Asymptotic Analysis of Double-Scattering Channels

Asymptotic Analysis of Double-Scattering Channels

Jakob Hoydis12, Romain Couillet3, and Mérouane Debbah2 1Department of Telecommunications,2Alcatel-Lucent Chair on Flexible Radio, Supélec, France 3EDF Chair on System Sciences and the Energy Challenge, Centrale Paris-Supélec, France
{jakob.hoydis, romain.couillet, merouane.debbah}@supelec.fr
Abstract

We consider a multiple-input multiple-output (MIMO) multiple access channel (MAC), where the channel between each transmitter and the receiver is modeled by the doubly-scattering channel model. Based on novel techniques from random matrix theory, we derive deterministic approximations of the mutual information, the signal-to-noise-plus-interference-ratio (SINR) at the output of the minimum-mean-square-error (MMSE) detector and the sum-rate with MMSE detection, which are almost surely tight in the large system limit. Moreover, we derive the asymptotically optimal transmit covariance matrices. Our simulation results show that the asymptotic analysis provides very close approximations for realistic system dimensions.

I Introduction

Most works on wireless multiple-input multiple-output (MIMO) systems share the underlying assumption of a rich scattering environment and, thus, Rayleigh or Rician fading channel matrices with full rank. However, several measurements of outdoor MIMO channels have shown that this assumption fails to hold in certain scenarios, where low-rank channels are observed despite low antenna correlation at the transmitter and receiver (see e.g. [1, 2]). Motivated by these observations, a generalized fading MIMO channel model, the so-called “doubly-scattering model” [3], was proposed and has since then attracted significant research interest. A special case of the doubly-scattering model is the keyhole channel [4, 5] which exhibits null correlation between the entries of the channel matrix but only a single degree of freedom. The existence of such channels in reality was confirmed by measurements in [5].

Several theoretical works have studied the doubly-scattering model so far. The authors of [6] derive capacity upper-bounds for the general model and a closed-form expression for the keyhole channel. An asymptotic study of the outage capacity of the multi-keyhole channel was presented in [7]. The diversity order of the doubly-scattering model was considered in [8] and it was shown that a MIMO system with transmit antennas, receive antennas and scatterers achieves the diversity of order . A closed-from expression of the diversity-multiplexing trade-off (DMT) was derived in [9]. Beamforming along the strongest eigenmode over Rayleigh product MIMO channels, i.e., the doubly-scattering model without any form of correlation, was considered in [10]. Here, the authors derive exact expressions of the cumulative distribution function (cdf) and the probability density function (pdf) of the largest eigenvalue of the Gramian of the channel matrix and compute closed-form results for the ergodic capacity, outage probability and signal-to-noise-plus-interference-ratio (SINR) distribution. In a later paper [11], the MIMO multiple access channel (MAC) with doubly-scattering fading is analyzed. The authors obtain closed-form upper-bounds on the sum-capacity and prove that the transmitters should send their signals along the eigenvectors of the transmit correlation matrices in order to maximize capacity.

Despite the significant interest in the doubly-scattering channel model, little work has been done to study its asymptotic performance when the channel dimensions grow large. We are only aware of [2], in which a model without transmit and receive correlation is studied relying on tools from free probability theory. Implicit expressions of the asymptotic mutual information and the SINR of the minimum-mean-square-error (MMSE) detector are found.

In this paper, we consider a MIMO MAC with double-scattering fading in its most general form and derive deterministic approximations of the (ergodic) mutual information, the (ergodic) sum-rate with MMSE detection and the SINR at the output of the MMSE detector. The approximations become almost surely exact as the dimensions of all channel matrices grow large and can be easily numerically computed with negligible computing complexity. In addition, we provide the asymptotically capacity maximizing transmit covariance matrices and present an iterative water-filling algorithm for their computation. Our numerical results suggest that the asymptotic approximations are already very tight for channel dimensions with as little as four transmit and receive antennas and are therefore of clear practical value.

