Multi-Keyhole Effect in MIMO AF Relay Downlink ...

IEEE GLOBECOM 2011 - Wireless Communications Symposium

Multi-Keyhole Effect in MIMO AF Relay Downlink Transmission with Space-Time Block Codes Trung Q. Duong∗, Himal A. Suraweera†, Chau Yuen†, and Hans-J¨urgen Zepernick∗ ∗ Blekinge Institute of Technology, Sweden (email: {quang.trung.duong,hans-jurgen.zepernick}@bth.se) † Singapore University of Technology and Design, Singapore (e-mail: {himalsuraweera,yuenchau}@sutd.edu.sg)

Abstract—Multi-keyhole bridges the gap between single keyhole and full-scattering multiple-input multiple-output (MIMO) channels. In this paper, we therefore investigate the multikeyhole effect on the MIMO amplify-and-forward (AF) relay downlink transmission with orthogonal space-time block codes. In particular, we derive the analytical symbol error rate (SER) expression for the considered system with arbitrary number of keyholes. Moreover, SER approximations in the high SNR regime for several important special scenarios of multi-keyhole channels are further derived. These asymptotic results provide important insights into the impact of system parameters on the SER performance. Our analysis is confirmed by comparing with Monte-Carlo simulations.

I. I NTRODUCTION In recent years, a large body of research has studied the transmission of orthogonal space-time block code (OSTBC) in relay networks to explore the spatial diversity gain in a distributed manner (see, e.g., [1]–[4] and the references therein). In particular, by incorporating the line-of-sight effect into the second hop, the symbol error rate (SER) over dualhop Rayleigh-Rician fading channels has been investigated in [1]. The error probability of multiple-input multiple-output (MIMO) systems using OSTBCs and semi-blind amplify-andforward (AF) relays over dual-hop Rayleigh fading channels has been presented in [2]. For CSI-assisted AF relay, the bit error rate (BER) of MIMO system with OSTBC transmission has been presented in [3]. Although rich-scattering conditions have been widely employed in the existing literature on MIMO, the assumption fails to model some practical scenarios with lack of scattering objects. The later propagation phenomenon can be characterized as a keyhole or a pinhole MIMO channel. The keyhole channel has been validated by conducting many measurement campaigns [5]. The keyhole modeling describes the most extreme case of channel fading, i.e., the channel matrix is unitrank regardless of the number of transmit/receive antennas. As a consequence, keyhole effects substantially decrease the performance of MIMO systems. Despite of this important practical phenomenon, however, to the best of the authors’ knowledge, little is known on the effects of keyhole propagation in dual-hop relay systems. Very recently, the effect of keyhole on the performance of relay systems has been investigated in [6]–[8]. In [6], [7], the authors have shown that using the decode-and-forward relay can mitigate the deleterious effect of keyhole. In [8], we adopted the same practical system model assumed in [6],

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[7] and investigated the effect of antenna correlation on the performance of single keyhole MIMO AF relay systems. It has been shown that the antenna correlation has no impact on the diversity order. In this paper, we take a step further and investigate the effects of multi-keyhole propagation for OSTBC transmission in AF relay systems. In many practical scenarios, the degeneration effect of poor-scattering environment can be more generally modeled as multi-keyhole channels. This generalization embraces both rich-scattering and keyhole conditions in which the rank of channel matrix now can vary from unit to full-rank [9], [10]. Our new contributions are as follows. We first characterize the end-to-end instantaneous received signal-to-noise ratio (SNR), which allows us to obtain the exact moment generating function (MGF) of end-to-end SNR. By utilizing this result, we derive an analytical expression for the exact SER and an approximate expression for a downlink system where an nS -antenna base station (source) communicates with an nD antenna mobile station (destination) through the assistance of an nR -antenna relay, i.e., nS > min(nR , nD ). The derived MGF expression is also useful to study additional important performance criteria such as the outage probability and the ergodic capacity. We have also presented a high SNR analysis where both diversity and array gain are quantified in several special cases of interest. It is demonstrated that under the considered downlink scenario, the diversity order is min(nK , nR ) for a multi-keyhole MIMO/multiple-input singleoutput (MISO) channel, i.e., nD = 1, and nK is the number of keyholes. Notation: Vectors and matrices are denoted in lower case/upper case boldface, respectively. I n represents the n × n AkF defines the Frobenius norm of the identity matrix and kA A ) means the matrix A . E {.} is the expectation operator. det(A determinant of the matrix A . Kν (·) is the modified Bessel function of the second kind [11, Eq. (8.432.3)] and B (·, ·) is the Beta function [11, Eq. (8.380.1)]. II. S YSTEM

