The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

On the Performance of Cognitive Underlay Alamouti Space-Time Coding Schemes Nguyen Van Chinh∗ , Vo Nguyen Quoc Bao† , and Nguyen Luong Nhat† ∗ Telecommunications University, Nha Trang, Vietnam Email: [email protected] † Posts and Telecommunications Institute of Technology, Vietnam. E-mail: {baovnq,nhatnl}@ptithcm.edu.vn

to-noise ratios (SNRs), the a tight lower bound for the system outage probability was derived over Rayleigh fading channels. To provide spatial diversity gain, the best relay selection for DF [7] and AF [8] were considered for cognitive underlay networks. In [9], the problem of relay position optimization was solved for Rayleigh fading channels. To reduce the hardware complexity at secondary destination, underlay distributed switch-and-stay combining networks was proposed recently in [10]. At the same time, the effect of imperfect channel state information of interference links on cognitive networks are investigated in [11]. Numerical results show that the power back-off technique is an efficient approach to improve the performance of secondary underlay networks while guaranteeing the quality-of-service of primary networks. In this paper, we, for the first time, study the performance of the Alamouti scheme under a peak interference constraint allowed at the primary receiver. Since it is very hard to directly employ the conventional moment generating function (MGF) approach to investigate such networks, we propose a new derivation approach, where interference links are taken into account. In particular, we derive the closed-form expressions for outage probability and the Shannon capacity over Rayleigh fading channels. The remainder of this paper is organized as follows. In the next section, we present the system and channel model. In Section III, the system performance in terms of outage probability and Shannon capacity is derived for Rayleigh fading channels. Numerical results are presented in Section IV and conclusions are drawn in Section V.

Abstract—In this paper, we study the performance of underlay Alamouti schemes over Rayleigh fading channels. In particular, the closed form expressions of outage probability and Shannon capacity are derived. We develop a new derivation approach, which allows to obtain the exact form of Shannon capacity, where two interference links to the primary receiver are taken into account. Monte Carlo simulations are performed to verify the correctness of the analysis results. The numerical results show that the system achieves full diversity and outperform underlay direct transmissions.

I. I NTRODUCTION Nowadays, the increasing demand of spectrum resources due to next generation wireless networks results in a collision with the comprehensive, well-established, but increasingly obsolescent policy for the allocation of the radio spectrum. Cognitive radio, proposed by Mitola [1], is a promising technology to improve the spectrum utilization by enabling enabling non-licensed (secondary) users to get dynamic access to the under-utilized spectrum of the licensed (primary) users. Cognitive radio has recently adopted for some international standards with tangible frequency bands in its operation, i.e., IEEE 802.22 [2]. In a spectrum sharing system, secondary users are allowed to utilize the spectrum licensed to primary users only if reliable operation of the primary network is guaranteed. In order to protect primary networks, three cognitive spectrum access paradigms, namely overlay, underlay and interweave, were introduced [3]. Among them, underlay has attracted a lot of attention due to its advantage on providing concurrent transmission for both primary and secondary networks. Recently, there have been considerable cognitive relay networks on underlay cognitive networks, (e.g., see [4]–[11]). In [4], the effect of multi-user diversity for cognitive underlay system was investigated in terms of Shannon capacity over Rayleigh fading channels. Numerical results show that the interference channels from secondary users to the primary user have high impact on secondary networks. Considering Nakagami-𝑚 channels, the authors in [5] derived the outage probability of underlay dual-hop decode-and-forward (DF) networks showing that the impact of the fading parameters of the interfering links on the outage probability depends on not only the fading parameters of the secondary transmission links but also the interference temperature constraint. In [6], taking into account the dependence among the received signal-

978-1-4799-1089-2/13/$31.00 ©2013 IEEE

II. S YSTEM M ODEL We consider an Alamouti-based system with two transmit antennas and one receive antenna operating over spectrum sharing environment. The secondary transmitter (S-Tx) simultaneously transmits its data to the secondary receiver (S-Rx) in the same spectrum as that for a primary user (PU) as long as the interference power received at the primary receiver is strictly lower than the maximum allowable interference level. Adopting the Alamouti scheme across two transmit antennas [12], the un-coded data transmission occurs two consecutive symbol intervals, i.e., in the first symbol interval, two information-bearing symbols 𝑠1 and 𝑠2 are sent simultaneously on transmit antenna 1 and 2, respectively, then −𝑠∗2 and 𝑠1 continue to be transmitted in the second symbol interval.

