Hadamard upper bound on optimum joint decoding capacity of Wyner Gaussian cellular MAC
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Abstract
This article presents an original analytical expression for an upper bound on the optimum joint decoding capacity of Wyner circular Gaussian cellular multiple access channel (C-GCMAC) for uniformly distributed mobile terminals (MTs). This upper bound is referred to as Hadamard upper bound (HUB) and is a novel application of the Hadamard inequality established by exploiting the Hadamard operation between the channel fading matrix G and the channel path gain matrix Ω. This article demonstrates that the actual capacity converges to the theoretical upper bound under the constraints like low signal-to-noise ratios and limiting channel path gain among the MTs and the respective base station of interest. In order to determine the usefulness of the HUB, the behavior of the theoretical upper bound is critically observed specially when the inter-cell and the intra-cell time sharing schemes are employed. In this context, we derive an analytical form of HUB by employing an approximation approach based on the estimation of probability density function of trace of Hadamard product of two matrices, i.e., G and Ω. A closed form of expression has been derived to capture the effect of the MT distribution on the optimum joint decoding capacity of C-GCMAC. This article demonstrates that the analytical HUB based on the proposed approximation approach converges to the theoretical upper bound results in the medium to high signal to noise ratio regime and shows a reasonably tighter bound on optimum joint decoding capacity of Wyner GCMAC.
Keywords
Time Division Multiple Access Multiple Input Multiple Output Hadamard Product Path Gain Hadamard Inequality1. Introduction
The ever growing demand for communication services has necessitated the development of wireless systems with high bandwidth and power efficiency [1, 2]. In the last decade, recent milestones in the information theory of wireless communication systems with multiple antenna and multiple users have offered additional newfound hope to meet this demand [3, 4, 5, 6, 7, 8, 9, 10, 11]. Multiple input multiple output (MIMO) technology provides substantial gains over single antenna communication systems, however the cost of deploying multiple antennas at the mobile terminals (MTs) in a cellular network can be prohibitive, at least in the immediate future [3, 8]. In this context, distributed MIMO approach is a means of realizing the gains of MIMO with single antenna terminals in a cellular network allowing a gradual migration to a true MIMO cellular network. This approach requires some level of cooperation among the network terminals which can be accomplished through suitably designed protocols [4, 5, 6, 12, 13, 14, 15, 16]. Toward this end, in the last few decades, numerous articles have been written to analyze various cellular models using information theoretic argument to gain insight into the implications on the performance of the system parameters. For an extensive survey on this literature, the reader is referred to [5, 6, 17, 18, 19] and the references there in.
The analytical framework of this article is inspired by analytically tractable model for multicell processing (MCP) as proposed in [7], where Wyner incorporated the fundamental aspects of cellular network into the framework of the well known Gaussian multiple access channel (MAC) to form a Gaussian cellular MAC (GCMAC). The majority of the MCP models preserve fundamental assumptions which has initially appeared in Wyner's model, namely (i) interference is considered only from two adjacent cells; (ii) path loss variations among the MTs and the respective base stations (BSs) are ignored; (iii) the interference level at a given BS from neighboring users in adjacent cells is characterized by a deterministic parameter 0 ≤ Ω ≤ 1, i.e., the collocation of MTs (users).^{a}
A. Background and related study
In [7], Wyner considered optimal joint processing of all BSs by exploiting cooperation among the BSs. It has been shown that intra-cell time division multiple access (TDMA) scheme is optimal and achieves capacity. Later, Shamai and Wyner considered a similar model with frequency flat fading scenario and more conventional decoding schemes, e.g., single-cell processing (SCP) and two-cell-site processing schemes [5, 6]. It has also been shown that the optimum joint decoding strategy is distinctly advantageous over intra-cell TDMA scheme and fading between the terminals in a communication link increases the capacity with the increase in the number of jointly decoded users. Later, in [20] Wyner model has been modified by employing multiple transmitting and receiving antennas at both ends of the communication link in the cellular network where each BS is also composed of multiple antennas. Recently, new results have been published by further modifying the Wyner model with shadowing [21].
