IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 1, JANUARY 2011
13
Performance of Equal Gain Combining with Quantized Phases in Rayleigh Fading Channels Umar H. Rizvi, Student Member, IEEE, Ferkan Yilmaz, Mohamed-Slim Alouini, Fellow, IEEE, Gerard J. M. Janssen, Member, IEEE, and Jos H. Weber, Senior Member, IEEE
Abstract—In this paper, we analyze the error probability of equal gain combining with quantized channel phase compensation for binary phase shift keying signalling over Rayleigh fading channels. The probability density and characteristic functions of the combined signal amplitude are derived and used to compute the analytic expressions for the bit error probability in dependance of the number of quantization levels 𝐿, the number of diversity branches 𝑁𝑅 and the average received signal-to-noise ratio. The analysis is utilized to outline the trade-off between 𝑁𝑅 and 𝐿 and to compare the performance with non-coherent binary frequency shift keying and differential binary phase shift keying schemes under diversity reception. Index Terms—Quantization noise, BER analysis, Rayleigh fading, Equal gain combining and RF system design.
I. I NTRODUCTION
T
HE deleterious effects of fading in wireless communications can be effectively mitigated with diversity combining (DC) [1]. Diversity can be realized in time by sending the same signal over various time slots, in frequency by sending the same signal over multiple frequencies or in space by employing multiple receive antennas. Equal gain combining (EGC) is one particular example of DC which offers a good trade-off between performance and implementation complexity [1]. EGC when implemented with multiple receive antennas has the disadvantage of requiring two mixers (downconverters) and two analog-to-digital converters (ADCs) per antenna element. This makes the system design inefficient in terms of cost and power consumption and is difficult to realize, especially in portable terminals or at very high carrier frequencies such as 60 GHz. Recently, radio frequency (RF) level antenna DC [2], [3] has been proposed to overcome these drawbacks. In RF level EGC, the channel phase compensation is performed in the hardware by making use of circuit elements such as phase shifters, while channel estimation is performed Paper approved by G. K. Karagiannidis, the Editor for Fading Channels and Diversity of the IEEE Communications Society. Manuscript received July 28, 2009; revised May 18, 2010. U. H. Rizvi was formerly at the Wireless and Mobile Communications Group, Delft University of Technology, Delft, The Netherlands. He is currently working at ASML, Veldhoven, The Netherlands (e-mail:
[email protected]). F. Yilmaz is with the Electrical Engineering Program, KAUST, Saudi Arabia (e-mail:
[email protected]). M.-S. Alouini was formerly at the Department of Electrical and Computer Engineering, Texas A & M University at Qatar, Doha, Qatar. He is currently with the Electrical Engineering Program, KAUST, Saudi Arabia (e-mail:
[email protected]). G. J. M. Janssen and J. H. Weber are with the Wireless and Mobile Communications Group, Delft University of Technology, Delft, The Netherlands (e-mail: {g.janssen, j.h.weber}@ewi.tudelft.nl). Digital Object Identifier 10.1109/TCOMM.2011.121410.090129
in the digital baseband. This significantly reduces the system hardware and power requirements. Since channel phase compensation is realized at RF level, it is cumbersome to be accomplished with infinite (or high finite) precision because continuous phase shifters are difficult to manufacture, especially at very high carrier frequencies such as 60 GHz [2]. One alternative is to carry out phase compensation in small discrete steps which is equivalent to phase quantization. From a hardware complexity and cost perspective, the number of steps in the phase shifters should be kept as small as possible. On the other hand, larger steps imply a higher degradation in the system performance. Therefore, it is crucial to determine the required number of steps 𝐿 for the phase shifters to achieve acceptable performance. Furthermore, since quantized channel phase compensation makes