Capacity Bounds on the Ergodic Capacity of Distributed MIMO Systems over K Fading Channels

Abstract

The performance of D-MIMO systems is not only affected by multipath fading but also from shadowing fading, as well as path loss. In this paper, we investigate the ergodic capacity of D-MIMO systems operating in non-correlated K fading (Rayleigh/Gamma) channels. With the aid of majorization and Minkowski theory, we derive analytical closed-form expressions of the upper and lower bounds on the ergodic capacity for D-MIMO systems over non-correlated K fading channels, which are quite general and applicable for arbitrary signal-to-noise ratio (SNR) and the number of transceiver antennas. To intuitively reveal the impacts of system and fading parameters on the ergodic capacity, we deduce asymptotic approximations in the high and low SNR regimes. Finally, we pursue the massive MIMO systems analysis for the lower bound and derive closed-form expressions when the number of antennas at BS grows large, and when the number of antennas at transceivers becomes large with a fixed and finite ratio. It is demonstrated that the proposed expressions on the ergodic capacity accurately match with the theoretical analysis.

1. Introduction

Recently, distributed multiple-input multiple-out (D-MIMO) wireless communication systems have received significant attention as they can combine the advantages of point-to-point MIMO with distributed antenna system (DAS) [1]-[4]. These gains are achieved by deploying multiple antennas at the radio ports (RPs) that are geographically distributed. Contrary to collocated MIMO (C-MIMO) system, D-MIMO system suffers from different degrees of shadowing fadings and path losses caused by different geographical positions and access distances. This makes the performance analysis of D-MIMO systems more challenging. However, large-scale fading (shadowing fading and path loss) is a crucial factor to assess the performance of D-MIMO systems [5][6]. For this reason, we herein investigate the capacity of D-MIMO systems over composite fading channels.

In the context of composite fading channels, Rayleigh/Lognormal (RLN) model is known as the most prevalent model, which has been extensively used to characterize the effects of composite fading in terrestrial and satellite wireless communication [7]-[11]. The main drawback of the RLN model is that the probability density function (PDF) of the composite fading model involves complicated mathematical formulas, which renders them inconvenient for analytical performance evaluations. To solve this issue, the friendlier Gamma distribution was used to approximate the lognormal distribution leading to the K composite distribution model (Rayleigh/Gamma distribution) [12]. Some empirical measurements reveal a general consensus that K fading model can capture diverse scattering phenomena such as tropospheric propagation of radio waves [5][13], various types of radar clutter [14], and optical scintillation from the atmosphere [15].

Motivated by the previous discussion, a plethora of recent works focus on the performance of K fading MIMO systems. In [16], authors investigate the outage probability performance of correlated-K fading channels with arbitrary and not necessarily identical parameters, while [17] provides the exact expressions on outage probability for exponentially correlated K fading channels. Finally, [18][18] consider the performance of correlated K fading MIMO channels with zero-forcing (ZF) receivers. To the best of authors’ knowledge, there is no work about ergodic capacity bounds of D-MIMO systems over spatially non-correlated K fading channels. In this light, we herein try to bridge this gap by presenting the upper and lower bounds on the ergodic capacity for spatially non-correlated K fading channel. In particular, the main contributions of this paper are summarized as follows:

(1) The analytical upper and lower bounds of ergodic capacity for spatially non-correlated K fading channel are derived by virtue of majorization and Minkowski theory. The proposed upper bound can be obtained by investigating the relationships between eigenvalues and diagonal elements of the Wishart matrix. The proposed lower bound can be obtained with the aid of the Minkowski’s inequality.

(2) In order to reveal intuitive insights into the impacts of system parameters on the ergodic capacity, we pursue the asymptotic analysis in the low and high signal-to-noise ratio (SNR) regimes. In the low SNR regime, we explore the asymptotic performance by two metrics of the minimum normalized energy per information bit to reliably convey any positive rate and the wideband slope. In the high SNR regime, the effects of small and large-scale fading on the ergodic capacity are decoupled.

(3) Based on the proposed lower bound, we explore the asymptotic system performance for the massive MIMO system by deploying a large number of antennas at the BS and at both ends with a fixed and finite ratio. It is demonstrated that the effect of small-scale fading is canceled and the sum rate is affected by the large-scale shadowing fading and path loss.

The remainder of the paper is organized as follows: In Section 2, we introduce the D-MMO fading model and provide the definition of ergodic capacity. In Section 3, some mathematical preliminaries are provided for analysis. In Section 4, we derive closed-form upper and lower bounds on the ergodic capacity of the spatially non-correlated K fading channel and perform the asymptotic analysis in the low and high SNR regimes and massive MIMO systems. Some numerical results and corresponding analysis are presented in Section 5. Section 6 concludes the paper and summarizes the key findings.

