Symbol and Bit Error Probabilities of Orthogonal ... - Semantic Scholar
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 12, DECEMBER 2008
Symbol and Bit Error Probabilities of Orthogonal Space-Time Block Codes with Antenna Selection over Keyhole Fading Channels Nghi H. Tran, Member, IEEE, Ha H. Nguyen, Senior Member, IEEE, and Tho Le-Ngoc, Fellow, IEEE
Abstract—The symbol error rate (SER) and the bit error rate (BER) of orthogonal space-time block codes (OSTBCs) with antenna selection over keyhole fading channels are examined. Considered are receive antenna selection, transmit antenna selection, and joint antenna selection at both the transmitter and the receiver. The exact SERs of OSTBCs for M -PSK and square M QAM constellations are obtained using the technique of moment generating function (MGF). By applying the Bonferroni-type bounds, tight lower and upper bounds for both the SER and BER are provided in closed-form expressions with finite-range single integrals. The bounds can be applied to arbitrary constellations and mappings. Numerical results show that the bounds can be used to provide practically the exact SER and BER over a wide range of the signal-to-noise ratio. Index Terms—Antenna selection, keyhole fading, orthogonal space-time block codes, Bonferroni-type bounds, symbol error rate (SER), bit error rate (BER).
I. I NTRODUCTION
S
PACE-TIME code is a promising modulation technique to maximize the diversity gain over multiple-input and multiple-output (MIMO) channels with multiple antennas at the transmitter and the receiver [1], [2]. Due to its simple implementation while providing high diversity gains, orthogonal space-time block codes (OSTBCs) [2], [3] have received a special attention, both in research community and industry. However, the deployment of OSTBCs in particular and MIMO systems in general can be quite costly since the transmitter and/or the receiver require multiple radio frequency chains and low-noise amplifiers. To overcome this challenge, performing antenna selection is an attractive technique. This is because with antenna selection the system can still retain most of the advantages offered by multiple antennas while the complexity of hardware implementation is reduced [4]. Although transmit antenna selection Manuscript received September 19, 2007; revised January 6, 2008 and April 13, 2008; accepted April 21, 2008. The associate editor coordinating the review of this paper and approving it for publication was B. S. Rajan. This work was supported in part by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada (NSERC). The first author would also like to acknowledge the University of Saskatchewan’s Graduate Scholarship and the Fellowship received from TRlabs-Saskatoon. A part of this work was presented at the IEEE International Conference on Communications (ICC), Beijing, China, May 19-23 2008. N. H. Tran was with the Department of Electrical & Computer Engineering, University of Saskatchewan, Saskatoon, SK, Canada S7N 5A9. He is now with the Department of Electrical & Computer Engineering, McGill University, Montreal, Quebec, Canada H3A 2A7. (e-mail: [email protected]). H. H. Nguyen is with the Department of Electrical & Computer Engineering, University of Saskatchewan, Saskatoon, SK, Canada S7N 5A9. (e-mail: [email protected].). T. Le-Ngoc is with the Department of Electrical & Computer Engineering, McGill University, Montreal, Quebec, Canada H3A 2A7. (e-mail: [email protected].). Digital Object Identifier 10.1109/T-WC.2008.071041
and receive antenna selection has been well-investigated over the conventional independent and identically distributed (i.i.d.) fading channel (see, for example, [5], [6] and many references therein), only a few efforts have been devoted to the case of MIMO wireless channels with the existence of keyhole fading, where scattering processes are present at both the transmitter and the receiver [7], [8]. In particular, capacity and diversity analysis for antenna selection over keyhole fading was investigated in [9], which shows that the performance of systems using antenna selection can match to that of a baseline full-antenna system. Regarding OSTBCs, to the best of our knowledge, only the approximation of the BER of a system employing Alamouti code and BPSK modulation at the transmitter and the selection of one best receive antenna was provided in [10]. This paper examines the error performance, both the SER and the BER, of OSTBCs over keyhole fading channels with joint antenna selection at both the transmitter and the receiver, which includes receive antenna selection and transmit antenna selection as special cases. At first, by deriving the moment generating function (MGF) of an instantaneous signal-to-noise ratio (SNR) at the receiver, the exact SERs of OSTBCs for M -PSK and square M -QAM constellations are computed. Then the Bonferroni-type bounds are applied to obtain the tight lower and upper bounds on both the SER and BER. These bounds, which can be effectively computed with finiterange single integrals, are applicable to any constellations and mappings. It is shown that the bounds can be used to provide practically the exact SER and BER over a wide range of SNR. It should be noted that this paper only considers the case that the channel state information is perfectly estimated at the receiver. The analysis with antenna selection in the presence of channel estimation error, as similar to [11]–[13], would be an interesting topic for further studies. II. S YSTEM M ODEL AND T HE T ECHNIQUE OF M OMENT G ENERATING F UNCTION (MGF) A. System Model Consider a MIMO system with Lt transmit and Lr receive antennas. The channel model for keyhole fading is [9] H = hLt h Lr where hLt and hLr are column vectors of size Lt and Lr , respectively, whose entries are circularly symmetric complex Gaussian random variables with variance 1/2 per dimension CN (0, 1). For receive antenna selection, the receiver will select Nr out of Lr antennas. If antenna selection is performed at the transmitter, it is assumed that the transmitter knows only the indices of Nt out of Lt transmit antennas sent back from the receiver via a rate-limited feedback with error-free channel. Moreover, joint antenna
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selection with Nt transmit antennas and Nr receive antenna can also be implemented. In a space-time block coded system, with Nt and Nr selected antennas, each block of K · b information bits is first mapped to a sequence of K symbols {v1 , v2 , . . . , vK }, where each vk can be one of M = 2b signal points {su }M u=1 in a M -ary constellation Φ. The mapping rule ξ defines the correspondence between b bits and a signal point in Φ. The sequence {v1 , v2 , . . . , vK } is then encoded into an N × Nt space-time block code matrix G. Here, N is the number of rows of the space-time code matrix and the code rate is R = K/N symbol per channel use. In the case of OSTBCs, the columns of G are orthogonal. The receive signals at the Nr selected receive antennas over N time periods are collectively given as: ρ ˜ R= GH + W . (1) Nt ˜ = hNt h with hNt and hNr correspond to Nt Here, H Nr and Nr selected components of hLt and hLr , respectively. The matrix W is a N × Nr matrix representing additive white Gaussian noisewhose entries are also CN (0, 1). The normalization factor Nρt in (1) ensures that the average SNR at each receive antenna is ρ, independent of Nt . Due to the orthogonal structure of OSTBCs, it can be shown that a MIMO fading channel can be equivalently represented by a single-input single-output (SISO) channel [14]. Let Y be ˜ The achievable SNR per the squared Frobenius norm of H. symbol is given as: ρs =
ρR ρR ·Y = · U · V, Nt Nt
(2)
where U = ||hNt ||2 and V = ||hNr ||2 . From (2), it is obvious that the problem of joint transmit and receive antenna selection to maximize ρs can be decomposed into the problems of transmit antenna selection and receive antenna selection independently. In particular, Nt and Nr transmit and receive antennas should be selected to maximize U and V , respectively.