the system partially coherent a natural question to ask then is: how many steps of compensation are necessary for a given number of receive antenna branches to outperform non-coherent and differentially-coherent schemes? The performance analysis for EGC, with perfect and imperfect channel state information at the receiver, has been extensively reported in the literature [4]–[12]. In [4], the probability density function (PDF) for the sum of Rayleigh random variables is represented in the form of a series and used for error rate evaluation. A decision variable based approach is taken in [5] to evaluate the performance of the EGC combiner and simplify the analysis presented in [4]. In [6] the characteristic function (CHF) based approach is applied to alleviate the need for computing the PDF of the combined signal amplitude. A similar but slightly different approach is adopted in [7], the difference being in the calculation of the conditional error probability (CEP). Closed form error probability expressions for the case of dual branch EGC over Rayleigh fading channels were presented in [8]. In [9], the performance of dual branch EGC with unequal branch SNRs for correlated Rayleigh fading channels was investigated. In [10], the impact of Gaussian weighting errors on the performance of EGC with binary phase shift keying (BPSK) modulation over Rayleigh fading channels is analyzed. In [11], the impact of carrier phase error on EGC under uncorrelated Rayleigh fading channels using Gram-Charlier series expansion was carried out. This work was extended for uncorrelated Nakagami-𝑚 fading channels for dual branch EGC in [12]. In both [11], [12] the carrier phase error was assumed to be Tikhonov distributed. However, to the best of our knowledge no performance analysis for the case of quantized channel phase compensation with EGC has been reported in the literature. In this paper, we investigate the impact of quantized phase
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 1, JANUARY 2011
II. S YSTEM M ODEL The model for a communication system employing RF level EGC is depicted in Figure 1. It is assumed that channel state estimation is carried out in the digital baseband domain, while phase compensation is implemented at the RF level by hardware phase shifters. This system has obvious advantages in terms of cost, complexity and power consumption in comparison with the conventional antenna diversity combining system. To reduce hardware complexity the phase compensation can be performed in discrete steps which is equivalent to phase quantization. The complex baseband signal after RF level EGC is given as 𝑁𝑅 ∑ ( 𝑗𝜃𝑖 ) 𝒬 ℎ𝑖 𝑒 𝛼 + 𝑛𝑖 𝛽𝑖 𝑒−𝑗𝜃𝑖 𝑟=
=
𝑖=1 𝑁𝑅 ∑ 𝑖=1
𝒬 ℎ𝑖 𝑒𝑗 (𝜃𝑖 −𝜃𝑖 ) 𝛼 + 𝑛 ˜𝑖 =
𝑁𝑅 ∑
0L[HU 4XDQWL]HG 3KDVH 6KLIWHU
compensation on the performance of EGC. Analytic expressions for the bit error rate (BER) of BPSK modulated signals over Rayleigh fading channels as a function of the number of phase quantization levels 𝐿, the signal-to-noise ratio (SNR) and the number of receive antennas 𝑁𝑅 , are derived based on approximate and exact expressions for the PDF and CHF of the combined signal amplitude. Two approximate expressions for the BER are derived. The first expression is very tight at all ranges of SNR and can be used to accurately evaluate the error rate performance, while the second approximation is valid only in the high SNR regime. This second approximation yields a simple closed form expression which can be used to determine the coding gain, the diversity order and the degradation in the system performance. The BER expressions are then used to outline the trade-off between 𝑁𝑅 and 𝐿 and to compare the performance with non coherent binary frequency shift keying (BFSK) and differential binary phase shift keying (DBPSK) under diversity reception. The paper is organized as follows. Section II presents the system model and notations. The derivation of exact and approximate PDFs and CHFs along with the error probability analysis is outlined in Section III. The numerical results are given in Section IV. Conclusions are drawn in Section V.
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Fig. 1.