Notation： Upper and lower case boldfaces are denoted the matrix and vector, respectively, while the notations ℂ and ℝ denote the sets of complex and integer numbers, respectively. Let ·T ·H ·-1, and ·† denote transpose, conjugate transpose, inverse, and pseudoinverse operations of a matrix, respectively. The notation Ip denotes p×p the identity matrix. The notation ε stands for the expectation operation, while the notation ≺ denotes the majorization relation. The notation ·ij denotes the i, j -th entry of a matrix and det · stands for the determinant of a square matrix. Finally, the notations d · and λ · represent the main diagonal elements and eigenvalues of a Hermitian matrix, respectively.

 

2. D-MIMO Fading Model and Ergodic Capacity

2.1 D-MIMO Fading Model

We consider a general uplink D-MIMO system with one base station (BS) connected with Nr receive antennas and L radio ports (RPs) each connected with Nt transmit antennas. As in [2][5][10][11], no channel station information (CSI) is assumed at the RPs, and full CSI is assumed at the BS. The optimum transmission strategy is that the total power is equally split by the LNt antennas. Thus, the corresponding input-output relation is

where y ∈ ℂNrx1 and x ∈ ℂLNtx1 are the received and transmitted signal vector, respectively, whereas n ∈ ℂNrx1 is the additive white Gaussian noise (AWGN) with zero mean and covariance ε[nnH] = N0INr, where N0 is the noise power.

The random matrix H ∈ ℂNrxLNt captures small-scale fading, whose elements are modeled as independent and identically distributed (i.i.d.) CN 0,1 random variables (RVs). Therefore, the envelope of r = |hij| follows a Rayleigh distribution [19]

where Ω = ε[r2] is the average power, and U r is the unit step function. In our case, the value of the parameter Ω is assumed to be equal to unity.

The diagonal matrix Ξ ∈ ℝ LNt×LNt captures large-scale fading, which includes shadowing fading and path loss. It can be expressed as

where Dm denotes the distance between BS and the m-th RP, m = 1,⋯,L, υ ∈ 2,5 is the path loss exponent, which is a key metric to characterize the rate of decay of the signal with the distance [20]. The shadowing fading is captured by coefficient ξm, which is modeled as a Gamma RV. In this case, the PDF of coefficient ξm is given as

where km and Ωm = ε ξm/km represent the shape and scale parameters of the Gamma distribution respectively, whereas Γ · is the Gamma function as defined in [21].

2.2 Ergodic Capacity

As stated previously, we consider that BS has perfect knowledge of CSI, and an equal-power allocation strategy across all the RP’s antennas is executed. Then, the ergodic capacity of the D-MIMO systems is given as [5]

where γ = P/N0 is the average SNR, W = Ξ1/2HHHΞ1/2 and the expectation operation is taken over all random matrices H and Ξ (or likewise W). For the Hermitian matrix W, we define that

In this correspondence, we focus on the case Nr > LNt. All the results can be extended to the case of Nr ≤ LNt by employing the identity

By using the singular value decomposition, the ergodic capacity in (5) can be re-written as

where λm is the m-th eigenvalue of the Hermitian matrix W.

 

3. Mathematical Preliminaries

In this section, RV distribution properties and majorization theory are presented to execute the bounds analysis for non-correlated K fading D-MIMO system. These conclusions will be used in Section 4.

3.1 RV Distribution

Lemma 1: [19] Let X ~ R Ω be a Rayleigh RV. Then, the RV Y = X2 follows an exponential distribution with mean Ω. That is, the PDF of Y is

Lemma 2: [19] Let be a set of n i.i.d. exponential RVs with mean Ω. Then, the RV follows a Gamma distribution with the shape parameter n and scale parameter Ω. Then, the PDF of the RV Y is

Lemma 3: [19] Let be a set of n independent Gamma RV with the same scale parameter Ω but possibly different shape parameters μ1,⋯,μn, respective. Then

3.2 Majorization Theory

Majorization theory is an extremely useful and powerful tool for the theory of inequalities [22] and recently has been extensively used in wireless communication field [23]0. The relevant results of majorization theory are provided for our analysis.

Lemma 4: [22][23] Let R be a Hermitian matrix with diagonal elements denoted by the vector d and eigenvalues denoted by the vector λ. Then

Lemma 5: [22][23] Let ϕ be the real-valued function on ℝn. If g : ℝ → ℝ is concave. Then ϕ, which is defined as

is Schur-concave. In the same way, if g is convex, then ϕ is Schur-convex.

