Performance of SC Receiver over TWDP Fading ...

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Performance of SC Receiver over TWDP Fading Channels Rupaban Subadar and Aheibam Dinamani Singh

Abstract Performance of selection combining (SC) receiver has been derived over two wave diffused power fading channels (TWDP) with arbitrary and non-identical fading parameters. An expression of CDF of SNR for TWDP fading channel has been derived, which is used to obtain the PDF of SNR at the output of SC receiver. The PDF of output SNR has been used to derive expressions of outage probability and average bit error rate for coherent and non-coherent modulation schemes. Effect of number of branches L and the fading parameters K and ∆ on the system performance has been studied. The obtained results are verified with the special case results available in literatures and by Monte carlo simulation.

Index Terms ABER, Diversity combining, Outage probability, SC receiver, TWDP fading channel.

I. I NTRODUCTION Diversity technique is a well known method to remove the adverse effect of fading in wireless channels. In diversity receiver transmitted signal is received through multiple number of independent fading paths and combining them the signal-to-noise ratio (SNR) of the received signal is improved. Out of all combining techniques the selection combining (SC) is the most simple from implementation point of view and hence has more practical importance. Two-wave diffuse power (TWDP) fading model consist of two specular Authors are with the Department of Electronics and Communication Engineering, North Eastern Regional Institute of Science and Technology, INDIA (e-mail:[email protected], [email protected]).

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multipath components in the presence of diffusely propagating waves [1] and can better represent the realworld frequency-selective fading data obtained from wireless sensor networks [2]. The TWDP fading is observed in a variety of propagation scenarios and may occur for typical narrow-band receiver operation, directional antennas and wide-band signals increase the likelihood of TWDP small-scale fading [1]. Also, Rayleigh, Rician and One-wave fading models are special cases of TWDP fading model [1]. Although, TWDP fading model can better represent real-world fading scenario and useful to understand performance of wireless system, but, so far, only a few works have been published [3]–[6] for this fading model. In [3], bit error rate performance expressions for an uncoded binary phase-shift keying (BPSK) system is derived using alternate expression of Gaussian Q function [7] and for maximal ratio combining (MRC) system, performance of BPSK is presented in [5]. Average bit error rate (ABER) performance of Gray coded QAM signalling in TWDP environment is presented in [4] using the cumulative distribution function of TWDP fading and that for MRC system is derived in [6]. Performance of SC receiver over TWDP fading channels is not known, this generates a motive to derive the performance measures of SC receiver over TWDP fading channels. The rest of this paper is organized as follows. In Section II, the channel and system considered for analysis has been discussed. Probability density function (PDF) of combiner output has been derived in Section III. In Section IV, performance of SC receiver have been obtained over TWDP fading channels and in Section V, numerical results and discussion are presented. Finally, the paper is concluded in Section VI.

II. C HANNELS AND S YSTEMS The channel has been assumed to be slow, frequency nonselective, with TWDP fading statistics. The complex low pass equivalent of the received signal over one symbol duration Ts can be expressed as r0 (t) = re jϕ s(t) + n(t),

(1)

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where s(t) is the transmitted symbol with energy Es and n(t) is the complex Gaussian noise having zero mean and two sided power spectral density 2N0 . Random variable (RV) φl represents the phase and r is the TWDP distributed fading amplitude having approximate PDF given by [1] µ 2 ¶ L ³r ´ r −r fR (r) = 2 exp − K ; K; α a D i ∑ i σ σ 2σ2 i=1

(2)

p p where D (z; K, αi ) = (1/2) exp(αi K)I0 (z 2K(1 − αi ))+(1/2) exp(−αi K)I0 (z 2K(1 + αi )), αi = ∆ cos(π(i− 1)/(2L − 1)), I0 (·) is the modified Bessel function of the first kind and zeroth order, K is the ratio of total specular power to diffused power, ∆ indicates the relative strength of the two specular component, L is the order of the approximate TWDP PDF and L ≥ (1/2)K∆ should be used so that (2) does not deviate significantly from the exact PDF. In the SC receiver, the signal-to-noise ratio (SNR) of received signals from all diversity branches are monitored and the branch having highest SNR is selected for detection [7]. Mathematically, the output SNR can be given as γSC = Max(γ1 , γ2 , . . . , γn ),

where γi =

Eb 2 N0 αi , i = 1, 2, . . . , n

(3)

is the instantaneous input branch SNR of the SC receiver.