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The pdf of U is also expressed in a similar form. Since Y = U · V and U and V are two independent random variables, one has: ∞ y 1 dv. (6) pV (v)pU pY (y) = v v 0 The moment generating function of Y with joint receive and (r,t) transmit antenna selection can then be computed as φY (s) = ∞ E[exp(−sY )] = 0 exp(−sy)pY (y)dy. As shown in Appendix A, the pdf of Y , pY (y), and (r,t) the MGF of Y , φY (s), can be written in closed-form expressions. In particular, from (28), it can be seen that the MGF of Y is expressed in a closed form as the sum of various functions, each having the same form of f (s; A, B, C, D) = As−B U(B, C, D/s), where U(a, b, x) is the confluent hypergeometric function of the second kind [16] and A, B, C, D are some constants. For example, in the case of receive antenna selection only, since Lt − Nt = 0 and the fact that β0 = 0 and ηi,0 = 0, the MGF of Y in (28) can be written in a more compact form as: 2 Γ(Lt ) −Nr Lr s U(Nr , Nr − Lt + 1, 1/s) Γ(Lt ) Nr 2 L −1 Lr −Nr Nr + i Γ(Lt ) Nr + i t + βi s−Lt U Lt , Lt 2 Nr Nr s i=1 N r −1 Γ(j)Γ(Lt ) −j ηi,j − s U (j, 1 + j − Lt , 1/s) . 2 j=1 (7)
(r)
φY (s) =
With the closed-form expressions of the MGF of Y presented in (28), the exact SER of OSTBCs for standard M PSK and square M -QAM constellations can be computed straightforwardly. More specifically, by averaging the conditional SERs for M -PSK and square M -QAM constellations given in [17] over the pdf of the instantaneous SNR Y , one obtains the following expressions for the general case of joint receive and transmit antenna selection: ρR 1 π−π/M (r,t) Nt gMPSK MPSK φY Psymbol (error) = dθ (8) π 0 sin2 θ
B. MGF of Y and the Exact SER In order to derive the SER and BER, this subsection first evaluates the MGF of the random variable Y . Following the analysis in [15], the probability density function (pdf) of the random variable V is given as: Lr −Nr Lr exp(−v) + βi exp(−v)· v Nr −1 pV (v) = Nr Γ(Nr ) i=0 ⎛ ⎞⎤ N r −1 iv ⎝exp − ηi,j v j−1 ⎠⎦ , − (3) Nr j=0 where βi =
−Nr Nr −1 r (−1)i Lr −N i>0 i i 0 Otherwise j−1 1 −i j>0 Γ(j) Nr ηi,j = 0 Otherwise
(4)
(5)
MQAM (error) Psymbol
=
4q π
−
4q 2 π
π/2
0
(r,t) φY
π/4
0
ρR
(r,t) φY
Nt gMQAM 2
sin θ
ρR
Nt gMQAM 2
sin θ
dθ dθ (9)
(r,t)
where φY (·) is the MGF of Y , given in (28). It can be seen that the single integrals in (8) and (9) are over finite ranges. Therefore, the exact SERs for M -PSK and square M QAM can be computed easily. This fact holds for transmit antenna selection and receive antenna selection as well, since the MGFs of Y in all three cases have similar expressions. III. T IGHT U PPER AND L OWER B OUNDS ON THE SER AND BER BASED ON B ONFERRONI -T YPE B OUNDS Bonferroni-type bounds are the bounds on the probability of a union. Let A1 , A2 , . . . , AX be X events defined on a given
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probability space. Then the probability of the union event is X X PU = P Ax = (−1)x+1 Sx , (10) x=1
x=1
where Sx = P (Ax1 ∩ · · · ∩ Axk ) and the sum is taken over all 1 ≤ x1 < · · · < xk ≤ X. The interesting result is that by truncating the sum in (10) at any Sx , a lower or upper bound is obtained, depending on the sign of the last term. Specifically, PU ≤ S1 , PU ≥ S1 −S2 , PU ≤ S1 −S2 +S3 and so on. Two specific versions of the Bonferroni-type bounds are briefly described next. The Kounias lower bound is given by, ⎤ ⎡ P (Ax ) − P (Ay ∩ Ax )⎦ , (11) PU ≥ maxΓ ⎣ x∈Γ
y,x∈Γ,y<x
where Γ is any arbitrary subset of IX = {1, 2, . . . , X}. As pointed out in [18], one can apply an efficient stepwise algorithm, whose complexity is linear in the number of events, to find the best subset that gives the tightest bound. For X ≥ 2, the tightest Hunter upper bound is: PU ≤
N
P (Ax ) − maxT0 ∈T
P (Ax ∩ Ay ),
(12)
(x,y)∈T0
x=1
where T0 is any tree spanning the indices of the set A1 , . . . , AX , (x, y) is an edge in T0 . The greedy algorithm [18] can then be applied to construct the optimal spanning tree T0 for (12). The Bonferroni-type bounds have been applied to analyze the performance of OSTBCs in various scenarios [19], [20], but without antenna selection. In the following, the Bonferroni-type bounds are applied to derive the tight upper and lower bounds on the SER and BER of OSTBCs with antenna selection.
(r)
where φY (·) is either φY (·) given in (7) for receive antenna (r,t) selection, or φY (·) given in (28) for joint receive and transmit antenna selection. Clearly, the unconditional PEP Pu ( p,u ) can be easily calculated using finite-range single integrals. Following the analysis in [18], [19], one has the conditional two-dimensional PEP expressed in terms of two-dimensional joint Gaussian function as follows: √ √ (16) Pu ( p,u ∩ q,u |Y ) = Ψ ρp,q,u , δpu Y , δqu Y , where ρp,q,u =< sp − su , sq − su > /dp,u dq,u and Ψ(r, a, b) is the two-dimensional joint Gaussian function, defined as: Ψ(r, a, b) =
1 2π 1 − r 2 √
∞
a
∞ b
exp
x2 − 2rxy + y 2 dxdy. 2(1 − r 2 ) (17)
√ √ Furthermore, since δpu Y and δqu Y are non-negative, one can use the single integral representation of (17) to obtain the conditional PEP: √ √ Ψ ρp,q,u , δpu Y , δqu Y ϕ(dp,u /dq,u ,ρp,q,u ) 2 δp,u Y 1 dθ exp − = 2π 0 2 sin2 θ ϕ(dq,u /dp,u ,ρp,q,u ) 2 δq,u Y 1 + dθ (18) exp − 2π 0 2 sin2 θ √ where ϕ(x, r) = tan−1 x 1 − r2 /(1 − rx) . Taking the average of (18) over Y using the MGF of Y , one arrives at: ϕ(dp,u /dq,u ,ρp,q,u ) 2 δp,u 1 dθ Pu ( p,u ∩ q,u ) = φY 2π 0 2 sin2 θ ϕ(dq,u /dp,u ,ρp,q,u ) 2 δq,u 1 + dθ φY 2π 0 2 sin2 θ (19)
A. Bonferroni-Type Bounds on the SER ⎛ In general, the SER can be expressed as [19]:
⎞ 1 Psymbol (error) = P (|su )P (su ) = Pu ⎝ p,u ⎠ , M u=1 u=1 p=u M
(13)
where P ( |su ) is the conditional error probability given that su was sent and p,u is the event that sp has a larger metric than su [19]. In order to obtain the two Bonferronitype bounds for the SER, one needs to compute the pairwise error probability (PEP), Pu ( p,u ), and the two dimensional PEP, Pu ( p,u ∩ q,u ), for given q, p, and u. Given Y , the conditional PEP can be computed as follows [19]: 2 δp,u Y 1 π/2 exp − dθ (14) Pu ( p,u |Y ) = π 0 2 sin2 θ where δp,u = Rρ/2Nt dp,u and dp,u = su −sp . Averaging (14) over Y , one has: 2 δp,u 1 π/2 Pu ( p,u ) = φY dθ (15) π 0 2 sin2 θ
Clearly, the two-dimensional PEP Pu ( p,u ∩ q,u ) can also be effectively computed by single finite-range integrals. Using (15) and (19), one obtains the lower and upper bounds, based on the Kounias and Hunter bounds, respectively, for the SER, which shall be referred to simply as the stepwise lower bound and the greedy upper bound. It should be noted that these bounds are applicable for any arbitrary constellations. B. Bonferroni-Type Bounds on the BER The upper and lower bounds on the BER can also be obtained using the MGF of Y . In particular, the BER can be computed as follows [18], [19]: Pbit (error) =
M 1 Pb (su ), M u=1
(20)
where Pb (su ), the bit error probability when su is sent, is ⎛ ⎞⎤ ⎡ M 1 Pb (su ) = (21) H(q, u) ⎣1 − Pu ⎝ p,q ⎠⎦ . b q=1
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p=q
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It should be mentioned that this computation is only valid when γp,q,q ≥ 0. When γp,q,u < 0, one can use the relationship Q(x) = 1−Q(−x) to obtain a similar expression. On the other hand, given Y , the two-dimensional PEP, Pu ( p,q ∩ l,q ) is given as [18]: √ √ Pu ( p,q ∩ l,q |Y ) = Ψ ρp,l,q , γp,q,u Y , γl,q,u Y . (24) Therefore, the unconditional two-dimensional PEP can be expressed in the following closed form:
0
10
−1
10
r
IV. I LLUSTRATIVE R ESULTS This section provides the numerical results to confirm our analysis. In particular, the cases of receive antenna selection and joint antenna selection at the transmitter and the receiver are considered. Furthermore, we concentrate on Alamouti v1 v2 scheme [3] with Nt = N = 2 and G = . −v2 v1 Unless stated otherwise, the number of receive antennas is always Lr = 6. In the case of receive antenna selection only, one has Lt = Nt = 2. For joint receive and transmit antenna selection, the number of transmit antenna is assumed to be Lt = 4. The SNR is defined as ρ (dB). A. Receive antenna selection 1) SER Performance: Figures 1 and 2 show the simulation results and the exact analysis of the SER presented in (8) and (9) for the Alamouti scheme employing 8-PSK and 16QAM constellations, respectively. Three values of Nr , namely Nr = 1, Nr = 2, and Nr = 6, are investigated. For
r
−2
10
Lr=6,Nr=1 L =6,N =2
−3
10
−4
10
−5
10
0
r
r
Simulation, 8−PSK Exact analysis, 8−PSK Greedy UB, 8−PSK Stepwise LB, 8−PSK 5
10
15 ρ (dB)
Lr=6,Nr=6 20
25
30
Fig. 1. The simulation results, exact analysis, greedy upper bound (UB), and stepwise lower bound (LB) for 2 × 2 Alamouti scheme using 8-PSK constellation with receive antenna selection for Lt = Nt = 2 and various values of Nr . The performances with Lr = Nr = 2 are provided as references.