Lemma 6: [25][26] Let ϕ be the real-valued function on ℝn, which is defined as

Then, ϕ is a Schur-concave function.

 

4. Ergodic Capacity Bounds for D-MIMO Systems

In this section, we elaborate on the ergodic capacity of spatially non-correlated K fading D-MIMO systems, where multipath, shadowing fading and path loss are considered. In view of majorization [22]-[26] and Minkowski theory [27][28], the analytical upper and lower bounds on the ergodic capacity of D-MIMO systems are derived. In order to obtain the insightful insights into the impacts of the system parameters on the ergodic capacity, the asymptotic approximations on the ergodic capacity at high and low SNR regimes are provided.

4.1 Ergodic Capacity Upper Bound

Using the results of [25], we provide a novel upper bound of the ergodic capacity for non-correlated K fading channels. The result will be given in the following theorem.

Theorem 1: For non-correlated K fading channels, the upper bound of the ergodic capacity for D-MIMO systems is gives as

where is G · the Meijer’s G function as defined in [21].

Proof: A detail proof is provided Appendix I. ■

It is noteworthy that the proposed upper bound involves Meijer’s G-function which can be efficiently evaluated by standard mathematical software packages like Mathematica or Maple. Moreover, the formula (15) reduces to the result of [25,Th.1] under the case of L = 1 and Ξ = INt.

Although the formula in (15) presents a closed-form analytical expression, it does not give useful insights into the impacts of systems parameters on the ergodic capacity. In this light, we perform the asymptotic analysis in the high and low SNR regimes.

Corollary 1: For high SNR regime, the upper bound in (15) becomes

where φ · = d ln Γ · /dx is the digamma function as defined in [21].

Proof: In high SNR (large γ), the dominant term of logarithmic function is . Thus, the function can log2 1+ can be approximated by log2 . In turn, we use the following integral identities [21].

After some simplification, we complete the proof. ■

The above corollary reveals that the effects of large- and small-scale fading can be decoupled for high SNRs. The similar result appears in [2]. Moreover, we can observe that the proposed upper bound increases logarithmically with transmit power γ. More important, Nr, km and Ωm have a beneficial impact on the upper bound where a large transceivers distance Dm effectively reduces it since large distance yields large path loss.

In general, it is straightforward to study low SNR performance by deriving the first-order Taylor expansion of the proposed upper bound around γ→0. The recent publications, e.g., [29][30] have shown that this approach can not adequately reflect the impact of the system parameters on D-MIMO performance and lead to misleading results in the low-SNR regime. In this light, it is beneficial to analyze the upper bound at low SNR regime in terms of the normalized transmit energy per information bit Eb/N0 rather than SNR, which is originally proposed in [29]. Hence, the upper bound at low-SNRs is formulated as

where Eb/N0min is minimum normalized energy per information bit required to convey any non-negative rate reliably, while S0 is the capacity versus SNR slope. 0 and 0 are the first and second derivatives of the proposed upper bound.

Corollary 2: For low SNR, the metrics of the minimum energy per information bit and the wideband slope are given respectively

Proof: The proof starts by rewrite (49)

By taking the first and second derivatives of (23) with respect to γ → 0. We can derive as

With the aid of the definition of expectation and successively applying (18), the formulas (24) and (25) can be further simplified as

Finally, the result of (21) and (22) can be derived by substituting (26) and (27) into (20). After some manipulations, the proof of the proposition is completed. ■

The proposition reveals that the result of (21) is determined by the number of receiver antennas Nr, the number of RP L, shadowing fading parameters km and Ωm, and path loss parameter dm and υ, while the result is independent of the number of RP’s antenna Nt. Note that the similar results have been appeared in [31]. For Ξ = ILNt, L = 1, the two metrics reduce to Eb/N0min = L ln 2/Nr and S0 = 2NrNt/Nr + 1 , which is consistent with [18][31].

4.2 Ergodic Capacity Lower Bound

In this subsection, we derive an analytical lower bound expression on the ergodic capacity of D-MIMO system over non-correlated K fading channel. The key result is summarized in the following theorem.

Theorem 2: For non-correlated K fading D-MIMO channels, the lower bound of the ergodic capacity for D-MIMO systems is

Proof: The proof is provided in Appendix II. ■

It is observed that the lower bound of the ergodic capacity monotonically increases with the number of BS antenna Nr, the fading parameter km and the transmit power γ while decreases with the transceiver distances Dm. In addition, there are similarities between (28) and (16). Finally, the similar conclusions are also made in [10].