A. PDF of Signal-to-Noise Ratio From (2), PDF of SNR for TWDP fading channel can be obtained by performing square transformation followed by transformation multiplying by factor as

Es N0

=

γ¯ , 2σ2 (K+1)

K+1 γ ,

Using [1, (42)] and [7] the factor

Es N0

can be expressed

where, γ¯ is the average SNR. The final expression of PDF of SNR can be given as

fγ (γ) =

where, η =

Es N0 .

¢ ¡ p η L 1 ai e−{ηγ+p2i− j } I0 2 p2i− j ηγ , ∑ ∑ 2 i=1 j=0

h i p2i− j = K 1 + (−1) j ∆Cos π(i−1) 2L−1 .

(4)

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III. PDF

OF

S ELECTION C OMBINER O UTPUT SNR

For a SC receiver working over independent TWDP fading channels, an expression for PDF of received SNR at the l th input branch can be expressed as

fγl (γl ) = where ηl =

Kl +1 γl .

¡ p ¢ ηl L 1 ai e−{ηl γl +p2i− j } I0 2 p2i− j ηl γl , ∑ ∑ 2 i=1 j=0

(5)

Kl and γl is the fading parameter and average branch SNR of l th branch, respectively.

The cumulative distribution function (CDF) expression of SNR for l th receive branch can be expressed as ZΓl

FΓl (Γl ) =

fγl (γl ) dγl 0

Z∞

ηl L 1 = 1 − ∑ ∑ ai e−p2i− j −ηl γl 2 i=1 j=0 Γl ¢ ¡ p ×I0 2 p2i− j ηl γl dγl

(6)

Solving the integral in (6) using [8, (1)], an expression for the CDF can be obtained as

FΓl (Γl ) = 1 −

³p ´ p 1 L 1 a Q 2p , 2η Γ 2i− j l l ∑∑ i 1 2 i=1 j=0

(7)

where Qx (a, b) is the marcum Q function [8]. For independent input branch fading signals, the joint CDF of M input SNRs can be obtained by the product of M CDFs in (7). Thus, the CDF of the combiner output SNR can be obtained (by substituting Γl = γsc , ∀l) as M

(

FγSC (γSC ) = ∏

l=1

) ´ ³p p 1 L 1 1 − ∑ ∑ ai Q1 2p2i− j , 2ηl γsc . 2 i=1 j=0

(8)

For identical branch fading parameter i.e. ηl = η, ∀l and equal average branch SNRs i.e. γ¯ l = γ¯ , ∀l (8) can be simplified as

"

# ´ M ³p p 1 L 1 FγSC (γSC ) = 1 − ∑ ∑ ai Q1 2p2i− j , 2ηγsc 2 i=1 j=0

(9)

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Differentiating (9) w. r. t. γsc , an expression for the PDF of the output SNR can be obtained as " # ³p ´ M−1 p Mη 1 L 1 fγsc (γsc ) = 1 − ∑ ∑ a i Q1 2p2i− j , 2ηγ 2 2 i=1 j=0 L

×∑

1

∑ aie−( p2i− j +ηγsc)I0

¡ p ¢ 2 p2i− j ηγsc .

(10)

i=1 j=0

For ∆ = 0 and K = 0, which corresponds to Rayleigh fading channels, it is easy to verify that (10) can be simplified to the results in [9, (7.60)]. For non-identical fading, the PDF of output SNR can be obtain by differentiating (8) as

fγsc (γsc )

¡ p ¢ 1 M L 1 ai ηl e−( p2i− j +ηl γsc ) I0 2 p2i− j ηl γsc ∑ ∑ ∑ 2 l=1 i=1 j=0 ) ( M p ¢ ¡p 1 L 1 2p2i− j , 2η j γsc × ∏ 1 − ∑ ∑ a i Q1 2 i=1 j=0 j=1

=

( j6=l)

(11)

IV. P ERFORMANCE OF SC

RECEIVER

A. Outage probability Outage probability, Pout is a standard performance measure used for diversity systems operating over fading channels. It is defined as the probability that the instantaneous error probability exceeds a specified value or equivalently, the output SNR, γ, falls below a certain specified threshold γth [7]. This can be obtained from the CDF of output SNR by substituting γsc = γth . For identical fading parameter the expression of Pout can be obtained from (9) as "

Pout

´ ³p p 1 L 1 = 1 − ∑ ∑ a i Q1 2p2i− j , 2ηγth 2 i=1 j=0

#M (12)

Similarly, for non-identical fading parameter expression of Pout can be obtained by substituting γsc = γth in (8) as M

Pout = ∏

l=1

(

) ³p ´ p 1 L 1 1 − ∑ ∑ a i Q1 2p2i− j , 2ηl γth 2 i=1 j=0

(13)

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B. Average bit error rate An expression for the ABER can be obtained by averaging the conditional bit error rate (BER), pe (ε |γ ), for the modulation scheme used, over the PDF of the SC output SNR. Mathematically, it can be given as Z∞

pe (ε |γsc ) fγ (γsc )dγsc .