Pu ( p,q ∩ l,q ) ϕ(γp,q,u /γl,q,u ,ρp,l,q ) 2 γp,q,u 1 = φY dθ 2π 0 2 sin2 θ ϕ(γl,q,u /γp,q,u ,ρp,l,q ) 2 γl,q,u 1 + φY dθ (25) 2π 0 2 sin2 θ
0
10
−1
10
Lr=2,Nr=2 −2
10
Lr=6,Nr=2
Lr=6,Nr=1
SER
The expression of Ψ (r, a, b) in (18) is only valid for nonnegative a and b. When at least one of a and b is negative, Ψ (r, a, b) can be expressed in the form of other Ψ(·) function with non-negative arguments [19]. Hence, a similar expression of Pu ( p,q ∩ l,q ) can still be obtained Using (23) and (25), the Kounias and Hunter bounds can be calculated, which lead to the upper and lower bounds, respectively, for the BER in (20). They are simply referred to as the stepwise upper bound and the greedy lower bound.
L =2,N =2
SER
In (21), H(q, u) is the Hamming distance between the label of sq and su (which depends on the mapping rule ξ). Given Y , the one-dimensional PEP is computed as [18], [19]: 2 √ Y γp,q,u 1 π/2 Pu ( p,q |Y ) = Q γp,q,u Y = exp − dθ π 0 2 sin2 θ (22) δ 2 −δ 2 where γp,q,u = p,uδp,q q,u . Similar to the previous sections, averaging (22) over Y and using the MGF of Y , one obtains: 2 γp,q,u 1 π/2 Pu ( p,q ) = dθ. (23) φY π 0 2 sin2 θ
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−3
10
L =6,N =6 r
−4
10
−5
10
5
r
Simulation, 16−QAM Exact analysis, 16−QAM Greedy UB, 16−QAM Stepwise LB, 16−QAM 10
15
20 ρ (dB)
25
30
35
Fig. 2. The simulation results, exact analysis, greedy upper bound (UB), and stepwise lower bound (LB) for 2 × 2 Alamouti scheme using 16-QAM constellation with receive antenna selection for Lt = Nt = 2 and various values of Nr . The performances with Lr = Nr = 2 are provided as references.
comparison, the derived greedy upper bound and the stepwise lower bound are provided to show the tightness of the bounds. The performances of the system with Lt = Nt = Lr = Nr = 2 antennas (without antenna selection) are also plotted as references. It can be observed from Figs. 1 and 2 that the simulation results, the exact analysis and the bounds cannot be distinguished over a wide range of SNR for both constellations. This means that the bounds can be used as an effective tool to predict the actual SER of OSTBCs with antenna selection over keyhole fading channels. The results agree with [18], [19] that the two Bonferroni-type lower and upper bounds are very accurate in estimating the probability of a union.
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0
0
10
10
−1
Simulation, 8−PSK Exact analysis, 8−PSK Greedy UB, 8−PSK Stepwise LB, 8−PSK
−1
10
10 Lr=2,Nr=2
−2
SER
SER
10 L =6,N =1
−3
−4
−5
10
0
r
Lr=6,Nr=2
10
10
L =2,N =2, t t L =6,N =1
−2
10
10
−3
10
r
15 ρ (dB)
r
L =2,N =2, t t L =6,N =2
r
−4
Lr=6,Nr=6
r
Lt=4,Nt=2, L =6,N =1
r
r
L =4,N =2, t t L =6,N =2
10
Simulation, (1,7) Greedy UB, (1,7) Stepwise LB, (1,7) 5
r
r
r
−5
20
25
30
Fig. 3. The simulation results, greedy upper bound (UB), and stepwise lower bound (LB) for 2 × 2 Alamouti scheme using (1,7) constellation with receive antenna selection for Lt = Nt = 2 and various values of Nr . The performances for the case of Lr = Nr = 2 are provided as references.
10
0
5
10
15 ρ (dB)
20
25
30
Fig. 5. The simulation results, exact analysis, greedy upper bound (UB), and stepwise lower bound (LB) for 2 × 2 Alamouti scheme using 8-PSK constellation with joint receive and transmit antenna selection for Lt = 4, Nt = 2, and various values of Nr selected from Lr = 6 receive antennas. The performances with receive antenna selection only for Lt = Nt = 2 are provided as references.
0
10
−1
10
Lr=2,Nr=2 −2
BER
10
Lr=6,Nr=2
−3
Lr=6,Nr=1
10
−4
10
−5
10
0
Simulation, natural mapping Greedy LB, natural mapping Stepwise UB, natural mapping Simulation, Gray mapping Greedy LB, Gray mapping Stepwise UB, Gray mapping 5
10
15
ρ (dB)
20
25
30
Fig. 4. The simulation results, greedy lower bound (LB), and stepwise upper bound (UB) on the BER for 2×2 Alamouti scheme using 8-PSK constellation with receive antenna selection, employing Gray and natural mappings for Lt = Nt = 2 and various values of Nr . The performances for the case of Lr = Nr = 2 are provided for comparison.
As mentioned before, the exact computations of the SER in (8) and (9) are only applicable for standard M -PSK and square M -QAM. The bounds, on the other hand, do not have this limitation, since they can be applied for arbitrary constellations. As an example, Fig. 3 plots the simulation results, the greedy UB, and the stepwise LB for 2 × 2 Alamouti scheme using (1,7) constellation [21] and with Lt = Nt = 2, Lr = 6, and various values of Nr . It can be seen that the bounds and the simulation results almost coincide, which means that they can practically provide the exact SER. Similar results were also observed with different non-regular constellations. 2) BER Performance: Figure 4 plots the simulation results, the greedy lower bound (LB), and the stepwise upper bound
(UB) on the BER for the 2 × 2 Alamouti scheme with 8PSK constellation that employs Gray and natural mappings and with Lt = Nt = 2, Lr = 6. Two cases of Nr = 1 and Nr = 2 are examined. The bounds for the case of Lt = Nt = Lr = Nr = 2 antennas (i.e., without antenna selection) are also provided for comparison. To the authors’ best knowledge, general analytical evaluation of the BER of OSTBCs with antenna selection under keyhole fading is not available in the literature. Similar to the results obtained for the SER performance, it can be observed that the bounds practically yield the exact BER over a wide range of the SNR, for both Gray and natural mappings. Again, it should be noted that the bounds are applicable for arbitrary constellations and mappings. B. Joint receive and transmit antenna selection 1) SER Performance: To further confirm the accuracy of our analysis in the case of joint receive and transmit antenna selection, Fig. 5 shows the simulation results and the exact analysis of the SER presented in (8) for the Alamouti scheme that employs 8-PSK constellation. In particular, shown in Fig. 5 are the performances for the cases of Lt = 4, Nt = 2, Lr = 6, and various values of Nr . The two derived lower and upper bounds are also plotted for comparison. Furthermore, the performances of the systems performing receive antenna selection only with Lt = Nt = 2, Lr = 6, and different values of Nr presented earlier are provided as references. As in the case of receive antenna selection, similar observations regarding the tightness of the greedy upper bound and the stepwise lower bound can be made from Fig. 5. Given the same number of selected transmit antennas Nt and selected receive antennas Nr , the performance advantage of joint transmit and receive antenna selection over receive antenna selection only can also be clearly observed.
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the general case of joint receive and transmit antenna selection is given as:
0
10
Simulation, natural mapping Greedy LB, natural mapping Stepwise UB, natural mapping Simulation, Gray mapping Greedy LB, Gray mapping Stepwise UB, Gray mapping
−1
10
+
−2
BER
10
Lt=2,Nt=2, L =6,N =1
−3
10
r
−4
−
r
+
r
0
5
10
1 Γ(Nt )
⎡
Lr −Nr
p=0
Nr + i βi ⎣ y Nr
i=0
15
ρ (dB)
20
25
2) BER Performance: Finally, Fig. 6 provides the simulation results, the greedy lower bound (LB), and the stepwise upper bound (UB) on the BER for the 2 × 2 Alamouti scheme with 8-PSK constellation that employs Gray and natural mappings and for joint receive and transmit antenna selection with Lt = 4, Nt = 2, Lr = 6. As before, two different values of Nr , namely Nr = 1 and Nr = 2 are considered. The bounds for the case of receive antenna selection only with Lt = Nt = 2 are also provided for comparison. The results shown in Fig. 6 again validate our analytical analysis on the bit error probability for the general case of joint antenna selection at both the transmitter and the receiver.