In order to obtain the intuitive insights into the impact of systems parameter, the asymptotic performance at high SNR regime is examined in the following proposition

Corollary 3: For high SNR, the lower bound in (28) converges to

Proof: After some simplification, we can complete the proof by taking transmit power γ large (γ→∞). ￭

Comparing Proposition 1 and Proposition 3, we can conclude that the two propositions have the similar conclusion except the small-scale fading. Moreover, (16) and (28) converge the same result for large antennas systems by using the property of digamma function.

In order to obtain the diversity order of the D-MIMO system, the expression on the lower bound of ergodic capacity is studied at high SNR. In this corresponding, we execute the analysis by invoking the affine expansion of the lower bound of the ergodic capacity [11]

where S∞ and L∞ represent the high-SNR slope in bits/s/Hz per 3-dB units and the high-SNR power offset, in 3-dB units, respectively, which are formulated as

Corollary 4: For high SNR, the metrics of the high-SNR slope and high-SNR power offset are provided

Proof: The process of the proof references proposition 2 of [11]. Omitting explicit details, we complete the proof. ■

It is intuitively observed that high-SNR slope S∞ in (34) is independent of the BS antenna number Nr, shadowing fading parameters km and Ωm, and path loss parameters Dm and υ. The similar observation is appeared in [11]. For high-SNR power offset L∞, the effects of the large and small-scale can be decoupled. Finally, the larger distances between BS and RPs Dm, the much more effectively reduce the system performance due to the increased path loss.

4.3 Massive MIMO Analysis

Recently, massive MIMO has emerged as one of the most promising technologies since it has the potential to improvement spectral and energy efficiency [32]-[34]. In the following, we study the asymptotic performance of massive MIMO system for the lower bound of theorem 2.

In order to obtain the intuitive insights into the massive MIMO analysis, the three separate cases are considered:

(1) Fixed L and Nt, whilst Nr → ∞: Directly, when the number of the BS antennas grows without bound, whilst L and Nt are kept fixed, the ergodic capacity lower bound of (28) tends to infinity.

Corollary 5: Fixed L and Nt, while Nr → ∞, the lower bound of the ergodic capacity becomes

Proof: The proof starts by introducing the follow identity [35]

Substituting (36) into (28), we can obtain the desired result of (36) after some simplifications. ■

From the result of (35), we can observe that the effects of small-scale fading can be omitted for the number of the BS antennas grows without bound. The similar appears in [2]. Moreover, the ergodic capacity grows into infinity with BS antennas.

(2) Fixed L, κ = Nr/LNt > 1, whilst Nt,Nr → ∞ : In this case, the number of the BS and each user’s antennas grows large with a fixed and finite ratio, whilst the number of users is kept fixed.

Corollary 6: Fixed L, κ, whilst Nt,Nr → ∞, the lower bound of the ergodic capacity becomes

Proof: The proof starts by re-writing (29) as follows

By using (36), the first sum term in (38) can be expressed as

The second sum term in (39) can be written as in integral form

Recalling the following integral identity [36]

Combining (39), (40), (41) with (38), we can conclude the proof after some algebraic manipulations. ■

Through corollary 6, we can conclude that the ergodic capacity linear increases with transmit antennas and logarithmically increases with transmit power γ.

(3) Fixed Nt, κ = Nr/LNt > 1, whilst L, Nr → ∞: In this case, it is equivalent that all users are mutually independent and uniformly distributed in the circle of the cell. It is assumed that the shadow fading parameters are fixed constant, i.e., km = k, Ωm = Ω0. Thus, the PDF of the distance between the BS and users is given by [37]

Corollary 7: Fixed Nt, κ, whilst L, Nr → ∞, the lower bound of the ergodic capacity becomes

where a = γΩ0κκ κ – 11-κ exp φ k – 1, R0 is the radius of the cell, r0 is the closest distance between the users and BS.

Proof: Using the shadow fading parameter assumption, the lower bound of corollary 6 can be further simplified as

By using the relation between the sum and expectation, the sum term in (44) is expressed integral form

Based on the definition of expectation and the PDF of Dm in (42), ① can be further expressed as

We now recall the following integral identity [36]

Substituting (47) into (44), we can conclude the proof after some manipulations ￭

Corollary 7 reveals that the result of (43) is quite general, which can be applied to arbitrary number of the transceiver antennas. Moreover, it is inferred that a large value R0 reduces the ergodic capacity while a large value r0 increases it.

 

5. Numerical Results

In this section, some numerical results are presented to furtherly validate the derived analytical results. For all simulations, it is assumed that there are L =3, Nt = 2 except Fig. 3. We reap the multipath and shadowing fading matrix H and Ξ according (2) and (4) by generating 100000 random realizations and thereafter get the simulated ergodic capacity via (5).

