Cooperative Diversity Can Mitigate Keyhole Effects in Wireless MIMO Systems Oussama Souihli

Tomoaki Ohtsuki

Graduate School of Science and Technology Keio University 3-14-1 Hiyoshi, Yokohama 223-8522, Japan Email: [email protected]

Faculty of Science and Technology Keio University 3-14-1 Hiyoshi, Yokohama 223-8522, Japan Email: [email protected]

Abstract—A MIMO keyhole is a propagation environment such that the channel gain matrix has unit rank (single degree of freedom), irrespective of the number of deployed antennas or their correlations (spacing), thereby reducing the MIMO channel capacity to that of a SISO channel. Related literature seems to consider such degeneration hopeless. Contrary to this general belief, this paper demonstrates that cooperative diversity can mitigate keyhole effects. Precisely, provided that the source-relay channel is keyhole-free, we show that there exists a “cutoff” relay transmit power above which keyhole effects can be mitigated even when both the source-destination and the relay-destination channels incur keyhole effect. We explicit the closed form of this power threshold as function of the source transmit power and the channel matrices brought into play in the relay channel. Numerical examples confirm the relevance of our claim.

I. I NTRODUCTION One reason behind the popularity of multiple-inputmultiple-output (MIMO) systems is their high spectral efficiency [1], [2]. Indeed, assuming channel state information (CSI) availability, it is known that the capacity of a MIMO channel between an NT -antenna transmitter and an NR antenna receiver is n times that of a single-input-singleoutput (SISO) channel, where n is the rank of the MIMO channel gain matrix, less than or equal to min{NT , NR } [1]– [3]. In particular, if the channel matrix has full rank (a socalled rich-scattered environment), channel capacity scales with min{NT , NR }, i.e. n = min{NT , NR }. From this perspective, it is interesting to determine conditions under which channel matrix has full rank, and to ensure that these conditions are met so that channel capacity is maximized. While it has been long believed that decorrelating transmit antennas (e.g. by sufficiently spacing them) amply ensures a full-rank channel matrix [4], [5], recent works [4]–[6] demonstrated that in a so-called keyhole/pinhole scenario, channel matrix has unit rank, even when its entries have zero correlations between each other. Subsequently, the benefits of rich scattering are suppressed, and channel capacity scales as that of a SISO channel (i.e. n = 1). Such a frustrating result was first theoretically predicted in [4], [5], and later verified through an experimental testbed in [6]. MIMO keyholes occur when radio waves come across metal obstacles with small holes only through which they can

propagate (spatial keyholes, see Fig. 1). They are also encountered in urban environments with so-called street canyons (narrow streets bordered by tall buildings), and in some indoor environments such as corridors, hallways and subway tunnels, settings which may act as single-moded waveguides at large distance from the source, thereby allowing only a single electromagnetic mode to pass through (modal keyholes). Finally, outdoor keyholes may also occur owing to a diffraction at rooftop edges (diffraction-induced keyholes). Because channel has unit rank irrespective of fading correlations or the number of deployed antennas, such degeneration seems to have been thought of as irremediable. Indeed, related literature has been limited to system performance analyses under keyhole effect: achievable outage capacity regions [8], [9], performance analyses in multiple keyhole scenarios [10], [11], performance of STBC codes in keyhole environments [12]–[14], evaluation of level crossing rate (LCR) [15] and pairwise error probability [16], to name a few. There seems to be no related work that has attempted to provide a solution to such issue, perhaps with the exception of [4] where the authors pointed out that a horizontal arrangement of rooftop antenna arrays mitigates diffraction-induced keyholes. Regrettably, this solution proves inadequate to combat other kinds of keyholes (e.g. spatial keyholes or modal keyholes). To the best of our knowledge, there has been no universal remedy againt MIMO keyholes, to date. In this paper, we investigate whether cooperative diversity (relay deployment) can mitigate keyhole effects. A MIMO relay channel is depicted in Fig. 2. It brings into play three nodes (source, relay and destination) and three channel gain matrices denoted by F, G and H. In downlink channels of cellular networks, source and relay nodes are fixed base stations (BS) arbitrarily positioned by the network operator. Therefore, it seems reasonable to assume the channel between them, F, to enjoy rich scattering. Cases where either of the channels involving the destination (i.e. G or H) is keyholefree are trivial, as either the direct link (source-destination) or the relayed link (source-relay-destination) are keyhole-free. Hence, we focus in this paper on the more challenging scenario where both channels involving the destination (i.e. G and H) suffer from keyhole effects (i.e. have unit rank). This chal-

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Fig. 2.

The general MIMO relay channel

in Section VI. II. S YSTEM M ODEL AND P RELIMINARIES

Fig. 1.

