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High-SNR power offset in multiantenna communication.
... Despite the many research advances on multiuser broadcast channels (see e.g., [4]- [6] and references therein), understanding and quantifying the fundamental difference in the expected sum capacity with optimal and sub-optimal precoding techniques remains a sparsely investigated area. The focus of the paper is to close this research gap. ...
... The focus of the paper is to close this research gap. The authors of [6], [7] present some preliminary results on this topic for point-to-point and multiuser MIMO systems for simple uncorrelated Rayleigh fading channels. Nevertheless, such analysis do not capture, to the full extent, the heterogeneity present in multiuser channels due to wide ranging propagation conditions. ...
... In reality, some users may experience the presence of dominant line-of-sight (LOS) components, while others may be experience heavy non-LOS (NLOS) conditions. In contrast to [6], [7], we consider the general case of a heterogeneous Ricean fading channel, where each terminal has a unique Ricean K-factor and LOS component steering angle. For single-stream transmission, we analyze the expected sum capacity loss between DPC and ZF, and for multistream transmission, we consider BD precoding expected sum capacity performance relative to DPC over a wide range of operating signal-to-noise ratios (SNRs), BS/user antenna numbers, and Ricean K-factors. ...
We analytically approximate the expected sum capacity loss between the optimal downlink precoding technique of dirty paper coding (DPC), and the sub-optimal technique of zero-forcing precoding, for multiuser channels. We also consider the most general case of multi-stream transmission to multiple users, where we evaluate the expected sum capacity loss between DPC and block diagonalization precoding. Unlike previously, assuming heterogeneous Ricean fading, we utilize the well known affine approximation to predict the expected sum capacity difference between both precoder types (optimal and sub-optimal) over a wide range of system and propagation parameters. Furthermore, for single-stream transmission, we consider the problem of weighted sum capacity maximization, where a similar quantification of the sum capacity difference between the two precoder types is presented. In doing so, we disclose that power allocation to different users proportional to their individual weights asymptotically maximizes the weighted sum capacity. Numerical simulations are presented to demonstrate the tightness of the developed expressions relative to their simulated counterparts.
... Spatially correlated MIMO channels have been well characterized for a variety of transmit correlation models [3]–[6]. Traditionally, transmit correlation has been considered to be a detrimental source, thereby incurring power loss at high signal-to-noise ratio (SNR) (e.g., [7]). Some exceptions where transmit correlation helps capacity are the case of low SNR [5], [6] , where the capacity-achieving input covariance is nonisotropic , and the case where channel state information (CSI) is not available at all [8], for which knowing the statistics of the channel may effectively help. ...
... Remark 1. An alternative expression of the asymptotic capacity behavior for r ≥ K can be found by utilizing the approach in [7], [32] 6 . Comparing with the alternative characterization and other previous results [5], [6] for the point-to-point MIMO case, we can see that (8) in ...
... It immediately follows from [7] and [29] that, for M ≥ K, the high-SNR capacity of the i.i.d. 5 Since users in a particular group are often closely located, the intra-group cooperation within such a group is more feasible than the full cooperation across all users over the entire BS coverage. 6 Although these point-to-point results assume that only the distribution of a channel is accessible at the transmitter, the difference from the perfect CSIT case that we are assuming vanishes at high SNR when the number of receive antennas is greater than or equal to the total number of transmit antennas (this is the case of the dual MAC in (6) when r ≥ K ). ...
Correlation across transmit antennas, in multiple antenna systems (MIMO), has
been studied in various scenarios and has been shown to be detrimental or
provide benefits depending on the particular system and underlying assumptions.
In this paper, we investigate the effect of transmit correlation on the
capacity of the Gaussian MIMO broadcast channel (BC), with a particular
interest in the large-scale array (or massive MIMO) regime. To this end, we
introduce a new type of diversity, referred to as transmit correlation
diversity, which captures the fact that the channel vectors of different users
may have different, and often nearly mutually orthogonal, large-scale channel
eigen-directions. In particular, when taking the cost of downlink training
properly into account, transmit correlation diversity can yield significant
capacity gains in all regimes of interest. Our analysis shows that the system
multiplexing gain can be increased by a factor up to ,
where M is the number of antennas and is the common rank of the
users transmit correlation matrices, with respect to standard schemes that are
agnostic of the transmit correlation and treat the channels as if they were
isotropically distributed. Thus, this new form of diversity reveals itself as a
valuable "new resource" in multiuser communications.
... This enables the direct comparison of different strategies with equal diversity gain under different channel models and CSI assumptions. A similar high-SNR affine characterization under ergodic channel conditions is proposed in [29], being, thus, the ergodic capacity the performance measure of interest. The authors in [29] point out the importance of extending their results to the nonergodic case when some of the degrees of freedom of the channel are sacrificed to increase the diversity gain, i.e., in different operational points of the DMT curve, and this is exactly the contribution of this paper. ...
... A similar high-SNR affine characterization under ergodic channel conditions is proposed in [29], being, thus, the ergodic capacity the performance measure of interest. The authors in [29] point out the importance of extending their results to the nonergodic case when some of the degrees of freedom of the channel are sacrificed to increase the diversity gain, i.e., in different operational points of the DMT curve, and this is exactly the contribution of this paper. Our approach is also similar to that in [18], where an affine approximation of the high-SNR outage capacity is derived. ...
... (38) The SNR gap for uncorrelated Rician channels coincides with that derived in [36, Th. 1], while the strictly negative nature of the SNR gap for semicorrelated Rayleigh channels recalls the known fact that the outage probability with isotropic inputs is increased by antenna correlation [37]. As in the ergodic case addressed in [29], this penalty can be arbitrarily large if one or some of the eigenvalues of the correlation matrix are small. Finally, it is worth remarking that assumption (34) does not include all rate adaptation strategies with . ...
Following the seminal work by Zheng and Tse on the diversity and multiplexing tradeoff (DMT) of multiple-input multiple-output (MIMO) channels, in this paper, we introduce the array gain to investigate the fundamental relation between transmission rate and reliability in MIMO systems. The array gain gives information on the power offset that results from exploiting channel state information at the transmitter or as a consequence of the channel model. Hence, the diversity, multiplexing, and array gain (DMA) analysis is able to cope with the limitations of the original DMT and provide an operational meaning in the sense that the DMA gains of a particular system can be directly translated into a parameterized characterization of its associated outage probability performance. In this paper, we derive the best DMA gains achievable by any scheme employing isotropic signaling in uncorrelated Rayleigh, semicorrelated Rayleigh, and uncorrelated Rician block-fading MIMO channels. We use these results to analyze the effect of important channel parameters on the outage performance at different points of the DMT curve.
... In particular, (15) becomes asymptotically tight with O(1) replaced by o(1), when K g = r g . An alternative expression of (15) can be found in (107) of Appendix B, based on the approach in [7], [27] 5 . Comparing with the alternative characterization, we can see that (15) in Theorem 2 is more intuitive and insightful, even though the remaining term is O(1) instead of o(1). ...
... which approaches close to the high-SNR capacity of the i.i.d. Rayleigh fading MIMO channel [7] C ...
... Using Lemma 2 in [7] (See also [27]), we can get ...
We investigate the impact of transmit correlation on the capacity of
correlated fading MIMO broadcast channels (BCs) and establish capacity
characterizations in various regimes of system parameters, with a particular
interest in the large-scale array (or massive MIMO) regime. It is advocated in
this paper that transmit correlation can be of use to increase both
multiplexing gain and power gain in multiuser MIMO systems. We show the
potential gains and succinctly characterize fundamental limits in correlated
fading MIMO BCs, assuming an ideal condition for which \emph{transmit
correlation diversity} is well defined. In particular, transmit correlation is
shown to improve the system multiplexing gain up to by a factor of the degrees
of transmit correlation diversity. Not relying on the ideal condition on
transmit correlations, we further propose a joint spatial division and
multiplexing (JSDM) scheme based on opportunistic beamforming that, for a large
number of users, can achieve near optimal sum-rate scaling with very limited
channel state feedback in realistic channels.
... The resulting zero-order term or high-load offset captures the impact of other channel features, such as scheduling and coding. The high-load refinement is inspired by the seminal works of Shamai et al. [22] and Lozano et al. [19], dealing (among other things) with the power offset in the high-SNR regime. In fact, the definitions of the high-load slope and offset are similar to the respective high-SNR slope and offset, where the term in the latter is replaced by the term . ...
... The difference lies in the fact that while the SNR is related to the input signals and additive noise vector statistics, the number of users is related to the resulting fading statistics via the user selection scheme. This fact makes the high-load parameters hard to calculate and much more setting dependent than the general expressions derived for the high-SNR parameters [19]. ...
... Proof: See Appendix B. Evidently, is a decreasing function of the interference factor . Comparing (11) to (18), it is clear that the ZFBF scheme is superior to the ICTS scheme, when the SNR is above a certain threshold (19) which is an increasing function of . It is noted that for , the circulant channel transfer matrix is singular and channel inversion methods, such as ZFBF are not applicable. ...
For a multiple-input single-output (MISO) downlink channel with M transmit antennas, it has been recently proved that zero-forcing beamforming (ZFBF) to a subset of (at most) M "semi-orthogonal" users is optimal in terms of sum-rate, asymptotically with the number of users. However, determining the subset of users for transmission is a complex optimization problem. Adopting the ZFBF scheme in a cooperative multi-cell scenario renders the selection pr ocess even more difficult since more users are likely to be involved. In this paper, we consider a multi-cel l cooperative ZFBF scheme combined with a simple sub-optimal users selection procedure for the Wyner downlink channel setup. According to this sub-optimal procedure, the user with the "best" local channel is selected for transmission in each cell. The performance of this sub-optimal scheme is investigated in terms of both, the conventional scaling law of the sum-rate with the number of users, and a sum-rate offset. We term this characterization of the sum-rate for large number of users as high-load regime characterization, and point out the similarity of this approach with the standard affine approximation used in the high-SNR regime. It is shown that under an overall power constraint, the sub-optimal cooperative multi-cell ZFBF scheme achieves the same sum-rate growth rate and slightly degraded offset law, when compared to an optimal scheme deploying joint multi-cell dirty-paper coding (DPC) techniques, asymptotically with the number of users
... given by (37). To solve the integral I 5 in (37) which involves an incomplete gamma function with a shift parameter, we first apply [31, eq. ...
... To gain insight on the average secrecy capacity at high SNR, the slope and power offset at the high SNR are also needed to be analyzed. To facilitate the asymptotic analysis, we adopt the general form in [37] to express C ∞ s as ...