The key idea behind the proofs in this paper is that the doubly-scattering channel model can be interpreted as a Kronecker channel [12] with a random receive correlation matrix, which itself is modeled by the Kronecker model. This observation allows us to build upon [12] which provides an asymptotic analysis of the Kronecker channel model with deterministic correlation matrices. We then extend this work by allowing the correlation matrices to be random. The results in this paper are obtained through advanced tools from random matrix theory (inspired by [13, 14], see also the textbook [15] for a comprehensive introduction and a contemporary overview of recent research results) and are hence not only a novel contribution to the field of wireless communications but also to the field of large random matrix theory. We also believe that the developed techniques can be successfully applied to the study of even more involved channel models, such as channels with line-of-sight (LOS) components or MIMO product channels with an arbitrary number of matrices.

Ii System model

Consider a discrete-time MIMO channel from transmitters, equipped with () antennas, respectively, to a receiver with antennas. The channel output vector at a given time reads

 y =K∑k=1Hkxk+n (1)

where and , , are the channel matrix and the transmit vector associated with the th transmitter, and is a noise vector. The channel matrices are modeled by the double-scattering model [3]

 Hk=1√NknkR12kW1,kS12kW2,kT12k (2)

where , and are deterministic correlation matrices, while and are independent standard complex Gaussian matrices. Since the distributions of and are unitarily invariant we can assume without loss of generality to be diagonal matrices. Denote the instantaneous normalized mutual information of the channel (1) in nats/s/Hz, defined as

 IN(ρ)=1Nlogdet(IN+1ρK∑k=1HkQkHHk). (3)

Iii Main results

The notation denotes in the sequel that and all , grow infinitely large, satisfying , . These conditions ensure that all matrix dimensions grow at a similar speed. Additionally we need the following technical assumptions:

A 1

For all , , and , where is the spectral norm.

Our first theorem introduces a set of implicit equations which uniquely determines the quantites (). These quantities will be used in the sequel to provide deterministic approximations of which become almost surely arbitrarily tight as .

Theorem 1 (Fundamental equations)

The following system of implicit equations in , and ():

 ¯gk =1nktrT12kQkT12k(gkT12kQkT12k+Ink)−1 gk =1nkNk∑j=1sk,jδk1+¯gksk,jδk (4) δk =1NktrRk(K∑k=1nkNk¯gkgkδkRk+ρIN)−1

has a unique solution satisfying for all and .

Remark III.1

One can also prove that , and can be computed by a classical fixed-point algorithm which iteratively computes (4), starting from some arbitrary initialization . This algorithm generally converges in a few iterations (depending on the system size) and does not pose any computational challenge.

The next theorem provides a deterministic, asymptotically tight approximation of the (ergodic) mutual information based on the quantites as provided by Theorem 1.

Theorem 2 (Mutual information)
 (i) IN(ρ)−¯IN(ρ)a.s.% −−−−→N→∞0 (ii) EIN(ρ)−¯IN(ρ)−−−−→N→∞0

where

 ¯IN(ρ) = 1Nlogdet(IN+1ρK∑k=1nkNk¯gkgkδkRk) +1NK∑k=1[logdet(INk+¯gkδkSk) +logdet(Ink+gkT12kQkT12k)−2nkgk¯gk]

and are the unique positive solutions to (4).

The following result allows us to compute the asymptotically optimal precoding matrices which maximize under individual transmit power constraints.

Theorem 3 (Optimal power allocation)

The solution to the following optimization problem:

 (¯Q∗1,…,¯Q∗K) = argmaxQ1,…,Qk¯IN(ρ) s.t.1nktrQk≤Pk ∀k

is given as , where is defined by the spectral decomposition of and is given by the water-filling solution:

 ¯p∗k,j=(μk−1g∗ktk,j)+ (5)

where is chosen to satisfy and is given by Theorem 1 for .

Remark III.2

The optimal power allocation matrices can be calculated by the iterative water-filling Algorithm 1 (see [12, Remark 2] and [13, Remark 5] for a discussion of the convergence of this algorithm).