A MIMO (spatial) keyhole scenario

lenging assumption is also in line with practical scenarios as it makes no assumptions on the destination’s mobility pattern or localization (in downlink channels of cellular networks, the destination is traditionally a mobile station (MS)). Finally, we only consider degraded relay channels, i.e. channels where the source-relay signal is better than the source-destination signal, as it is often the case [17]. Under such framework, we take aim at determining necessary and sufficient conditions for keyhole mitigation to be feasible. We make the following findings: 1) There exists a “cutoff” relay transmit power P1∗ above which keyhole effects can be mitigated (i.e. MIMO capacity is n = min{NT , NR } times that of a SISO system). 2) Decreasing the relay transmit power below P1∗ renders keyhole effect unresolvable. Besides, we provide a closed form expression for the relay transmit power threshold P1∗ , which we find to be function of the source transmit power and the channel matrices brought into play in the considered relaying scenario. Hence, assuming appropriate power allocation, cooperative diversity is put forward as an efficient way to mitigate MIMO keyhole effects, even when both the source-destination and the relay-destination channels are unit-rank. Furthermore, as we tackle the problem from an information-theoretic perspective, the proposed solution is universal, being indifferent to the physical origins behind the keyhole phenomenon. The remainder is organized as follows. We start by presenting in Section II system model and preliminaries related to our study. Then, we derive in Section III a closed form for the capacity of the MIMO degraded relay channel. From this closed form, we infer in Section IV necessary and sufficient conditions for keyhole mitigation to be possible, and we provide a keyhole mitigation scheme for downlink channels in closed-loop MIMO cellular networks. Ultimately, we provide numerical examples in Section V and we conclude our work

We start by introducing the system model and relevant notations related to MIMO communications over the Rayleigh fading channel. Then, we briefly explain the capacity degradation owing to keyhole effects, and we review the definition and capacity theorem relative to the general degraded relay channel. A. Notations Throughout the paper, the following notations will be used. Vectors will be denoted in bold, and matrices in capital bold letters. ⊗ denotes the outer product, † the Hermitian (conjugate transpose) operator, tr (·) the trace operator, E {·} the mathematical expectation (expected value), h the Shannon entropy and I (x; y) the mutual information between input x and output y [18]. When a is a real number, let a+ max (a, 0). When A is a complex matrix, let A denote its Frobenius norm: A tr {AA† }. When x, y are complex random vectors, let Qxy denote their cross-covariance matrix: Qxy

†

E{(x − E{x}) (y − E{y}) }

(1)

which, when x, y are zero-mean, simplifies to: Qxy

=

E{xy† }

(2)

B. System Model A generic MIMO relay channel is presented in Fig. 2, where the transmitter, the receiver and the relay are equipped with multiple antennas. For simplicity, we assume same number N of transmit/receive antennas for all three entities, i.e. NT = NR = N . Source-relay, relay-destination and sourcedestination channels are denoted by F, G, H, respectively. These are assumed to be frequency-flat, complex-valued and with Rayleigh block fading, obeying Dent’s model [22]. Signals transmitted by the source and the relay are resp. denoted by x, x1 and are assumed to be zero-mean complex random vectors. Signals received by the relay and the destination are resp. denoted by y1 , y. These are corrupted by additive zero mean white Gaussian (ZMWG) noises z1 , z of variances σz21 , σz2 , respectively. Thus, the MIMO relay channel can be modeled by the following equations: y1 = Fx + z1 (3) y = Hx + Gx1 + z

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Finally, source and relay transmissions are subject to power constraints that can be modeled as follows: ⎧

⎨ tr (Qxx ) tr E xx† ≤ P

(4) ⎩ tr (Qx1 x1 ) tr E x1 x†1 ≤ P1 where Qxx , Qx1 x1 denote the covariance matrices of signals x, x1 , respectively. C. On Keyhole Effect Consider, a MIMO scenario made up by an N -antenna transmitter and an N -antenna receiver, and assume perfect CSI to be available to both channel endpoints. Then, it is widely acknowledged that the achievable capacity is given by [1], [3]: C(μ)

N

=

+

(log (μλi ))

(5)

i=1

where μ denotes the waterfilling level, a parameter chosen to meet a given power constraint and (λi )1≤i≤N denote the squared eigenvalues of the channel matrix H. From (5), one may infer that the higher the channel rank (i.e. the more non-null eigenvalues the channel matrix has), the greater the channel capacity. In a keyhole environment, however, channel matrix H is the outer product of two random vectors f , g [4]–[6]: H

=

f ⊗g

†

f ·g ⎛ f1 g1† ⎜ .. ⎝ .

=

fM g1†

... .. . ...

† ⎞ f1 gN .. ⎟ . ⎠

(6) (7) (8)

† fM gN

Theorem 1 ( [17]): The capacity C of the degraded relay channel is given by: C = max min {I (x, x1 ; y) , I (x; y1 |x1 )} p(x,x1 )

E. On the Relaying Strategy In [17], it was proved that the relaying strategy that achieves the aforementioned capacity region is Decode-and-Forward (DF). Therefore, this strategy will be considered in our proposed relaying scheme. For further details about this strategy, the interested reader is kindly referred to [17] or to our extended work [21]. F. Capacity of the Degraded MIMO Relay Channel Indubitably, Cooperative diversity in MIMO systems has been intensively studied in literature, recently, as relays allow for many benefits to wireless networks (such as reduced transmit power, reduced interference and increased throughputs) owing to cell radius reduction [19]. However, though many papers investigated achievable capacity in MIMO relay channels, these results seem to be unfit to our study as, to our knowledge, they either made restrictive channel assumptions (such as [?] where no direct link between source and destination was assumed) or considered general (non-necessarily degraded) relays (see, e.g., [20]), for which they could only provide capacity lower/upper bounds. Hence, out of concern for completeness, we derived a closed form for the channel capacity of the MIMO degraded relay channel, which is given as follows: Lemma 1: The capacity of the degraded MIMO channel is given by: Cd

1

which clearly has unit rank . Therefore, under such circumstances, MIMO channel capacity is no better than that of a SISO channel, as only one channel eigenvalue is non-null. D. The Degraded Relay Channel The relay channel of Fig. 2 is completely defined by specifying the probability density function p (y, y1 |x, x1 ) [17]. We start by recalling the definition and capacity theorem relative to the general degraded relay channel. Definition 1 ( [17]): The relay channel is said to be degraded if: p (y, y1 |x, x1 )