Recently, Wyner model has been investigated to account for randomly distributed users, i.e., non-collocated users [21, 22, 23, 24]. In [22], an instant signal-interference-ratio (SIR) and averaged throughout for randomly distributed users have been derived by employing TDMA and code division multiple access (CDMA) schemes. It has been shown that the Wyner model is accurate only for the system with sufficient number of simultaneous users. It has also been shown that for MCP scenario, the CDMA outperforms the inter-cell TDMA which is opposite to the original results of Wyner, where inter-cell TDMA is shown to be capacity achieving [7]. Later in the article, similar kind of analysis has also been presented for downlink case which is out of scope of this article. The readers are referred to [22] and references there in.
Although the Wyner model is mathematically tractable, but still it is unrealistic with respect to practical cellular systems that the users are collocated with the BSs and offering deterministic level of interference intensity to the respective BS. As a consequence, another effort has been made to derive an analytical capacity expression based on random matrix theory [21, 23]. Despite the fact that the variable-user density is used in this article, the analysis is only valid under the asymptotic assumptions of large number of MTs K, i.e., K → ∞ and infinite configuration of number of cooperating BSs N, i.e., N → ∞ such that $\frac{K}{N}\to c\in \left(0,1\right)$[17, 21, 23, 24]. On the contrary, the main contribution of our article is to offer non-asymptotic approach to derive information theoretic bound on Wyner GCMAC model where finite number of BSs arranged in a circle are cooperating to jointly decode the user's data.
B. Contributions
In this article, we consider a circular version of Wyner GCMAC (by wrap around the linear Wyner model to form a circle) which we refer to as circular GCMAC (C-GCMAC) throughout the article [12]. We consider an architecture where the BSs can cooperate to jointly decode all user's data, i.e., macro-diversity. Thus, we dispense with cellular structure altogether and consider the entire network of the cooperating BSs and the users as a network-MIMO system [12]. Each user has a link to each BS and BSs cooperate to jointly decode all user's data. The summary of main contributions of this article are described as follows. We derive a non-asymptotic analytical upper bound on the optimum joint decoding capacity of Wyner C-GCMAC by exploiting the Hadamard inequality for finite cellular network-MIMO setup. The bound is referred to as Hadamard upper bound (HUB). In this study, we alleviate the Wyner's original assumption by assuming that the MTs are uniformly distributed across the cells in Wyner C-GCMAC.
In first part of this article, we introduce the derivation of Hadamard inequality and its application to derive information theoretic bound on optimum joint decoding capacity which we referred to as theoretical HUB. The theoretical results of this article are exploited further to study the effect of variable path gains offered by each user in adjacent cells to the BS of interest (due to variable-user density). The performance analysis of first part of this article includes the presentation of capacity expressions over multi-user and single-user decoding strategies with and without intra-cell and inter-cell TDMA schemes to determine the existence of the proposed upper bound. In the second part of this article, we derive the analytical form of HUB by approximating the probability density function (PDF) of Hadamard product of channel fading matrix G and channel path gain matrix Ω. The closed form representation of HUB is presented in the form of Meijer's G-Function. The performance and comparison description of analytical approach includes the presentation of information theoretic bound over the range of signal-to-noise ratios (SNRs) and the calculation of mean area spectral efficiency (ASE) over the range of cell radii for the system under consideration.
This article is organized as follows. In Section II, system model for Wyner C-GCMAC is recast in Hadamard matrix framework. Next in Section III, the Hadamard inequality is derived as Theorem 3.3 based on Theorem 3.1 and Corollary 3.2. While in Section IV, a novel application of the Hadamard inequality is employed to derive the theoretical upper bound on optimum joint decoding capacity. This is followed by the several simulation results for a single-user and the multi-user scenarios that validate the analysis and illustrate the effect of various time sharing schemes on the performance of the optimum joint decoding capacity for the system under consideration. In Section V, we derive a novel analytical expression for an upper bound on optimum joint decoding capacity. This is followed by numerical examples and discussions in Section VI that validate the theoretical and analytical results, and illustrate the accuracy of the proposed approach for realistic cellular network-MIMO systems. Conclusions are presented in Section VII.