A PPENDIX A T HE PDF AND MGF OF Y FOR JOINT RECEIVE AND
ηi,j y
j+Nt 2
−1
2
√ Kj−Nt (2 y)
! Nr + i Nt + k βi βk K0 2 y Nt Nr k=0 L −N L −N N −1 p−1 r t t t r 2 Nr + i βi βk ηk,p y N r i=0 p=0 k=0 Nr + i ·Kp−1 2 y Nr L −N L −N N −1 j−1 t t r r r 2 Nt + k βk βi ηi,j y Nt i=0 j=0 k=0 Nt + k ·Kj−1 2 y Nt
+
i=0
−
−
+
Lr −Nr N r −1 L t −Nt N t −1 i=0
j=0
βi ηi,j βk ηk,p y
p+j −1 2
p=0
k=0
√ Kp−j (2 y) . (26)
By using [22, 6.643.3] and [16, 13.1.33], one obtains the following integral regarding the modified Bessel function of the second kind: 0
∞
√ y μ−1/2 exp(−sy)K2ν (2β y) dy
1 = Γ(μ + ν + 1/2)Γ(μ − ν + 1/2)β 2ν s−(μ+ν+1/2) 2 β2 · U μ + ν + 1/2, 1 + 2ν, , (27) s From (26) and (27) and after some manipulations, the MGF of Y can be computed as follows: (r,t)
(s) φ LYr Lt = s−Nr U (Nr , Nr − Nt + 1, 1/s) Nr
+
TRANSMIT ANTENNA SELECTION
Using (6) and based on the expression of the modified Bessel function second kind, namely Kν (x · z) = −x of the ν ∞ z 2 exp (v + z /v) v (−ν+1) dv [22], the pdf of Y for 2 0 2
KNt −1
Nr + i y Nr
Lr −Nr Lt −Nt
V. C ONCLUSION This paper investigated the SER and BER of OSTBCs over keyhole fading channels with antenna selection. For standard M -PSK and square M -QAM constellations, closed-form expressions with finite-range single integrals were provided to compute exactly the SER. Furthermore, very tight lower and upper bounds on the SER and the BER were developed. These bounds are applicable for arbitrary constellations and mappings and can provide practically the exact SER and BER over a wide range of the SNR. The studies in this paper therefore provide a very effective tool to predict the SER and BER performances of OSTBCs over keyhole fading channels with antenna selection, without the need of time-consuming simulations.
2
j=0
30
Fig. 6. The simulation results, greedy lower bound (LB), and stepwise upper bound (UB) on the BER for 2 × 2 Alamouti scheme using 8-PSK constellation and Gray and natural mappings, with joint receive and transmit antenna selection for Lt = 4, Nt = 2, and Nr = 1 selected from Lr = 6 receive antennas. The performances for the case of receive antenna selection only with Lt = Nt = 2 are provided for comparison.
Nt −1
N r −1
−
−5
10
Nt +Nr √ 1 1 Lr Lt pY (y) = y 2 −1 KNr −Nt (2 y) Γ(Nr )Γ(Nt ) 2 Nr Nt Nr −1 L t −Nt 2 Nt + k Nt + k 1 βk y KNr −1 2 y Γ(Nr ) Nt Nt k=0 Nt −1 p+Nr √ −1 2 ηk,p y Kp−Nr (2 y)
r
Lt=4,Nt=2, L =6,N =1
10
4823
L t −Nt k=0
−
βk
Nt
Nt + k Nt N t −1
Nr −1 s
−Nr
Nt + k U Nr , Nr , Nt s
ηk,p Γ(p)s−p U (p, 1 + p − Nr , 1/s)
p=0
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+
Lr −Nr
βi
i=0
Nr + i Nr N r −1
−
Nt −1
s−Nt U
Nr + i Nt , Nt , Nr s ⎤
ηi,j Γ(j)s−j U (j, 1 + j − Nt , 1/s)⎦
j=0
+ −
−
+
(Nr + i)(Nt + k) 1, 1, Nr Nt s i=0 k=0 L −N L −N N −1 p−1 r t t t r Nr + i βi βk ηk,p Γ(p) s−p N r i=0 k=0 p=0 (Nr + i) ·U p, p, Nr s ⎛ j−1 L −Nr N t −Nt Lr r −1 Nt + k ⎝ βk βi ηi,j Γ(j) s−j N t i=0 j=0 k=0 (Nt + k) ·U j, j, Nt s ⎛ N Lr −Nr N −1 −N −1 L r t t t ⎝ βi ηi,j βk ηk,p Γ(j)Γ(p)s−p Lr −Nr L t −Nt
i=0
j=0
βi βk s−1 U
k=0
p=0
1 ·.U p, 1 + p − j, . s (28)
R EFERENCES [1] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [2] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, pp. 1456–1467, July 1999. [3] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, pp. 1451– 1458, Oct. 1998. [4] S. Sanayei and A. Nosratinia, “Antenna selection in MIMO systems,” IEEE Trans. Commun., vol. 42, pp. 68–73, Mar. 2004. [5] D. J. Love, “On the prebability of error of antenna-subset selection with space-time block code,” IEEE Trans. Commun., vol. 53, pp. 1799–1803, Nov. 2005.
[6] S. Kaviani and C. Tellambura, “Closed-form BER analysis for antenna selection using orthogonal space-time block codes,” IEEE Commun. Lett., vol. 10, pp. 704–706, 2006. [7] D. Gesbert, H. Bolcskei, D. A. Gore, and A. J. Paulraj, “Outdoor MIMO wireless channels: models and performance prediction,” IEEE Trans. Commun., vol. 50, pp. 1926–1934, Dec. 1997. [8] D. Chizhik, G. J. Foschini, M. J. Gans, and R. A. Valenzuela, “Keyholes, correlations, and capacities of multielement transmit and receive antennas,” IEEE Trans. Wireless Commun., vol. 1, pp. 361–368, Apr. 2002. [9] S. Sanayei and A. Nosratinia, “Antenna selection in keyhole channels,” IEEE Commun. Mag., vol. 55, pp. 404–408, Mar. 2007. [10] L. Yang and J. Qin, “Performance of STBCs with antenna selection: spatial correlation and keyhole effects,” IEE Proc. Commun., vol. 153, pp. 15–20, Feb. 2006. [11] P. Garg, R. K. Mallik, and H. M. Gupta, “Performance analysis of spacetime coding with imperfect channel estimation,” IEEE Trans. Wireless Commun., vol. 4, pp. 257–265, Jan. 2005. [12] P. Garg, R. K. Mallik, and H. M. Gupta, “Exact error performance of square orthogonal space- time block coding with channel estimation,” IEEE Trans. Commun., vol. 54, pp. 430–437, Mar. 2006. [13] R. K. Mallik and Q. T. Zhang, “A tight upper bound on the PEP of a space-time coded system,” IEEE Trans. Wireless Commun., vol. 6, pp. 3238–3247, 2007. [14] S. Sandhu and A. Paulraj, “Space-time block codes: a capacity perspective,” IEEE Commun. Lett., vol. 4, pp. 384–386, Dec. 2000. [15] A. M¨uller and J. Speidel, “Capacity of multiple-input multiple-output keyhole channels with antenna selection,” in Proc. European Wireless Conference, Paris, France, Apr. 2007. [16] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover Publications, Inc., 1965. [17] M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels: A Unified Approach to Performance Analysis. New York: Wiley, 2000. [18] H. Kuai, F. Alajaji, and G. Takahara, “Tight error bounds for nonuniform signaling over AWGN channels,” IEEE Trans. Inform. Theory, vol. 46, pp. 2712–2718, Nov. 2000. [19] F. Behnamfar, F. Alajaji, and T. Linder, “Tight error bounds for spacetime orthogonal block codes under slow Rayleigh flat fading,” IEEE Trans. Commun., vol. 53, pp. 952–956, 2005. [20] N. H. Tran, H. H. Nguyen, and T. Le-Ngoc, “Performance bounds of orthogonal space-time block codes over keyhole Nakagami-m channels,” IEEE Signal Processing Lett., vol. 14, pp. 605–608, Sept. 2007. [21] N. H. Tran and H. H. Nguyen, “Signal mappings of 8-ary constellations for bit interleaved coded modulation with iterative decoding,” IEEE Trans. Broadcasting, vol. 52, pp. 92–99, Mar. 2006. [22] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. Academic Press, 2000.