In this paper, we investigate the secrecy performance against eavesdropping of a land mobile satellite (LMS) system, where the satellite employs the spot beam technique, and both the terrestrial user and eavesdropper are equipped with multiple antennas and utilize maximal ratio combining (MRC) to receive the confidential message. Specifically, in terms of the availability of the eavesdropper’s CSI at the satellite, we consider both passive (Scenario I) and active (Scenario II) eavesdropping. For Scenario I where the eavesdropper’s channel state information (CSI) is unknown to the satellite, closed-form expressions for the probability of non-zero secrecy capacity and secrecy outage probability are derived. Furthermore, expressions for the asymptotic secrecy outage probability are also presented to reveal the secrecy diversity order and array gain of the considered system. For Scenario II where the eavesdropper’s CSI is available at the satellite, novel expressions for the exact and asymptotic average secrecy capacity are obtained. Based on a simple asymptotic formula, we can characterize the high signalto- noise ratio (SNR) slope and high SNR power offset of the LMS systems. Finally, simulations are provided to validate our theoretical analysis and show the effect of different parameters on the system performance.
... Interpreting the 1/ e term in (18) as a high-SNR offset 3 [10] gives a detailed discussion of the importance of high-SNR offset in the capacity analysis of MIMO systems. Note that this offset is missing in (5). ...
... (similarly to [10]), the diversity gain in (14) becomes ( ) ( ) d d r γ ≈ ∆ γ . Using (18), the differential diversity gain (17) can be expressed as ...
Diversitymultiplexing tradeoff (DMT) presents a compact framework to compare various MIMO systems and channels in terms of the two main advantages they provide (i.e. high data rate and/or low error rate). This tradeoff was characterized asymptotically (SNR-> infinity) for i.i.d. Rayleigh fading channel by Zheng and Tse (2003). The SNR-asymptotic DMT overestimates the finite-SNR one (R. Narasimhan, 2006). In this paper, using the recent results on the size-asymptotic (in the number of antennas) outage capacity distribution, we derive and analyze the finite-SNR DMT for a broad class of channels (not necessarily Rayleigh fading). Systems with unequal number of Tx and Rx antennas exhibit qualitatively-different behavior from those with equal number of antennas: while the size-asymptotic DMT of the latter converges to the SNR- asymptotic DMT as the SNR grows, that of the former does not. However, the size-asymptotic DMT does provide an accurate approximation of the true DMT at low to moderately-high SNR, even for modest number of antennas, and hence is complementary of the SNR-asymptotic DMT of Zheng and Tse. Combining these two, a new DMT is obtained that is accurate over the whole SNR range. A number of generic properties of the DMT that hold at any SNR, for any number of antennas (i.e. not only asymptotic, either in size or in SNR) and for any fading channel are given. In particular, we demonstrate that the linear interpolation of the DMT for fractional multiplexing gain in (Zheng and Tse 2003) does not hold at finite SNR. Extensive Monte-Carlo simulations validate the analysis and the conclusions.
... This suggests that the constant c (high-SNR offset) should also be included in the DMT if the error rate performance is of importance. This problem is somewhat eliminated by using the multiplexing gain definition in (16), as 3 [8] gives a detailed discussion of the importance of high-SNR offset in the capacity analysis of MIMO systems. Note that this offset is missing in (5). ...
... Interpreting the 1/ e term in (19) as a high-SNR offset (similarly to [8]), the diversity gain in (14) becomes ( ) ( ) d d r γ ≈ ∆ γ . Using (19), the differential diversity gain (17) can be expressed as ...
Diversity-multiplexing tradeoff (DMT) was characterized asymptotically (SNR- > infinity) for i.i.d. Rayleigh fading channel by Zheng and Tse . The SNR-asymptotic DMT overestimates the finite-SNR one . This paper outlines a number of additional limitations and difficulties of the DMT framework and discusses their implications. Using the recent results on the size-asymptotic (in the number of antennas) outage capacity distribution, the finite-SNR, size-asymptotic DMT is derived for a broad class of fading distributions. The SNR range over which the finite-SNR DMT is accurately approximated by the SNR-asymptotic one is characterized. The multiplexing gain definition is shown to affect critically this range and thus should be carefully selected, so that the SNR-asymptotic DMT is an accurate approximation at realistic SNR values and thus has operational significance to be used as a design criterion. The finite-SNR diversity gain is shown to decrease with correlation and power imbalance in a broad class of fading channels, and such an effect is described in a compact, closed form. Complete characterization of the outage probability (or outage capacity) requires not only the finite-SNR DMT, but also the SNR offset, which is introduced and investigated as well. This offset, which is not accounted for in the DMT framework, is shown to have a significant impact on the outage probability for a broad class of fading channels, especially when the multiplexing gain is small. The analytical results and conclusions are validated via extensive Monte Carlo simulations. Overall, the size-asymptotic DMT represents a valuable alternative to the SNR-asymptotic one.
... Interestingly, it can be found that (29) can be also derived from (23) by setting ρ as infinity, which suggests that this lower bound is exact in the high-SNR regime. The ECC in the high-SNR regime can be generally approximated as [18] C ≈ S ∞ (log 2 ρ − L ∞ ) , (30) where S ∞ = lim ρ→∞ C log 2 ρ denotes the high-SNR slope in bits/s/Hz/(3 dB), and L ∞ = lim ρ→∞ log 2 ρ − C S∞ denotes the high-SNR power offset in 3 dB units. Based on (15) and (29), we can evaluate S ∞ and L ∞ in closed-form as follows. ...
Reconfigurable intelligent surfaces (RISs) have emerged as a promising technique to enhance the system spectral efficiency. This letter investigates the ergodic channel capacity (ECC) of an RIS-aided multiple-input multiple-output channel under the assumption that the transmitter-RIS, RIS-receiver, and transmitter-receiver channels contain deterministic line-of-sight paths. Novel expressions are derived to characterize the upper and lower bounds of the ECC. To unveil more system insights, asymptotic analyses are performed to the system ECC in the limit of large signal-to-noise ratio (SNR) and number of reflecting elements (REs). Theoretical analyses suggest that the RIS's deployment can shape the ECC curve by influencing its high-SNR power offset and the ECC can get improved by increasing the number of REs.
... These key parameters explicitly capture the impact of antenna configurations and channel fading coefficients on the achievable secrecy rate. Therefore, they have previously been examined for non-secrecy scenarios in [32]–[34]. We commence the asymptotic analysis by presenting the first order expansion of F γ M (γ ) in the high SNR regime. ...
This paper advocates physical layer security of maximal ratio combining (MRC) in wiretap two-wave with diffuse power fading channels. In such a wiretap channel, we consider that confidential messages transmitted from a single antenna transmitter to an M-antenna receiver are overheard by an N-antenna eavesdropper. The receiver adopts MRC to maximize the probability of secure transmission, whereas the eavesdropper adopts MRC to maximize the probability of successful eavesdropping. We derive the secrecy performance for two practical scenarios: 1) the eavesdropper's channel state information (CSI) is available at the transmitter and 2) the eavesdropper's CSI is not available at the transmitter. For the first scenario, we develop a new analytical framework to characterize the average secrecy capacity as the principal security performance metric. Specifically, we derive new closed-form expressions for the exact and asymptotic average secrecy capacity. Based on these, we determine the high signal-to-noise ratio power offset to explicitly quantify the impacts of the main channel and the eavesdropper's channel on the average secrecy capacity. For the second scenario, the secrecy outage probability is the primary security performance metric. Here, we derive new closed-form expressions for the exact and asymptotic secrecy outage probability. We also derive the probability of nonzero secrecy capacity. The asymptotic secrecy outage probability explicitly indicates that the positive impact of M is reflected in the secrecy diversity order and the negative impact of N is reflected in the secrecy array gain. Motivated by this, we examine the performance gap between N and N+1 antennas based on their respective secrecy array gains.
... However, the paper's general upper bound is given as an infinite summation of Hayakawa polynomials of one matrix argument, which the authors acknowledge as quite involved and computationally inefficient. We also refer to the work presented by Lozano et al. [20], who considered the high-SNR capacity offsets in order to establish the key effect of the so-called power offset on MIMO performance. Summarizing, it appears that no tractable analytical results exist in the literature for the upper bound of double-sided correlated MIMO systems in Ricean fading with arbitrary-rank of the deterministic component. ...
In this paper, a novel analytical upper bound on the ergodic capacity of Multiple-Input Multiple-Output (MIMO) communication systems is derived, based on a key power normalization. Given their high practical usability, we are particularly interested in dual-branch configurations where both the transmitter (Tx) and receiver (Rx) deploy two antenna elements. Contrary to the majority of related studies, where only the common case of Rayleigh fading was considered, our analysis is extended to account for the generalized case of Ricean fading where a deterministic Line-of-Sight (LoS) component exists in the communication link and both ends are affected by spatial correlation. In the following, it is clearly shown that the proposed bound is not only remarkably simple and efficient but also applicable for any arbitrary system Signal-to-Noise Ratio (SNR) and rank of the mean channel matrix. The tightness of the bound is also explored where it is demonstrated that as the SNR tends to zero the bound becomes asymptotically tight; at high SNRs, the offset between empirical capacity and the proposed bound is analytically computed which implies that an explicit asymptotic capacity expression can ultimately be obtained.
... We remark that (38) and (39) can also be directly used to obtain tight bounds on the high SNR power offset of DSC- ARM MIMO channels, as defined in [21] 2 . ...
This paper presents new statistical properties of complex noncentral matrix-variate quadratic forms. In contrast to previous results, the expressions do not involve infinite sums over partitions, or matrix-variate polynomials, and are easily and efficiently computable. These properties are used to derive new upper and lower bounds on the ergodic mutual information of double-sided correlated Rician MIMO channels with arbitrary-rank channel mean matrices. The bounds are shown to be tighter than previous reported bounds in the literature
... In addition, [5] provides also the high-SNR characterization of the output-erasure channel rate, albeit the calculation of the high-SNR power offset involves a fixed point-equation. Applying definitions of [12] to the rate expression (6), we provide an alternative expression for the high-SNR power offset as well. ...