System architecture for an RF level diversity combiner.
where 𝜃𝑖ℰ represents the phase error which is uniformly distributed between (−𝜋/𝐿, 𝜋/𝐿). In this paper, we consider the case of BPSK signalling, which implies that only the real part of the signal has an impact on the detection performance. This reduces (1) to 𝑟=
𝑁𝑅 ∑
𝑁𝑅 ∑ ( ) 𝑛𝑖 ] = ℎ𝑖 cos 𝜃𝑖ℰ 𝛼 + ℜ[˜ ℎ𝑖 𝑧𝑖 𝛼 + ℜ[˜ 𝑛𝑖 ],
𝑖=1
(3)
𝑖=1
where ℜ[.] denotes the real part of a complex quantity. The instantaneous signal-to-noise ratio (SNR) is defined as 𝛾 = 𝑥2 where √ √ 𝑁𝑅 𝑁𝑅 𝐸𝑠 ∑ 𝐸𝑠 ∑ 𝑥= ℎ𝑖 𝑧 𝑖 = 𝑦𝑖 𝑁𝑅 𝑁0 𝑖=1 𝑁𝑅 𝑁0 𝑖=1 √ 𝑁𝑅 1 ∑ 𝑤 = 𝑣𝑖 = √ , (4) 𝑁𝑅 𝑖=1 𝑁𝑅 and 𝐸𝑠 denotes the average signal constellation energy, √ 𝐸𝑠 , 𝑣𝑖 = 𝑦𝑖 𝑁0 𝑦 𝑖 = ℎ𝑖 𝑧 𝑖 , and 𝑤=
𝑁𝑅 ∑
𝑣𝑖 .
(5) (6)
(7)
𝑖=1
ℰ
ℎ𝑖 𝑒−𝑗𝜃𝑖 𝛼 + 𝑛 ˜𝑖,
(1)
𝑖=1
˜ 𝑖 represent zero mean complex additive white where 𝑛𝑖 and 𝑛 Gaussian noise (AWGN) with variance 𝑁0 /2 per dimension and 𝑁𝑅 is the number of receive antennas. The channel amplitudes ℎ𝑖 are assumed to be Rayleigh distributed with 𝜃𝑖 uniformly distributed between [−𝜋, 𝜋) . The variable gain amplifier amplitudes are denoted by 𝛽𝑖 and are set to unity for EGC. It is assumed that the antenna elements are sufficiently separated for each branch to undergo independent and identical distributed (i.i.d) fading. The quantized channel phase is denoted by 𝜃𝑖𝒬 and 𝛼 represents the baseband modulated symbol chosen from the signal constellation set 𝒜 = {−1, +1}. The last equality in (1) follows from the fact that for an 𝐿−level uniform quantizer the signal phase can be modeled as (2) 𝜃𝑖𝒬 = 𝜃𝑖 + 𝜃𝑖ℰ ,
The channel estimation for an RF level EGC can be accomplished as outlined in [3]. Assuming slow fading and two receive diversity branches, the channel phase estimation sequence can be explained as: ∙ Choose the first diversity branch by setting 𝛽1 = 1, 𝛽2 = 0 and estimate 𝜃1 . ∙ Choose the second diversity branch by setting 𝛽1 = 0, 𝛽2 = 1 and estimate 𝜃2 . This method can be applied for determining the channel phase for arbitrary number of branches.