The last two results of this correspondence provide deterministic approximations of the SINR at the output of the MMSE detector and the sum-rate with MMSE detection.

Theorem 4 (SINR of the MMSE detector)

Assume and for all and let be the SINR at the output of the MMSE detector related to the transmit symbol , given by

 γk,j=hHk,j(K∑i=1HiHHi−hk,jhHk,j+ρIN)−1hk,j.

Then

 γk,j−¯γk,ja.s% .−−−−→N→∞0

where and is by given by Theorem 1.

Remark III.3

The theorem is also valid under the more general assumptions and . We can then simply define the matrices and for which the theorem holds.

Corollary 1 (Sum-rate with MMSE decoding)

Under the same assumptions as in Theorem 4, let

 R(ρ)=1NK∑k=1nk∑j=1log(1+γk,j).

Then,

 (i) R(ρ)−1NK∑k=1nk∑j=1log(1+tk,jgk)a.s.−−−−→N→∞0 (ii) ER(ρ)−1NK∑k=1nk∑j=1log(1+tk,jgk)a.s.−−−−→N→∞0

where is given by Theorem 1.

Iii-a The Rayleigh product channel

A special case of the double-scattering channel is the Rayleigh product MIMO channel [10] which does not exhibit any form of correlation between the transmit and receive antennas or the scatterers. For this model, the Theorems 1, 2 and 4 can be given in closed-from as shown in the next corollary.

Corollary 2 (Rayleigh product channel)

For all , let , and assume , and . Then,

 ¯IN(ρ) = log(1+1ρNKS¯g(¯g+SN−1)) − KSNlog(1+NS(¯g−1)) −Klog(¯g)−2K(1−¯g)

and

 ¯γk,j=1−¯g¯g

where is the unique solution to

 ¯g3−¯g2(2−SN−1K) +¯g(1−SN−1K+SNK(1+ρ))−SNKρ = 0 (6)

such that and .

Note that similar expressions for the asymptotic mutual information and MMSE-SINR have been obtained in [2] by means of free probability theory. However, their results require the numerical solution of a third order differential equation.

Iv Numerical examples

As a first example, we consider the “multi-keyhole channel”, i.e., , , , , for . The signal-to-noise-ratio (SNR) is denoted by . Fig. 1 depicts the normalized ergodic mutual information and its asymptotic approximation versus SNR for a single () and multiple () “keyholes”. Surprisingly, the match between both results is almost perfect although the channel dimensions are very small.

As a second example, we consider a multiple access channel from transmitters, assuming the double-scattering model in [3]. Under this model, the correlation matrices are given as , and , where is defined as

 [G(ϕ,d,n)]k,l=1nn−12∑j=1−n2exp(i2πd(k−l)sin(jϕ1−n)).

The values and determine the angular spread of the radiated and received signals, and are the antenna spacings at the th transmitter and receiver in multiples of the signal wavelength, and can be seen as the number and as the spacing of the scatterers. For simplicity, we assume , , , , and for all . We further assume for all , with and . Fig. 2 shows and with uniform and optimal power allocation versus SNR. Again, our asymptotic results yield very tight approximations for even small system dimensions. For comparison, we also provide the sum-rate with MMSE detection and its deterministic approximation . We observe a good fit between both results at low SNR values, but a slight mismatch for higher values. This is due to a slower convergence of the SINR to its deterministic approximation .

V Conclusion

We have studied a MIMO MAC with doubly-scattering fading channels. Under the assumption that the dimensions of all channel matrices grow infinitely large, we have derived almost surely tight deterministic approximations of the mutual information, the SINR of the MMSE detector and the sum-rate with MMSE detection. In addition, we have provided an iterative water-filling algorithm to compute the asymptotically optimal transmit covariance matrices. Our numerical results show that the asymptotic analysis provides very close approximations for very small system dimensions with as little as four transmit and received antennas. We believe that the techniques used in this paper could be succesfully applied to the study of even more involved channel models.