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Symbol and Bit Error Probabilities of Orthogonal Space-Time Block Codes with Antenna Selection over Keyhole Fading Channels Nghi H. Tran, Member, IEEE, Ha H. Nguyen, Senior Member, IEEE, and Tho Le-Ngoc, Fellow, IEEE
Abstract—The symbol error rate (SER) and the bit error rate (BER) of orthogonal space-time block codes (OSTBCs) with antenna selection over keyhole fading channels are examined. Considered are receive antenna selection, transmit antenna selection, and joint antenna selection at both the transmitter and the receiver. The exact SERs of OSTBCs for M -PSK and square M QAM constellations are obtained using the technique of moment generating function (MGF). By applying the Bonferroni-type bounds, tight lower and upper bounds for both the SER and BER are provided in closed-form expressions with finite-range single integrals. The bounds can be applied to arbitrary constellations and mappings. Numerical results show that the bounds can be used to provide practically the exact SER and BER over a wide range of the signal-to-noise ratio. Index Terms—Antenna selection, keyhole fading, orthogonal space-time block codes, Bonferroni-type bounds, symbol error rate (SER), bit error rate (BER).
I. I NTRODUCTION
S
PACE-TIME code is a promising modulation technique to maximize the diversity gain over multiple-input and multiple-output (MIMO) channels with multiple antennas at the transmitter and the receiver [1], [2]. Due to its simple implementation while providing high diversity gains, orthogonal space-time block codes (OSTBCs) [2], [3] have received a special attention, both in research community and industry. However, the deployment of OSTBCs in particular and MIMO systems in general can be quite costly since the transmitter and/or the receiver require multiple radio frequency chains and low-noise amplifiers. To overcome this challenge, performing antenna selection is an attractive technique. This is because with antenna selection the system can still retain most of the advantages offered by multiple antennas while the complexity of hardware implementation is reduced [4]. Although transmit antenna selection Manuscript received September 19, 2007; revised January 6, 2008 and April 13, 2008; accepted April 21, 2008. The associate editor coordinating the review of this paper and approving it for publication was B. S. Rajan. This work was supported in part by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada (NSERC). The first author would also like to acknowledge the University of Saskatchewan’s Graduate Scholarship and the Fellowship received from TRlabs-Saskatoon. A part of this work was presented at the IEEE International Conference on Communications (ICC), Beijing, China, May 19-23 2008. N. H. Tran was with the Department of Electrical & Computer Engineering, University of Saskatchewan, Saskatoon, SK, Canada S7N 5A9. He is now with the Department of Electrical & Computer Engineering, McGill University, Montreal, Quebec, Canada H3A 2A7. (e-mail: [email protected]). H. H. Nguyen is with the Department of Electrical & Computer Engineering, University of Saskatchewan, Saskatoon, SK, Canada S7N 5A9. (e-mail: [email protected].). T. Le-Ngoc is with the Department of Electrical & Computer Engineering, McGill University, Montreal, Quebec, Canada H3A 2A7. (e-mail: [email protected].). Digital Object Identifier 10.1109/T-WC.2008.071041
and receive antenna selection has been well-investigated over the conventional independent and identically distributed (i.i.d.) fading channel (see, for example, [5], [6] and many references therein), only a few efforts have been devoted to the case of MIMO wireless channels with the existence of keyhole fading, where scattering processes are present at both the transmitter and the receiver [7], [8]. In particular, capacity and diversity analysis for antenna selection over keyhole fading was investigated in [9], which shows that the performance of systems using antenna selection can match to that of a baseline full-antenna system. Regarding OSTBCs, to the best of our knowledge, only the approximation of the BER of a system employing Alamouti code and BPSK modulation at the transmitter and the selection of one best receive antenna was provided in [10]. This paper examines the error performance, both the SER and the BER, of OSTBCs over keyhole fading channels with joint antenna selection at both the transmitter and the receiver, which includes receive antenna selection and transmit antenna selection as special cases. At first, by deriving the moment generating function (MGF) of an instantaneous signal-to-noise ratio (SNR) at the receiver, the exact SERs of OSTBCs for M -PSK and square M -QAM constellations are computed. Then the Bonferroni-type bounds are applied to obtain the tight lower and upper bounds on both the SER and BER. These bounds, which can be effectively computed with finiterange single integrals, are applicable to any constellations and mappings. It is shown that the bounds can be used to provide practically the exact SER and BER over a wide range of SNR. It should be noted that this paper only considers the case that the channel state information is perfectly estimated at the receiver. The analysis with antenna selection in the presence of channel estimation error, as similar to [11]–[13], would be an interesting topic for further studies. II. S YSTEM M ODEL AND T HE T ECHNIQUE OF M OMENT G ENERATING F UNCTION (MGF) A. System Model Consider a MIMO system with Lt transmit and Lr receive antennas. The channel model for keyhole fading is [9] H = hLt h Lr where hLt and hLr are column vectors of size Lt and Lr , respectively, whose entries are circularly symmetric complex Gaussian random variables with variance 1/2 per dimension CN (0, 1). For receive antenna selection, the receiver will select Nr out of Lr antennas. If antenna selection is performed at the transmitter, it is assumed that the transmitter knows only the indices of Nt out of Lt transmit antennas sent back from the receiver via a rate-limited feedback with error-free channel. Moreover, joint antenna
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selection with Nt transmit antennas and Nr receive antenna can also be implemented. In a space-time block coded system, with Nt and Nr selected antennas, each block of K · b information bits is first mapped to a sequence of K symbols {v1 , v2 , . . . , vK }, where each vk can be one of M = 2b signal points {su }M u=1 in a M -ary constellation Φ. The mapping rule ξ defines the correspondence between b bits and a signal point in Φ. The sequence {v1 , v2 , . . . , vK } is then encoded into an N × Nt space-time block code matrix G. Here, N is the number of rows of the space-time code matrix and the code rate is R = K/N symbol per channel use. In the case of OSTBCs, the columns of G are orthogonal. The receive signals at the Nr selected receive antennas over N time periods are collectively given as: ρ ˜ R= GH + W . (1) Nt ˜ = hNt h with hNt and hNr correspond to Nt Here, H Nr and Nr selected components of hLt and hLr , respectively. The matrix W is a N × Nr matrix representing additive white Gaussian noisewhose entries are also CN (0, 1). The normalization factor Nρt in (1) ensures that the average SNR at each receive antenna is ρ, independent of Nt . Due to the orthogonal structure of OSTBCs, it can be shown that a MIMO fading channel can be equivalently represented by a single-input single-output (SISO) channel [14]. Let Y be ˜ The achievable SNR per the squared Frobenius norm of H. symbol is given as: ρs =
ρR ρR ·Y = · U · V, Nt Nt
(2)
where U = ||hNt ||2 and V = ||hNr ||2 . From (2), it is obvious that the problem of joint transmit and receive antenna selection to maximize ρs can be decomposed into the problems of transmit antenna selection and receive antenna selection independently. In particular, Nt and Nr transmit and receive antennas should be selected to maximize U and V , respectively.