This paper considers the input-erasure Gaussian channel. In contrast to the output-erasure channel where erasures are applied to the output of a linear time-invariant (LTI) system, here erasures, known to the receiver, are applied to the inputs of the LTI system. Focusing on the case where the input symbols are independent and identically distributed (i.i.d)., it is shown that the two channels (input- and output-erasure) are equivalent. Furthermore, assuming that the LTI system consists of a two-tap finite impulse response (FIR) filter, and using simple properties of tri-diagonal matrices, an achievable rate expression is presented in the form of an infinite sum. The results are then used to study the benefits of joint multicell processing (MCP) over single-cell processing (SCP) in a simple linear cellular uplink, where each mobile terminal is received by only the two nearby base-stations (BSs). Specifically, the analysis accounts for ergodic shadowing that simultaneously blocks the mobile terminal (MT) signal from being received by the two BS. It is shown that the resulting ergodic per-cell capacity with optimal MCP is equivalent to that of the two-tap input-erasure channel. Finally, the same cellular uplink is addressed by accounting for dynamic user activity, which is modelled by assuming that each MT is randomly selected to be active or to remain silent throughout the whole transmission block. For this alternative model, a similar equivalence results to the input-erasure channel are reported.
... As part of the prior art in this respect, the analytical tools to study the power-bandwidth tradeoff in the power-limited regime have been previously developed in the context of point-to-point single-user communications [15][16] , and were extended to multiuser (point-to-multipoint and multipoint-to-point) set- tings [17][20], as well as to adhoc wireless networking examples of single-relay channels [21][22], AWGN multihop networks [23][25] and dense multi-antenna relay networks [26]. In the bandwidth-limited regime, the necessary tools to perform the power-bandwidth tradeoff analysis were developed by [17] in the context of codedivision multiple access (CDMA) systems and were later used by [27] and [28] to characterize fundamental limits in multi-antenna channels over point-to-point and broadcast communication, respectively, and by [24], [26] to study a variety of adhoc networking scenarios. We assume that all terminals in the multiuser relay channel are supplied with finite total average transmit power P (in Watts (W)) over unconstrained bandwidth B (in Hertz (Hz)). ...
The goal of this paper is to understand the key merits of multihop relaying techniques jointly in terms of their energy efficiency and spectral efficiency advantages in the presence of multiuser diversity gains from opportunistic (i.e., channel-aware) scheduling and identify the regimes and conditions in which relay-assisted multiuser communication provides a clear advantage over direct multiuser communication. For this purpose, we use Shannon-theoretic tools to analyze the tradeoff between energy efficiency and spectral efficiency (known as the power-bandwidth tradeoff) over a fading multiuser relay channel with K users in the asymptotic regime of large (but finite) number of users (i.e., dense network). Benefiting from the extreme-value theoretic results of [1], we characterize the power-bandwidth tradeoff and the associated energy and spectral efficiency measures of the bandwidth-limited high signal-to-noise ratio (SNR) and power-limited low SNR regimes, and utilize them in investigating the large system behavior of the multiuser relay channel as a function of the number of users and physical channel SNRs. Our analysis results in very accurate closed-form formulas in the large (but finite) K regime that quantify energy and spectral efficiency performance, and provides insights on the impact of multihop relaying and multiuser diversity techniques on the power-bandwidth tradeoff.
... The structure of rate-optimal designs for scenarios in which the channel is zero-mean but the covariance has a more general structure than the Kronecker model was considered in [13]. While obtaining closed-form expressions for ergodic rate-optimal input covariance matrices seems to be quite difficult in general [14], significant insight into the structure of those matrices can be drawn by studying asymptotically low-SNR and high-SNR regimes, [2], [3], [15] and [16], [17], respectively. In addition to ergodic rate objectives, the design of the input covariance matrices can be tailored to meet other pragmatic objectives; e.g., outage capacity [18] and various error rate perspectives [19], [20]. ...
... Rayleigh channel [2], followed by results for the semi-correlated [3]- [8] and doubly-correlated Rayleigh channels [9]- [11]. The mean MI has also been studied in limiting cases, either in terms of antennas or SNRs [12]- [17]. ...
In this paper, we present a general analytical framework for the exact mutual information (MI) cumulants of multiple-input multiple-output (MIMO) systems with perfect receiver channel state information (CSI) and no transmitter CSI. Our derivation is based on a recent parameterization of the joint ordered eigenvalue probability density function (PDF), that encompasses both uncorrelated/semi-correlated Rayleigh channels as well as uncorrelated Rician channels. In addition, we extend our framework to account for the cumulants of doubly-correlated Rayleigh channels and also to deduce tractable expressions in the high Signal-to-Noise ratio (SNR) regime. The cumulants are particularly useful to study all high-order statistics (HOS) of the MI; in fact, they can be used to express the MI mean and variance as a finite sum of determinants. Our analytical expressions are then validated via Monte-Carlo simulations with the attained accuracy being excellent in all cases.
... Argumentation: For the argumentation, we use the concept of the high-SNR slope. In [18] the high-SNR slope is defined as ...
Secrecy on the physical layer receives increased research interest. Especially in the wireless communication scenarios, the interest in confidentiality of messages increases. In this paper, we discuss the effectiveness of the beamforming strategies maximum ratio transmission (MRT), zero forcing (ZF), and an optimized beamforming (BF) strategy for the transmission under secrecy constraints. The applied channel model is the flat-fading MISO (multiple-input single-output) wiretap channel, where the channel to the eavesdropper is only partly known at the transmitter. The scenario with perfect and no channel information to the eavesdropper are included as special cases. We show that under certain conditions the achievable secrecy rates for the transmission with optimized beam-forming can be increased by transmitting artificial noise (AN) in the null space of the channel to the intended receiver, which produces additional interference only at the eavesdropper. We provide a characterization of the optimal beamforming and artificial noise beamforming and analyze the optimal power allocation for asymptotic cases (low and high SNR as well as no and perfect channel information). Numerical simulations illustrate the results.
... The MIMO capacity at high and low SNR for interference-limited scenarios is addressed in [16], [17]. A worst-case analysis for MIMO capacity with CSI at the transmitter and at the receiver, conditioned on the channel matrix, can be found in [18]. ...
Multiple-input multiple-output (MIMO) systems are being considered as one of the key enabling technologies for future wireless networks. However, the decrease in capacity due to the presence of interferers in MIMO networks is not well understood. In this paper, we develop an analytical framework to characterize the capacity of MIMO communication systems in the presence of multiple MIMO co-channel interferers and noise. We consider the situation in which transmitters have no channel state information, and all links undergo Rayleigh fading. We first generalize the determinant representation of hypergeometric functions with matrix arguments to the case when the argument matrices have eigenvalues of arbitrary multiplicity. This enables the derivation of the distribution of the eigenvalues of Gaussian quadratic forms and Wishart matrices with arbitrary correlation, with application to both single-user and multiuser MIMO systems. In particular, we derive the ergodic mutual information for MIMO systems in the presence of multiple MIMO interferers. Our analysis is valid for any number of interferers, each with arbitrary number of antennas having possibly unequal power levels. This framework, therefore, accommodates the study of distributed MIMO systems and accounts for different spatial positions of the MIMO interferers.
... For the low-SNR regime, [8] shows that the fixed-point equations characterizing the rate can be approximated to a closed-form solution. Using this result, the low-SNR parameters of this achievable rate, namely the minimum energy per-bit required for reliable communication and respective energy per-bit slope at zero rate [29], can be expressed as a function of the low-SNR parameters of the unlimited setup and the link capacity C. In addition [8] provides the rate in which the limited backhaul capacity should scale with the SNR for the achievable rate to maintain the multiplexing gain and high- SNR power offset (see [30]) of the unlimited setup [18]. In the case of cluster CI (case (ii) above), an achievable rate is derived in [7][8] by allowing partial local decoding at the base stations. ...
Recent information theoretic results on cooperation in cellular systems are reviewed, addressing both multicell processing (cooperation among base stations) and relaying (cooperation at the user level). Two central issues are addressed, namely, first multicell processing is studied with either limited-capacity backhaul links to a central processor or only local (and finite-capacity) cooperation among neighboring cells. The role of codebook information, decoding delay and network planning (frequency reuse) are specifically highlighted along with the impact of different transmission/ reception strategies. Next, multicell processing is considered in the presence of cooperation at the user level, focusing on both out-of-band relaying via conferencing users and in-band relaying by means of dedicated relays. Non-fading and fading uplink and downlink channels adhering to simple Wyner-type, cellular system models are targeted.
... In addition, [5] provides also the high-SNR characterization of the output-erasure channel rate, albeit the calculation of the high-SNR power offset involves a fixed point-equation. Applying definitions of [12] to the rate expression (6), we provide an alternative expression for the high-SNR power offset as well. ...
This paper considers the input-erasure Gaussian channel. In contrast to the output-erasure channel where erasures are applied to the output of a linear time-invariant (LTI) system, here erasures, known to the receiver, are applied to the inputs of the LTI system. Focusing on the case where the input symbols are independent and identically distributed (i.i.d)., it is shown that the two channels (input- and output-erasure) are equivalent. Furthermore, assuming that the LTI system consists of a two-tap finite impulse response (FIR) filter, and using simple properties of tri-diagonal matrices, an achievable rate expression is presented in the form of an infinite sum. The results are then used to study the benefits of joint multicell processing (MCP) over single-cell processing (SCP) in a simple linear cellular uplink, where each mobile terminal is received by only the two nearby base-stations (BSs). Specifically, the analysis accounts for ergodic shadowing that simultaneously blocks the mobile terminal (MT) signal from being received by the two BS. It is shown that the resulting ergodic per-cell capacity with optimal MCP is equivalent to that of the two-tap input-erasure channel. Finally, the same cellular uplink is addressed by accounting for dynamic user activity, which is modelled by assuming that each MT is randomly selected to be active or to remain silent throughout the whole transmission block. For this alternative model, a similar equivalence results to the input-erasure channel are reported. Comment: Presented at the Forty-Sixth Annual Allerton Conference on Communication, Control, and Computing, Sep. 2008, Monticello, Illinois
... Second, SNR is directly related to the transmitted power, which is an important design parameter. A significant amount of efforts have been made in order to achieve high values of SNR [3] [5]. However, none of these methods relate the complexity of multiuser receivers for achieving high SNR values. ...