III. E RROR P ROBABILITY A NALYSIS The error probability for EGC is given as ∫ ∞ 𝑃𝑒 = 𝑃 (𝜀 ∣𝑥 ) 𝑝𝑥 (𝑥) 𝑑𝑥,
(8)
0
where 𝑃 (𝜀 ∣𝑥 ) and 𝑝𝑥 (𝑥) represent the CEP and the PDF of 𝑥, respectively. To alleviate the need for 𝑝𝑥 (𝑥) computation,
RIZVI et al.: PERFORMANCE OF EQUAL GAIN COMBINING WITH QUANTIZED PHASES IN RAYLEIGH FADING CHANNELS
which is generally cumbersome, we can write the BER using the CHF based approach as [6] ∫ ∞ 1 𝑃𝑒 = 𝒢 (𝜔) 𝜙∗𝑥 (𝜔) 𝑑𝜔, (9) 2𝜋 −∞ where 𝜙∗𝑥 (𝜔) denotes the complex conjugate of the CHF of 𝑥 and ∫ ∞ 𝒢 (𝜔) = 𝑃 (𝜀 ∣𝑥 ) 𝑒𝑗𝜔𝑥 𝑑𝑥 (10) −∞
denotes the Fourier transform (FT) of the CEP. The FT of the CEP for BPSK signalling is given as [6] ( ) ( ) 𝜔2 𝜔 3 𝜔2 1 √ 1 𝐹1 1; ; − 𝒢 (𝜔) = + 𝑗 − 𝑗𝑒− 4 , (11) 2𝜔 𝜋 2 4 where 1 𝐹1 (⋅; ⋅; ⋅) represents the confluent hypergeometric function of the first kind. After some manipulation (9) can be written as [6] ∫ 2 𝜋/2 ℜ [tan (𝜂) 𝒢 (tan (𝜂)) 𝜙∗𝑥 (tan (𝜂))] 𝑑𝜂. (12) 𝑃𝑒 = 𝜋 0 sin (2𝜂) The CHF of 𝑥 is given by ( √ ) 𝑁 ( ) 𝑁𝑅 𝑅 ∏ ∏ 𝜔 𝐸𝑠 𝜙𝑥 (𝜔) = 𝜙𝑦𝑖 𝜔 𝜙𝑣𝑖 √ = . (13) 𝑁𝑅 𝑁0 𝑁𝑅 𝑖=1 𝑖=1 The average SNR per branch is denoted by [ ] [ 2 ] 𝐸𝑠 𝐸 𝑦𝑖2 𝛾 𝑖 = 𝐸 𝑣𝑖 = = 𝛾, 𝑁0
(14)
where 𝐸 [.] denotes the expectation operation. The last equality follows from the fact that the branches have identical average SNR because of the i.i.d fading. Thus, to evaluate the error probability we need to determine the CHF of 𝑦𝑖 , which is a product of two random variables, i.e., ℎ𝑖 and 𝑧𝑖 . Since 𝜃𝑖ℰ is uniformly ( )distributed between (−𝜋/𝐿, 𝜋/𝐿), the PDF of 𝑧𝑖 = cos 𝜃𝑖ℰ is given as 𝐿 , 𝑝𝑧𝑖 (𝑧) = √ 𝜋 1 − 𝑧2
cos (𝜋/𝐿) ≤ 𝑧 ≤ 1.
(15)
15
where erf (⋅) denotes the error function. For the case when 𝐿 → ∞, 𝑝𝑦𝑖 (𝑦) can be written as [ ( ( 𝜋 ) )] 𝑦2 𝑦 tan 𝐿 1 −Ω √ lim 𝑝𝑦𝑖 (𝑦) = √ 𝑒 𝑖 lim 𝐿erf 𝐿→∞ 𝐿→∞ 𝜋Ω𝑖 Ω𝑖 2 2𝑦 − Ω𝑦 = 𝑒 𝑖, (20) Ω𝑖 which is the PDF of Rayleigh fading and the system behaves as conventional EGC. This has been used in [6] to compute the CHF and the error probability in conjunction with (12). It can be easily shown that ( (𝜋 )) 𝑣 tan 𝐿 𝐿 − 𝑣𝛾2 √ 𝑝𝑣𝑖 (𝑣) = √ 𝑒 erf . (21) 𝜋𝛾 𝛾 The CHF of 𝑣𝑖 can be written as ∫ ∞ 𝜙𝑣𝑖 (𝜔) = 𝑝𝑣𝑖 (𝑣) 𝑒𝑗𝜔𝑣 𝑑𝑣.