4819
The pdf of U is also expressed in a similar form. Since Y = U · V and U and V are two independent random variables, one has: ∞ y 1 dv. (6) pV (v)pU pY (y) = v v 0 The moment generating function of Y with joint receive and (r,t) transmit antenna selection can then be computed as φY (s) = ∞ E[exp(−sY )] = 0 exp(−sy)pY (y)dy. As shown in Appendix A, the pdf of Y , pY (y), and (r,t) the MGF of Y , φY (s), can be written in closed-form expressions. In particular, from (28), it can be seen that the MGF of Y is expressed in a closed form as the sum of various functions, each having the same form of f (s; A, B, C, D) = As−B U(B, C, D/s), where U(a, b, x) is the confluent hypergeometric function of the second kind [16] and A, B, C, D are some constants. For example, in the case of receive antenna selection only, since Lt − Nt = 0 and the fact that β0 = 0 and ηi,0 = 0, the MGF of Y in (28) can be written in a more compact form as: 2 Γ(Lt ) −Nr Lr s U(Nr , Nr − Lt + 1, 1/s) Γ(Lt ) Nr 2 L −1 Lr −Nr Nr + i Γ(Lt ) Nr + i t + βi s−Lt U Lt , Lt 2 Nr Nr s i=1 N r −1 Γ(j)Γ(Lt ) −j ηi,j − s U (j, 1 + j − Lt , 1/s) . 2 j=1 (7)
(r)
φY (s) =
With the closed-form expressions of the MGF of Y presented in (28), the exact SER of OSTBCs for standard M PSK and square M -QAM constellations can be computed straightforwardly. More specifically, by averaging the conditional SERs for M -PSK and square M -QAM constellations given in [17] over the pdf of the instantaneous SNR Y , one obtains the following expressions for the general case of joint receive and transmit antenna selection: ρR 1 π−π/M (r,t) Nt gMPSK MPSK φY Psymbol (error) = dθ (8) π 0 sin2 θ
B. MGF of Y and the Exact SER In order to derive the SER and BER, this subsection first evaluates the MGF of the random variable Y . Following the analysis in [15], the probability density function (pdf) of the random variable V is given as: Lr −Nr Lr exp(−v) + βi exp(−v)· v Nr −1 pV (v) = Nr Γ(Nr ) i=0 ⎛ ⎞⎤ N r −1 iv ⎝exp − ηi,j v j−1 ⎠⎦ , − (3) Nr j=0 where βi =
−Nr Nr −1 r (−1)i Lr −N i>0 i i 0 Otherwise j−1 1 −i j>0 Γ(j) Nr ηi,j = 0 Otherwise
(4)
(5)
MQAM (error) Psymbol
=
4q π
−
4q 2 π
π/2
0
(r,t) φY
π/4
0
ρR
(r,t) φY
Nt gMQAM 2
sin θ
ρR
Nt gMQAM 2
sin θ
dθ dθ (9)
(r,t)
where φY (·) is the MGF of Y , given in (28). It can be seen that the single integrals in (8) and (9) are over finite ranges. Therefore, the exact SERs for M -PSK and square M QAM can be computed easily. This fact holds for transmit antenna selection and receive antenna selection as well, since the MGFs of Y in all three cases have similar expressions. III. T IGHT U PPER AND L OWER B OUNDS ON THE SER AND BER BASED ON B ONFERRONI -T YPE B OUNDS Bonferroni-type bounds are the bounds on the probability of a union. Let A1 , A2 , . . . , AX be X events defined on a given
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probability space. Then the probability of the union event is X X PU = P Ax = (−1)x+1 Sx , (10) x=1
x=1
where Sx = P (Ax1 ∩ · · · ∩ Axk ) and the sum is taken over all 1 ≤ x1 < · · · < xk ≤ X. The interesting result is that by truncating the sum in (10) at any Sx , a lower or upper bound is obtained, depending on the sign of the last term. Specifically, PU ≤ S1 , PU ≥ S1 −S2 , PU ≤ S1 −S2 +S3 and so on. Two specific versions of the Bonferroni-type bounds are briefly described next. The Kounias lower bound is given by, ⎤ ⎡ P (Ax ) − P (Ay ∩ Ax )⎦ , (11) PU ≥ maxΓ ⎣ x∈Γ
y,x∈Γ,y<x
where Γ is any arbitrary subset of IX = {1, 2, . . . , X}. As pointed out in [18], one can apply an efficient stepwise algorithm, whose complexity is linear in the number of events, to find the best subset that gives the tightest bound. For X ≥ 2, the tightest Hunter upper bound is: PU ≤
N
P (Ax ) − maxT0 ∈T
P (Ax ∩ Ay ),
(12)
(x,y)∈T0
x=1
where T0 is any tree spanning the indices of the set A1 , . . . , AX , (x, y) is an edge in T0 . The greedy algorithm [18] can then be applied to construct the optimal spanning tree T0 for (12). The Bonferroni-type bounds have been applied to analyze the performance of OSTBCs in various scenarios [19], [20], but without antenna selection. In the following, the Bonferroni-type bounds are applied to derive the tight upper and lower bounds on the SER and BER of OSTBCs with antenna selection.
(r)
where φY (·) is either φY (·) given in (7) for receive antenna (r,t) selection, or φY (·) given in (28) for joint receive and transmit antenna selection. Clearly, the unconditional PEP Pu ( p,u ) can be easily calculated using finite-range single integrals. Following the analysis in [18], [19], one has the conditional two-dimensional PEP expressed in terms of two-dimensional joint Gaussian function as follows: √ √ (16) Pu ( p,u ∩ q,u |Y ) = Ψ ρp,q,u , δpu Y , δqu Y , where ρp,q,u =< sp − su , sq − su > /dp,u dq,u and Ψ(r, a, b) is the two-dimensional joint Gaussian function, defined as: Ψ(r, a, b) =
1 2π 1 − r 2 √
∞
a
∞ b
exp
x2 − 2rxy + y 2 dxdy. 2(1 − r 2 ) (17)
√ √ Furthermore, since δpu Y and δqu Y are non-negative, one can use the single integral representation of (17) to obtain the conditional PEP: √ √ Ψ ρp,q,u , δpu Y , δqu Y ϕ(dp,u /dq,u ,ρp,q,u ) 2 δp,u Y 1 dθ exp − = 2π 0 2 sin2 θ ϕ(dq,u /dp,u ,ρp,q,u ) 2 δq,u Y 1 + dθ (18) exp − 2π 0 2 sin2 θ √ where ϕ(x, r) = tan−1 x 1 − r2 /(1 − rx) . Taking the average of (18) over Y using the MGF of Y , one arrives at: ϕ(dp,u /dq,u ,ρp,q,u ) 2 δp,u 1 dθ Pu ( p,u ∩ q,u ) = φY 2π 0 2 sin2 θ ϕ(dq,u /dp,u ,ρp,q,u ) 2 δq,u 1 + dθ φY 2π 0 2 sin2 θ (19)
A. Bonferroni-Type Bounds on the SER ⎛ In general, the SER can be expressed as [19]:
⎞ 1 Psymbol (error) = P (|su )P (su ) = Pu ⎝ p,u ⎠ , M u=1 u=1 p=u M
(13)
where P ( |su ) is the conditional error probability given that su was sent and p,u is the event that sp has a larger metric than su [19]. In order to obtain the two Bonferronitype bounds for the SER, one needs to compute the pairwise error probability (PEP), Pu ( p,u ), and the two dimensional PEP, Pu ( p,u ∩ q,u ), for given q, p, and u. Given Y , the conditional PEP can be computed as follows [19]: 2 δp,u Y 1 π/2 exp − dθ (14) Pu ( p,u |Y ) = π 0 2 sin2 θ where δp,u = Rρ/2Nt dp,u and dp,u = su −sp . Averaging (14) over Y , one has: 2 δp,u 1 π/2 Pu ( p,u ) = φY dθ (15) π 0 2 sin2 θ
Clearly, the two-dimensional PEP Pu ( p,u ∩ q,u ) can also be effectively computed by single finite-range integrals. Using (15) and (19), one obtains the lower and upper bounds, based on the Kounias and Hunter bounds, respectively, for the SER, which shall be referred to simply as the stepwise lower bound and the greedy upper bound. It should be noted that these bounds are applicable for any arbitrary constellations. B. Bonferroni-Type Bounds on the BER The upper and lower bounds on the BER can also be obtained using the MGF of Y . In particular, the BER can be computed as follows [18], [19]: Pbit (error) =
M 1 Pb (su ), M u=1
(20)
where Pb (su ), the bit error probability when su is sent, is ⎛ ⎞⎤ ⎡ M 1 Pb (su ) = (21) H(q, u) ⎣1 − Pu ⎝ p,q ⎠⎦ . b q=1
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It should be mentioned that this computation is only valid when γp,q,q ≥ 0. When γp,q,u < 0, one can use the relationship Q(x) = 1−Q(−x) to obtain a similar expression. On the other hand, given Y , the two-dimensional PEP, Pu ( p,q ∩ l,q ) is given as [18]: √ √ Pu ( p,q ∩ l,q |Y ) = Ψ ρp,l,q , γp,q,u Y , γl,q,u Y . (24) Therefore, the unconditional two-dimensional PEP can be expressed in the following closed form:
0
10
−1
10
r
IV. I LLUSTRATIVE R ESULTS This section provides the numerical results to confirm our analysis. In particular, the cases of receive antenna selection and joint antenna selection at the transmitter and the receiver are considered. Furthermore, we concentrate on Alamouti v1 v2 scheme [3] with Nt = N = 2 and G = . −v2 v1 Unless stated otherwise, the number of receive antennas is always Lr = 6. In the case of receive antenna selection only, one has Lt = Nt = 2. For joint receive and transmit antenna selection, the number of transmit antenna is assumed to be Lt = 4. The SNR is defined as ρ (dB). A. Receive antenna selection 1) SER Performance: Figures 1 and 2 show the simulation results and the exact analysis of the SER presented in (8) and (9) for the Alamouti scheme employing 8-PSK and 16QAM constellations, respectively. Three values of Nr , namely Nr = 1, Nr = 2, and Nr = 6, are investigated. For
r
−2
10
Lr=6,Nr=1 L =6,N =2
−3
10
−4
10
−5
10
0
r
r
Simulation, 8−PSK Exact analysis, 8−PSK Greedy UB, 8−PSK Stepwise LB, 8−PSK 5
10
15 ρ (dB)
Lr=6,Nr=6 20
25
30
Fig. 1. The simulation results, exact analysis, greedy upper bound (UB), and stepwise lower bound (LB) for 2 × 2 Alamouti scheme using 8-PSK constellation with receive antenna selection for Lt = Nt = 2 and various values of Nr . The performances with Lr = Nr = 2 are provided as references.