= p (y1 |x, x1 ) p (y|y1 , x1 )

(9)

This definition of degradation can be interpreted as physical degradation. In other words, a relay channel is degraded if the signal received by the relay is better than the signal received by the destination [17]. This is the case, for instance, when the relay is closer to the source node than the destination node, owing to pathloss attenuation (assuming similar signal to noise ratios (SNRs) in both channels). 1 Observe, for instance, that any two rows are linearly dependent, since: f ∀ 1 ≤ i, j ≤ M, rowj = fj rowi i

(10)

= min {C1 , C2 }

where: ⎧ σz21 1 † † ⎪ ⎪ = log I + GQ G + HQ H C ⎨ 1 x1 x1 xx N 2 2 σz σz ⎪ 1 ⎪ ⎩ C2 = log IN + FQxx F† 2 σz1

(11)

(12)

Proof: see Appendix A. III. O N K EYHOLE E FFECT M ITIGATION We start by stating the main goal behind our proposal. Then, we present our main result and we outline the proposed keyhole-mitigation scheme. A. Problem Statement As explained in the introduction, we focus on the more challenging case where both the source-destination channel, H, and the relay-destination channel, G, incur keyhole effect (i.e. have unit rank). From previous capacity results, we would like to infer necessary and sufficient conditions under which the keyhole effect can be mitigated (i.e. capacity scales linearly with n = N ), assuming the source-relay channel, F, is richscattered (i.e. full-rank).

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B. Necessary and Sufficient Conditions for Keyhole Effect Mitigation We consider equal power allocation, for simplicity. Theorem 2: Consider a degraded keyhole channel where the source relay-channel F is full-rank and both the sourcedestination H and the relay-destination G channels are unitrank. Then, there exists a relay transmit power threshold P1∗ such that: ∗ • For any relay transmit power P1 ≥ P1 , keyhole effect can be mitigated, i.e. capacity scales linearly with N ; ∗ • For any relay transmit power P1 < P1 , keyhole effect cannot be mitigated. Moreover, when the source equally allocates its transmit power P among its transmit antennas, P1∗ is given by: N σz2 P P ∗ 1 + N σ 2 φi − N σ 2 η − 1 P1 = γ (13) z1

i=1

z

where γ, η resp. denote the only non-null squared eigenvalues of G, H and, ∀ 1 ≤ i ≤ N , φi denotes the ith squared (nonnull) eigenvalue of F. Proof: From Theorem 1, we know that the capacity of the general degraded relay channel is given by: = min {C1 , C2 }

Cd From Lemma 1, C1 ⎧ ⎪ ⎪ ⎨ C1 = log IN + ⎪ ⎪ ⎩ C2 = log IN +

(14)

and C2 are given by:

1 1 † † GQx1 x1 G + 2 HQxx H 2 σz σz 1 FQxx F† σ2

(15)

z1

Of all three channel matrices, only F is full-rank. Therefore, only C2 scale linearly with N . It follows that channel capacity C d scales linearly with the number of antennas iff min {C1 , C2 } = C2 , which is equivalent to: IN + 1 GQx x G† + 1 HQxx H† ≥ IN + 1 FQxx F† 1 1 σz2 σz2 σz21 (16) As we assumed equal power allocation, the right expression in (16) reads [1], [3]: N P IN + 1 FQxx F† = 1 + (17) φ i σz21 N σz21 i=1 where φi denotes the ith squared eigenvalue of matrix F. On the other hand, because channels H and G are unit-rank, the left expression in (16) simplifies to: IN + 1 GQx x G† + 1 HQxx H† = 1+ P1 γ + P η 1 1 σz2 σz2 σz2 N σz2 (18) where γ, η are the only non-null squared eigenvalues of G, H, respectively. Therefore: N P P1 P 1 + (19) η ≥ φ C1 ≥ C2 ⇔ 1 + 2 γ + i σz N σz2 N σz21 i=1

⇔ P1 ≥

σz2 γ

N

1+

i=1

P N σz21

φi −

P N σz2 η

−1

(20)

Q.E.D. The interested reader is referred to [21] for the closedform expression of P1∗ when power is allocated by means of waterfilling. Finally, it is noteworthy that for a fixed relay power P1 = P1∗ , (20) can be viewed as an upper bound on the source transmit power P instead. C. A Keyhole Mitigation Scheme We assume a block-fading channel model, with block duration T . In closed-loop MIMO systems, it is custom to divide the block fade duration T into two periods: • A training phase of duration Tp , during which the BS sends pilots to intended receivers who estimate channels from the received training sequence and feed back their respective CSI estimates; • A transmission phase of duration Td = T − Tp Tp during which the BS uses the fed-back CSI to send data to scheduled users by means of adaptive transmission. The proposed scheme to mitigate keyhole effects through cooperative diversity in closed-loop MIMO systems is as follows: • During Tp : – BS sends training sequence (pilots symbols) to MS and to the relay; The relay sends pilot symbols to MS – From received pilots, the relay estimates the channel F and MS estimates channels H, G – Finally, the relay feeds back channel estimation to BS; MS feeds back channel estimations to the relay and to BS • During Td : Decode and Forward (DF) – First time slot: BS sends data to MS and relay, with power P (equally allocated among the eigenmodes of F) and at rate R up to I (x; y1 |x1 ). Since R > I (x; y), only the relay can successfully decode x. The MS has to wait for the assistance of the relay during the following time slot. – Second time slot: The relay computes minimum transmit power P1∗ (required to mitigate the keyhole effect) using (13). Then, the relay sends x1 with power P1∗ at rate R1 up to I (x1 ; y). – Finally, MS decodes x1 , then uses it to decode BS’s message x. Fig. 3 provides a flowchart of the proposed scheme. IV. N UMERICAL E XAMPLES In this Section, we numerically evaluate the achievable rates for different degradation assumptions and relaying strategies. papered results are the average of 10,000 channel realizations. Our aim is to confirm that, owing to the proposed scheme, capacity scales with the number of antennas, N , when F is full rank, even if H, G are unit-rank. For this sake, channels

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Fig. 3. Proposed relay-assisted keyhole mitigation scheme for downlink channels of MIMO-based cellular networks.