(22)
0
Substituting (21) in (22) and using the alternative representation of the error function ∫ 2 𝜋/2 − sin𝑥22(𝜙) erf (𝑥) = 1 − 𝑒 𝑑𝜙, (23) 𝜋 0 we obtain the CHF as given in (24). The CHF given in (24) can be efficiently estimated using the Gauss-Chebyshev quadrature (GCQ) [16, p. 889, Eq. (25.4.38)] as given in (25), where 𝑎𝑘 represent the roots of the Gauss-Chebyshev polynomial of degree 𝑁𝑔 and are given as ( ) 𝜋 (2𝑘 − 1) . (26) 𝑎𝑘 = cos 2𝑁𝑔 The series converges very fast and typically 𝑁𝑔 = 64 is seen to be sufficient for excellent accuracy. Thus (24) or (25) together with (12) can be used to evaluate the BER for arbitrary 𝐿 and 𝑁𝑅 . Using GCQ, i.e., (25) in conjunction with (12) leads to significant computational advantages as compared to the case when (24) is used with (12).
A. PDF and CHF as
Using [13, Eq. (4)], the PDF of 𝑦𝑖 = ℎ𝑖 𝑧𝑖 can be written ∫ ∞ ( 𝑦 )] 𝑑 [ 𝐹ℎ𝑖 𝑝𝑧𝑖 (𝑧) 𝑑𝑧, 𝑝𝑦𝑖 (𝑦) = (16) 𝑑𝑦 𝑧 0
where 𝐹ℎ𝑖 (ℎ) denotes the cumulative distribution function (CDF) of ℎ𝑖 . The CDF for the Rayleigh fading case is given as 2 (17) 𝐹ℎ𝑖 (ℎ) = 1 − 𝑒−ℎ /Ω𝑖 , [ 2] where Ω𝑖 = 𝐸 ℎ𝑖 . Substituting (17) and (15) in (16), we obtain ∫ 1 −𝑦2 /Ω𝑖 𝑧2 𝑦𝑒 2𝐿 √ 𝑑𝑧, (18) 𝑝𝑦𝑖 (𝑦) = 𝜋Ω𝑖 𝐿𝑐 𝑧 2 1 − 𝑧 2 where 𝐿𝑐 = cos (𝜋/𝐿). By substituting 𝑧 = sin (𝜃) and using the identities [14] and [15], the integral given in (18) can be written in closed form as ( ( 𝜋 )) 𝑦2 𝑦 tan 𝐿 𝐿 −Ω √ 𝑝𝑦𝑖 (𝑦) = √ 𝑒 𝑖 erf , (19) 𝜋Ω𝑖 Ω𝑖
B. High SNR Analysis The BER at high SNRs is dependent on the behavior of the PDF around the origin, i.e., the first non-zero term in the Maclaurin series expansion is sufficient to quantify the performance [17]. Using the first non-zero term in the Maclaurin series expansion of (21), 𝑝𝑣𝑖 (𝑣) can be approximated as (𝜋) 2𝐿𝑣 tan 𝐿 𝑝𝑣𝑖 (𝑣) ≈ . (27) 𝜋𝛾 The Laplace Transform (LT) of the approximated 𝑝𝑣𝑖 (𝑣) is now given by (𝜋) 2𝐿 tan 𝐿 . (28) 𝐿𝑣𝑖 (𝑠) = 𝜋𝑠2 𝛾 Based on this, the LT of the approximated 𝑝𝑤 (𝑤) is found as ( ( 𝜋 ) )𝑁𝑅 2𝐿 tan 𝐿 𝐿𝑤 (𝑠) = . (29) 𝜋𝛾𝑠2