Wireless Multiuser receivers suffer from their relatively higher computational complexity that prevents widespread use of this technique. In addition, one of the main characteristics of multi-channel communications that can severely degrade the performance is the inconsistent and low values of SNR that result in high BER and poor channel capacity. It has been shown that the computational complexity of a multiuser receiver can be reduced by using the transformation matrix (TM) algorithm [4]. In this paper, we provide quantification of SNR based on the computational complexity of TM algorithm. We show that the reduction of complexity results high and consistent values of SNR that can consequently be used to achieve a desirable BER performance. In addition, our simulation results suggest that the high and consistent values of SNR can be achieved for a desirable BER performance. The performance measure adopted in this paper is the consistent values of SNR. Comment: 9 pages IEEE format, International Journal of Computer Science and Information Security, IJCSIS July 2009, ISSN 1947 5500, Impact Factor 0.423
Intelligent reflecting surfaces (IRSs) are envisioned to provide reconfigurable wireless environments for future communication networks. In this paper, both downlink and uplink IRS-aided non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) networks are studied, in which an IRS is deployed to enhance the coverage by assisting a cell-edge user device (UD) to communicate with the base station (BS). To characterize system performance, new channel statistics of the BS-IRS-UD link with Nakagami-
m
fading are investigated. For each scenario, the closed-form expressions for the outage probability and ergodic rate are derived. To gain further insight, the diversity order and high signal-to-noise ratio (SNR) slope for each scenario are obtained according to asymptotic approximations in the high-SNR regime. It is demonstrated that the diversity order is affected by the number of IRS reflecting elements and Nakagami fading parameters, but the high-SNR slope is not related to these parameters. Simulation results validate our analysis and reveal the superiority of the IRS over the full-duplex decode-and-forward relay.
Due to the underlying sparse structure of the mmWave channels, which indeed makes the exact closed-form capacity expressions inherently hard to derive, there has been less research on the ergodic capacity of mmWave systems. To overcome this problem, by means of the majorization theory, this paper analyzes the ergodic capacity of point-to-point mmWave communication systems under finite-dimensional channel model. In particular, we derive several closed-form ergodic capacity approximations, which exhibit excellent tightness in spite of whether the steering matrices are singular or not. Then, several Jensen’s approximations and bounds of the ergodic capacity are also derived. The results indicate that the ergodic capacity seems to increase logarithmically with the number of antennas, the transmit SNR per antenna, and the eigenvalues of the steering matrix products. Besides, the DFT matrices can effectively characterize the spatial directions of mmWave channels when the number of antennas grows large. After that, high-SNR ergodic capacity, high-SNR slope, and power offset are also analyzed. It indicates that for a finite-dimensional channel, the maximum multiplexing gain increases with the number of paths instead of the number of antennas in Rayleigh channels. Numerical simulations are performed to validate the results.
Shannon’s mutual information of a random multiple antenna and multipath time varying channel is studied in the general case where the process constructed from the channel coefficients is an ergodic and stationary process which is assumed to be available at the receiver. From this viewpoint, the channel can also be represented by an ergodic self-adjoint block-Jacobi operator, which is close in many aspects to a block version of a random Schr(odinger operator. The mutual information is then related to the so-called density of states of this operator. In this paper, it is shown that under the weakest assumptions on the channel, the mutual information can be expressed in terms of a matrix-valued stochastic process coupled with the channel process. This allows numerical approximations of the mutual information in this general setting. Moreover, assuming further that the channel coefficient process is a Markov process, a representation for the mutual information offset in the large Signal to Noise Ratio regime is obtained in terms of another related Markov process. This generalizes previous results from Levy et.al. [18], [19]. It is also illustrated how the mutual information expressions that are closely related to those predicted by the random matrix theory can be recovered in the large dimensional regime.
This paper investigates the achievable sum rates of uplink massive multiple-input multiple-output systems with
matched-filter
(MF) and
zero-forcing
(ZF) receivers, under the spatial–temporal correlation channel scenario. First, we give the lower and upper bounds of the system capacity. Then, the
asymptotic equivalent
(AE) expressions of the bounds are obtained based on the random matrix theory. Moreover, the
power scaling laws
of MF and ZF receivers are exploited. Specifically, it is shown that the transmit power of each user can be scaled down by both MF and ZF receivers. Besides, the existence of the optimal number of users for the ZF receiver is proved based on the AE lower bound, and an efficient algorithm is proposed to obtain the near-optimal number of users when the number of receive antennas is fixed. Meanwhile, some simplified expressions of key parameters are obtained in the low-signal-to-noise-ratio (SNR) regime (i.e., the minimum normalized transmit energy per information bit and wideband slope) and the high-SNR regime (i.e., high-SNR slope and power offset). The simulation results and complexity analysis validate the proposed bounds and AE expressions.
Cochannel interference is one of the inevitable deleterious components in designing and analyzing of a wireless network. The use of multiple antennas at both transmitting and receiving nodes is a promising technique to suppress and/or alleviate the effect of cochannel interference on capacity. In this paper, we assess the effects of both antenna correlation and cochannel interference on the ergodic capacity of multiple-input multiple-output channels with covariance feedback. In particular, we consider a general family of spatial fading correlation model-called
unitary-independent-unitary
-which encompasses most of zero-mean channels with arbitrary fading profiles including the popular separable correlation channel models. We derive the average minimum mean-square error and signal-to-interference-plus-noise ratio of the parallel spatial streams using Berezin’s supermathematics. We then put forth the structure of optimal input covariance matrix maximizing the mutual information connected with the necessary and sufficient conditions as a generalization of the noise-limited case, which is tested by a simple iterative algorithm. Together with the powerful supermathematical framework, the result in the paper enables us to quantify the multiuser MIMO interference effects on the capacity in terms of spatial correlation and interference power heterogeneity.
In this letter, we investigate the problem of classification with high dimensional data using low dimensional random projections in the presence of inter- and intra- signal correlations. Each sensor is assumed to compress its high dimensional (Gaussian) signal vector using random projections in a multi-sensor setting. In order to quantify the classification performance with compressed data, we consider the Bhattacharya distance as the performance metric. In the presence of intra-signal correlation at a given sensor, the degradation in the Bhattacharya distance with compressed data is shown to be non-linear with the compression ratio in contrast to the case when there is no intra-signal correlation. In the presence of inter-signal correlation, the degradation in the Bhattacharya distance with compressed data depends on whether or not an identical projection matrix is used to compress data at multiple sensors. IEEE
We introduce a novel analytical framework for the end-to-end (e2e) maximum throughput under delay constraints, namely as effective capacity (EC), for multi-source and multidestination amplify-and-forward cooperative networks. The network operates in the presence of Rayleigh fading and employs frequency-division duplex nodes having the ability to simultaneously transmit as sources and receive as relays. Co-channel interference and noise is present at the relay nodes, whereas the destination nodes are noise-limited. A linear precoding technique is applied during reception to combine the input signals. As precoding, zero forcing (ZF), maximal-ratio combining (MRC), and minimum mean-squared error (MMSE) are studied. Each relay forwards the received signal to the destination by employing the maximum-ratio transmission scheme (MRT). Both exact analytical expressions and tight high signal-to-noise ratio bounds of e2e EC are obtained for the ZF/MRT scheme, while the optimal power allocation problem maximizing the e2e EC is also addressed. For the MRC/MRT and MMSE/MRT schemes, we derive approximate, yet highly accurate EC analytical expressions, as well as asymptotically tight closed-form expressions. Selected numerical and simulation results show that MMSE/MRT always yields the best performance followed by the ZF/MRT and MRC/MRT schemes. Moreover, it is shown that as the number of relay nodes increases, the ZF/MRT and MMSE/MRT schemes achieve almost identical and always better e2e EC performance than the MRC/MRT one.
We investigate the capacity scaling of MIMO systems with the system dimensions. To that end we quantify how the mutual information varies when the number of antennas (at either the receiver or transmitter side) is altered. For a system comprising R receive and T transmit antennas with R > T, we find the following: By removing as many receive antennas as needed to obtain a square system (provided the channel matrices before and after the removal have full rank) the maximum resulting loss of mutual information over all signal-to-noise ratios (SNRs) depends only on R, T and the matrix of left-singular vectors of the initial channel matrix, but not on its singular values. In particular, if the latter matrix is Haar distributed the ergodic rate loss is given by ∑Tt=1 ∑Rr=T+1 1/r-t nats. Under the same assumption, if T,R → ∞ with the ratio ϕ ≜ T/R fixed, the rate loss normalized by R converges almost surely to H(ϕ) bits with H(∙) denoting the binary entropy function. We also quantify and study how the mutual information as a function of the system dimensions deviates from the traditionally assumed linear growth in the minimum of the system dimensions at high SNR.
Epilepsy is one of the major neurological disorders occurring in the brain which causes sudden and recurrent seizures. These seizures cause loss of motor control in the human brain thereby bringing dangerous situations in the life of an epileptic patient. The patients tend to develop anxiety problems throughout their life because the seizures occur without any preliminary warnings. Electroencephalogram (EEG) is used to measure the recordings of the human brain and it helps greatly as a tool for analyzing, recognizing, diagnosing and classifying the epilepsy from EEG signals. Since the recordings are pretty huge, it is quite hectic to process the entire data and therefore certain dimensionality reduction techniques are required to reduce the dimensions of the data. In this paper, the dimensions of the data is reduced with the help of Linear Discriminant Analysis (LDA) and then the dimensionally reduced data is transmitted with the help of Space Time Trellis Coded Multiple Input Multiple Output Orthogonal Frequency Division Multiplexing (STTC MIMO-OFDM) System. At the receiver side, the Expectation Maximization Based Gaussian Mixture Model (EM-GMM) is engaged to classify the epilepsy from EEG signals. Also the Bit Error Rate (BER) is analyzed at the receiver side along with the classification accuracy. The performance measures analyzed here are Specificity, Sensitivity, Time Delay, Quality Values, Performance Index and Accuracy.
In this paper, we consider a point-to-point multiinput multi-output wireless powered communication system, where the source S is powered by a dedicated power beacon (PB) with multiple antennas. Employing the time splitting protocol, the energy constrained source S first harvests energy through the radio-frequency signals sent by the PB, and then uses this energy to transmit information to the destination D. Unlike several prior works, we assume that the energy transfer link is subjected to Rician fading, Which is a real fading environment, due to relatively short range power transfer distance and the existence of a strong line of sight path.We present a comprehensive analysis of the achievable ergodic capacity in two scenarios, depending on the availability of channel state information (CSI) at PB, namely, absence of CSI and partial CSI. For the former case, equal power allocation is used, while for the later one, energy beamforming is used to enhance the energy transfer efficiency. For both cases, closed-form expressions for the upper and lower bounds of the ergodic capacity are derived. Furthermore, the optimal time split is discussed, and the capacity in the low and high signal-to-noise ratio regimes is studied through simple closed-form expressions. Numerical results and simulations are provided to validate the theoretical analysis. The results show that the Rician factor K has a significant impact on the ergodic capacity performance, and this impact strongly depends on the availability of CSI in the PB.