Vi Appendix

Proof:

The key idea is that the doubly scattering model can be considered as the Kronecker channel model [12] with random correlation matrices. For the Kronecker model, the matrices are given as

 Hk=1√nkZkW2,kT12k (7)

where is a deterministic matrix and and are as defined in (2). The fundamental equations (cf. Theorem 1) for this model are given by [12, Corollary 1]

 ¯ek =1nktrT12kQkT12k(ekT12kQkT12k+Ink)−1 ek =1nktrZkZHk(K∑i=1¯eiZiZHi+ρIN)−1. (8)

Assume now to be random and modeled as

 Zk=1√NkR12kW1,kS12k. (9)

Notice first that the expressions of the quantities are unaffected by this assumption. Second, have become random quantities and it is our goal to find deterministic approximations of , such that as . Following similar steps as in the proof of [13, Theorem 4], one can now show that

 maxk|¯ek−¯gk| a.s.−−−−→N→∞0 maxk|ek−gk| a.s.−−−−→N→∞0 (10)

where and satisfy (4). The proof of uniqueness of such solutions relies on arguments of so called standard functions [16] and follows similar steps as in [13, Proof of Theorem 3] or [14, Proof of Theorem 1]. \qed

Proof:

We rely again on the observation that the doubly scattering model can be considered as the Kronecker channel model [12] with random correlation matrices (cf. (7) and (9)). A deterministic equivalent of the mutual information for the Kronecker model (7) was provided in [12, Theorem 2]. Here, is defined as a function of the quantities and (given as the unique solutions to (8)) which reads:

 ¯IKronN(ρ)=1Nlogdet(IN+1ρK∑k=1¯ekZkZHk) +K∑k=11Nlogdet(Ink+ekT12kQkT12k)−1NK∑k=1nkek¯ek. (11)

Due to the almost sure convergence of and established in Theorem 1, we can simply replace and by and , respectively. It remains now to find a deterministic equivalent of the first term , which is random since the matrices are random. However, this term is nothing but the mutual information of another Kronecker channel with matrices . Hence we can apply again [12, Theorem 2] to find its deterministic equivalent. Combining both results yields and terminates the proof of . Denote the probability space engendering the sequences . Then, on a sub-space of of measure , we have by : as . Integrating this expression over implies by the dominated convergence theorem [17] part . \qed

Proof:

Similar to the proof of [12, Proposition 3], one can show that the covariance matrices should align to the eigenvectors of the transmit correlation matrices to maximize , i.e., . Note that it was also proved in [11, Theorem 1] that these signaling directions are optimal to maximize . One can then further show that

 d¯IN(ρ)d¯p∗k,j=g∗ktk,j1+g∗ktk,j¯p∗k,j∀k,j

and . Since is hence strictly concave in it follows from the KKT conditions [18] that is given by the water-filling solution (5) with power constraint . \qed

Proof:

Assume that are given by the Kronecker model (7) with deterministic matrices . It follows from standard lemmas of random matrix theory (see e.g. [13, Appendix C], that the following limit holds:

 γk,j−tk,j1nktrZkZHk(K∑i=1HiHHi+ρInk)a.s.−−−−→N→∞0.

Applying [12, Theorem 1] to the second term yields

 γk,j−tk,j1nktrZkZHk(K∑i=1¯eiZiZHi+ρInk)a.s.−−−−→N→∞0

where are given as the solutions to (8). Notice from (8) that the second term is equal to . Assume now the matrices random and given by (9). Thus, we have by (10)

 γk,j−tk,jgka.s.−−−−→N→∞0.
\qed
Proof:

Part follows directly from Theorem 4 and the continuous mapping theorem [19, Theorem 2.3]. Part follows from the same arguments as in the proof of Theorem 3 . \qed

Proof:

One can show by straight-forward but tedious calculations that the fundamental equations (4) can be reduced to a single implicit equation (2) if , and for all . Replacing and by and in the expressions of and , respectively, leads to the desired result. \qed

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