16
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 1, JANUARY 2011
𝜔2 𝛾
− ( √ ) ∫ 𝜋 2 2 𝜋 𝜔 𝛾 𝑒 4+4 csc (𝜙) tan ( 𝐿 ) 𝐿 − 𝐿 − 𝐿 2 √ 𝜙𝑣𝑖 (𝜔) = 𝑒 4 − 𝑒 4 erf − ( 𝜋 ) 𝑑𝜙 + 2 2 2𝑗 𝜋 0 1 + csc2 (𝜙) tan2 𝐿 ⎛ ⎞ 𝜔2 𝛾 − √ ∫ 𝜋2 4+4 csc2 (𝜙) tan2 ( 𝜋 ) 𝐿 𝐿 𝜔 𝑒 𝛾 ⎝ √ ⎠ √ ( ) erf ( ) 𝑑𝜙. 𝜋 0 2 𝜋 2 𝜋 2 2 1 + csc (𝜙) tan 𝐿 2𝑗 1 + csc (𝜙) tan 𝐿 𝜔2 𝛾
𝜔2 𝛾
(24)
2
𝜔 𝛾 − ( √ ) 𝑁𝑔 √ 2 𝜋 2 𝜋 𝐿 − 𝐿 − 𝜔 𝛾 𝐿𝜋 ∑ 1 − 𝑎2𝑘 𝑒 4+4 csc ( 4 (𝑎𝑘 +1)) tan ( 𝐿 ) 4 4 √ 𝜙𝑣𝑖 (𝜔) ≈ 𝑒 − 𝑒 erf − ( (𝜋) + ) 2 2 2𝑗 4𝑁𝑔 1 + csc2 𝜋4 (𝑎𝑘 + 1) tan2 𝐿 𝑘=1 ⎛ ⎞ 𝜔2 𝛾 − √ 𝑁𝑔 √ 4+4 csc2 ( 𝜋 (𝑎𝑘 +1)) tan2 ( 𝜋 ) 4 𝐿 𝜔 𝐿𝜋 ∑ 1 − 𝑎2𝑘 𝑒 𝛾 ⎝ √ ⎠ √ ( ( 𝜋 ) erf ( (𝜋) , ) ) 4𝑁𝑔 1 + csc2 𝜋4 (𝑎𝑘 + 1) tan2 𝐿 2𝑗 1 + csc2 𝜋4 (𝑎𝑘 + 1) tan2 𝐿 𝑘=1 𝜔2 𝛾
𝜔2 𝛾
Taking the inverse LT we can write the PDF of 𝑤 as ( ( 𝜋 ) )𝑁 𝑅 2𝐿 tan 𝐿 𝑤2𝑁𝑅 −1 , 𝑝𝑤 (𝑤) ≈ 𝜋𝛾 Γ (2𝑁𝑅 )
(30)
where Γ (.) denotes the gamma function. Using (8) we can formulate the BER at high SNR as ( ( 𝜋 ) )𝑁𝑅 2𝐿 tan 𝐿 1 𝑃𝑒 ≈ ⋅ 2Γ (2𝑁𝑅 ) 𝜋𝛾 ( ) ∫ ∞ 𝑤 erfc √ (31) 𝑤2𝑁𝑅 −1 𝑑𝑤, 𝑁𝑅 0 where we have used
( √ ) 𝑃 ( 𝜀∣ 𝑤) = (1/2) erfc 𝑤/ 𝑁𝑅 .
Solving the above integral yields ( ( 𝜋 ) )𝑁𝑅 𝑁𝑅 𝐿 tan 𝐿 1 𝑃𝑒 ≈ . 2Γ (1 + 𝑁𝑅 ) 2𝜋𝛾
(32)
(33)
This can now be compared with [17, Eq. (1)], i.e., 𝑃𝑒 ≈ (𝐺𝑐 𝛾)−𝐺𝑑 ,
(34)
where 𝐺𝑐 denotes the coding gain and 𝐺𝑑 represents the diversity order. We get 𝐺𝑐 =
1/𝑁𝑅
2𝜋 (2Γ (1 + 𝑁𝑅 )) (𝜋) 𝑁𝑅 𝐿 tan 𝐿
and
𝐺𝑑 = 𝑁𝑅 .
(35)
Due to the presence of tan (𝜋/𝐿), (33) holds for 𝐿 > 2. For the conventional case, i.e., when 𝐿 → ∞ we get ( )𝑁𝑅 𝑁𝑅 1 𝑃𝑒𝐶 = lim 𝑃𝑒 = 𝑁𝑅 +1 , (36) 𝐿→∞ 2 Γ (1 + 𝑁𝑅 ) 𝛾 which gives us 𝐺𝐶 𝑐 =
2 (2Γ (1 + 𝑁𝑅 ))1/𝑁𝑅 𝑁𝑅
and
𝐺𝐶 𝑑 = 𝑁𝑅 .