Despite the importance of Rayleigh-product multiple-input multiple-output (MIMO) channels and their experimental validations, there is no work investigating their performance in the presence of residual additive transceiver hardware impairments, which arise in practical scenarios. Hence, this paper focuses on the impact of these residual imperfections on the ergodic channel capacity for optimal receivers, and on the ergodic sum-rates for linear minimum mean-squared-error (MMSE) receivers. Moreover, the low and high-signal-to-noise ratio (SNR) cornerstones are characterized for both types of receivers. Simple closed-form expressions are obtained that allow the extraction of interesting conclusions. For example, the minimum transmit energy per information bit for optimal and MMSE receivers are not subject to any additive impairments. In addition to the exact analysis, we also study the Rayleigh-Product channels in the large system regime, and we elaborate on the behavior of the ergodic channel capacity with optimal receivers by varying the severity of the transceiver additive impairments.
Seizure is a kind of transient abnormal behaviour of neurons occurring within the brain which disturbs the mental and physical activities of the patient. Due to the synchronized activity of large group of neurons, epileptic seizures occur. These epileptic seizures cause numerous changes in the behaviour and perception of the patient. Even if the epileptic seizures are quite rare in a given patient, the fear of the occurrence of next seizure and helplessness feeling will have a very strong influence on the daily life of a patient. To improve the quality of life of these epileptic patients, a systematic method to predict the occurrence of seizures should be reliable. In biomedical science, EEG signals provide very valid contributions and hence the careful analysis of EEG recordings is required to understand the valuable information. To capture the brain signals of the epileptic patient, Electroencephalography (EEG) is widely used. EEG has a good temporal resolution and is non invasive in nature with a low maintenance cost. Since the EEG recordings are too huge to process, certain form of dimensionality reduction technique is required to reduce the dimensions of the EEG data in order to process the data. In this paper, Power Spectral Density (PSD) is used a dimensionality reduction technique to reduce the dimensions. It is then transmitted through the Space Time Trellis Coded Multiple Input Multiple Output Orthogonal Frequency Division Multiplexing (STTC MIMO-OFDM) System. As the system suffers from a high PAPR, Quadrature Mirror Filtering Based Partial Transmit Sequence (QMF-PTS) is proposed to reduce the PAPR. At the receiver, the classifier engaged here is Polynomial Kernel Based Support Vector Machine (PK-SVM) to classify the epilepsy from EEG signals. The performance metrics is analyzed in terms of performance index, sensitivity, specificity, time delay, quality value and accuracy.
To assess the neurological conditions, non-invasive techniques for recording activities of the brain are used widely. The recording of the brain activity helps us to understand the brain functions properly. The scalp Electroencephalography (EEG) recordings act as a fundamental tool to trace the pathological brain activity and so it is used widely to treat the patients affected with epilepsy. Since the recordings of the EEG are quite long and hectic to process, Linear Graph Embedding (LGE) is used to reduce the dimensions of the EEG data in this paper. It is then transmitted to the Space Time Trellis Coded Multiple Input Multiple Output Orthogonal Frequency Division Multiplexing (STTC MIMO OFDM) System. As the system suffers from a high PAPR, Principal Component Analysis Based Selective Mapping Technique (PCA-SLM) is used to reduce the PAPR. At the receiver the Gaussian Kernel Based Support Vector Machine (GK-SVM) is employed to act as a post classifier to classify the epilepsy risk levels from EEG signals. Along with the classification results, the Bit Error Rate (BER) is also analyzed in the receiver. The performance metrics here are Specificity, Sensitivity, Time Delay, Quality Values, Accuracy and Performance Index.
We provide nonasymptotic upper and lower bounds on the sum-rate capacity of
Rayleigh block-fading multiple-access channels for the setup where a priori
channel state information is not available. The upper bound relies on a dual
formula for channel capacity and on the assumption that the users can cooperate
perfectly. The lower bound is derived assuming a noncooperative scenario, where
each user employs unitary space-time modulation (independently from the other
users). Numerical results show that the gap between the upper and the lower
bound is small already at moderate SNR values. This suggests that the sum-rate
capacity gains obtainable through user cooperation are minimal.
In this paper, we investigate the ergodic sum rate of K fading MIMO channels with linear MMSE receiver, which is an analytically friendly fading model to describe both the small scale fading and shadowing. The exact closed-form expression of the ergodic sum rate was derived, which is applicable for arbitrary number of transmit and receive antennas and any signal to noise ratio (SNR) range. Moreover, simple expressions were derived for the ergodic sum rate in the high and low SNR regimes. The analytical results enable us to gain insights on how the key system parameters affect the ergodic sum rate of the system. Numerical results are presented and verified via Monte Carlo simulations.
Using the Coulomb Fluid method, this paper derives central limit theorems
(CLTs) for linear spectral statistics of three "spiked" Hermitian random matrix
ensembles. These include Johnstone's spiked model (i.e., central Wishart with
spiked correlation), non-central Wishart with rank-one non-centrality, and a
related class of non-central F matrices. For a generic linear statistic, we
derive simple and explicit CLT expressions as the matrix dimensions grow large.
For all three ensembles under consideration, we find that the primary effect of
the spike is to introduce an O(1) correction term to the asymptotic mean of
the linear spectral statistic, which we characterize with simple formulas. The
utility of our proposed framework is demonstrated through application to three
different linear statistics problems: the classical likelihood ratio test for a
population covariance, the capacity analysis of multi-antenna wireless
communication systems with a line-of-sight transmission path, and a classical
multiple sample significance testing problem.
Land mobile satellite can exploit multiple input multiple output techniques to achieve high transmission rates. This article evaluates, theoretically, the capacity of the single input multiple output system utilizing uniform linear arrays at the receiver terminal for satellite applications. The theoretical study is performed at C-band and accounts for different shadowing conditions. Additionally, polarization effects are introduced and capacity results are presented that take into account the depolarization. For this investigation, a model for the scattering caused depolarization based on Stokes parameters is applied. Decrease of channel capacity is determined for some special cases both for Rayleigh fading and for the ULA with different number of receive antennas.
A second-order approximation is derived for the ergodic capacity of a MIMO communication system in which the mean and covariance of the channel are known at transmitter. This approximation is then used to efficiently obtain an almost optimal input covariance. More specifically, the design of the input covariance matrix that optimizes the second-order approximation is cast as a convex optimization problem. Using the Karush-Kuhn-Tucker optimality conditions for that problem, an efficient algorithm is devised for obtaining second-order optimal input covariance matrices. Our design examples show that the resulting input covariances yield ergodic rates that are close to the ergodic capacity of the system.
We investigate the sum capacity of Block Diagonalization precoding Multiple Input Multiple Output Broadcast Channels (BD MIMO
BC) with imperfect Channel State Information (CSI) at the base station. Since it is difficult to obtain the exact expression,
a lower and an upper bounds of the sum capacity under Gaussian channel estimation errors are drived instead. Analyses show
that the gap between two bounds is considerably tight at all Signal to Noise Ratio (SNR) region. From the lower bound of the
sum capacity, we can see that the multiplexing gain tends to be zero at high SNR region, which indicates that the BD MIMO
BC system with channel estimation errors is interference-limited at high SNR.
A compact model for the evaluation of scaling laws in random wireless networks is proposed and studied. The model allows the information-theoretic characterization of both point-to-point as well as distributed communications. It is analyzed under several assumptions about spatial correlation and the utilized channel state information and transmission schemes.
Cooperative decoding at the base stations (or access points) of an infrastructure wireless network is currently well recognized as a promising approach for intercell interference mitigation, thus enabling high frequency reuse. Deployment of cooperative multicell decoding depends critically on the tolopology and quality of the available backhaul links connecting the base stations. This work studies a scenario where base stations are connected only if in adjacent cells, and via finite-capacity links. Relying on a linear Wyner-type cellular model with no fading, achievable rates are derived for the two scenarios where base stations are endowed only with the codebooks of local (in-cell) mobile stations, or also with the codebooks used in adjacent cells. Moreover, both uni- and bidirectional backhaul links are considered. The analysis sheds light on the impact of codebook information, decoding delay, and network planning (frequency reuse) on the performance of multicell decoding as enabled by local and finite-capacity backhaul links. Analysis in the high-signal-to-noise ratio (SNR) regime and numerical results validate the main conclusions.
Selected mapping (SLM) is a popular scheme for peak power reduction in orthogonal frequency-division multiplexing (OFDM) systems. In this letter, the performance of various versions of SLM, among them ordinary and directed SLM, in single- and multi-antenna point-to-point OFDM systems is assessed. Analytic expressions for the distribution of the PAR are derived. Numerical results cover that significant gains over conventional SLM can be achieved by directed SLM.
In the last few years, the asymptotic distribution of the singular values of certain random matrices has emerged as a key tool in the analysis and design of wireless communication channels. These channels are characterized by random matrices that admit various statistical descriptions depending on the actual application. The goal of this paper is the investigation and application of random matrix theory with particular emphasis on the asymptotic theorems on the distribution of the squared singular values under various assumption on the joint distribution of the random matrix entries
We consider K links operating concurrently in the same spectral band. Each
transmitter has multiple antennas, while each receiver uses a single antenna.
This setting corresponds to the multiple-input single-output interference
channel. We assume perfect channel state information at the single-user
decoding receivers whereas the transmitters only have estimates of the true
channels. The channel estimation errors are assumed to be bounded in elliptical
regions whose geometry is known at the transmitters. Robust beamforming
optimizes worst-case received power gains, and a Pareto optimal point is a
worst-case achievable rate tuple from which it is impossible to increase a
link's performance without degrading the performance of another. We
characterize the robust beamforming vectors necessary to operate at any Pareto
optimal point. Moreover, these beamforming vectors are parameterized by
real-valued parameters. We analyze the system's spectral efficiency at
high and low signal-to-noise ratio (SNR). Zero forcing transmission achieves
full multiplexing gain at high SNR only if the estimation errors scale linearly
with inverse SNR. If the errors are SNR independent, then single-user
transmission is optimal at high SNR. At low SNR, robust maximum ratio
transmission optimizes the minimum energy per bit for reliable communication.
Numerical simulations illustrate the gained theoretical results.
A contemporary perspective on the tradeoff between transmit antenna diversity and spatial multiplexing is provided. It is argued that, in the context of most modern wireless systems and for the operating points of interest, transmission techniques that utilize all available spatial degrees of freedom for multiplexing outperform techniques that explicitly sacrifice spatial multiplexing for diversity. In the context of such systems, therefore, there essentially is no decision to be made between transmit antenna diversity and spatial multiplexing in MIMO communication. Reaching this conclusion, however, requires that the channel and some key system features be adequately modeled and that suitable performance metrics be adopted; failure to do so may bring about starkly different conclusions. As a specific example, this contrast is illustrated using the 3GPP Long-Term Evolution system design.
