the low SNR region.
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