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The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

the PDF of 𝛾𝑘 , we have 𝛾 𝛼𝑘 , 𝑓𝛾𝑘 (𝛾) = 𝐹𝛾𝑘 (𝛾) = 2, 𝛾 + 𝛼𝑘 (𝛾 + 𝛼𝑘 )

Assuming that fading channel gains are constant during two consecutive symbol durations, we have the received signal at S-Rx as √ √ (1) 𝑟1 = 𝑃1 ℎ1 𝑠1 + 𝑃2 ℎ2 𝑠2 + 𝑛1 , √ √ ∗ ∗ 𝑟2 = 𝑃1 ℎ1 (−𝑠2 ) + 𝑃2 ℎ2 𝑠1 + 𝑛2 , (2)

ℐ 𝜆

where 𝛼1 = 𝛼2 = 𝛼 = 𝒩𝑝0 𝜆ℎ𝑓 . Having the PDF and CDF of 𝛾𝑘 in hands allows us to directly determine the CDF of 𝛾Σ . Making use the fact that are always non-negative, we can write

where 𝑛𝑘 with 𝑘 = 1, 2 is the additive white Gaussian noise (AWGN) at S-Rx with the variance 𝑁0 and ℎ𝑘 denotes the complex channel coefficients from transmit antenna 𝑘 of STx to S-Rx, which are modelled as independent samples of complex Gaussian random variables. We further denote 𝑓𝑘 as interference channel coefficients of the link from antenna 𝑘 of S-Tx to PU. Under Rayleigh fading channels, the PDF and CDF of 𝑋 ∈ {ℎ𝑘 , 𝑓𝑘 } with 𝑘 = 1, 2 is given as 1 − 𝜆𝑥 𝑒 𝑋 𝜆𝑋

𝑓𝑋 (𝑥) =

𝐹𝛾Σ (𝛾) = Pr (𝛾1 + 𝛾2 < 𝛾) ∫𝛾 = 𝐹𝛾1 (𝛾 − 𝛾2 )𝑓𝛾2 (𝛾2 )𝑑𝛾2 .

Substituting (3) and (4) into (13), 𝐹𝛾Σ (𝛾) can be derived with the help of partial fraction technique as ∫𝛾 𝐹𝛾Σ (𝛾) = 0

and 𝐹𝑋 (𝑥) = 1 − 𝑒

=

(4)

with 𝜆𝑋 ∈ {𝜆ℎ , 𝜆𝑓 }. Assuming that perfect CSI of interference links is available at S-Tx [3], [4], the transmit power of S-Tx, 𝑃𝑘 , is set at 𝐼p

𝑃𝑘 =

∣𝑓𝑘 ∣

2,

where 𝐼p denotes the maximum tolerable interference power. At S-Rx, after processed by an Alamouti decoder [12], the estimated version of 𝑠1 and 𝑠2 are given by √ √ 𝑃1 ℎ∗1 𝑟1 + 𝑃2 ℎ2 𝑟2∗ , (6) 𝑠+ 1 = √ √ + ∗ ∗ 𝑠2 = 𝑃2 ℎ2 𝑟1 − 𝑃1 ℎ1 𝑟2 , (7)

𝑃𝑘 2 𝑁0 ∣ℎ𝑘 ∣

𝛾Σ =

2

𝑁0 ∣𝑓1 ∣

2

+

𝐼p ∣ℎ2 ∣

(8) (9)

A. Outage Probability The system outage probability is defined as the probability that the total instantaneous SNR falls below a predetermined threshold, 𝛾th . Mathematically, we have OP =Pr (𝛾Σ < 𝛾th ) 𝛾th 2𝛼2 𝛼 = + . log 2 𝛾th + 2𝛼 (𝛾th + 2𝛼) 𝛾th + 𝛼

2

𝑁0 ∣𝑓2 ∣

2.

(14)

In this section, we investigate the system performance in terms of outage probability and ergodic capacity. In addition, the asymptotic analysis is provided revealing the system behaviour at high SNR regime. We first consider the system outage probability.

with 𝑘 = 1, 2, 𝛾Σ is re-expressed as 𝐼p ∣ℎ1 ∣

2𝛼2 𝛾 𝛼 + 2 ln 𝛾 + 𝛼 . 𝛾 + 2𝛼 (𝛾 + 2𝛼)

III. P ERFORMANCE A NALYSIS

It has been well known that the post-processing effective SNR for the Alamouti scheme is given by ) 𝑃𝑘 ( 2 2 ∣ℎ1 ∣ + ∣ℎ2 ∣ . (10) 𝛾Σ = 𝑁0 Denoting 𝛾𝑘 = follows:

𝛼2 𝛾 − 𝛾2 𝑑𝛾2 (𝛾 − 𝛾2 + 𝛼1 ) (𝛾2 + 𝛼2 )2

The PDF of 𝛾Σ is related to its cumulative distribution function as 𝑑𝐹𝛾Σ (𝛾) 𝑓𝛾Σ (𝛾) = 𝑑𝛾 4𝛼2 ln 𝛼 4𝛼2 ln(𝛾 + 𝛼) = − 3 + 3 (𝛾 + 2𝛼) (𝛾 + 2𝛼) 2 4𝛼 2 (15) + 2 − 𝛾 + 𝛼 + 𝛾 + 2𝛼 . (𝛾 + 2𝛼)

(5)

which results in (after some simplifications) ) ( 2 2 𝑠1 + ℎ∗1 𝑛1 + ℎ2 𝑛∗2 , 𝑠+ = 𝑃 ∣ℎ ∣ + 𝑃 ∣ℎ ∣ 1 1 2 2 1 ) ( 2 2 𝑠+ 𝑠2 − ℎ∗1 𝑛∗2 + ℎ∗2 𝑛∗2 , 2 = 𝑃1 ∣ℎ1 ∣ + 𝑃2 ∣ℎ2 ∣

(13)

0

(3)

− 𝜆𝑥 𝑋

(12)

(11)

(16)

In what follows, we shall be interested in the OP at high SNR regime, which provides in Proposition 1. Proposition 1: For given channel statistics 𝛼1 and 𝛼2 , the system outage probability is tightly approximated as

Recognizing that the PDF of the effective SNR, 𝛾Σ , is the key to the unified analysis of the system under consideration, our immediate intention will be to derive the desired the probability density function (PDF) of 𝛾Σ . Denoting 𝐹𝛾𝑘 (𝛾) and 𝑓𝛾𝑘 (𝛾) as the cumulative distribution function (CDF) and

𝛾th 2 (17) 2𝛼2 and the system achieves full diversity of two. Proof: At high SNR regime, 𝐹𝛾𝑘 (𝛾) is well-approximated as 𝐹𝛾𝑘 (𝛾) → 𝛼𝛾 leading to 𝑓𝛾𝑘 (𝛾) → 𝛼1 . Making use the same OP →

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The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

steps as for (13), we have

by parts, we obtain

∞ ∫∞ 𝑑𝛾 ln(1 + 𝛾)  1 + 𝒥1 (𝛼) = − 2  2 2 2(𝛾 + 2𝛼) 𝛾=0 (𝛾 + 1)(𝛾 + 2𝛼) 0 { 1−2𝛼+2𝛼 ln 2𝛼 , 𝛼 ∕= 0.5 4𝛼(2𝛼−1)2 . (22) = 1 𝛼 = 0.5 4,

∫𝛾 𝐹𝛾Σ (𝛾) =

𝐹𝛾1 (𝛾 − 𝛾2 )𝑓𝛾2 (𝛾2 )𝑑𝛾2 0

𝛾th 2 = 2. 2𝛼

(18)

𝐼p 𝑁0 ,

i.e., )2 ( 1 𝜆𝑓 𝛾th ( 𝐼p )−2 , 𝐹𝛾Σ (𝛾) = 𝑁0 2 𝜆ℎ

Rewriting (18) in terms of

To solve 𝒥2 , using integration by parts again, we obtain ∫∞ ln(𝛾 + 𝛼) ln(1 + 𝛾) 𝑑𝛾 𝒥2 (𝛼) = 3 (𝛾 + 2𝛼) 0 ∞ ln(1 + 𝛾) ln(𝛾 + 𝛼)  =−  2  (𝛾 + 2𝛼) 𝛾=0 



(19)

by now we have established that diversity of order two is mathematically visible, which completes the proof.

+

1 2

B. Ergodic Capacity + In the Shannon’s sense, channel capacity is a standard performance measure, defined as the highest achievable transmission rate under which the errors are recoverable. Over AWGN channels, the normalized Shannon capacity is given by 𝐶AGWN = log2 (1 + 𝛾Σ ) leading to the average channel capacity of Alamouti scheme over fading channels as ∫∞ (20) 𝒞 = log2 (1 + 𝛾)𝑓𝛾Σ (𝛾)𝑑𝛾.

4𝛼 ln 2

+

(𝛾 + 2𝛼)

0

 4𝛼 + ln 2

2 − ln 2

∫∞

𝒥2 (𝛼)

∫∞ [ 0

𝒥3 (1,2𝛼)

𝑑𝛾 

2 𝑑𝛾.