Notation: Throughout the article, ℝ^{N × 1}and ℂ^{N × 1}denote N dimensional real and complex vector spaces, respectively. Furthermore, ℙ^{N × 1}denotes N dimensional permutation vector spaces which has 1 at some specific position in each column. Moreover, the matrices are represented by an uppercase boldface letters, as an example, the N × M matrix A with N rows and M columns are represented as A^{N × M}. Similarly, the vectors are represented by a lowercase boldface italic version of the original matrix, as an example, a N × 1 column vector a is represented as a^{N × 1}. An element of the matrix or a vector is represented by the non-boldface letter representing the respective vector structure with subscripted row and column indices, as an example a_{n,m}refers to the element referenced by row n and column m of a matrix A^{N × M}. Similarly, a_{ k } refers to element k of the vector a^{N × 1}. Scalar variables are always represented by a non-boldface italic characters. The following standard matrix function are defined as follows: (·) ^{ T } denotes the non-Hermitian transpose; (·) ^{ H } denotes the Hermitian transpose; tr (·) denotes the trace of a square matrix; det (·) and | · | denote the determinant of a square matrix; ||A|| denotes the norm of the matrix A; $E\left[\cdot \right]$ denotes the expectation operator and (∘) denotes the Hadamard operation (element wise multiplication) between the two matrices.
2. Wyner Gaussian cellular Mac model
A. System model
where G_{N,K}∈ ℂ^{N×NK}such that ${G}_{N,K}~\mathcal{C}\mathcal{N}\left(0,{I}_{N}\right)$ and Ω_{N,K}∈ ℝ^{N×NK}such that ${\Omega}_{N,K}~\mathcal{U}\left(0,1\right)$. The modeling of channel path gain matrix Ω_{N,K}for a single-user and the multi-user environments can be well understood from the following Lemma.
Lemma 2.1: (Modeling of Channel Path Gain Matrix)
The matrix S is real and orthogonal, hence S^{-1} = S^{ T } and also the basis vectors are orthogonal for ℝ ^{ N } .
- Symmetrical channel path gain matrix: In this scenario, the structure of the channel path gain matrix is typically circular for a single-user case. Therefore, the path gains between the MTs T_{j+i}for {i = 0, ±1} and the respective BSs B_{ j }are deterministic and can be viewed as a row vector of the resultant N × N circular channel path gain matrix Ω. Mathematically, the first row of the channel matrix may be expressed as^{d}$\Omega \left(1,:\right)=\left({\Omega}_{{B}_{j}{T}_{j}},{\Omega}_{{B}_{j}{T}_{j+i}}0,0,0,{\Omega}_{{B}_{j}{T}_{j-i}}\right)$, where ${\Omega}_{{B}_{j}{T}_{j}}$ is the path gain between the intra-cell MTs T_{ j }and the respective BSs in j th cell and ${\Omega}_{{B}_{j}{T}_{j+i}}$ for i = ± 1 is the channel path gain between the MTs T_{j+i}for i = ± 1 in the adjacent cells and the respective BSs in j th cell. In this context, it is known that the circular matrix Ω can be expressed as a linear combination of powers of the shift operator S[27, 28]. Therefore, the resultant circular channel path gain matrix (symmetrical) for K = 1 active user in each cell can be expressed as${\Omega}_{N,1}={I}_{N}+{\Omega}_{{B}_{j}{T}_{j+1}}S+{\Omega}_{{B}_{j}{T}_{j-1}}{S}^{T},$(8)
where 1_{ K } denotes 1 × K all ones vector and (⊗) denotes the Kronecker product.