Pu ( p,q ∩ l,q ) ϕ(γp,q,u /γl,q,u ,ρp,l,q ) 2 γp,q,u 1 = φY dθ 2π 0 2 sin2 θ ϕ(γl,q,u /γp,q,u ,ρp,l,q ) 2 γl,q,u 1 + φY dθ (25) 2π 0 2 sin2 θ
0
10
−1
10
Lr=2,Nr=2 −2
10
Lr=6,Nr=2
Lr=6,Nr=1
SER
The expression of Ψ (r, a, b) in (18) is only valid for nonnegative a and b. When at least one of a and b is negative, Ψ (r, a, b) can be expressed in the form of other Ψ(·) function with non-negative arguments [19]. Hence, a similar expression of Pu ( p,q ∩ l,q ) can still be obtained Using (23) and (25), the Kounias and Hunter bounds can be calculated, which lead to the upper and lower bounds, respectively, for the BER in (20). They are simply referred to as the stepwise upper bound and the greedy lower bound.
L =2,N =2
SER
In (21), H(q, u) is the Hamming distance between the label of sq and su (which depends on the mapping rule ξ). Given Y , the one-dimensional PEP is computed as [18], [19]: 2 √ Y γp,q,u 1 π/2 Pu ( p,q |Y ) = Q γp,q,u Y = exp − dθ π 0 2 sin2 θ (22) δ 2 −δ 2 where γp,q,u = p,uδp,q q,u . Similar to the previous sections, averaging (22) over Y and using the MGF of Y , one obtains: 2 γp,q,u 1 π/2 Pu ( p,q ) = dθ. (23) φY π 0 2 sin2 θ
4821
−3
10
L =6,N =6 r
−4
10
−5
10
5
r
Simulation, 16−QAM Exact analysis, 16−QAM Greedy UB, 16−QAM Stepwise LB, 16−QAM 10
15
20 ρ (dB)
25
30
35
Fig. 2. The simulation results, exact analysis, greedy upper bound (UB), and stepwise lower bound (LB) for 2 × 2 Alamouti scheme using 16-QAM constellation with receive antenna selection for Lt = Nt = 2 and various values of Nr . The performances with Lr = Nr = 2 are provided as references.
comparison, the derived greedy upper bound and the stepwise lower bound are provided to show the tightness of the bounds. The performances of the system with Lt = Nt = Lr = Nr = 2 antennas (without antenna selection) are also plotted as references. It can be observed from Figs. 1 and 2 that the simulation results, the exact analysis and the bounds cannot be distinguished over a wide range of SNR for both constellations. This means that the bounds can be used as an effective tool to predict the actual SER of OSTBCs with antenna selection over keyhole fading channels. The results agree with [18], [19] that the two Bonferroni-type lower and upper bounds are very accurate in estimating the probability of a union.
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0
0
10
10
−1
Simulation, 8−PSK Exact analysis, 8−PSK Greedy UB, 8−PSK Stepwise LB, 8−PSK
−1
10
10 Lr=2,Nr=2
−2
SER
SER
10 L =6,N =1
−3
−4
−5
10
0
r
Lr=6,Nr=2
10
10
L =2,N =2, t t L =6,N =1
−2
10
10
−3
10
r
15 ρ (dB)
r
L =2,N =2, t t L =6,N =2
r
−4
Lr=6,Nr=6
r
Lt=4,Nt=2, L =6,N =1
r
r
L =4,N =2, t t L =6,N =2
10
Simulation, (1,7) Greedy UB, (1,7) Stepwise LB, (1,7) 5
r
r
r
−5
20
25
30
Fig. 3. The simulation results, greedy upper bound (UB), and stepwise lower bound (LB) for 2 × 2 Alamouti scheme using (1,7) constellation with receive antenna selection for Lt = Nt = 2 and various values of Nr . The performances for the case of Lr = Nr = 2 are provided as references.
10
0
5
10
15 ρ (dB)
20
25
30
Fig. 5. The simulation results, exact analysis, greedy upper bound (UB), and stepwise lower bound (LB) for 2 × 2 Alamouti scheme using 8-PSK constellation with joint receive and transmit antenna selection for Lt = 4, Nt = 2, and various values of Nr selected from Lr = 6 receive antennas. The performances with receive antenna selection only for Lt = Nt = 2 are provided as references.
0
10
−1
10
Lr=2,Nr=2 −2
BER
10
Lr=6,Nr=2
−3
Lr=6,Nr=1
10
−4
10
−5
10
0
Simulation, natural mapping Greedy LB, natural mapping Stepwise UB, natural mapping Simulation, Gray mapping Greedy LB, Gray mapping Stepwise UB, Gray mapping 5
10
15
ρ (dB)
20
25
30
Fig. 4. The simulation results, greedy lower bound (LB), and stepwise upper bound (UB) on the BER for 2×2 Alamouti scheme using 8-PSK constellation with receive antenna selection, employing Gray and natural mappings for Lt = Nt = 2 and various values of Nr . The performances for the case of Lr = Nr = 2 are provided for comparison.
As mentioned before, the exact computations of the SER in (8) and (9) are only applicable for standard M -PSK and square M -QAM. The bounds, on the other hand, do not have this limitation, since they can be applied for arbitrary constellations. As an example, Fig. 3 plots the simulation results, the greedy UB, and the stepwise LB for 2 × 2 Alamouti scheme using (1,7) constellation [21] and with Lt = Nt = 2, Lr = 6, and various values of Nr . It can be seen that the bounds and the simulation results almost coincide, which means that they can practically provide the exact SER. Similar results were also observed with different non-regular constellations. 2) BER Performance: Figure 4 plots the simulation results, the greedy lower bound (LB), and the stepwise upper bound
(UB) on the BER for the 2 × 2 Alamouti scheme with 8PSK constellation that employs Gray and natural mappings and with Lt = Nt = 2, Lr = 6. Two cases of Nr = 1 and Nr = 2 are examined. The bounds for the case of Lt = Nt = Lr = Nr = 2 antennas (i.e., without antenna selection) are also provided for comparison. To the authors’ best knowledge, general analytical evaluation of the BER of OSTBCs with antenna selection under keyhole fading is not available in the literature. Similar to the results obtained for the SER performance, it can be observed that the bounds practically yield the exact BER over a wide range of the SNR, for both Gray and natural mappings. Again, it should be noted that the bounds are applicable for arbitrary constellations and mappings. B. Joint receive and transmit antenna selection 1) SER Performance: To further confirm the accuracy of our analysis in the case of joint receive and transmit antenna selection, Fig. 5 shows the simulation results and the exact analysis of the SER presented in (8) for the Alamouti scheme that employs 8-PSK constellation. In particular, shown in Fig. 5 are the performances for the cases of Lt = 4, Nt = 2, Lr = 6, and various values of Nr . The two derived lower and upper bounds are also plotted for comparison. Furthermore, the performances of the systems performing receive antenna selection only with Lt = Nt = 2, Lr = 6, and different values of Nr presented earlier are provided as references. As in the case of receive antenna selection, similar observations regarding the tightness of the greedy upper bound and the stepwise lower bound can be made from Fig. 5. Given the same number of selected transmit antennas Nt and selected receive antennas Nr , the performance advantage of joint transmit and receive antenna selection over receive antenna selection only can also be clearly observed.
