2011 17th Asia-Pacific Conference on Communications (APCC) 2nd – 5th October 2011 | Sutera Harbour Resort, Kota Kinabalu, Sabah, Malaysia

Comparison of Diversity Combining Techniques for MIMO Systems Luu Pham Tuyen

Vo Nguyen Quoc Bao

Binh Dinh Operation and Maintenance Center (OMC) Vietnam Posts and Telecommunications (VNPT) Email: [email protected]

Telecom. Dept. Posts and Telecom. Inst. of Tech. (PTIT), Vietnam Email: [email protected]

Abstract—Several diversity combining techniques have been proposed for multiple-input multiple-output (MIMO) systems, but only isolated performance comparisons have been reported. In this paper, we aim at comparing three MIMO schemes using selection combining (SC), maximal ratio combining (MRC) and switch-and-stay combining (SSC). It is analytically demonstrated that all the systems achieve full diversity gain with different coding gain. Besides, the performance gap between SC and MRC approaches to the limit and tends to increase linearly proportional to the number of transmit antennas. In case of SSC, the loss between SSC and SC, as well as between SSC and MRC, is not bounded as increasing signal-to-noise ratio (SNR).

I. I NTRODUCTION Multiple-input multiple-output (MIMO) is a wireless system that uses multiple antenna elements at both ends of a wireless communication link. For the same transmission power, MIMO technique can be used for increasing system capacity and diversity gain, i.e. reliability of a wireless link, as compared to a conventional single-input single-output (SISO) system [1]. The revolutionary idea behind MIMO technology is that contrary to SISO where fading is treated as one of the largest obstacles, fading in MIMO is viewed as an opportunity by using appropriate combining and decoding technique at the receivers. At present, MIMO has been adopted in some industry wireless standards, e.g. WiFi IEEE 802.11n, and the proposed for 3GPP Long Term Evolution/Long Term Evolution Advanced, 3GPP2 Ultra Mobile Broadband [2]. The main difficulty in MIMO practical implementation is its high complexity relative to the use of separate radio frequency (RF) chains for each employed antenna [3]. While antenna elements are usually cheap and demand simple manufacture technique, a receive RF chain normally comprising a low noise amplifier, frequency down-converters, analog-to-digital converters and several filters is a key factor which increases implementation cost significantly. Moreover, employing many RF chains, especially on mobile devices with limited size and battery capacity, results in difficult installation and/or more power consumption. The use of antenna selection (AS) technique in MIMO systems has recently gained high attention [3]–[8]. The advantage of this technique is that it can mitigate the hardware complexity while retaining diversity gain offered by MIMO technique. In particular, by choosing only the best signals for decoding, the number of RF chains needed are smaller than the

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number of available antennas [3]–[5]. The AS technique can be applied in MIMO at the transmitters, receivers, or both, corresponding to be called as TAS (Transmitter AS), RAS (Receiver AS) or T-RAS (Transmitter-Receiver AS). Among theses, RAS is the most practical one due to its simplicity, while TAS and T-RAS require channel state information (CSI) known at the transmitters. Besides, they only work well when the channel varies slowly. In designing AS-based MIMO networks, two common AS criteria have been proposed including maximizing system capacity and maximizing system quality. A nice overview for AS technique in MIMO system can be found in [3], [4], [9]. So far, many research works involving the performance derivation of MIMO have been reported in the literature, e.g. see [10]–[13]. However, to our best knowledge, the performance comparison of MIMO systems using maximal ratio combining (MRC), selection combining (SC) and switch-andstay combining (SSC) in terms of diversity gain and coding gain has not been considered and demands contributions. In this paper, motivated by all of the above, we study the system performance loss of MIMO systems employing either SC or SSC instead of MRC. We also investigate the impact of the number of transmit antennas on the system performance. Some discussions in case of varying the number of receive antennas are also represented. II. S YSTEM M ODEL In this paper, we consider a MIMO N × 2 system using N transmit antennas and two receive antennas, as shown in Fig. 1. The received signal vector on the ith receive antenna is written as r ρ yi = Xhi + wi . (1) N where ρ is the average signal-to-noise ratio (SNR) on each receive antenna, i ∈ {1, 2} is the receive antenna index. yi and wi are L × 1 column vectors representing the received signal and AWGN noise on the ith receive antenna, respectively; X is a L × N matrix representing a transmitted space-time codeword spanning N transmit antennas and L time instants; hi = [hi1 , hi2 , . . . , hiN ]T contains the channel coefficients between N transmit antennas and the ith receive antenna.