AND

C HANNEL M ODEL

A. Protocol Description We consider a dual-hop MIMO relay system with source, relay and destination terminals having nS , nR and nD antennas respectively. Let H 1 ∈ CnR ×nS be the channel matrix between the source and the relay, and H 2 ∈ CnD ×nR be the channel matrix between the relay and the destination. The channel gains are assumed to be fixed during a block of Tc symbols

and slowly changed over independent blocks. The system under consideration is downlink in which the source acts as a base station and the destination is a mobile station, i.e., nS > min(nR , nD ). Moreover, we assume that the direct link does not exist in this system possibly due to high shadowing and large path loss effects. An OSTBC is generated at the source before transmission over the dual-hop AF relay network. A sequence of information symbols x1 , x2 , ..., xN is selected from a constellation  S with average transmit power per symbol Es , i.e., E |xi |2 = Es . These symbols are then encoded into an OSTBC matrix denoted by an Tc × nS transmission matrix G with the property that columns of G are orthogonal. The inputoutput relationship for the communication of the source-relay link is expressed as Y 1 = H 1X + W 1 ,

(1)

where σk is the power gain of the k-th keyhole and A is √ the diagonal matrix whose k-th diagonal element is σk . hr,1 , . . . , h r,nK ] and H t = [h ht,1 , . . . , h t,nK ] Moreover, H r = [h are mutually independent matrices. Note that, depending on the keyhole number nK , the rank of matrix H 2 , can vary from one to full-rank. III. P ERFORMANCE A NALYSIS A. The MGF of the end-to-end instantaneous SNR In this subsection, we will derive the MGF of γD , which will be utilized to obtain the system performance. By the definition, the MGF is the inverse Laplace transform of the probability density function (PDF), i.e., ΦγD (s) = EγD {exp(−sγD )}. As can be observed from (3), the MGF of γ2 can be expressed as

2    −1/2

† 2

H 2H 1 GH ΦγD (s) = E a I nD + G H 2H 2

. (5) F

where the OSTBC transmission matrix at the source is defined as X = G T and W 1 denotes the additive white Gaussian noise (AWGN) matrix at the relay. The relay then multiplies the received signal Y 1 with a fixed amplifying gain G, and retransmits the resulting signal to the destination. At the destination, the input-output relationship is given by Y 1 + W 2, Y 2 = H 2 GY

(2)

where W 2 is the AWGN matrix at the destination. In this paper, without loss of generality, we assume that the relay terminal operates in semi-blind mode and consumes the same amount p of power as the source, leading to the amplifying gain G = γ¯ /[nR (1 + γ¯ )], where γ¯ = Es /N0 is the average SNR. The end-to-end instantaneous SNR is written as [1], [12]

 −1/2

2 † 2

, γD = a¯ γ I + G H H H (3) 2 2

nD

F

Since H 1 follows matrix-variate complex Gaussian distribution, by applying results from random matrix theory [1], [12], after some algebraic manipulations, we obtain   nS  det I nD + G2H 2H †2    . (6) ΦγD (s) = E  det I nD + G2 (1 + sαΩ1 ) H 2H †2 From (6), and using the property that H 2H 2 † is a positivedefinite matrix, (6) can be rewritten as ( q  nS ) Y 1 + G2 λi ΦγD (s) = EΛ , (7) 1 + G2 (1 + a¯ γ s)λi i=1

where λi , i = 1, . . . , q, are the q non-zero ordered eigenvalues of H 2H †2 and Λ = diag (λ1 , . . . , λq ). Averaging in (7) requires the joint PDF of λ1 , . . . , λq for which a combined general expression is presented in the appendix.

where a = Tc /(N nS ).

B. Channel Model We consider a downlink cellular network where the source and the relay terminals are fixed base stations and can be installed at strategic locations by network operators. In contrast, the destination is a mobile station. In the considered system model, it is reasonable to assume that the sourcerelay link enjoys a rich-scattering environment yielding its corresponding channel matrix H 1 to have Rayleigh fading with entries CN (0, 1). On the other hand, we consider a practical scenario in which the mobile station is located in a poor scattering environment, e.g. due to mobility. In order to characterize this scenario, the channel matrix H 2 is assumed to undergo multi-keyhole fading with nK number of keyholes. The multi-keyhole channel model extends the widely used single keyhole channel model (see e.g., [6]–[8]). In fact, the multi-keyhole channel can be viewed as a generalized model which bridges the gap between single-keyhole and rich scattering MIMO channels. The multi-keyhole fading channel can be mathematically represented as nK X √ H2 = σkh r,kh †t,k = H rAH †t , (4) k=1