H, G were generated according to block fading model [22] at a frequency of 2.4 GHz, and such that they have unit rank, while channel F was generated such that it has full rank. Identity matrix was chosen as input covariance matrix (i.e. equal power allocation). Fig. 4 portrays the capacity scaling with the number of antennas, for DF relaying strategy. First, we observe that the mutual information I(x, x1 ; y) does not scale linearly with the number of antennas. This is owing to the fact that both channels H, G incur keyhole effect. Contrarily, mutual information I(x; y1 |x1 ) scales linearly with N , and it also exceeds the former mutual information. Under such settings, we observe that satisfying (13) ensures the capacity linear scaling with the number of antennas, despite channels H, G incurring keyhole effect. Therefore, the relevance of our proposal is validated. Furthermore, Fig. 5 portrays the capacity scaling with SNR, using DF relaying strategy. A 4 × 4 MIMO system was considered and, out of concern for fairness, equal SNRs were set for the different channels involved. Again, we observe that, under the aforementioned settings, mutual information I(x; y1 |x1 ) exceeds I(x, x1 ; y). Indeed, as SNR increases, noise variances (σz ) and (σz1 ) decrease. However, how much this decrease benefits to channel capacities differs between both mutual informations. Precisely, I(x, x1 ; y) would benefit less from this decrease as noise variances in (18) increase only one eigenvalue each (since channels G, H are unit-rank). Contrarily, I(x; y1 |x1 ) benefits from the noise power decrease by a factor of as many eigenvalues as the number of antennas (see (17)), hence the significant capacity gap.

Fig. 4. Comparison of the two mutual information expressions’ scaling with N , the number of transmit antennas, for SNR = 10 dB. In this example, H, G incur keyhole effect (i.e. have unit rank), while F is full-rank.

Fig. 5. Comparison of the two mutual information expressions’ scaling with SNR. Same assumptions as in Fig. 4, MIMO 4 × 4 system.

exists a “cutoff” relay transmit power above which keyhole effects can be mitigated irrespective of the source-destination and relay-destination channels. We explicited the closed form of this power threshold as function the source transmit power and the channels brought into play in the relaying scenario. Simulation results confirmed the relevance of our proposal. A PPENDIX A P ROOF OF L EMMA 1 From Theorem 1, we have:

V. C ONCLUSION In this paper, we considered the problem of ensuring MIMO capacity linear scaling with the number of transmit antennas when the destination suffers from a keyhole effect. We demonstrated that cooperative diversity (relay deployment) can mitigate such phenomenon, to some extent. Precisely, if the source-relay channel is full rank, we proved that there

Cd

=

max min {I (x, x1 ; y) , I (x; y1 |x1 )}

p(x,x1 )

(21)

Since we have already specified capacity maximizing p (x, x1 ), capacity simplifies to: Cd

= min {I (x, x1 ; y) , I (x; y1 |x1 )}

(22)

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We shall develop each mutual information separately. The first mutual information is given by [23]: I (x, x1 ; y)

We have:

= h (y) − h (y|x, x1 ) (23) = h (y) − h (Hx + Gx1 + z|x, x1 ) (24) = h (y) − h (z) (25)

N h (y) = log (2πe) E yy†

(26)

The fact that x, x1 are zero-mean and independent of z yields: = 0 (27) E (Hx + Gx1 ) z†

† E z (Hx + Gx1 ) = 0 (28) Meanwhile:

† E (Hx + Gx1 ) (Hx + Gx1 ) = HQxx H† +GQx1 x1 G† + HQxx1 G† + GQx1 x H†

(29)

Therefore:

N 1 h (y) = log (2πe) 2 IN HQxx H† + GQx1 x1 G† (30) σz

where Qxx1 Qx x †1 E zz

E xx†1 = 0 E x1 x† = 0 =

(31) (32)

1 σz2 IN

(33)

On the other hand, z is zero-mean Gaussian with variance σz2 , therefore: N (34) h (z) = log (2πe) σz2 IN Finally, we get: I (x, x1 ; y) = log IN +

1 † σz2 GQx1 x1 G

+

1 † σz2 HQxx H

(35)

As for the second mutual information, we have [23]: I (x; y1 |x1 )

h (y1 |x1 ) − h (y1 |x, x1 ) (36) = h (Fx + z1 |x1 ) − h (Fx + z1 |x, x1 ) = h (y1 ) − h (z1 ) (37)

since x1 is independent of x and z1 . Besides,

N h (y1 ) = log (2πe) E y1 y1† N = log (2πe) σz21 IN + FQxx F† and:

(39)

N h (z1 ) = log (2πe) σz21 IN

(40)

FQxx F†

(41)

Therefore:

I (x; y1 |x1 ) = log IN +

Q.E.D.