203 0 . 01014 

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Fig. 1.


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The system under consideration.

Here, the channels are assumed to be independent and identically distributed (i.i.d.) flat, quasi-static Rayleigh fading, so the entries of hi are modeled as i.i.d. complex Gaussian random variables with zero mean and variance 0.5 per dimension. It is further assumed that they keep constant over a entire codeword and fade independently from one to another, i.e the channel coherence time Tc is equal to L instants; the elements of wi are modeled as i.i.d. complex Gaussian random variables with zero mean and variance 0.5 per dimension. The CSI is unknown at the transmitter but is known exactly at the receiver and a power constraint, E[tr(X H X)] = N L, where tr(.) is the trace operator, is imposed on transmitted symbols. The full-rank space-time code will be used at the transmitter ensuring that the system will obtain full transmit diversity gain. The signals at the receiver are combined by using MRC, SC or SSC technique. In case of using MRC, there are two available RF chains at the receiver, while using SC or SSC, only one RF chain is required at the receiver. As mentioned in [9], the complexity of SC and SSC is lower than that of MRC. After combining, the signals (at the corresponding combiner output) will be decoded by a maximum likelihood decoder. With SC, the antenna having maximum instantaneous SNR will be selected for decoding [14]. The instantaneous SNR on each receive antenna is written as r ρ 2 khi k . (2) γi = N With SSC, we use the switching algorithm mentioned in [15]–[18]. That is, the currently connected antenna will be used for decoding until its instantaneous SNR falls below a predetermined threshold, and the receiver will switch to another antenna regardless of whether or not the SNR on that antenna is above or below the predetermined threshold. In SSC systems, choosing switching threshold for minimizing system performance metrics (i.e. outage probability, bit error rate, symbol error rate or pairwise error probability) is an optimization problem. With optimal switching thresholds, the performance of systems will be minimized.

III. P ERFORMANCE A NALYSIS In this section, we investigate the system performance through the pairwise error probability (PEP). The PEP is defined as the probability that the receiver incorrectly decodes codeword X1 while the transmitter sends codeword X2 from a codebook containing X1 and X2 [19]. By using full-rank space-time codes at the transmitter and maximum likelihood decoder at the receiver, the average pairwise error probability PEP is upper-bounded [8] 1 PEP (ρ) ≤ 2

Z∞



 λmin exp − u fz (u)du, 4

(3)

0

where λmin is the smallest eigenvalue of a codeword distance H matrix (X1 − X2 ) (X1 − X2 ). fz (u) is the probability density function (PDF) of SNR at the combiner output. A. Switch and Stay Combining For SSC at the receivers, the upper bound of average PEP, PEPSSC (ρ) using optimal switching threshold (in minimizing PEP sense) is given by [8] where C := λmin /4+N/ρ and ϕ := 4N 2 /λmin . By using conventional diversity gain definition as
d = lim (− ln PEP(ρ) / ln ρ), the work in [8] shows that ρ→∞

MIMO-SSC systems achieve full diversity gain as 2N with optimal switching thresholds. In case of using fixed switching threshold, MIMO-SSC systems achieve only transmit diversity gain as N . B. Selection Combining When the receiver employs SC technique, the PDF function of SNR at the SC combiner output is given by [9] fSC (u) = 2fs (u)Fs (u),