(37)
(25)
For 𝑁𝑅 = 1, (36) reduces to 1/ (4𝛾), which is a well known result for BPSK over single branch Rayleigh fading [18, p. 819]. It is interesting to note here that the diversity order is not affected by the phase quantization 𝐿. The coding gain on the other hand is scaled by a factor of 𝜋/ (𝐿 tan (𝜋/𝐿)). Therefore, we can write the system performance degradation Δ as the(difference ) in coding gains (in dB), which gives Δ = at high SNR (𝛾 ≫ 1) as 10 log10 𝐺𝑐 /𝐺𝐶 𝑐 ( ) 𝜋 Δ = 10 log10 . (38) 𝐿 tan (𝜋/𝐿) IV. N UMERICAL R ESULTS Any of the CHFs (24) or (25) can be used in conjunction with (12) to numerically evaluate the performance of BPSK signalling with RF level EGC over Rayleigh fading channels. In this section we use (25) with 𝑁𝑔 = 64. A comparison of the analytic and Monte Carlo simulated BER for 𝑁𝑅 = 2 and using various phase quantization levels 𝐿, is shown in Figure 2. The simulation and analysis is seen to be in good agreement. At a BER of 10−2 for 𝑁𝑅 = 2 there is a performance improvement of more than 6 dB, when 𝐿 is increased from 2 to 3. This is significant in comparison to the case when 𝐿 is increased from 3 to 4, in which case the gain is only about 1 dB. It can also be seen that for only 𝐿 = 4, the performance is within 1.1 dB of the conventional EGC. This can also be observed with (38), which for 𝐿 = 4 gives Δ = 1.05 dB. A comparison with non-coherent BFSK and DBPSK for various values of 𝑁𝑅 and 𝐿 is shown in Figure 3. The results for BFSK and DBPSK are obtained as outlined in [18, p. 825, Eqs. (14.4-15), (14.4-27), (14.4-30)]. It can be seen that both BFSK and DBPSK exhibit a superior performance to BPSK when 𝐿 = 2, i.e., for two level phase compensation. The performance on the other hand is much worse as compared to DBPSK for 𝐿 = 2. In order to achieve superior performance to a conventional DBPSK diversity receiver at least 𝐿 = 3 is required. A comparison between the exact and approximate BER expressions is shown in Figure 4. The approximation is very tight in the high SNR regime especially for 𝛾 > 15 dB.
RIZVI et al.: PERFORMANCE OF EQUAL GAIN COMBINING WITH QUANTIZED PHASES IN RAYLEIGH FADING CHANNELS
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0
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Eq. (33) Eq. (12) L=3 L=6
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Simulation Analytic, L=2 Analytic, L=3 Analytic, L=4 Analytic, L=6 Conventional (L → ∞)
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Fig. 4. A comparison of tight (12) and high SNR (33) BER for various values of 𝑁𝑅 and 𝐿.
A comparison of simulated and analytic (12) BER for 𝑁𝑅 = 2.
Fig. 2.
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γ[dB]
γ [dB]
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L=2 BFSK DBPSK L=3 L=4 Conventional (L → ∞)
Average Bit Error Probability
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Fig. 3.
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Fig. 5.
N
R
BER for various values of 𝑁𝑅 and 𝐿 at 𝛾 = 5 dB.
BER comparison of BPSK with BFSK and DBPSK at 𝛾 = 10 dB.