(38)

1 σz21

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Tomoaki Ohtsuki

Graduate School of Science and Technology Keio University 3-14-1 Hiyoshi, Yokohama 223-8522, Japan Email: [email protected]

Faculty of Science and Technology Keio University 3-14-1 Hiyoshi, Yokohama 223-8522, Japan Email: [email protected]

Abstract—A MIMO keyhole is a propagation environment such that the channel gain matrix has unit rank (single degree of freedom), irrespective of the number of deployed antennas or their correlations (spacing), thereby reducing the MIMO channel capacity to that of a SISO channel. Related literature seems to consider such degeneration hopeless. Contrary to this general belief, this paper demonstrates that cooperative diversity can mitigate keyhole effects. Precisely, provided that the source-relay channel is keyhole-free, we show that there exists a “cutoff” relay transmit power above which keyhole effects can be mitigated even when both the source-destination and the relay-destination channels incur keyhole effect. We explicit the closed form of this power threshold as function of the source transmit power and the channel matrices brought into play in the relay channel. Numerical examples confirm the relevance of our claim.

I. I NTRODUCTION One reason behind the popularity of multiple-inputmultiple-output (MIMO) systems is their high spectral efficiency [1], [2]. Indeed, assuming channel state information (CSI) availability, it is known that the capacity of a MIMO channel between an NT -antenna transmitter and an NR antenna receiver is n times that of a single-input-singleoutput (SISO) channel, where n is the rank of the MIMO channel gain matrix, less than or equal to min{NT , NR } [1]– [3]. In particular, if the channel matrix has full rank (a socalled rich-scattered environment), channel capacity scales with min{NT , NR }, i.e. n = min{NT , NR }. From this perspective, it is interesting to determine conditions under which channel matrix has full rank, and to ensure that these conditions are met so that channel capacity is maximized. While it has been long believed that decorrelating transmit antennas (e.g. by sufficiently spacing them) amply ensures a full-rank channel matrix [4], [5], recent works [4]–[6] demonstrated that in a so-called keyhole/pinhole scenario, channel matrix has unit rank, even when its entries have zero correlations between each other. Subsequently, the benefits of rich scattering are suppressed, and channel capacity scales as that of a SISO channel (i.e. n = 1). Such a frustrating result was first theoretically predicted in [4], [5], and later verified through an experimental testbed in [6]. MIMO keyholes occur when radio waves come across metal obstacles with small holes only through which they can

propagate (spatial keyholes, see Fig. 1). They are also encountered in urban environments with so-called street canyons (narrow streets bordered by tall buildings), and in some indoor environments such as corridors, hallways and subway tunnels, settings which may act as single-moded waveguides at large distance from the source, thereby allowing only a single electromagnetic mode to pass through (modal keyholes). Finally, outdoor keyholes may also occur owing to a diffraction at rooftop edges (diffraction-induced keyholes). Because channel has unit rank irrespective of fading correlations or the number of deployed antennas, such degeneration seems to have been thought of as irremediable. Indeed, related literature has been limited to system performance analyses under keyhole effect: achievable outage capacity regions [8], [9], performance analyses in multiple keyhole scenarios [10], [11], performance of STBC codes in keyhole environments [12]–[14], evaluation of level crossing rate (LCR) [15] and pairwise error probability [16], to name a few. There seems to be no related work that has attempted to provide a solution to such issue, perhaps with the exception of [4] where the authors pointed out that a horizontal arrangement of rooftop antenna arrays mitigates diffraction-induced keyholes. Regrettably, this solution proves inadequate to combat other kinds of keyholes (e.g. spatial keyholes or modal keyholes). To the best of our knowledge, there has been no universal remedy againt MIMO keyholes, to date. In this paper, we investigate whether cooperative diversity (relay deployment) can mitigate keyhole effects. A MIMO relay channel is depicted in Fig. 2. It brings into play three nodes (source, relay and destination) and three channel gain matrices denoted by F, G and H. In downlink channels of cellular networks, source and relay nodes are fixed base stations (BS) arbitrarily positioned by the network operator. Therefore, it seems reasonable to assume the channel between them, F, to enjoy rich scattering. Cases where either of the channels involving the destination (i.e. G or H) is keyholefree are trivial, as either the direct link (source-destination) or the relayed link (source-relay-destination) are keyhole-free. Hence, we focus in this paper on the more challenging scenario where both channels involving the destination (i.e. G and H) suffer from keyhole effects (i.e. have unit rank). This chal-

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Fig. 2.

The general MIMO relay channel

in Section VI. II. S YSTEM M ODEL AND P RELIMINARIES

Fig. 1.

A MIMO (spatial) keyhole scenario

lenging assumption is also in line with practical scenarios as it makes no assumptions on the destination’s mobility pattern or localization (in downlink channels of cellular networks, the destination is traditionally a mobile station (MS)). Finally, we only consider degraded relay channels, i.e. channels where the source-relay signal is better than the source-destination signal, as it is often the case [17]. Under such framework, we take aim at determining necessary and sufficient conditions for keyhole mitigation to be feasible. We make the following findings: 1) There exists a “cutoff” relay transmit power P1∗ above which keyhole effects can be mitigated (i.e. MIMO capacity is n = min{NT , NR } times that of a SISO system). 2) Decreasing the relay transmit power below P1∗ renders keyhole effect unresolvable. Besides, we provide a closed form expression for the relay transmit power threshold P1∗ , which we find to be function of the source transmit power and the channel matrices brought into play in the considered relaying scenario. Hence, assuming appropriate power allocation, cooperative diversity is put forward as an efficient way to mitigate MIMO keyhole effects, even when both the source-destination and the relay-destination channels are unit-rank. Furthermore, as we tackle the problem from an information-theoretic perspective, the proposed solution is universal, being indifferent to the physical origins behind the keyhole phenomenon. The remainder is organized as follows. We start by presenting in Section II system model and preliminaries related to our study. Then, we derive in Section III a closed form for the capacity of the MIMO degraded relay channel. From this closed form, we infer in Section IV necessary and sufficient conditions for keyhole mitigation to be possible, and we provide a keyhole mitigation scheme for downlink channels in closed-loop MIMO cellular networks. Ultimately, we provide numerical examples in Section V and we conclude our work