Pe (γ) =

(14)

0

In this paper, we consider binary, coherent and non-coherent modulation schemes, and obtain the ABER expressions for these schemes, as shown below: 1) Binary Coherent Modulations: For binary coherent modulations, the expression for the conditional √ BER can be given as pe, ch (ε |γ) = Q ( 2aγ), where a = 0.5, 1 for coherent frequency shift-keying (CFSK) and coherent phase shift-keying (CPSK) modulations, respectively [10]. Putting pe, ch (ε |γsc ) and fγsc (γsc ) from (10) together into (14), an expression for the ABER for identical fading parameter can be obtained as

Pe,ch (γ¯ ) =

Mη 2

Z∞

1 L 1 1 − ∑ ∑ ai 2 i=1 j=0

0

× Q1 ×Q

"

´iM−1 ³p p 2p2i− j , 2ηγsc ´

³p

2aγSC

L

1

∑ ∑ aie−( p2i− j +ηγsc)

i=1 j=0

¡ p ¢ ×I0 2 p2i− j ηγsc dγSC

(15)

Putting pe, ch (ε |γsc ) and fγsc (γsc ) from (11) together into (14), an expression for the ABER for non-identical fading parameter can be obtained as

M

×

∏

j=1 ( j6=l)

Z∞ ³p ´ 1 M L 1 ¯ Pe,ch (γ) = ∑ ∑ ∑ ai ηl Q 2aγSC 2 l=1 i=1 j=0 0 ( ) L 1 p ¢ ¡p 1 1 − ∑ ∑ a i Q1 2p2i− j , 2η j γsc 2 i=1 j=0

¡ p ¢ ×e−( p2i− j +ηl γsc ) I0 2 p2i− j ηl γsc dγSC

(16)

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2) Binary Non-coherent Modulations: For binary non-coherent modulations, the conditional BER is given as pe, nch (ε |γsc ) = 12 exp(−aγsc ), where a = 0.5, 1 for noncoherent frequency shift-keying (NCFSK) and differential phase shift-keying (DPSK) modulations, respectively [10]. By putting pe, nch (ε |γsc ) and fγsc (γsc ) from (10) together into (14), an expression for the ABER for identical fading parameter can be obtained as Mη Pe,nch (γ¯ ) = 4

Z∞

" 1−

0

1 L 1 ∑ ∑ ai 2 i=1 j=0

´iM−1 L ³p p 2p2i− j , 2ηγ ×Q1 ∑

1

∑ ai

i=1 j=0

¡ p ¢ ×e−[ p2i− j +(a+η)γsc ] I0 2 p2i− j ηγsc dγSC

(17)

Putting pe, ch (ε |γsc ) and fγsc (γsc ) from (11) together into (14), an expression for the ABER for non-identical fading parameter can be obtained as Z∞

1 M L 1 Pe,nch (γ¯ ) = ∑ ∑ ∑ ai ηl e−(aγsc +p2i− j +ηl γsc ) 4 l=1 i=1 j=0 0 ( ) L 1 M p ¡p ¢ 1 × ∏ 1 − ∑ ∑ a i Q1 2p2i− j , 2η j γsc 2 i=1 j=0 j=1 ( j6=l)

¡ p ¢ ×I0 2 p2i− j ηl γsc dγSC

(18)

V. N UMERICAL R ESULTS AND D ISCUSSION The expressions of outage probability and ABER obtained in Section IV are numerically evaluated and plotted for different values of M, K and ∆ for the purpose of illustration. In Figs. 1 and 2, outage probability (Pout ) vs. average received SNR per branch (γ¯ ) have been plotted for identical and nonidentical fading parameter, respectively. From the figures it is clear that with increase in diversity branch M for constant K and ∆ improves the outage performance due to decrease in the total probability of deep fading in the channel. When ∆ = 1, two direct wave presents in TWDP fading are in antiphase [3]. Hence, increases in ∆ indicates the increase in phase difference, which degrades the the outage performance

8

0

10

L=2

∆=0 ∆=0.5

−1

10

* Simulation Result

Outage Probability

K=2 −2

10

L=3

−3

10

−4

10

K=6

2

Fig. 1.