(23)

2(2𝛼 − 1) ∫∞ 0

2 0

ln(𝛾 + 𝛼) ln(𝛾 + 𝛼) − 𝛾+1 𝛾 + 2𝛼

ln(𝛾 + 1)

1 2 − 2(2𝛼 − 1) (𝛾 + 2𝛼)

∫∞ 0

]

ln(𝛾 + 𝛼) (𝛾 + 2𝛼)

2.

(24)

With the help of the identity [13, Eq. (4.291.17)], we have { 𝑎 ln 𝑎−𝑏 ln 𝑏 , 𝑎 ∕= 𝑏 𝑎𝑏−𝑏2 𝒥3 (𝑎, 𝑏) = . (26) 1+ln 𝑎 𝑎=𝑏 𝑎 ,

2 𝑑𝛾

We are now in a position to derive 𝒥4 (𝑎, 𝑏, 𝑐). By recognizing the integral representation of the dilogarithm function [14, Eq. ∫𝑥 𝑡𝑑𝑡 , after some manipulations, (27.7.1)], i.e., Li2 (−𝑥) = ln𝑡−1



] ln(𝛾 + 1) ln(𝛾 + 1) − 𝑑𝛾 , 𝛾+𝛼 𝛾 + 2𝛼



2 𝑑𝛾+

Recalling 𝒥3 and 𝒥4 , we can see that (24) can be expressed as 𝐽4 (1, 𝛼, 2𝛼) 𝒥3 (1, 2𝛼) − 𝒥2 (𝛼) = 2𝛼2 2𝛼 𝒥4 (𝛼, 1, 2𝛼) 𝒥3 (𝛼, 2𝛼) (25) + 2 − 2(2𝛼 − 1) . 2(2𝛼 − 1)

ln(𝛾 + 1)

(𝛾 + 2𝛼) 0





3

ln(𝛾 + 𝛼)

∫∞ [

1

1 − 2𝛼

𝒥1 (𝛼)

ln(𝛾 + 𝛼) ln(1 + 𝛾)

(𝛾 + 𝛼)(𝛾 + 2𝛼)

(𝛾 + 1)(𝛾 + 2𝛼)

0

0

The goal of this section is to find a tractable statistical description of the ergodic capacity. As such, plugging (15) into (20), 𝒞 is re-expressed as ∫∞ ln(1 + 𝛾) 4𝛼 ln 𝛼 𝒞 =− 3 𝑑𝛾 ln 2 (𝛾 + 2𝛼) 0

  +

0 ∫∞

ln(𝛾 + 1)

Employing partial fraction decomposition, we can rewrite 𝒥2 as ] ∫∞ [ ln(𝛾 + 1) ln(𝛾 + 1) 1 − 𝒥2 (𝛼) = 2 2𝛼 𝛾+𝛼 𝛾 + 2𝛼

0

∫∞

1 2

→0

∫∞

1

(21)

we find out that 𝒥4 (𝑎, 𝑏, 𝑐) can be derived as (27) shown at the top of the next page.

𝒥4 (1,𝛼,2𝛼)

Pulling everything together, i.e., (22), (25), (26), (27), and (21), we can obtain the exact closed-form expression for 𝒞. It should be noted that the dilogarith function is available as a

where 𝒥𝑘 with 𝑘 = 1, . . . , 4 are auxiliary functions, which are derived as follows. Starting with 𝒥1 and based on integration

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The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

𝐽4 (𝑎, 𝑏, 𝑐) =

⎧ ln2 𝑏  ⎨ − 2 +

) ( ) ( ) ( ) ( ) ( − ln 1 − 𝑎𝑏 ln 𝑎𝑏 + ln 1 − 𝑎𝑐 ln 𝑎𝑐 + Li2 𝑎𝑏 − Li2 𝑎𝑐 , ( ) 2 𝜋2 − 1) − ln2 𝑐 − Li2 1𝑐 , 6 2+ ln 𝑐 ln(𝑐 ) ( ) ( ) ( 2 2 − 𝜋6 + ln2 𝑎 − ln2 𝑏 − ln 1 − 𝑎𝑏 ln 𝑎𝑏 + Li2 𝑎𝑏 ,

ln2 𝑐 2

 ⎩

Simulation Analysis

Shannon Capacity

10

and

Alamouti

Direct Transmission 6

5

10

15 Average SNRs

20

25

30

Fig. 2. Shannon capacity versus average SNRs.

0

10

Simulation Exact Analysis Approximation

Outage Probability

Direct Transmission −2

10

Alamouti −3

10

−4

5

10

15 Average SNRs

20

25

(29)

In this paper, we have derived the important performance metrics of cognitive underlay Alamouti schemes. New, tractable expressions for outage probability and Shannon capacity were derived over Rayleigh fading channels providing a computationally efficient alternative to Monte Carlo simulations in investigating behaviours of Alamouti schemes under spectrum sharing constraints. Although our analysis is limited on one receive antenna, this approach can be extended straightforwardly for the case of receive antennas.