We start by introducing the system model and relevant notations related to MIMO communications over the Rayleigh fading channel. Then, we briefly explain the capacity degradation owing to keyhole effects, and we review the definition and capacity theorem relative to the general degraded relay channel. A. Notations Throughout the paper, the following notations will be used. Vectors will be denoted in bold, and matrices in capital bold letters. ⊗ denotes the outer product, † the Hermitian (conjugate transpose) operator, tr (·) the trace operator, E {·} the mathematical expectation (expected value), h the Shannon entropy and I (x; y) the mutual information between input x and output y [18]. When a is a real number, let a+ max (a, 0). When A is a complex matrix, let A denote its Frobenius norm: A tr {AA† }. When x, y are complex random vectors, let Qxy denote their cross-covariance matrix: Qxy

†

E{(x − E{x}) (y − E{y}) }

(1)

which, when x, y are zero-mean, simplifies to: Qxy

=

E{xy† }

(2)

B. System Model A generic MIMO relay channel is presented in Fig. 2, where the transmitter, the receiver and the relay are equipped with multiple antennas. For simplicity, we assume same number N of transmit/receive antennas for all three entities, i.e. NT = NR = N . Source-relay, relay-destination and sourcedestination channels are denoted by F, G, H, respectively. These are assumed to be frequency-flat, complex-valued and with Rayleigh block fading, obeying Dent’s model [22]. Signals transmitted by the source and the relay are resp. denoted by x, x1 and are assumed to be zero-mean complex random vectors. Signals received by the relay and the destination are resp. denoted by y1 , y. These are corrupted by additive zero mean white Gaussian (ZMWG) noises z1 , z of variances σz21 , σz2 , respectively. Thus, the MIMO relay channel can be modeled by the following equations: y1 = Fx + z1 (3) y = Hx + Gx1 + z

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Finally, source and relay transmissions are subject to power constraints that can be modeled as follows: ⎧

⎨ tr (Qxx ) tr E xx† ≤ P

(4) ⎩ tr (Qx1 x1 ) tr E x1 x†1 ≤ P1 where Qxx , Qx1 x1 denote the covariance matrices of signals x, x1 , respectively. C. On Keyhole Effect Consider, a MIMO scenario made up by an N -antenna transmitter and an N -antenna receiver, and assume perfect CSI to be available to both channel endpoints. Then, it is widely acknowledged that the achievable capacity is given by [1], [3]: C(μ)

N

=

+

(log (μλi ))

(5)

i=1

where μ denotes the waterfilling level, a parameter chosen to meet a given power constraint and (λi )1≤i≤N denote the squared eigenvalues of the channel matrix H. From (5), one may infer that the higher the channel rank (i.e. the more non-null eigenvalues the channel matrix has), the greater the channel capacity. In a keyhole environment, however, channel matrix H is the outer product of two random vectors f , g [4]–[6]: H

=

f ⊗g

†

f ·g ⎛ f1 g1† ⎜ .. ⎝ .

=

fM g1†

... .. . ...

† ⎞ f1 gN .. ⎟ . ⎠

(6) (7) (8)

† fM gN

Theorem 1 ( [17]): The capacity C of the degraded relay channel is given by: C = max min {I (x, x1 ; y) , I (x; y1 |x1 )} p(x,x1 )

E. On the Relaying Strategy In [17], it was proved that the relaying strategy that achieves the aforementioned capacity region is Decode-and-Forward (DF). Therefore, this strategy will be considered in our proposed relaying scheme. For further details about this strategy, the interested reader is kindly referred to [17] or to our extended work [21]. F. Capacity of the Degraded MIMO Relay Channel Indubitably, Cooperative diversity in MIMO systems has been intensively studied in literature, recently, as relays allow for many benefits to wireless networks (such as reduced transmit power, reduced interference and increased throughputs) owing to cell radius reduction [19]. However, though many papers investigated achievable capacity in MIMO relay channels, these results seem to be unfit to our study as, to our knowledge, they either made restrictive channel assumptions (such as [?] where no direct link between source and destination was assumed) or considered general (non-necessarily degraded) relays (see, e.g., [20]), for which they could only provide capacity lower/upper bounds. Hence, out of concern for completeness, we derived a closed form for the channel capacity of the MIMO degraded relay channel, which is given as follows: Lemma 1: The capacity of the degraded MIMO channel is given by: Cd

1

which clearly has unit rank . Therefore, under such circumstances, MIMO channel capacity is no better than that of a SISO channel, as only one channel eigenvalue is non-null. D. The Degraded Relay Channel The relay channel of Fig. 2 is completely defined by specifying the probability density function p (y, y1 |x, x1 ) [17]. We start by recalling the definition and capacity theorem relative to the general degraded relay channel. Definition 1 ( [17]): The relay channel is said to be degraded if: p (y, y1 |x, x1 )