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 12, DECEMBER 2008
the general case of joint receive and transmit antenna selection is given as:
0
10
Simulation, natural mapping Greedy LB, natural mapping Stepwise UB, natural mapping Simulation, Gray mapping Greedy LB, Gray mapping Stepwise UB, Gray mapping
−1
10
+
−2
BER
10
Lt=2,Nt=2, L =6,N =1
−3
10
r
−4
−
r
+
r
0
5
10
1 Γ(Nt )
⎡
Lr −Nr
p=0
Nr + i βi ⎣ y Nr
i=0
15
ρ (dB)
20
25
2) BER Performance: Finally, Fig. 6 provides the simulation results, the greedy lower bound (LB), and the stepwise upper bound (UB) on the BER for the 2 × 2 Alamouti scheme with 8-PSK constellation that employs Gray and natural mappings and for joint receive and transmit antenna selection with Lt = 4, Nt = 2, Lr = 6. As before, two different values of Nr , namely Nr = 1 and Nr = 2 are considered. The bounds for the case of receive antenna selection only with Lt = Nt = 2 are also provided for comparison. The results shown in Fig. 6 again validate our analytical analysis on the bit error probability for the general case of joint antenna selection at both the transmitter and the receiver.
A PPENDIX A T HE PDF AND MGF OF Y FOR JOINT RECEIVE AND
ηi,j y
j+Nt 2
−1
2
√ Kj−Nt (2 y)
! Nr + i Nt + k βi βk K0 2 y Nt Nr k=0 L −N L −N N −1 p−1 r t t t r 2 Nr + i βi βk ηk,p y N r i=0 p=0 k=0 Nr + i ·Kp−1 2 y Nr L −N L −N N −1 j−1 t t r r r 2 Nt + k βk βi ηi,j y Nt i=0 j=0 k=0 Nt + k ·Kj−1 2 y Nt
+
i=0
−
−
+
Lr −Nr N r −1 L t −Nt N t −1 i=0
j=0
βi ηi,j βk ηk,p y
p+j −1 2
p=0
k=0
√ Kp−j (2 y) . (26)
By using [22, 6.643.3] and [16, 13.1.33], one obtains the following integral regarding the modified Bessel function of the second kind: 0
∞
√ y μ−1/2 exp(−sy)K2ν (2β y) dy
1 = Γ(μ + ν + 1/2)Γ(μ − ν + 1/2)β 2ν s−(μ+ν+1/2) 2 β2 · U μ + ν + 1/2, 1 + 2ν, , (27) s From (26) and (27) and after some manipulations, the MGF of Y can be computed as follows: (r,t)
(s) φ LYr Lt = s−Nr U (Nr , Nr − Nt + 1, 1/s) Nr
+
TRANSMIT ANTENNA SELECTION
Using (6) and based on the expression of the modified Bessel function second kind, namely Kν (x · z) = −x of the ν ∞ z 2 exp (v + z /v) v (−ν+1) dv [22], the pdf of Y for 2 0 2
KNt −1
Nr + i y Nr
Lr −Nr Lt −Nt
V. C ONCLUSION This paper investigated the SER and BER of OSTBCs over keyhole fading channels with antenna selection. For standard M -PSK and square M -QAM constellations, closed-form expressions with finite-range single integrals were provided to compute exactly the SER. Furthermore, very tight lower and upper bounds on the SER and the BER were developed. These bounds are applicable for arbitrary constellations and mappings and can provide practically the exact SER and BER over a wide range of the SNR. The studies in this paper therefore provide a very effective tool to predict the SER and BER performances of OSTBCs over keyhole fading channels with antenna selection, without the need of time-consuming simulations.
2
j=0
30
Fig. 6. The simulation results, greedy lower bound (LB), and stepwise upper bound (UB) on the BER for 2 × 2 Alamouti scheme using 8-PSK constellation and Gray and natural mappings, with joint receive and transmit antenna selection for Lt = 4, Nt = 2, and Nr = 1 selected from Lr = 6 receive antennas. The performances for the case of receive antenna selection only with Lt = Nt = 2 are provided for comparison.
Nt −1
N r −1
−
−5
10
Nt +Nr √ 1 1 Lr Lt pY (y) = y 2 −1 KNr −Nt (2 y) Γ(Nr )Γ(Nt ) 2 Nr Nt Nr −1 L t −Nt 2 Nt + k Nt + k 1 βk y KNr −1 2 y Γ(Nr ) Nt Nt k=0 Nt −1 p+Nr √ −1 2 ηk,p y Kp−Nr (2 y)
r
Lt=4,Nt=2, L =6,N =1
10
4823
L t −Nt k=0
−
βk
Nt
Nt + k Nt N t −1
Nr −1 s
−Nr
Nt + k U Nr , Nr , Nt s
ηk,p Γ(p)s−p U (p, 1 + p − Nr , 1/s)
p=0
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 12, DECEMBER 2008
+
Lr −Nr
βi
i=0
Nr + i Nr N r −1
−
Nt −1
s−Nt U
Nr + i Nt , Nt , Nr s ⎤
ηi,j Γ(j)s−j U (j, 1 + j − Nt , 1/s)⎦
j=0
+ −
−
+
(Nr + i)(Nt + k) 1, 1, Nr Nt s i=0 k=0 L −N L −N N −1 p−1 r t t t r Nr + i βi βk ηk,p Γ(p) s−p N r i=0 k=0 p=0 (Nr + i) ·U p, p, Nr s ⎛ j−1 L −Nr N t −Nt Lr r −1 Nt + k ⎝ βk βi ηi,j Γ(j) s−j N t i=0 j=0 k=0 (Nt + k) ·U j, j, Nt s ⎛ N Lr −Nr N −1 −N −1 L r t t t ⎝ βi ηi,j βk ηk,p Γ(j)Γ(p)s−p Lr −Nr L t −Nt
i=0
j=0
βi βk s−1 U
k=0
p=0
1 ·.U p, 1 + p − j, . s (28)
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[6] S. Kaviani and C. Tellambura, “Closed-form BER analysis for antenna selection using orthogonal space-time block codes,” IEEE Commun. Lett., vol. 10, pp. 704–706, 2006. [7] D. Gesbert, H. Bolcskei, D. A. Gore, and A. J. Paulraj, “Outdoor MIMO wireless channels: models and performance prediction,” IEEE Trans. Commun., vol. 50, pp. 1926–1934, Dec. 1997. [8] D. Chizhik, G. J. Foschini, M. J. Gans, and R. A. Valenzuela, “Keyholes, correlations, and capacities of multielement transmit and receive antennas,” IEEE Trans. Wireless Commun., vol. 1, pp. 361–368, Apr. 2002. [9] S. Sanayei and A. Nosratinia, “Antenna selection in keyhole channels,” IEEE Commun. Mag., vol. 55, pp. 404–408, Mar. 2007. [10] L. Yang and J. Qin, “Performance of STBCs with antenna selection: spatial correlation and keyhole effects,” IEE Proc. Commun., vol. 153, pp. 15–20, Feb. 2006. [11] P. Garg, R. K. Mallik, and H. M. Gupta, “Performance analysis of spacetime coding with imperfect channel estimation,” IEEE Trans. Wireless Commun., vol. 4, pp. 257–265, Jan. 2005. [12] P. Garg, R. K. Mallik, and H. M. Gupta, “Exact error performance of square orthogonal space- time block coding with channel estimation,” IEEE Trans. Commun., vol. 54, pp. 430–437, Mar. 2006. [13] R. K. Mallik and Q. T. Zhang, “A tight upper bound on the PEP of a space-time coded system,” IEEE Trans. Wireless Commun., vol. 6, pp. 3238–3247, 2007. [14] S. Sandhu and A. Paulraj, “Space-time block codes: a capacity perspective,” IEEE Commun. Lett., vol. 4, pp. 384–386, Dec. 2000. [15] A. M¨uller and J. Speidel, “Capacity of multiple-input multiple-output keyhole channels with antenna selection,” in Proc. European Wireless Conference, Paris, France, Apr. 2007. [16] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover Publications, Inc., 1965. [17] M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels: A Unified Approach to Performance Analysis. New York: Wiley, 2000. [18] H. Kuai, F. Alajaji, and G. Takahara, “Tight error bounds for nonuniform signaling over AWGN channels,” IEEE Trans. Inform. Theory, vol. 46, pp. 2712–2718, Nov. 2000. [19] F. Behnamfar, F. Alajaji, and T. Linder, “Tight error bounds for spacetime orthogonal block codes under slow Rayleigh flat fading,” IEEE Trans. Commun., vol. 53, pp. 952–956, 2005. [20] N. H. Tran, H. H. Nguyen, and T. Le-Ngoc, “Performance bounds of orthogonal space-time block codes over keyhole Nakagami-m channels,” IEEE Signal Processing Lett., vol. 14, pp. 605–608, Sept. 2007. [21] N. H. Tran and H. H. Nguyen, “Signal mappings of 8-ary constellations for bit interleaved coded modulation with iterative decoding,” IEEE Trans. Broadcasting, vol. 52, pp. 92–99, Mar. 2006. [22] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. Academic Press, 2000.
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