10

0

𝛼 ∕= 1 . 𝛼=1

V. C ONCLUSIONS

−1

10

𝛼log2 𝛼 𝛼−1 , 1 ln 2 ,

Fig. 1 shows the system outage probability under consideration. As we can see, the simulation results are in excellent agreement with the analysis ones. As a reference, the outage probability for direct transmission are also plotted. Observing the slope of the two curves, we can see that the Alamouti scheme achieves full spatial diversity of two while diversity of one is for underlay direct transmissions. We also observe that the approximated expression is asymptotically tight at high SNR regime. In Fig. 2, we investigate the system capacity. Similar to the outage probability, the Shannon capacity of underlay Alamouti schemes outperforms that of underlay direct transmissions.

4

0

{ 𝒞DT =

8

2

(27)

facilitate the comparison, we provide here (without derivation) the outage probability and the Shannon capacity of underlay direct transmissions, receptively 𝛾th 𝛾th → (28) OPDT = 𝛾th + 𝛼 𝛼

14

12

𝑎 ∕= 𝑏 ∕= 𝑐 𝑎=𝑏=1 𝑎=𝑐

ACKNOWLEDGMENT

30

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2012.20.

Fig. 1. Outage probability versus average SNRs.

R EFERENCES build-in function in most well-known mathematical softwares such as Matlab and Mathematica. Besides, there exists efficient approaches to directly calculate the dilogarith as in, e.g., see [15], [16].

[1] I. Mitola, J. and J. Maguire, G. Q., “Cognitive radio: making software radios more personal,” IEEE Personal Commun. Mag., vol. 6, no. 4, pp. 13–18, Aug. 1999. [2] M. Sherman, A. N. Mody, R. Martinez, C. Rodriguez, and R. Reddy, “IEEE standards supporting cognitive radio and networks, dynamic spectrum access, and coexistence,” IEEE Commun. Mag., vol. 46, no. 7, pp. 72–79, Jul. 2008. [3] A. Goldsmith, S. A. Jafar, I. Maric, and S. Srinivasa, “Breaking spectrum gridlock with cognitive radios: An information theoretic perspective,” Proc. IEEE, vol. 97, no. 5, pp. 894–914, May 2009. [4] B. Tae Won, C. Wan, J. Bang Chul, and S. Dan Keun, “Multi-user diversity in a spectrum sharing system,” IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 102–106, Jan. 2009.

IV. N UMERICAL RESULTS The purpose of this section is to verify the analysis in Sect. III as well as to show the advantage of the Alamouti scheme as compared with the underlay direct transmission. The channel settings are as follows: 𝜆ℎ = 3 and 𝜆𝑓 = 1. To

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The 2013 International Conference on Advanced Technologies for Communications (ATC'13)

[5] Z. Caijun, T. Ratnarajah, and W. Kai-Kit, “Outage analysis of decodeand-forward cognitive dual-hop systems with the interference constraint in Nakagami-𝑚 fading channels,” IEEE Trans. Veh. Technol., vol. 60, no. 6, pp. 2875–2879, 2011. [6] L. Luo, P. Zhang, G. Zhang, and J. Qin, “Outage performance for cognitive relay networks with underlay spectrum sharing,” IEEE Commun. Lett., vol. 15, no. 7, pp. 710–712, Jul. 2011. [7] V. N. Q. Bao and D. Q. Trung, “Exact outage probability of cognitive underlay DF relay networks with best relay selection,” IEICE Trans. Commun., vol. E95.B, no. 6, pp. 2169–2173, Jun. 2012. [8] V. N. Q. Bao, T. Duong, D. Da Costa, G. Alexandropoulos, and A. Nallanathan, “Cognitive amplify-and-forward relaying with best relay selection in non-identical rayleigh fading,” IEEE Commun. Lett., vol. 17, no. 3, pp. 475 – 478, Mar. 2013. [9] T.-T. Tran, V. N. Q. Bao, V. Dinh Thanh, and T. Q. Duong, “Performance analysis and optimal relay position of cognitive spectrum-sharing dualhop decode-and-forward networks,” in Proc. 2013 International Conference on Computing, Management and Telecommunications (ComManTel’13), 2013, pp. 269–273. [10] V. N. Q. Bao, T. Q. Duong, A. Nallanathan, and G. K. Karagiannidis,

[11]

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