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B. Exact Symbol Error Rate From (7), (36), and using Lemma 1 in [13], the MGF of γD can be readily derived as ΦγD (s) =

S) det(S K det(σij−1 )

,

(8)

Qq where K = i=1 Γ(nD − i + 1)Γ(nR − i + 1) and S is an nK × nK matrix whose entities are given by  i−1 σj , 1 ≤ i ≤ t,     R ∞ nK − nR +n D −1 nR +nD +i−t−q−1 2 2 S i,j = 0 2σj  q x −nS  2   ×Kn −n 2 x 1 + G a¯γ2xs , t < i ≤ nK .  D

σj

R

1+G x

In this paper, we consider M -PSK modulated source symbols. Therefore, the average SER can be expressed as [14] Pe =

1 π

Z

0

π π− M

Φγ D



g sin2 θ



dθ,

(9)


π 2 where g = sin M . Unfortunately, to the best of our knowledge, (9) can not be evaluated in closed-form. However, note that it is a convenient result and allows us to numerically evaluate the system’s average SER performance.

C. High SNR Symbol Error Rate To provide further insight into the impact of multi-keyhole phenomenon on the performance of the system, we will further scrutinize two important special scenarios: i) Single keyhole MIMO/MIMO relay channels (nK = 1) and ii) Multi-keyhole MIMO/MISO relay channels (nD = 1). An asymptotic expression for ΦγD (s) for arbitrary combinations of nS , nR , nD , and nK appears prohibitively complicated to obtain, if not impossible. 1) Single keyhole MIMO/MIMO Channels (nK = 1): In this special scenario, ΦγD (s) given in (7) becomes nR +nD Z ∞ nR +nD 2σ 2 ΦγD (s) = x 2 −1 Γ(n )Γ(n ) R D 0  r  −nS x G2 a¯ γ xs × KnD −nR 2 1+ . (10) σ 1 + G2 x To approximate ΦγD (s) in the high SNR, we employ [15]  √
(small z) for n = 0, − 12 ln (z) ≈ Kn 2 z (11) Γ(|n|) −|n|/2 z for n 6= 0. 2 •

nR = nD : By substituting (11) into (10) and exchanging the variable γ¯ x = y, we have Z ∞ (large γ ¯) (σ¯ γ )−nR ΦγD (s) ≈ ln(σ¯ γ )y nR −1 Γ(nR )Γ(nD ) 0  Z ∞
−nS y nR −1 ln(y) 1 + G2 asy dy − dy , (12) n (1 + aG2 sy) S 0

where (12) is obtained by neglecting the small term in (10). By applying the results of [11, Eq. (3.194.3)] and [16, Eq. (2.6.4.7)] with the condition that nS > nR , the MGF of γD for the case nR = nD can be expressed as ΦγD (s)

•

Γ(nS − nR ) Γ(nS )Γ(nR ) × [ln (a¯ γ s) − ψ(nR ) + ψ(nS − nR )] , (large γ ¯)

≈

−nR

(a¯ γ s)

(13)

where a1 = σG2 a. nR 6= nD : Similarly, we can find the MGF of γD as Z ∞ (large γ ¯) y m1 −1 (σ¯ γ )−m1 ΦγD (s) ≈ dy, Γ(nR )Γ(nD ) 0 (1 + aG2 sy)nS (14) where m1 = min(nR , nD ). Note that the integral in (14) converges when nS > min(nR , nD ) which then yields ΦγD (s)

(large γ ¯)

≈

Γ(nS − m1 )Γ(m1 )Γ(|nD − nR |) . (15) m Γ(nS )Γ(nR )Γ(nD ) (a1 γ¯ s) 1

By combining the two cases, the asymptotic SER can be expressed as Pe

(large γ ¯)

≈

Ξ1

Γ(nS − m1 )Γ(m1 )Γ(|nD − nR |) , Γ(nS )Γ(nR )Γ(nD ) (a1 γ¯ s)m1

2) Multi-keyhole MIMO/MISO Channels (nD = 1): The case of multi-keyhole MIMO/MISO system setup is reasonable in the context of current cellular networks where for e.g., due to space constraints mobile station is only equipped with a single antenna. In this special scenario, i.e., nD = 1, the elements of matrix S in (8) are represented as S ij = σji−1 for i = 1, . . . , nK − 1 and the last row of S is given by Z ∞ n +1 n − R −1 nR +1 S nK j = 2σj K 2 x 2 0 −nS  r  x G2 a¯ γ xs 1+ dx. (17) × KnR −1 2 σj 1 + G2 x Due to the multi-linear property of the determinant, to calcuS ) for large γ¯ , we use the series representation of the late det(S Bessel function as Kν (z) =