- Unsymmetrical channel path gain matrix: In this scenario, the MTs (users) in the adjacent cells are randomly distributed across the cells in the entire system. Therefore, the channel path gain matrix is not deterministic, and hence, the resultant matrix is no more circular. In this setup, the channel path gain matrix for single-user scenario can be mathematically modeled as follows:${\Omega}_{N,1}={I}_{N}+{\widehat{\Omega}}_{N,1}\circ S+{\widehat{\Omega}}_{N,1}\circ {S}^{T},$(10)
where ${\widehat{\Omega}}_{N,1}~\mathcal{U}\left(0,1\right)$.
B. Definitions
- i.
Intra-cell TDMA: a time sharing scheme where only one user in each cell in the system is allowed to transmit simultaneously at any time instant.
- ii.
Inter-cell TDMA: a time sharing scheme where only one cell in the system is active at any time instant such that each local user inside the cell is allowed to transmit simultaneously. The users in other cells in the system are inactive at that time instant.
- iii.
Channel path gain (Ω): normalized distance dependent path loss offered by intra-cell and inter-cell MTs to the BS of interest.
- iv.
MCP: a transmission strategy, where a joint receiver decodes all users data jointly (uplink); while the BSs can transmit information for all users in the system (downlink).
- v.
SCP: a transmission strategy where the BSs can only decode the data from their local users, i.e., intra-cell users and consider the inter-cell interference from the inter-cell users as a Gaussian noise (uplink); while the BSs can transmit information only for their local users, i.e., intra-cell users (downlink).
3. Information theory and Hadamard inequality
Theorem 3.1: (Hadamard Product)
where P_{ N } and ${\mathcal{P}}_{M}$ are N^{2} × N and M^{2} × M partial permutation matrices, respectively (in some of the literatures these matrices are referred to as selection matrices [29]). The j th column of P_{ N } and P_{ M } has 1 in its ((j - 1) N + j) th and ((j - 1) M + j) th positions, respectively, and zero elsewhere.
Proof: See [[31], Theorem 2.5].
Corollary 3.2: (Hadamard Product)
- i.
P _{ N } and P _{ M } are the only matrices of zeros and onces that satisfy (15) for all G and Ω.
- ii.
P ^{ T } P = I and PP ^{ T } is a diagonal matrix of zeros and ones, so 0 ≤ diag 0 (PP ^{ T } ) ≤ 1.
- iii.There exists a N ^{2} × (N ^{2} - N) matrix Q _{ N } and M ^{2} × (M ^{2} - M) matrix Q _{ M } of zeros and ones such that π ≜ (P Q) is the permutation matrix. The matrix Q is not unique but for any choice of Q, following holds:${P}^{T}Q=0;\phantom{\rule{1em}{0ex}}{Q}^{T}Q=I;\phantom{\rule{1em}{0ex}}Q{Q}^{T}=I-P{P}^{T}.$
- iv.Using the properties of a permutation matrix together with the definition of π in (iii); we have$\pi \phantom{\rule{0.3em}{0ex}}{\pi}^{T}=\left(P\phantom{\rule{1em}{0ex}}Q\right)\left(\begin{array}{c}\hfill {P}^{T}\hfill \\ \hfill {Q}^{T}\hfill \end{array}\right)=P{P}^{T}+Q{Q}^{T}=I.$
Theorem 3.3: (Hadamard Inequality)
This inequality is referred to as the Hadamard inequality which will be employed to derive the theoretical and analytical HUB on the capacity (14).
This completes the proof of Theorem 3.3. ■
An alternate proof of (18) is provided as Appendix A.
4. Theoretical Hub
Now, in the following sub-sections we analyze the validity of the HUB on optimum joint decoding capacity w.r.t a single-user and the multi-user environments under limiting constraints.