In Fig. 1, the tightness of the analytical upper bound in (15) and lower bound in (29) are investigated. In this simulation settings, we assume that Nr = 12,24,60, Nt = 2, L = 3, Ω = 1, km = 1, Ωm = 2, υ = 4, D1 = 1000m , D2 = 1500m D3 = 2000m , L = 1,⋯, L. We first get the simulated ergodic capacity via (5).

Fig. 1.Simulated ergodic capacity, analytical upper and lower bounds versus the SNR (L = 3, Nt = 2, Ω = 1, km = 1, Ωm = 2, υ = 4, D1 = 1000m , D2 = 1500m D3 = 2000m)

Clearly, the proposed upper and lower bounds tighten when the large number of BS antennas grows large. The upper bound matches the simulated ergodic capacity across the entire SNRs, while the lower bound converges to the exact high-SNR ergodic capacity, which is consistent with [18][38] . Then, we also observe that the lower bound is tighter that the upper bound for the high SNR regime. For Nr = 60, the curves of the upper bound and the simulated result are almost coincidence. For the high SNR regime, the curves of the lower bound and the simulated result are almost overlapped.

In Fig. 2, the asymptotic high SNR approximations for the upper bound in (16) and lower bound in (29) are compared with the simulated ergodic capacity in (5). The simulation parameters are the same of Fig. 1. As anticipated, the proposed bounds remains relatively tight for high SNR regime and large number of BS antennas Nr. From the zoomed figure, we can also observe that the lower bound is tighter that the upper bound at high SNRs. Fig. 2 also reveals that the proposed bounds approach the Monte-Carlo simulation for large value Nr.

Fig. 2.Simulated ergodic capacity and high SNR approximations for upper and lower bounds versus SNR (L = 3, Nt = 2, Ω = 1, km = 1, Ωm = 2, υ = 4, D1 = 1000m , D2 = 1500m , D3 = 2000m)

In Fig. 3, we investigate the simulated ergodic capacity and low SNR approximate capacity in (19) versus transmit energy per bit Eb/N0 (Proposition 2) for different transmit and receive antennas. For illustration purposes, we assume that the large scale fading matrix is set to the identity matrix Ξ = ILNt and L = 1. The Fig. 3 reveals that increasing the number of receive antennas Nr reduces the required Eb/N0min, and the same number of receive antennas Nr has the same required Eb/N0min. These confirm the analysis of Proposition 2 and coincide with the results of [31]. Moreover, the figure shows the larger wideband slope S0 is obtained with higher Nr and Nt . Finally, we can also observe that the analytical results sufficiently match the simulated results at low SNR regime.

Fig. 3.Low SNR capacity versus transmit Eb/N0 for different transceiver antennas (Nt = 1, Nr = 2; Nt = 2, Nr = 2 ; Nt = 1, Nr = 4)

The asymptotic capacity of large array systems is investigate in Fig. 4 for the case of Nt = 1 and Nt = 2, respectively. It is readily observed that a larger Nt increases the diversity and multiplexing gains, thereby yielding a large capacity. In addition, the capacity grows logarithmically with the number of BS antennas Nr without bound. Finally, we can also observe that the curves of lower bound in (28) and asymptotic capacity in (35) almost overlap with the simulated ergodic capacity in (5).

Fig. 4.Simulated ergodic capacity, lower bound and asymptotic capacity versus of the number of BS antennas (P = 20dB, L = 3, Ω = 1, km = 1, Ωm = 2, υ = 4, D1 = 1000m , D2 = 1500m D3 = 2000m)

 

6. Conclusion

In this paper, we elaborate on the ergodic capacity of D-MIMO systems over spatially non-correlated K fading channels. In particular, the closed-form upper and lower bounds on ergodic capacity are derived with the aid of majorization and Minkowski theories. It is demonstrated that the proposed upper bound remains relatively tight across the entire SNR regime and when the number of BS and transceiver antennas grows large, while proposed lower bound converges to the simulated results at high SNR regime. For high SNR regime, the lower bound is tighter than the upper bound, and vice versa. In order to obtain useful insights into the implications of the system parameters on the ergodic capacity, we also examine in detail the bounds of the asymptotic high and low SNR regimes. Finally, we explore the emerging area of massive MIMO systems in detail and the interesting results of the lower bound for the “large-systems” limits are derived. It is shown that the simulation results match theoretical analysis very well. The above analytical results encompass the Rayleigh multipath fading, Gamma shadowing fading and path loss of practical interest.