(5)

where fs (.) and Fs (.) are the PDF and cumulative distribution function (CDF) of the instantaneous SNR on each receive antenna, respectively. According to (2), over Rayleigh fading channels, the instantaneous SNR on each receive antenna

PEPSSC (ρ) ≤



1 N 2 Cρ

N



 1 −



  j  − ϕρ  N −1 ϕ ln Cρ  X ρ N Cρ N

  j=0

j!

follows Chi-square distribution with 2N degrees of freedom with the PDF and CDF expressed by [19]    N uN −1 Nu N exp − (6) fs (u) = ρ ρ (N − 1)! and   N −1 i N u X (N u/ρ) Fs (u) = 1 − exp − . ρ i! i=0

(7)

Combining (5), (6) and (7), and plugging the result into (3), we can express the upper bound of average PEP for SC (with the help of [20, eq. 3.351.3] and after some manipulations) as   N  Cρ N 1 − 1  ×  (N −1)! Cρ+N  N  NP  j  . (8) PEPSC (ρ) ≤ −1   (N −1+j)! N Cρ j=0

j!

Cρ+N

C. Maximal Ratio Combining When MRC is employed at the receiver, the SNR at the MRC combiner output has Chi-square distribution with 4N degrees of freedom [9], [21]. Utilizing the same derivation method as SC, the upper bound of the average PEP for MIMOMRC can be written as  2N 1 N PEPMRC ≤ . (9) 2 Cρ From (8) and (9), it is straightforward to show that not only MIMO-SC and MIMO-MRC but also MIMO-SSC achieves full spatial diversity gain as 2N . Therefore, it would be insightful to address the system performance loss in terms of coding gain. Before moving on the next section where we compare MIMO systems, it should be noted that the above PEP obtained for MRC, SC and SSC are derived in the same manner, thus it is likely fair to use the upper bound of PEP to compare their performance. D. Performance Comparison In order to address the behavior of the PEP loss, e.g. between SC and MRC, we take limit of the ratio PEPSC (ρ)/PEPMRC (ρ) as ρ approaches to infinity. Mathematically, the PEP loss between SC and MRC can be written as PLSC/MRC = lim

ρ→∞

PEPSC (ρ) . PEPMRC (ρ)

(10)

Using the same way and denoting the PEP loss between SSC and MRC as PLSSC/MRC , we can write PLSSC/MRC as PLSSC/MRC

PEPSSC (ρ) . = lim ρ→∞ PEPMRC (ρ)

(11)

+



Cρ N

−N NX −1 j=0

   j    N + ϕρ ln Cρ N  ,  j! 

(4)

Similarly, we have PLSSC/SC = lim

ρ→∞

PEPSSC (ρ) . PEPSC (ρ)

Replacing (8) and (9) into (10) yields  N PEPSC (ρ) Cρ =2 N PEPMRC (ρ)   N Cρ 1 ×  1 − (N −1)! Cρ+N   × −1 j  NP (N −1+j)! N j=0

j!

Cρ+N

(12)



 . 

(13)

For convenience, we define t := N/ (Cρ + N ). By using Maclaurin series expansion of the function f (t) = (N − N 1)!/(1 − t) , (13) can be further simplified as PEPSC (ρ) 2(2N −1)! 2N = (1−t) (N −1)!N ! PEPMRC (ρ) × 2 F1 (1, 2N ; N + 1; t) ,

(14)

where 2 F1 (.) is Gauss’s Hypergeometric series [22, eq. (15.1.1)]. Note that t tends to zero as ρ tends to infinity and + 2 F1 (a, b; c; 0) = 1 with a, b, c ∈ Z . Hence, from (14), it is straightforward to show that 2(2N − 1)! (1 − t)2N (N − 1)!N ! × 2 F1 (1, 2N ; N + 1; t) (2N )! = 2. (N !)