A. Single-user environment
i. Low inter-cell interference regime
It is to note that this is the scenario in cellular network when the MTs in adjacent cells are located far away from the BS of interest. Practically, the MTs in the adjacent cells which are located at the edge away from the BS of interest are offering negligible path gain.
This completes the proof of (21). ■
ii. Tightness of HUB--low SNR regime
This completes the proof of (25). ■
It is demonstrated in Figure 2 that as γ → 0, the gain inserted by the upper bound Δ = Δ_{0} ≈ 0 (compare the black solid curve with the red dashed curve). It can be seen from the figure that the theoretical HUB on optimum capacity is loose in the high range of SNR regime and comparably tight in the low SNR regime, and hence ${\stackrel{\u0304}{\mathsf{\text{C}}}}_{\mathsf{\text{opt}}}\left(p\left(H\right),\gamma \right)\approx {\mathsf{\text{C}}}_{\mathsf{\text{opt}}}\left(p\left(H\right),\gamma \right)$.
iii. Inter-cell TDMA scheme
The same has been shown in Figure 2. The black dashed-dotted curve and the curve with red square marker illustrate optimum capacity and theoretical HUB, respectively, when inter-cell interference is negligible, i.e., using (23). Next, the curve with green circle marker shows the capacity when inter-cell TDMA is employed, i.e., using (26).
B. Multi-user environment
In this section, we demonstrate the behavior of the theoretical HUB when two implementation forms of time sharing schemes are employed in multi-user environment. One is referred to as inter-cell TDMA, intra-cell narrowband scheme (TDMA, NB), and other is inter-cell TDMA, intra-cell wideband scheme [12]. We refer the later scheme as inter-cell time sharing, wideband scheme, (ICTS, WB) throughout the discussions. It is to note that SCP is employed only to determine the application of our bound for realistic cellular network.
i. Inter-cell TDMA, intra-cell narrow-band scheme (TDMA, NB)
ii. Inter-cell time sharing, wide-band scheme, (ICTS, WB)
Summary of theoretical Hadamard upper bound (HUB)
User(s) (K) | Constraints for ${C}_{opt}\left(p\left(H\right);\gamma \right)={\stackrel{\u0304}{C}}_{opt}\left(p\left(H\right);\gamma \right)$ | Constraints for ${C}_{opt}(p\left(H\right);\gamma )\; <{\stackrel{\u0304}{C}}_{opt}\left(p\left(H\right);\gamma \right)$ |
---|---|---|
K = 1 (Cooperative BS scenario) | i. Ω → 0, i.e., low level of inter-cell interference to the BS of interest. ii. γ → 0, i.e., the gain inserted by HUB Δ → 0 and is given by ${\Delta}_{0}=\underset{\gamma \to 0}{lim}\gamma E\left[\mathsf{\text{tr}}\left({\Gamma}_{\left(P,Q\right)}\right)\right]$. | $\Omega ~\mathcal{U}\left(0,1\right)$ (variable path gain among the MTs and the Bs of interest due to Uniformly distributed MTs across the cells). |
K > 1 (Non-cooperative BS scenario) | By employing intra-cell TDMA, intercell Narrowband (TDMA, NB) scheme. | By employing Inter-cell Time Sharing, Wideband (ICTS, WB) scheme. |
5. Analytical Hub
is the Shannon transform of a random square Hadamard composite matrix $\left(\stackrel{\u2323}{G}\circ \stackrel{\u2323}{\Omega}\right)$ and distributed according to the cumulative distribution function (CDF) denoted by ${\mathsf{\text{F}}}_{\stackrel{\u2323}{G}\circ \stackrel{\u2323}{\Omega}}\left(\mathsf{\text{tr}}\left(\stackrel{\u2323}{G}\circ \stackrel{\u2323}{\Omega}\right)\right)$[17], where $\stackrel{\u2323}{\gamma}=\gamma \phantom{\rule{0.3em}{0ex}}{N}^{2}$ and $\gamma =P/{\sigma}_{z}^{2}$ is the MT transmit power over receiver noise ratio.
where ${f}_{\stackrel{\u2323}{G}\circ \stackrel{\u2323}{\Omega}}\left(u\right)$ is the joint PDF of the $\mathsf{\text{tr}}\phantom{\rule{0.3em}{0ex}}\left(\stackrel{\u2323}{G}\right)$ and $\mathsf{\text{tr}}\phantom{\rule{0.3em}{0ex}}\left(\stackrel{\u2323}{\Omega}\right)$ which is evaluated as follows in the next sub-section.