4

6

8

10 12 SNR γ ¯ in dB

14

16

18

20

Outage Probability of SC Receiver with Identical fading (γth = 2dB) 0

10

∆1 = 0, ∆2 = 0 ∆1 = 0, ∆2 = 0.5 ∆1 = 0.5, ∆2 = 0

M =2 −1

Outage Probability

10

K1 = 2, K2 = 6

−2

10

−3

10

K1 = 1, K2 = 8

−4

10

* Simulation Result 2

Fig. 2.

4

6

8

10 12 SNR γ ¯ in dB

14

16

18

20

Outage Probability of 2 Branch SC Receiver with Non-identical fading (γth = 2dB)

as expected. Parameter K indicates the power of direct waves, increases in K indicates a better channel, hence, better performance. In Fig. 2, it can be observed that a 2-SC system with ∆1 = 0.5, ∆2 = 0 performs better than ∆1 = 0, ∆2 = 0.5, because of K2 is higher than K1 . In Figs. 3 and 4, ABER performances has been presented for BPSK and DPSK modulation schemes, respectively. The ABER performance of BPSK system is shown for constant value of K, whereas for DPSK system it is plotted for a constant value of ∆ for convenience of presentation. The observations are similar to that of outage probability. Computer simulated results has been included in the figures and also in close agreement with the numerical results.

9

−1

10

M =2

−2

ABER

10

∆=0 ∆ = 0.5 ∆ = 0.9 K=5

M =3

−3

10

−4

10

* Simulation Result 0

Fig. 3.

2

4

6

8

10 12 SNR γ ¯ in dB

14

16

18

20

ABER of SC BPSK System in TWDP Fading

−1

10

K=2 K=6 K = 10

M=2

∆ = 0.5 −2

ABER

10

M=3

−3

10

−4

10

* Simulation Result

2

Fig. 4.

4

6

8

10 12 SNR γ ¯ in dB

14

16

18

20

ABER of DPSK System in TWDP Fading

VI. C ONCLUSIONS In this paper, expressions for the outage probability and ABER of a SC receiver have been derived over TWDP fading channels. Mathematical expressions for the outage probability and ABER for binary coherent and non-coherent modulations have been obtained in terms of marcum Q function. Also, mathematical expressions for performance measures are obtained for non identical fading parameters and its effect on the performance is studied. Although the ABER performance in this presentation is shown for binary modulations, it can be extended for other known modulations using the approach demonstrated here. R EFERENCES [1] G. D. Durgin, T. S. Rappaport, And D. A. de Wolf, ”New analytical models and probability density functions for fading in wireless communication,” IEEE Trans. Commun., vol. 50, no. 6, pp. 1005-1015, Jun. 2002.

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[2] J. Frolik, ”A case for considering hyper-Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 6, pp. 1235-1239, Apr. 2007. [3] Soon H. Oh and Kwok H. Li, “BER performance of BPSK receivers over two-wave with diffuse power fading channels” IEEE Trans. Wireless Commun., vol. 4, no. 4 pp. 1448-1454, Jul. 2005. [4] Himal A. Suraweera, Wee S. Lee, and Soon H. Oh, “Performance analysis of QAM in a two-wave with diffuse power fading environment” IEEE Commun. Lett. vol.12, no. 2, pp. 109-111, Feb. 2008. [5] S. H. Oh, K. H. Li, and W. S. Lee, “Performance of BPSK pre-detection MRC systems over two-wave with diffuse power fading channels,” IEEE Trans. Wireless Commun., vol. 6, no. 8, pp. 27722775, Aug. 2007. [6] Yao Lu and Nan Yang, “Symbol error probability of QAM with MRC diversity in two-wave with diffuse power fading channels” IEEE Commun. Lett. vol.15, no. 1, pp. 10-12, Jan. 2011. [7] M. K. Simon and M. -S. Alouini, Digital Communications Over Fading Channels, 2nd ed. New York: Wiley, 2005. [8] Weisstein, Eric W. ”Marcum Q-Function.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/MarcumQFunction.html [9] T. S. Rappaport, Wireless Communications Principles and Practice, 2nd ed. PHI, 2007. [10] V. A. Aalo, “Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment,” IEEE Trans. on Commun., vol. 43, No. 8, pp. 2360-2369, Aug. 1995.