PLSC/MRC = lim

t→0

(15)

From (15), we conclude that the PEP loss between SC and MRC tends to a constant at high SNR and is directly proportional to the number of transmit antennas. Moreover, this also indicates that the coding gain loss between SC and MRC tends to a constant at high SNR regime. For PLSSC/MRC , from (4), (9) and using Maclaurin series expansion of the function ex with x := (ϕ/ρ) / ln (Cρ/N ), we arrive at   j ϕ Cρ N X  ∞ ρ ln N Cρ PLSSC/MRC = lim ρ→∞ N j! j=N    j N −1 N + ϕρ ln Cρ X N . (16) − j! j=0 It is easy to see that the RHS of (16) has an indeterminate form as ρ tends to infinity. However, the RHS of (16) can be lower bounded by dividing by ln ρN −1 at sufficient high SNR.

PEPSSC (ρ) = ∞. ρ→∞ PEPMRC (ρ)

PLSSC/MRC = lim

(17)

Using the same technique, we also have PLSSC/SC → +∞. Therefore, we can conclude that PEP loss between SSC and MRC is larger than that of between SSC and SC, and both of them are not bounded, i.e. they become larger at higher SNR. This is different to the PEP loss between SC and MRC. For this reason, it is shown that although SSC achieves full diversity gain as MRC and SC, SSC suffers an amount of coding gain loss that is unbounded. This loss quantity is considered as the cost of lower hardware complexity comparing to MRC and SC. Through the above-mentioned analyses, the behavior of performance loss among MRC, SC and SSC in case of the same number of transmit antennas can be investigated. In case of varying the number of transmit antennas, only the coding gain loss between SC and MRC can be visible from (12), that is, it is directly proportional to the number of transmit antennas. Thus, it is meaningful to observe the coding gain loss among SSC, SC and MRC at a specified PEP value and varying the number of transmit antennas. In doing so, letting PEPCOM as the specified common PEP value and solving the following equation system    PEPSSC (ρSSC ) = PEPCOM , (18) PEPSC (ρSC ) = PEPCOM   PEPMRC (ρMRC ) = PEPCOM

the coding gain loss (in dB) among SSC, SC and MRC can be written as    LossSSC−SC = 10 log 10(ρSSC /ρSC ) LossSSC−MRC = 10 log 10(ρSSC /ρMRC ) . (19)   LossSC−MRC = 10 log 10(ρSC /ρMRC )

The closed-form solution of (19) cannot be found in general, but it is easily solved by the numerical method because each individual equation in (18) has a unique root. The plots of (19) are illustrated and verified in Fig. 3. Additionally, with more than two receive antennas at the receiver over i.i.d. fading channels, the PDF function of SNR at SSC combiner output reduces to the case of two antennas [16]. That is, the upper bound PEP of SSC remains unchanged as observed in (4). This implies that using more than two antennas at the receivers with SSC is useless, i.e. no more receive diversity gain is obtained. On the contrary, in case of MRC and SC, many previous works show that full diversity gain is retained [3], [8], [23]. Hence, with SSC using more than two antennas at the receiver loses not only coding gain but also diversity gain as compared to MRC and SC.

phase shift keying (QPSK) modulation over i.i.d. flat, quasistatic Rayleigh fading channels. The corresponding value of λmin (with QPSK modulation) for N = 2 and 3 is 2 and 4. The plots of analysis upper bound PEP for SSC, SC and MRC from (4), (8), and (9) are shown in Fig. 2. By carefully observing this figure, it is shown that the PEP loss, as well as the coding gain loss, between MRC and SC tends to a constant at sufficient high SNR. Furthermore, the loss becomes larger since the number of transmit antennas is increased. However, this behavior is not true for the case of SSC. These validate the analytic results in (15) and (17). In Fig. 3, we plot the solutions of (19) in order to observe the coding gain loss when varying the number of transmit antennas. This figure again validates the above analyses. It is observed that as increasing N , the loss between SSC and MRC, as well as SC, becomes smaller, while in the case of SC and MRC, this becomes larger. Through this figure, a modest loss due to the use of SSC over SC is shown. At PEP of 10−10 , performance loss is about 3 dB for N = 2 and about 2.5 dB for N = 3. In Figs. 4, 5 and 6, the simulation results of the system bit error rate (BER) are represented. It is evident that the behavior of the PEP and the BER are the same. With two antennas employed at the receivers, MRC, SC and SSC achieve full diversity gain as 2N 1 . Fig. 4 also shows that a constant coding gain loss of 2 dB is hold in all ranges of operating SNRs. Furthermore, this loss gap becomes larger as increasing the number of transmit antennas (or the number of receive antennas) as illustrated in Fig. 5 and Fig. 6. However, the performance loss between SSC and MRC (as well as SSC and SC) does not keep constantly and seems to increases linearly as the increase of average SNR. As a final note, for MIMOSSC systems, there are no performance improvement when we increase the number of receive antennas as observed in Fig. 6. This agrees completely with the results reported previously in [8].