= p (y1 |x, x1 ) p (y|y1 , x1 )

(9)

This definition of degradation can be interpreted as physical degradation. In other words, a relay channel is degraded if the signal received by the relay is better than the signal received by the destination [17]. This is the case, for instance, when the relay is closer to the source node than the destination node, owing to pathloss attenuation (assuming similar signal to noise ratios (SNRs) in both channels). 1 Observe, for instance, that any two rows are linearly dependent, since: f ∀ 1 ≤ i, j ≤ M, rowj = fj rowi i

(10)

= min {C1 , C2 }

where: ⎧ σz21 1 † † ⎪ ⎪ = log I + GQ G + HQ H C ⎨ 1 x1 x1 xx N 2 2 σz σz ⎪ 1 ⎪ ⎩ C2 = log IN + FQxx F† 2 σz1

(11)

(12)

Proof: see Appendix A. III. O N K EYHOLE E FFECT M ITIGATION We start by stating the main goal behind our proposal. Then, we present our main result and we outline the proposed keyhole-mitigation scheme. A. Problem Statement As explained in the introduction, we focus on the more challenging case where both the source-destination channel, H, and the relay-destination channel, G, incur keyhole effect (i.e. have unit rank). From previous capacity results, we would like to infer necessary and sufficient conditions under which the keyhole effect can be mitigated (i.e. capacity scales linearly with n = N ), assuming the source-relay channel, F, is richscattered (i.e. full-rank).

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B. Necessary and Sufficient Conditions for Keyhole Effect Mitigation We consider equal power allocation, for simplicity. Theorem 2: Consider a degraded keyhole channel where the source relay-channel F is full-rank and both the sourcedestination H and the relay-destination G channels are unitrank. Then, there exists a relay transmit power threshold P1∗ such that: ∗ • For any relay transmit power P1 ≥ P1 , keyhole effect can be mitigated, i.e. capacity scales linearly with N ; ∗ • For any relay transmit power P1 < P1 , keyhole effect cannot be mitigated. Moreover, when the source equally allocates its transmit power P among its transmit antennas, P1∗ is given by: N σz2 P P ∗ 1 + N σ 2 φi − N σ 2 η − 1 P1 = γ (13) z1

i=1

z

where γ, η resp. denote the only non-null squared eigenvalues of G, H and, ∀ 1 ≤ i ≤ N , φi denotes the ith squared (nonnull) eigenvalue of F. Proof: From Theorem 1, we know that the capacity of the general degraded relay channel is given by: = min {C1 , C2 }

Cd From Lemma 1, C1 ⎧ ⎪ ⎪ ⎨ C1 = log IN + ⎪ ⎪ ⎩ C2 = log IN +

(14)

and C2 are given by:

1 1 † † GQx1 x1 G + 2 HQxx H 2 σz σz 1 FQxx F† σ2

(15)

z1

Of all three channel matrices, only F is full-rank. Therefore, only C2 scale linearly with N . It follows that channel capacity C d scales linearly with the number of antennas iff min {C1 , C2 } = C2 , which is equivalent to: IN + 1 GQx x G† + 1 HQxx H† ≥ IN + 1 FQxx F† 1 1 σz2 σz2 σz21 (16) As we assumed equal power allocation, the right expression in (16) reads [1], [3]: N P IN + 1 FQxx F† = 1 + (17) φ i σz21 N σz21 i=1 where φi denotes the ith squared eigenvalue of matrix F. On the other hand, because channels H and G are unit-rank, the left expression in (16) simplifies to: IN + 1 GQx x G† + 1 HQxx H† = 1+ P1 γ + P η 1 1 σz2 σz2 σz2 N σz2 (18) where γ, η are the only non-null squared eigenvalues of G, H, respectively. Therefore: N P P1 P 1 + (19) η ≥ φ C1 ≥ C2 ⇔ 1 + 2 γ + i σz N σz2 N σz21 i=1

⇔ P1 ≥

σz2 γ

N

1+

i=1

P N σz21

φi −

P N σz2 η

−1

(20)

Q.E.D. The interested reader is referred to [21] for the closedform expression of P1∗ when power is allocated by means of waterfilling. Finally, it is noteworthy that for a fixed relay power P1 = P1∗ , (20) can be viewed as an upper bound on the source transmit power P instead. C. A Keyhole Mitigation Scheme We assume a block-fading channel model, with block duration T . In closed-loop MIMO systems, it is custom to divide the block fade duration T into two periods: • A training phase of duration Tp , during which the BS sends pilots to intended receivers who estimate channels from the received training sequence and feed back their respective CSI estimates; • A transmission phase of duration Td = T − Tp Tp during which the BS uses the fed-back CSI to send data to scheduled users by means of adaptive transmission. The proposed scheme to mitigate keyhole effects through cooperative diversity in closed-loop MIMO systems is as follows: • During Tp : – BS sends training sequence (pilots symbols) to MS and to the relay; The relay sends pilot symbols to MS – From received pilots, the relay estimates the channel F and MS estimates channels H, G – Finally, the relay feeds back channel estimation to BS; MS feeds back channel estimations to the relay and to BS • During Td : Decode and Forward (DF) – First time slot: BS sends data to MS and relay, with power P (equally allocated among the eigenmodes of F) and at rate R up to I (x; y1 |x1 ). Since R > I (x; y), only the relay can successfully decode x. The MS has to wait for the assistance of the relay during the following time slot. – Second time slot: The relay computes minimum transmit power P1∗ (required to mitigate the keyhole effect) using (13). Then, the relay sends x1 with power P1∗ at rate R1 up to I (x1 ; y). – Finally, MS decodes x1 , then uses it to decode BS’s message x. Fig. 3 provides a flowchart of the proposed scheme. IV. N UMERICAL E XAMPLES In this Section, we numerically evaluate the achievable rates for different degradation assumptions and relaying strategies. papered results are the average of 10,000 channel realizations. Our aim is to confirm that, owing to the proposed scheme, capacity scales with the number of antennas, N , when F is full rank, even if H, G are unit-rank. For this sake, channels

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Fig. 3. Proposed relay-assisted keyhole mitigation scheme for downlink channels of MIMO-based cellular networks.