The trade-off between 𝑁𝑅 and 𝐿 is highlighted in Figure 5, in which the BER is presented as a function of 𝐿 and 𝑁𝑅 for 𝛾 = 5 dB. As an example, a target BER of 2 × 10−3 at an SNR of 5 dB can be achieved by either using 𝐿 = 2 with 𝑁𝑅 = 8 or using 𝐿 = 3 with 𝑁𝑅 = 4. The system designer can thus choose for the most cost effective solution. V. C ONCLUSIONS The impact of quantized channel phase compensation on the performance of RF level EGC was investigated. BER expressions for EGC and BPSK signalling over Rayleigh fading channels were derived and used to compare the performance with non coherent BFSK and DBPSK under diversity reception. It was shown that for only 3 level phase compensation BPSK exhibits superior performance to both BFSK and DBPSK. A closed form asymptotic error rate expression was used to determine the diversity order, coding gain and the degradation in system performance. It was shown that the
diversity order is independent of the phase quantization levels 𝐿 and the degradation in system performance in the high SNR regime is dependent only on 𝐿. The ability to make a system level trade-off between 𝐿 and 𝑁𝑅 was also demonstrated. ACKNOWLEDGEMENTS This work was supported in part by IOP GenCom under SiGi Spot project IGC.0503 and in part by Qatar National Research Fund (a member of the Qatar Foundation). R EFERENCES [1] M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels. John Wiley & Sons, 2004. [2] P. Baltus, P. Smulders, and Y. Yu, “Systems and architectures for very high frequency radio links," Analog Circuit Design (book chapter), 2008. [3] C. C. Ling and Z. Chunning, “Low-complexity antenna diversity receivers for mobile wireless applications," vol. 14, pp. 65-81, 2000. [4] N. C. Beaulieu and A. A. Abu-Dayya, “Analysis of equal diversity on Nakagami fading channels," IEEE Trans. Commun., vol. 39, no. 4, pp. 225-234, Feb. 1991.
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[5] Q. T. Zhang, “A simple approach to probability of error for equal gain combiners over Rayleigh fading," IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1151-1154, July 1999. [6] A. Annamalai, C. Tellambura, and V. K. Bhargava, “Equal-gain diversity receiver performance in wireless channels," IEEE Trans. Commun., vol. 48, no. 10, pp. 1732-1745, Oct. 2000. [7] M.-S. Alouini and M. K. Simon, “Performance analysis of coherent equal gain combining over Nakagami-𝑚 fading channels," IEEE Trans. Veh. Technol., vol. 50, no. 6, pp. 1449-1463, Nov. 2001. [8] X. Qi, M.-S. Alouini, and Y.-C. Ko, “Closed-form analysis of dualdiversity equal-gain combining over Rayleigh fading channels," IEEE Trans. Wireless Commun., vol. 2, no. 6, pp. 1120-1125, Nov. 2003. [9] R. K. Malik, M. Z. Win, and J. H. Winters, “Performance of dualdiversity predetection EGC in correlated Rayleigh fading with unequal branch SNRs," IEEE Trans. Commun., vol. 50, no. 7, pp. 1041-1044, July 2002. [10] R. Annavajjala and L. B. Milstein, “Performance of linear diversitycombining schemes on Rayleigh fading channels with binary signalling and Gaussian weighting errors," IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2267-2278, Sep. 2005.
[11] M. A. Najib and V. K. Prabhu, “Analysis of equal-gain diversity with partially coherent fading signals," IEEE Trans. Veh. Technol., vol. 49, no. 3, pp. 783-791, May 2000. [12] N. C. Sagias and G. K. Karagiannidis, “Effects of carrier phase error on EGC receivers in correlated Nakagami-𝑚 fading," IEEE Commun. Lett., vol. 9, no. 7, pp. 580-582, July 2005. [13] S. Nadarajah and S. Kotz, “On the product and ratio of gamma and Weibull random variables," vol. 22, pp. 338-344, 2006. [14] (2008) The wolfram functions site. [Online]. Available: http://functions.wolfram.com/01.03.21.0096.01 [15] (2008) The wolfram functions site. [Online]. Available: http://functions.wolfram.com/GammaBetaErf/Erfi/ [16] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th edition. Dover Publications, 1970. [17] Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels," IEEE Trans. Commun., vol. 51, no. 8, pp. 1389-1398, Aug. 2003. [18] J. G. Proakis, Digital Communications, 4th edition. McGraw-Hill, 2001.