A. Approximation of PDF of $\mathsf{\text{tr}}\phantom{\rule{0.3em}{0ex}}\left(\stackrel{\u2323}{G}\circ \stackrel{\u2323}{\Omega}\right)$
where we have made a use of Meijer's G-Function [35], available in standard scientific software packages, such as Mathematica, in order to transform the integral expression to the closed form and $\Theta =1\u221564\sqrt{2}{\pi}^{2}{\stackrel{\u2323}{\gamma}}^{2}$.
6. Numerical examples and discussions
7. Conclusion
The analytical upper bound referred to as HUB is derived on optimum joint decoding capacity for Wyner C-GCMAC under realistic assumptions: uniformly distributed MTs across the adjacent cells; and the finite number of cooperating BSs arranged in a circular configuration. New analytical approach have been reported to derive an information theoretic upper bound on the optimum joint decoding capacity of circular Wyner GCMAC. This approach is based on the approximation of the PDF of trace of composite Hadamard product matrix (G ∘ Ω) by employing the Hadamard inequality. A closed form expression has been derived to capture the effect of variable user-density in GCMAC. The proposed analytical approach has been validated by using Monte Carlo simulations for variable user-density cellular system. It has been shown that a reasonably tighter upper bound on optimum joint decoding capacity can be obtained by exploiting Hadamard inequality for realistic scenarios in cellular network. The importance of the methodology presented here lies in the fact that it allows a realistic representation of the MT's spatial arrangement. Therefore, this approach can be further exploited in order to investigate the various practical MT distributions and their effect on the optimum joint decoding capacity of system under consideration.
Appendix A
An Alternate Proof Of (18)
This completes the proof. ■
Remarks: The result (A.56) can be applied to the correlated scenario where the rank of the fading channel matrix may reduce to 1 [2, 15, 26]. Alternatively, the proof can be extended for channel matrix of any rank L. As an example, if A is a diagonalizable matrix of size N × N with rank L. Then, there are L square rank one matrices given as A_{1}, A_{2}, ..., A _{ L }, such that A = A_{1} + A_{2} + ... + A_{ L } and an alternative proof can be derived for such matrices.
ENDNOTES
^{a} MTs are also referred to as users and is interchangeably used throughout the article.
^{ b }T_{j+1}≜ T_{j+1}mod N.
^{c}Throughout this article, H_{N,K}, G_{N,K}and Ω_{N,K}refers to the channel matrices corresponding to N number of cells and K users per cell in a C-GCMAC. For brevity, the channel matrices will be expressed as H, G and Ω, respectively, unless it is necessary to emphasis the number of cells and the number of users.
^{d}Here, we used Matlab format to express row vector. For an example, Ω(1, :) shows First row vector of matrix Ω.
^{e}As an example, for N = 6 and K = 1, the partial permutation matrices are P ∈ ℙ^{36 × 6} and Q ∈ ℙ^{36 × 30}[26].
^{f}Terms with higher order of γ are ignored ⇔ γ^{ x } ≈ 0; ∀x = 2, 3, ... [33].
Notes
Acknowledgements
The authors would like to acknowledge the financial support of Picsel Technologies Ltd, Glasgow, UK and KAUST, KSA. This study was presented in part at IEEE International Symposium Wireless Communications and Systems, ISWCS'2010, York, UK, September 2010.
Supplementary material
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