A multiple transmit multiple receive (MTMR) capacity survey has
been performed in Manhattan, New York for mobile and pedestrian wireless
services, using 16 transmitters and 16 receivers. Shannon capacities
close to the Rayleigh i.i.d. have been measured for 2 Tx 2 Rx, 4 Tx 4
Rx, 16 Tx 16 Rx systems
We consider a narrowband point-to-point communication system with nT transmitters and nR receivers. We assume the receiver has perfect knowledge of the channel, while the transmitter has no channel knowledge. We consider the case where the receiving antenna array has uncorrelated elements, while the elements of the transmitting array are arbitrarily correlated. Focusing on the case where nT=2, we derive simple analytic expressions for the ergodic average and the cumulative distribution function of the mutual information for arbitrary input (transmission) signal covariance. We then determine the ergodic and outage capacities and the associated optimal input signal covariances. We thus show how a transmitter with covariance knowledge should correlate its transmissions to maximize throughput. These results allow us to derive an exact condition (both necessary and sufficient) that determines when beamforming is optimal for systems with arbitrary number of transmitters and receivers.
The dependence of the Gaussian input information rate on the line-of-sight (LOS) matrix in multiple-input multiple-output (MIMO) coherent Rician fading channels is explored. It is proved that the outage probability and the mutual information induced by a multivariate circularly symmetric Gaussian input with any covariance matrix are monotonic in the LOS matrix D, or more precisely, monotonic in D†D in the sense of the Loewner partial order. Conversely, it is also demonstrated that this ordering on the LOS matrices is a necessary condition for the uniform monotonicity over all input covariance matrices. This result is subsequently applied to prove the monotonicity of the isotropic Gaussian input information rate and channel capacity in the singular values of the LOS matrix. Extensions to multiple-access channels (MAC) are also provided.
This paper considers communication over coherent multiple-input multiple-output (MIMO) flat-fading channels where the channel is only known at the receiver. For this setting, we introduce the class of LAttice Space-Time (LAST) codes. We show that these codes achieve the optimal diversity-multiplexing tradeoff defined by Zheng and Tse under generalized minimum Euclidean distance lattice decoding. Our scheme is based on a generalization of Erez and Zamir mod-Lambda scheme to the MIMO case. In our construction the scalar "scaling" of Erez-Zamir and Costa Gaussian "dirty-paper" schemes is replaced by the minimum mean-square error generalized decision-feedback equalizer (MMSE-GDFE). This result settles the open problem posed by Zheng and Tse on the construction of explicit coding and decoding schemes that achieve the optimal diversity-multiplexing tradeoff. Moreover, our results shed more light on the structure of optimal coding/decoding techniques in delay-limited MIMO channels, and hence, open the door for novel approaches for space-time code constructions. In particular, 1) we show that MMSE-GDFE plays a fundamental role in approaching the limits of delay-limited MIMO channels in the high signal-to-noise ratio (SNR) regime, unlike the additive white Gaussian noise (AWGN) channel case and 2) our random coding arguments represent a major departure from traditional space-time code designs based on the rank and/or mutual information design criteria.
A technique is proposed for the derivation of upper bounds on channel capacity. It is based on a dual expression for channel capacity where the maximization (of mutual information) over distributions on the channel input alphabet is replaced with a minimization (of average relative entropy) over distributions on the channel output alphabet. We also propose a technique for the analysis of the asymptotic capacity of cost-constrained channels. The technique is based on the observation that under fairly mild conditions capacity achieving input distributions "escape to infinity." The above techniques are applied to multiple-antenna flat-fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown (or only partially known) at the receiver. It is demonstrated that, for high signal-to-noise ratio (SNR), the capacity of such channels typically grows only double-logarithmically in the SNR. To better understand this phenomenon and the rates at which it occurs, we introduce the fading number as the second-order term in the high-SNR asymptotic expansion of capacity, and derive estimates on its value for various systems. It is suggested that at rates that are significantly higher than the fading number, communication becomes extremely power inefficient, thus posing a practical limit on practically achievable rates. Upper and lower bounds on the fading number are also presented. For single-input-single-output (SISO) systems the bounds coincide, thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes. We also demonstrate that for memoryless multiple-input single-output (MISO) channels, the fading number is achievable using beam-forming, and we derive an expression for the optimal beam direction. This direction depends on the fading law and is, in general, not the direction that maximizes the SNR on the induced SISO channel. Using a new closed-form expression for the expectation of the logarithm of a noncentral chi-square distributed random variable we provide some closed-form expressions for the fading number of some systems with Gaussian fading, including SISO systems with circularly symmetric stationary and ergodic Gaussian fading. The fading number of the latter-
is determined by the fading mean, fading variance, and the mean squared error in predicting the present fading from its past; it is not directly related to the Doppler spread. For the Rayleigh, Ricean, and multiple-antenna Rayleigh-fading channels we also present firm (nonasymptotic) upper and lower bounds on channel capacity. These bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high SNR, and their ratio to capacity approaches one at low SNR.
The use of multiple-antenna arrays in both transmission and reception promises huge increases in the throughput of wireless communication systems. It is therefore important to analyze the capacities of such systems in realistic situations, which may include spatially correlated channels and correlated noise, as well as correlated interferers with known channel at the receiver. Here, we present an approach that provides analytic expressions for the statistics, i.e., the moments of the distribution, of the mutual information of multiple-antenna systems with arbitrary correlations, interferers, and noise. We assume that the channels of the signal and the interference are Gaussian with arbitrary covariance. Although this method is valid formally for large antenna numbers, it produces extremely accurate results even for arrays with as few as two or three antennas. We also develop a method to analytically optimize over the input signal covariance, which enables us to calculate analytic capacities when the transmitter has knowledge of the statistics of the channel (i.e., the channel covariance). In many cases of interest, this capacity is very close to the full closed-loop capacity, in which the transmitter has instantaneous channel knowledge. We apply this analytic approach to a number of examples and we compare our results with simulations to establish the validity of this approach. This method provides a simple tool to analyze the statistics of throughput for arrays of any size. The emphasis of this paper is on elucidating the novel mathematical methods used.
In this paper, we investigate the capacity distribution of spatially correlated, multiple-input-multiple-output (MIMO) channels. In particular, we derive a concise closed-form expression for the characteristic function (c.f.) of MIMO system capacity with arbitrary correlation among the transmitting antennas or among the receiving antennas in frequency-flat Rayleigh-fading environments. Using the exact expression of the c.f., the probability density function (pdf) and the cumulative distribution function (CDF) can be easily obtained, thus enabling the exact evaluation of the outage and mean capacity of spatially correlated MIMO channels. Our results are valid for scenarios with the number of transmitting antennas greater than or equal to that of receiving antennas with arbitrary correlation among them. Moreover, the results are valid for an arbitrary number of transmitting and receiving antennas in uncorrelated MIMO channels. It is shown that the capacity loss is negligible even with a correlation coefficient between two adjacent antennas as large as 0.5 for exponential correlation model. Finally, we derive an exact expression for the mean value of the capacity for arbitrary correlation matrices.
Wireless systems using multi-element antenna arrays simultaneously at both transmitter and receiver promise a much higher capacity than conventional systems. Previous studies have shown that single-user systems employing n-element transmit and receive arrays can achieve a capacity proportional to n, assuming independent Rayleigh fading between pairs of antenna elements. We explore the capacity of dual-antennaarray systems via theoretical analysis and simulation experiments, focusing particularly on the situation when there is correlation between the fading at different antennas. We derive and compare expressions for the asymptotic growth rate of capacity with n antennas for both independent and correlated fading cases; the latter is derived under some assumptions about the scaling of the fading correlation structure. We show that the capacity growth is linear in n in both the independent and correlated cases, but the growth rate is smaller in the latter case. We compare the ...
We generalize the Harish-Chandra-Itzykson-Zuber and certain other integrals (the Gross-Witten integral, the integrals over complex matrices and the integrals over rectangle matrices) using a notion of the tau function of the matrix argument. In this case one can reduce multi-matrix integrals to integrals over eigenvalues, which in turn are certain tau functions. We also consider a generalization of the Kontsevich integral.
Jacobians, Exterior Products, and Related Topics Kronecker Products
Abstract Inthispaperweconsiderinformationtheoreticproperties of ∞at fading channels with multiple antennas. In particular we derive maximum,information rates and asymptotic expressions for the error exponent in the general case. For the case of transmit diversity we deriveclosedformexpressionsfortheerrorexponentand cutofi rate.
We study the optimal transmission strategy of a multiple-inputsingle-output wireless communication link. The receiver has perfectchannel state information while the transmitter hasonly long-term channel state information in terms of the channelcovariance matrix. It was recently shown that the optimal eigenvectors of the transmitcovariance matrix correspond with the eigenvalues of the channelcovariance matrix. However, the optimal eigenvalues are difficult tocompute. We study the properties of these optimal capacity achieving eigenvalues, and present a necessary and sufficient condition for theoptimal eigenvalues of the transmit covariance matrix. Furthermore, we develop a necessary and sufficient condition forachieving capacity when transmitting in all directions. We compare thecapacity gain of an optimal diversity system with a system which works with beamforming, and we derive an upperbound. We answer the main questions regarding the system design using the developed results. Additionally, we show inwhich way the multiplexing gain can be computed in case the channel covariancematrix is given. We compute the maximum number of required paralleldata streams, and we define a multiplexing function inorder to obtain a measure for the available multiplexinggain. Furthermore, we show that the capacity gain is small considering theadditional complexity at the receiver. We illustrate allresults by numerical simulations.
This paper is motivated by the need for fundamental understanding of ultimate limits of bandwidth efficient delivery of higher bit-rates in digital wireless communications and to also begin to look into how these limits might be approached. We examine exploitation of multi-element array (MEA) technology, that is processing the spatial dimension (not just the time dimension) to improve wireless capacities in certain applications. Specifically, we present some basic information theory results that promise great advantages of using MEAs in wireless LANs and building to building wireless communication links. We explore the important case when the channel characteristic is not available at the transmitter but the receiver knows (tracks) the characteristic which is subject to Rayleigh fading. Fixing the overall transmitted power, we express the capacity offered by MEA technology and we see how the capacity scales with increasing SNR for a large but practical number, n, of antenna elements at both transmitter and receiver. We investigate the case of independent Rayleigh faded paths between antenna elements and find that with high probability extraordinary capacity is available. Compared to the baseline n = 1 case, which by Shannon’s classical formula scales as one more bit/cycle for every 3 dB of signal-to-noise ratio (SNR) increase, remarkably with MEAs, the scaling is almost like n more bits/cycle for each 3 dB increase in SNR. To illustrate how great this capacity is, even for small n, take the cases n = 2, 4 and 16 at an average received SNR of 21 dB. For over 99%
For a MIMO Ricean fading channel with perfect side information at the receiver we derive an analytic upper bound on the difference between capacity and the mutual information that is induced by an isotropic Gaussian input. We show that if the number of receiver antennas is at least equal to the number of transmitter antennas, then, as the signal-to-noise ratio tends to infinity, such an input is asymptotically optimal. But otherwise such an isotropic input might be suboptimal. We also propose an iterative algorithm to calculate the optimal power allocation.