H, G were generated according to block fading model [22] at a frequency of 2.4 GHz, and such that they have unit rank, while channel F was generated such that it has full rank. Identity matrix was chosen as input covariance matrix (i.e. equal power allocation). Fig. 4 portrays the capacity scaling with the number of antennas, for DF relaying strategy. First, we observe that the mutual information I(x, x1 ; y) does not scale linearly with the number of antennas. This is owing to the fact that both channels H, G incur keyhole effect. Contrarily, mutual information I(x; y1 |x1 ) scales linearly with N , and it also exceeds the former mutual information. Under such settings, we observe that satisfying (13) ensures the capacity linear scaling with the number of antennas, despite channels H, G incurring keyhole effect. Therefore, the relevance of our proposal is validated. Furthermore, Fig. 5 portrays the capacity scaling with SNR, using DF relaying strategy. A 4 × 4 MIMO system was considered and, out of concern for fairness, equal SNRs were set for the different channels involved. Again, we observe that, under the aforementioned settings, mutual information I(x; y1 |x1 ) exceeds I(x, x1 ; y). Indeed, as SNR increases, noise variances (σz ) and (σz1 ) decrease. However, how much this decrease benefits to channel capacities differs between both mutual informations. Precisely, I(x, x1 ; y) would benefit less from this decrease as noise variances in (18) increase only one eigenvalue each (since channels G, H are unit-rank). Contrarily, I(x; y1 |x1 ) benefits from the noise power decrease by a factor of as many eigenvalues as the number of antennas (see (17)), hence the significant capacity gap.

Fig. 4. Comparison of the two mutual information expressions’ scaling with N , the number of transmit antennas, for SNR = 10 dB. In this example, H, G incur keyhole effect (i.e. have unit rank), while F is full-rank.

Fig. 5. Comparison of the two mutual information expressions’ scaling with SNR. Same assumptions as in Fig. 4, MIMO 4 × 4 system.

exists a “cutoff” relay transmit power above which keyhole effects can be mitigated irrespective of the source-destination and relay-destination channels. We explicited the closed form of this power threshold as function the source transmit power and the channels brought into play in the relaying scenario. Simulation results confirmed the relevance of our proposal. A PPENDIX A P ROOF OF L EMMA 1 From Theorem 1, we have:

V. C ONCLUSION In this paper, we considered the problem of ensuring MIMO capacity linear scaling with the number of transmit antennas when the destination suffers from a keyhole effect. We demonstrated that cooperative diversity (relay deployment) can mitigate such phenomenon, to some extent. Precisely, if the source-relay channel is full rank, we proved that there

Cd

=

max min {I (x, x1 ; y) , I (x; y1 |x1 )}

p(x,x1 )

(21)

Since we have already specified capacity maximizing p (x, x1 ), capacity simplifies to: Cd

= min {I (x, x1 ; y) , I (x; y1 |x1 )}

(22)

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We shall develop each mutual information separately. The first mutual information is given by [23]: I (x, x1 ; y)

We have:

= h (y) − h (y|x, x1 ) (23) = h (y) − h (Hx + Gx1 + z|x, x1 ) (24) = h (y) − h (z) (25)

N h (y) = log (2πe) E yy†

(26)

The fact that x, x1 are zero-mean and independent of z yields: = 0 (27) E (Hx + Gx1 ) z†

† E z (Hx + Gx1 ) = 0 (28) Meanwhile:

† E (Hx + Gx1 ) (Hx + Gx1 ) = HQxx H† +GQx1 x1 G† + HQxx1 G† + GQx1 x H†

(29)

Therefore:

N 1 h (y) = log (2πe) 2 IN HQxx H† + GQx1 x1 G† (30) σz

where Qxx1 Qx x †1 E zz

E xx†1 = 0 E x1 x† = 0 =

(31) (32)

1 σz2 IN

(33)

On the other hand, z is zero-mean Gaussian with variance σz2 , therefore: N (34) h (z) = log (2πe) σz2 IN Finally, we get: I (x, x1 ; y) = log IN +

1 † σz2 GQx1 x1 G

+

1 † σz2 HQxx H

(35)

As for the second mutual information, we have [23]: I (x; y1 |x1 )

h (y1 |x1 ) − h (y1 |x, x1 ) (36) = h (Fx + z1 |x1 ) − h (Fx + z1 |x, x1 ) = h (y1 ) − h (z1 ) (37)

since x1 is independent of x and z1 . Besides,

N h (y1 ) = log (2πe) E y1 y1† N = log (2πe) σz21 IN + FQxx F† and:

(39)

N h (z1 ) = log (2πe) σz21 IN

(40)

FQxx F†

(41)

Therefore:

I (x; y1 |x1 ) = log IN +

Q.E.D.

(38)

1 σz21

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