0

MIMO−SSC, Fixed Threshold MIMO−SSC, Optimal Threshold MIMO−SC MIMO−MRC

−5

N=2

10

Pairwise Error Probability

The lower bound tends to infinity as ρ tends to infinity, hence, the final result of (16) is

10

−10

10

N=3 −15

10

0

Fig. 2.

10

20

30 40 50 60 Average SNR[dB] per receive antenna

70

80

Analysis upper bound of PEP of MRC, SC and SSC.

IV. N UMERICAL R ESULTS AND D ISCUSSION For all simulations, we use orthogonal spacetime block codes (OSTBC) [24]–[26] at the transmitters and quadrature

1 For

MIMO-SSC systems, it is only true with optimal switching threshold.

6

N=2 N=3

SSC vs. MRC

5

−2

Loss in Coding gain [dB]

10 SSC vs. SC Bit Error Rate

4

3

−4

10

2

1 −6

10

SC vs. MRC

0

0

−5

10

Fig. 3.

−10

10

Analysis coding gain loss among MRC, SC and SSC.

Bit Error Rate

−6

10

0

SISO MIMO−MRC 2x1 MIMO−SSC 2x2, Fixed Threshold MIMO−SSC 2x2, Optimal Threshold MIMO−SC 2x2 MIMO−MRC 2x2 5

Fig. 4.

Fig. 6.

10 15 20 Average SNR [dB] per receive antenna

25

30

Bit error rate of MIMO 2 × 3.

well as between SSC and MRC, is not bounded as increasing SNR. Among three combining technique considered, SSC is a attractive candidate for MIMO practical implementation, especially on mobile device. Furthermore, it should be noted that SSC technique is only suitable for MIMO system with two receive antennas.

−2

−4

5

−15

10 Pairwise Error Probability

10

10

MIMO−SSC 2x2 MIMO−SSC 2x3 MIMO−SC 2x3 MIMO−MRC 2x3

ACKNOWLEDGMENT

10 15 20 Average SNR [dB] per receive antenna

25

30

This research was supported by the Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) (No. 102.99-2010.10).

Bit error rate of MIMO 2 × 2.

R EFERENCES MIMO−SSC 2x2 MIMO−SSC 3x2 MIMO−SC 3x2 MIMO−MRC 3x2

−2

Bit Error Rate

10

−4

10

−6

10

0

5

Fig. 5.

10 15 20 Average SNR [dB] per receive antenna

25

30

Bit error rate of MIMO 3 × 2.

V. C ONCLUSION In this paper, we investigate and compare performance of MIMO N × 2 system using MRC, SC and SSC in terms of diversity gain and coding gain. Some discussions in case of having more than two receive antennas are also represented. We can conclude that with a N × 2 MIMO system using full-rank space-time codes at the transmitters, regardless of employing MRC, SC or SSC at the receiver, the system always achieves full diversity gain. About coding gain, the loss between SC and MRC tends to a constant value and this value becomes larger according to number of transmit antennas. In case of SSC, the loss between SSC and SC, as

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