First Page of the Article
The behavior of the multiple antenna broadcast channel at high SNR is investigated. The multiple antenna broadcast channel achieves the same multiplexing gain as the system in which all receivers are allowed to perfectly cooperate (i.e. transforming the system into a point-to-point MIMO system). However, the multiplexing gain alone is not sufficient to accurately characterize the behavior of sum rate capacity at high SNR. An affine approximation to capacity which incorporates the multiplexing gain as well as a power offset (i.e. a zero-order term) is a more accurate representation of high SNR behavior. The power offset of the sum rate capacity is shown to equal the power offset of the cooperative MIMO system when there are less receivers than transmit antennas. In addition, the power offset of using the sub-optimal strategy of beamforming is calculated. These calculations show that beamforming can perform quite well when the number of antennas is sufficiently larger than the number of receivers, but performs very poorly when there are nearly as many receivers as transmit antennas
Given a totally positive function K of two real variables, is there a method for establishing the total positivity of K in an “obvious” fashion? In the case in which K(x, y) = f(xy), where f is real-analytic in a neighborhood of zero, we obtain integral representations for the determinants which define the total positivity of K. The total positivity of K then follows immediately from positivity of the integrands. In particular, we analyze the total positivity of classical hypergeometric functions by these methods. The central theme of this work is the circle of ideas that relates total positivity to “spherical series” on the symmetric space , and classical hypergeometric functions to hypergeometric functions of matrix argument.
The analysis of flat-fading channels is often performed under the assumption that the additive noise is white and Gaussian, and that the receiver has precise knowledge of the realization of the fading process. These assumptions imply the optimality of Gaussian codebooks and of scaled nearest-neighbor decoding. Here we study the robustness of this communication scheme with respect to errors in the estimation of the fading process. We quantify the degradation in performance that results from such estimation errors, and demonstrate the lack of robustness of this scheme. For some situations we suggest the rule of thumb that, in order to avoid degradation, the estimation error should be negligible compared to the reciprocal of the signal-to-noise ratio (SNR)
The paper is largely expository, but some new results are included to round out the paper and bring it up to date. The following distributions are quoted in Section 7. 1. Type , exponential: (i) , (ii) Wishart, (iii) latent roots of the covariance matrix. 2. Type , binomial series: (i) variance ratio, F, (ii) latent roots with unequal population covariance matrices. 3. Type , Bessel: (i) noncentral , (ii) noncentral Wishart, (iii) noncentral means with known covariance. 4. Type , confluent hypergeometric: (i) noncentral F, (ii) noncentral multivariate F, (iii) noncentral latent roots. 5. Type , Gaussian hypergeometric: (i) multiple correlation coefficient, (ii) canonical correlation coefficients. The modifications required for the corresponding distributions derived from the complex normal distribution are outlined in Section 8, and the distributions are listed. The hypergeometric functions of matrix argument which occur in the multivariate distributions are defined in Section 4 by their expansions in zonal polynomials as defined in Section 5. Important properties of zonal polynomials and hypergeometric functions are quoted in Section 6. Formulae and methods for the calculation of zonal polynomials are given in Section 9 and the zonal polynomials up to degree 6 are given in the appendix. The distribution of quadratic forms is discussed in Section 10, orthogonal expansions of and in Laguerre polynomials in Section 11 and the asymptotic expansion of in Section 12. Section 13 has some formulae for moments.
Let be a matrix of random real variates such that the column vectors of X are independently and identically distributed as multivariate normals with zero mean vectors. Then a positive definite quadratic function in normal vectors is defined as XLX' where L is a symmetric positive definite (p.d.) matrix with real elements. In the analysis of variance, such functions appear. In the previous study, Khatri [14], [16], has established the necessary and sufficient conditions for the independence and the Wishartness of such functions. In this paper, we study the distribution of a positive definite quadratic function and the distribution of where is independently distributed of X and its columns are independently and identically distributed as multivariate normals with zero mean vectors. Moreover, we study the distribution of the characteristic (ch.) roots of and the similar related problems. When , the distribution of a p.d. quadratic function in normal variates (central or noncentral) has been studied by a number of people (see references). In the study of the above and related topics in multivariate distribution theory, we are using zonal polynomials. A. T. James [10], [11], [12], [13], and Constantine [1], [2], have used them successfully and have given the final results in a very compact form, using hypergeometric functions in matrix arguments. These functions are defined by \begin{equation*}\tag{1}_pF_q(a_1, \cdots, a_p; b_1, \cdots, b_q; Z) \end{equation*} where is a symmetric homogeneous polynomial of degree k in the latent roots of Z, called zonal polynomials (for more detail study of zonal polynomials, see the references of A. T. James and Constantine), are real or complex constants, none of the is an integer or half integer (otherwise some of the denominators in (1) will vanish), \begin{equation*}\tag{2}(a)_\kappa = \prod^m_{j = 1} (a - \frac{1}{2}(j - 1))_{kj} = \Gamma_m(a,\kappa)/\Gamma_m(a), \end{equation*} (x)_n = x(x + 1) \cdots (x + n - 1), (x)_0 = 1 and \begin{equation*}\tag{3}\Gamma_m(a) = \pi^{\frac{1}{4}m(m - 1)} \prod^m_{j = 1} \Gamma(a - \frac{1}{2}(j - 1)) \end{equation*} and \Gamma_m(a, \kappa) = \pi^{\frac{1}{4}m(m - 1)} \prod^m_{j = 1} \Gamma(a + k_j - \frac{1}{2}(j - 1)). In (1), Z is a complex symmetric matrix, and it is assumed that , otherwise the series may converge for . For , the series converge for , where denote the maximum of the absolute value of ch. roots of Z. For , the series converge for all Z. Similarly we define \begin{equation*}\tag{2b}_pF^{(m)}_q (a_1, a_2, \cdots, a_p; b_1, \cdots, b_q; S, R)\end{equation*} The Section 2 gives some results on integration with the help of zonal polynomials, the Section 3 derives the distributions based on p.d. quadratic functions, the Section 4 gives the moments of certain statistics arising in the study of multivariate distributions, and the Section 5 gives the results for complex multivariate Gaussian variates.
Let the column vectors of X: M x N, M < N, be distributed as independent complex normal vectors with the same covariance matrix Sigma. Then the usual quadratic form in the complex normal vectors is denoted by Z = XLXH where L: N x N is a positive definite hermitian matrix. This paper deals with a representation for the density function of Z in terms of a ratio of determinants. This representation also yields a compact form for the distribution of the generalized variance \Z\. (C) 2000 Academic Press.
Let X = {Xij:i, J = 1, 2,...} be an infinite dimensional random matrix, Tp be a p - p nonnegative definite random matrix independent of X, for p = 1, 2,.... Suppose (1/p) tr Tpk --> Hk a.s. as p --> [infinity] for k = 1, 2,..., and [Sigma]H2k-1/2k < [infinity]. Then the spectral distribution of Ap = (1/n) XpXp'Tp, where Xp = [Xij:i = 1,...,p; J = 1,...,n] tends to a nonrandom limit distribution as p --> [infinity], n --> [infinity], but p/n --> y > 0, under the mild conditions that Xy's are i.i.d. and EX112 < [infinity].
The high-signal to noise ratio (SNR) mutual information (MI) of the Ricean MIMO channel was investigated. The MI into the sum of the MIs of one single-input single-output (SISO) Ricean channel and several SISO rayleigh fading channels with different diversity orders was decomposed using an illustrative geometrical technique. An analytical approximation for the probability density function of the MI that reveals the gaussian nature of MI both in the Rayleigh and the rank-1 Ricean cases were derived. The accurate approximations for the mean and the variance of the MI which allow to analytically quantify the impact of the K-factor on capacity was also provided.
This paper considers reliable communication over a parallel (correlated) fading channel for short periods of time. We derive a code design criterion by taking a compound channel viewpoint of the outage capacity of the channel. Motivated by the criterion, we show existence of simple codes that achieve the optimal diversity-multiplexing tradeoff curve, introduced recently in (Zheng, L et al., 2003), simultaneously for every correlated parallel channel. We demonstrate a code with simple encoding and decoding for a parallel channel with two diversity branches. The codes for the parallel channel can be used on a correlated MIMO channel by using the DBLAST architecture to simultaneously achieve the diversity-multiplexing tradeoff curve for arbitrary fading channels.
In this paper, we show that the expected log determinant of a complex noncentral Wishart matrix is an increasing function of the noncentrality parameter. This demonstrates that the mutual information corresponding to an isotropically distributed Gaussian input to a multiantenna Ricean fading channel is nondecreasing in the line-of-sight component.
We present an analytical characterization of multi-antenna capacity in the limit of a large number of antennas. In contrast to previous studies, the entries of the channel matrix are not restricted to be identically distributed, thus incorporating diversity mechanisms that are otherwise excluded, such as those based on the use of antennas with distinct polarizations and radiation patterns. In addition to the capacity, first-order expressions in the low- and high-power regimes are also evaluated both asymptotically and non-asymptotically.
Asymptotic theorems are very commonly used in probability. For systems whose performance depends on a set of random variables, asymptotic analyses are often used to simplify calculations and obtain results yielding useful hints at the behavior of the system when the parameters take on finite values. These asymptotic analyses are especially useful whenever the convergence to the asymptotic results is so fat that even for moderate parameter values they yield results close to the true values. The goal of this paper is to illustrate this principle through a number of examples taken from multiple-antenna systems.