(−1)ν+1

ν−1 X

Γ(ν − k) (−1)k  z −ν+2k + Γ(k + 1) 2 2 k=0 h i ∞ ( z )ν+2k ln( z ) − ψ(k+1) − ψ(ν+k+1) X 2 2 2 2 Γ(k + 1)Γ(ν + k + 1)

k=0

From (17), (18), and after several calculations together with the help of [11, Eq. (3.194.3)] and [16, Eq. (2.6.4.7)], we can express the entries of the last row of matrix S as two terms I1 and I2 , i.e, S nK j = I1 + I2 , respectively shown as follows: I1 ≈

nX R −2 k=0

(−1)k Γ(nR − k − 1)Γ(nS − k − 1)σjnK −k−2 Γ(nS )(G2 a¯ γ s)k+1

I2 ≈ (−1)nR +1

∞ X σjnK −nR −k−1 B (nR + k, nS − nR − k)

Γ(k + 1)Γ(nR + k)(G2 a¯ γ s)k+1   × ln(G2 a¯ γ s) + ψ(nS − nR − k) + ψ(k + 1) . (19) k=0

To asymptotically approximate the determinant of matrix S , we consider two separate cases: i) nR > nK and ii) nR ≤ nK S) • nR > nK : The minimum exponent of γ ¯ to make det(S non-zero in I1 is k = nK − 1 and in I2 is k = nR − 1. Hence, I2 can be neglected as compared to I1 , which results in ΦγD (s)

(large γ ¯)

≈

S 1) Γ(nR − nK )Γ(nS − nK ) det(S KΓ(nS )(G2 a¯ γ s)nK det(σij−1 )

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, (20)

where S 1 is an nK × nK matrix whose entities are  i−1 σj , 1 ≤ i ≤ nK − 1, 1 S i,j = (−1)nK −1 σj−1 , i = nK . Since det(σij−1 ) =

(16)

with the condition that nS > min(nR , nD ) and Ξ1 is defined, for example with M -PSK modulation as  R π− π
−nR 
g γ ¯ a1 g 1 M  ln sin 2θ  π 0 sin2 θ −ψ(nR ) + ψ(nS −nR ) dθ, for nR = nD , Ξ1 = m1 π  sin2 θ  Γ(|nD −nR |) R π− M dθ, for nR 6= nD . π g 0

. (18)

1

S )= det(S

nK Q

1≤i nD .

0

10

-1

10

-2

Symbol Error Rate

10

A. The nD > nK Case Let 0 ≤ λ1 ≤ . . . ≤ λnK ≤ ∞ be the eigenvalues of H †2H 2 with H 2 ∈ CnD ×nR . Define Σ = AA† and B = A† H†r Hr A. Hence, we can write

-3

10

n =2,4,8,12 K

-4

10

8-PSK (5,3,2)-MIMO AF relay

H †2H 2 = H†t B Ht .

-5

10

-6

10

Keyhole Multi-keyhole Rayleigh

-7

10

0

5

10

15

SNR

20

25

30

(dB)

Fig. 4. SER of (5, 3, 2)-MIMO AF relay systems with multi-keyhole versus SNR for 8-PSK modulation and nK = 1, 2, 4, 8, 12.

minimum of nK and nR . Specifically, among the four cases, Case 7 surpasses Case 4, 5, and 6 since Case 7 has the highest value of min(nK , nR ). Finally, to further understanding the effect of multi-keyhole on the SER performance, Fig. 4 displays the performance of (5, 3, 2)-MIMO AF relay systems with various nK . For comparison, the case of MIMO AF relay channel where both hops undergo Rayleigh fading is also plotted in Fig. 4. It is observed that as nK ≤ 2 the diversity gain does not increase as expected. Moreover, the performance is significantly improved when nK is increased from 1 to 8, while additional keyholes, (nK > 8), has a diminishing impact on reducing the SER. When the number of keyholes increases in the limit nK → ∞, the performance approaches to that of Rayleigh MIMO AF relay channels [12].

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(26)

Let Dord = (x1 , . . . , xnK ) be the ordered eigenvalues of B with 0 ≤ x1 ≤ . . . ≤ xnK ≤ ∞. Conditioned on Dord , the joint PDF of the non-zero eigenvalues of H †2H 2 is given by 1 1 the joint PDF of the eigenvalues of B 2 Ht H†t B 2 . This has the same form as A† H†r Hr A and so the PDF comes from [18, Eq. (42)] as f (λ1 , . . . , λnK |x1 , . . . , xnK )  λ    QnK nR −nK − i nK −j det λi det e xj i=1 λi  . (27) ×= Q Q K 1 nK 1 B )nR n`