The analysis of fading channels is often done under the assumption
that the additive noise is white and Gaussian, and that the receiver has
precise knowledge of the realization of the fading process. These
assumptions imply the optimality of Gaussian codebooks and of a modified
nearest-neighbor decoding rule. Here we study the robustness of this
communication scheme with respect to errors in the estimation of the
fading process. We quantify the degradation in performance that results
from estimation errors, and demonstrate the lack of robustness of this
scheme. We treat the “flat fading” case exclusively
We study the optimal transmission strategy of a single-user multiple-input/multiple-output communication system with covariance feedback. We consider the situation with correlated receive and correlated transmit antennas in Rayleigh flat fading. Furthermore, we assume that the receiver has perfect channel state information, while the transmitter knows only the transmit correlation matrix and the receive correlation matrix. We show that transmitting in the direction of the eigenvectors of the transmit correlation matrix is the optimal transmission strategy. In addition to this, the optimal power allocation is studied and a necessary and sufficient condition for optimality of beamforming is derived. All theoretical results are illustrated by numerical simulations.
We solve the transmitter optimization problem and determine a necessary and sufficient condition under which beamforming achieves Shannon capacity in a linear narrowband point-to-point communication system employing multiple transmit and receive antennas with additive Gaussian noise. We assume that the receiver has perfect channel knowledge while the transmitter has only knowledge of either the mean or the covariance of the channel coefficients. The channel is modeled at the transmitter as a matrix of complex jointly Gaussian random variables with either a zero mean and a known covariance matrix (covariance information), or a nonzero mean and a white covariance matrix (mean information). For both cases, we develop a necessary and sufficient condition for when the Shannon capacity is achieved through beamforming; i.e., the channel can be treated like a scalar channel and one-dimensional codes can be used to achieve capacity. We also provide a waterpouring interpretation of our results and find that less channel uncertainty not only increases the system capacity but may also allow this higher capacity to be achieved with scalar codes which involves significantly less complexity in practice than vector coding.
We study the optimal transmission strategy of a multiple-input single-output (MISO) wireless communication link. The receiver has perfect channel state information (CSI), while the transmitter has different types of CSI, i.e., either perfect CSI, or no CSI, or long-term knowledge of the channel covariance matrix. For the case in which the transmitter knows the channel covariance matrix, it was recently shown that the optimal eigenvectors of the transmit covariance matrix correspond with the eigenvectors of the channel covariance matrix. However, the optimal eigenvalues are difficult to compute. We derive a characterization of the optimum power allocation. Furthermore, we apply this result to provide an efficient algorithm which computes the optimum power allocation. In addition to this, we analyze the impact of correlation on the ergodic capacity of the MISO system with different CSI schemes. At first, we justify the belief that equal power allocation is optimal if the transmitter is uninformed and the transmit antennas are correlated. Next, we show that the ergodic capacity with perfect CSI and without CSI at the transmitter is Schur-concave, i.e., the more correlated the transmit antennas are, the less capacity is achievable. In addition, we show that the ergodic capacity with covariance knowledge at the transmitter is Schur-convex with respect to the correlation properties. These results completely characterize the impact of correlation on the ergodic capacity in MISO systems. Furthermore, the capacity loss or gain due to correlation is quantified. For no CSI and perfect CSI at the transmitter, the capacity loss due to correlation is bounded by some small constant, whereas the capacity gain due to correlation grows unbounded with the number of transmit antennas in the case in which transmitter knows the channel covariance matrix. Finally, we illustrate all theoretical results by numerical simulations.
We derive performance limits for two closely related communication
scenarios involving a wireless system with multiple-element transmitter
antenna arrays: a point-to-point system with partial side information at
the transmitter, and a broadcast system with multiple receivers. In both
cases, ideal beamforming is impossible, leading to an inherently lower
achievable performance as the quality of the side information degrades
or as the number of receivers increases. Expected signal-to-noise ratio
(SNR) and mutual information are both considered as performance
measures. In the point-to-point case, we determine when the transmission
strategy should use some form of beamforming and when it should not. We
also show that, when properly chosen, even a small amount of side
information can be quite valuable. For the broadcast scenario with an
SNR criterion, we find the efficient frontier of operating points and
show that even when the number of receivers is larger than the number of
antenna array elements, significant performance improvements can be
obtained by tailoring the transmission strategy to the realized channel
Predictions of uplink space-diversity gain in the cellular and
personal communications systems (PCS) bands (near 850 MHz and 1.9 GHz,
respectively), suffer from incomplete modeling of multipath angular
spread σ. Using previously published measurements to reduce the
gap, we show that σ~[distance]-1/2 and that diversity
gains are about 2 dB higher for PCS than for cellular
The channel capacity in a Rayleigh fading environment is derived.
The result shows that the channel capacity in a Rayleigh fading
environment is always lower than that in a Gaussian-noise environment.
When operating a digital transmission in a mobile radio environment that
has Rayleigh fading statistics, it is very important to know the
degradations in channel capacity due to Rayleigh fading, and also to
what degree the diversity schemes can raise the channel capacity in a
Rayleigh fading environment. The curves are generated to show the
degradation of channel capacity in a Rayleigh fading environment and its
improvement by diversity schemes
This paper provides analytical characterizations of the impact on the multiple-antenna capacity of several important features that fall outside the standard multiple-antenna model, namely: (i) antenna correlation, (ii) Ricean factors, (iii) polarization diversity, and (iv) out-of-cell interference; all in the regime of low signal-to-noise ratio. The interplay of rate, bandwidth, and power is analyzed in the region of energy per bit close to its minimum value. The analysis yields practical design lessons for arbitrary number of antennas in the transmit and receive arrays.
The capacity of multiple-input multiple-output (MIMO) wireless channels is limited by both the spatial fading correlation and rank deficiency of the channel. While spatial fading correlation reduces the diversity gains, rank deficiency due to double scattering or keyhole effects decreases the spatial multiplexing gains of multiple-antenna channels. In this paper, taking into account realistic propagation environments in the presence of spatial fading correlation, double scattering, and keyhole effects, we analyze the ergodic (or mean) MIMO capacity for an arbitrary finite number of transmit and receive antennas. We assume that the channel is unknown at the transmitter and perfectly known at the receiver so that equal power is allocated to each of the transmit antennas. Using some statistical properties of complex random matrices such as Gaussian matrices, Wishart (1928) matrices, and quadratic forms in the Gaussian matrix, we present a closed-form expression for the ergodic capacity of independent Rayleigh-fading MIMO channels and a tight upper bound for spatially correlated/double scattering MIMO channels. We also derive a closed-form capacity formula for keyhole MIMO channels. This analytic formula explicitly shows that the use of multiple antennas in keyhole channels only offers the diversity advantage, but provides no spatial multiplexing gains. Numerical results demonstrate the accuracy of our analytical expressions and the tightness of upper bounds.
The capacity of wireless communication architectures equipped with multiple transmit and receive antennas and impaired by both noise and cochannel interference is studied. We find a closed-form solution for the capacity in the limit of a large number of antennas. This asymptotic solution, which is a sole function of the relative number of transmit and receive antennas and the signal-to-noise and signal-to-interference ratios (SNR and SIR), is then particularized to a number of cases of interest. By verifying that antenna diversity one can substitute for time and/or frequency diversity at providing ergodicity, we show that these asymptotic solutions approximate the ergodic capacity very closely even when the number of antennas is very small.
The use of channel feedback from receiver to transmitter is
standard in wireline communications. While knowledge of the channel at
the transmitter would produce similar benefits for wireless
communications as well, the generation of reliable channel feedback is
complicated by the rapid time variations of the channel for mobile
applications. The purpose of this paper is to provide an
information-theoretic perspective on optimum transmitter strategies, and
the gains obtained by employing them, for systems with transmit antenna
arrays and imperfect channel feedback. The spatial channel, given the
feedback, is modeled as a complex Gaussian random vector. Two extreme
cases are considered: mean feedback, in which the channel side
information resides in the mean of the distribution, with the covariance
modeled as white, and covariance feedback, in which the channel is
assumed to be varying too rapidly to track its mean, so that the mean is
set to zero, and the information regarding the relative geometry of the
propagation paths is captured by a nonwhite covariance matrix. In both
cases, the optimum transmission strategies, maximizing the information
transfer rate, are determined as a solution to simple numerical
optimization problems. For both feedback models, our numerical results
indicate that, when there is a moderate disparity between the strengths
of different paths from the transmitter to the receiver, it is nearly
optimal to employ the simple beamforming strategy of transmitting all
available power in the direction which the feedback indicates is the
strongest
The CDMA channel with randomly and independently chosen spreading
sequences accurately models the situation where pseudonoise sequences
span many symbol periods. Furthermore, its analysis provides a
comparison baseline for CDMA channels with deterministic signature
waveforms spanning one symbol period. We analyze the spectral efficiency
(total capacity per chip) as a function of the number of users,
spreading gain, and signal-to-noise ratio, and we quantify the loss in
efficiency relative to an optimally chosen set of signature sequences
and relative to multiaccess with no spreading. White Gaussian background
noise and equal-power synchronous users are assumed. The following
receivers are analyzed: (a) optimal joint processing, (b) single-user
matched filtering, (c) decorrelation, and (d) MMSE linear processing
We investigate the use of multiple transmitting and/or receiving antennas for single user communications over the additive Gaussian channel with and without fading. We derive formulas for the capacities and error exponents of such channels, and describe computational procedures to evaluate such formulas. We show that the potential gains of such multi-antenna systems over single-antenna systems is rather large under independence assumptions for the fades and noises at different receiving antennas. 1 Introduction We will consider a single user Gaussian channel with multiple transmitting and/or receiving antennas. We will denote the number of transmitting antennas by t and the number of receiving antennas by r. We will exclusively deal with a linear model in which the received vector y 2 C r depends on the transmitted vector x 2 C t via y = Hx+ n (1) where H is a r Theta t complex matrix and n is zero-mean complex Gaussian noise with independent, equal variance real and imaginary p...
The capacity of the randomly spread synchronous code-division multiple-access (CDMA) channel subject to frequency -flat fading is studied in the wide-band limit of large number of users. We find the spectral efficiency as a function of the number of users per chip, the distribution of the flat fading, and the signal-to-noise ratio (SNR), for the optimum receiver as well as linear receivers (single-user matched filter, decorrelator, and minimum mean-square errror (MMSE)). The potential improvements due to both decentralized transmitter power control and multiantenna receivers are also analyzed.
Multiple antennas can be used for increasing the amount of diversity or the number of degrees of freedom in wireless communication systems. In this paper, we propose the point of view that both types of gains can be simultaneously obtained for a given multiple antenna channel, but there is a fundamental tradeoff between how much of each any coding scheme can get. We give a simple characterization of the optimal tradeoff curve and use it to evaluate the performance of existing multiple antenna schemes.
- R A Horn
- C R Johnson
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.