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High-SNR Power Offset in Multiantenna Communication
Abstract
The analysis of the multiple-antenna capacity in the high-SNR regime has hitherto focused on the high-SNR slope (or maximum multiplexing gain), which quantifies the multiplicative increase as a function of the number of antennas. This traditional characterization is unable to assess the impact of prominent channel features since, for a majority of channels, the slope equals the minimum of the number of transmit and receive antennas. Furthermore, a characterization based solely on the slope captures only the scaling but it has no notion of the power required for a certain capacity. This paper advocates a more refined characterization whereby, as a function of SNR|dB, the high-SNR capacity is expanded as an affine function where the impact of channel features such as antenna correlation, unfaded components, etc., resides in the zero-order term or power offset. The power offset, for which we find insightful closed-form expressions, is shown to play a chief role for SNR levels of practical interest.
... 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 Theorem 1 is more intuitive and insightful. ...
... 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.
... For the more interesting case, with N r ≥ N t , we see that the high SNR power offset is non-zero. Very importantly, this case involves the same types of expectations as those required for the high SNR analysis of ergodic MIMO mutual information, for which closed-form solutions exist for a wide range of fading channel scenarios [5,15,37,38]. In the sequel, we will draw upon these previous results in order to derive new simple closed-form expressions for L mmse ∞ under a range of conditions. ...
... Proof: The result is obtained by substituting (41) and [38,Eq.15] ...
... Now consider the excess high SNR power offset. To evaluate this, we require the high SNR power offset with optimal receivers L opt ∞ , which for the transmit-receive correlated case was first presented in [38,Eq. 28]. ...
This paper investigates the achievable sum rate of multiple-input multiple-output (MIMO) wireless systems employing linear minimum mean-squared error (MMSE) receivers. We present a new analytic framework which unveils an interesting connection between the achievable sum rate with MMSE receivers and the ergodic mutual information achieved with optimal receivers. This simple but powerful result enables the vast prior literature on ergodic MIMO mutual information to be directly applied to the analysis of MMSE receivers. The framework is particularized to various Rayleigh and Rician channel scenarios to yield new exact closed-form expressions for the achievable sum rate, as well as simplified expressions in the asymptotic regimes of high and low signal to noise ratios. These expressions lead to the discovery of key insights into the performance of MIMO MMSE receivers under practical channel conditions.
... In terms of standard high SNR notation where [27], the multiplexing gain S ∞ = 1 and the rate offset ...
... Based on (6), L ∞ therefore is the Fig. 1 for ǫ = 0.01. As L → ∞ the offset converges to −E[log 2 |h| 2 ] ≈ 0.83, the offset of the ergodic Rayleigh channel [27]. ...
... Applying this into (27), the rate difference can be approximated as: and SNR = 10 dB, the negative term is much smaller than 0.5(1 − ǫ) for M < 5000) and thus can be reasonably neglected. By ignoring this negative term and replacing the denominator with the leading order M term, we get a further approximation of the rate gap: ...
This paper studies the performance of hybrid-ARQ (automatic repeat request) in Rayleigh block fading channels. The long-term average transmitted rate is analyzed in a fast-fading scenario where the transmitter only has knowledge of channel statistics, and, consistent with contemporary wireless systems, rate adaptation is performed such that a target outage probability (after a maximum number of H-ARQ rounds) is maintained. H-ARQ allows for early termination once decoding is possible, and thus is a coarse, and implicit, mechanism for rate adaptation to the instantaneous channel quality. Although the rate with H-ARQ is not as large as the ergodic capacity, which is achievable with rate adaptation to the instantaneous channel conditions, even a few rounds of H-ARQ make the gap to ergodic capacity reasonably small for operating points of interest. Furthermore, the rate with H-ARQ provides a significant advantage compared to systems that do not use H-ARQ and only adapt rate based on the channel statistics.
... Note that the high-SNR slope is defined as [37] ...
... where η is defined in (37). Proof: See Appendix I. ...
... (141) follows from the fact that g(0) = 1, and η is defined in (37). Finally, inserting (140) and (141) into (56), the wideband slope expression in (60) is readily obtained. ...
Throughput and energy efficiency in fading channels are studied in the presence of randomly arriving data and statistical queueing constraints. In particular, Markovian arrival models including discrete-time Markov, Markov fluid, and Markov-modulated Poisson sources are considered. Employing the effective bandwidth of time-varying sources and effective capacity of time-varying wireless transmissions, maximum average arrival rates in the presence of statistical queueing constraints are characterized. For the two-state (ON/OFF) source models, throughput is determined in closed-form as a function of the source statistics, channel characteristics, and quality of service (QoS) constraints. Throughput is further studied in certain asymptotic regimes. Furthermore, energy efficiency is analyzed by determining the minimum energy per bit and wideband slope in the low signal-to-noise ratio (SNR) regime. Overall, the impact of source characteristics, QoS requirements, and channel fading correlations on the throughput and energy efficiency of wireless systems is identified.
... For moderate to large system size only r = nR/C results in the convergence at realistic SNR values (see Fig. 4), so that the SNR-asymptotic DMT has operational significance only for this multiplexing gain definition. Comparing (33) and (34) to (29), one concludes that the SNR offsets for r = R/ ln(γ/e) and r = nR/C are the same, but there is an additional SNR offset factor e 2r(n−r) for r = R/ ln γ, which can be very significant for large n or r, as examples below demonstrate. Based on (29), (33) and (34), we remark that the SNR offset c γ is exponentially large in the diversity gain d(r) for various multiplexing gain definitions. ...
... Comparing (33) and (34) to (29), one concludes that the SNR offsets for r = R/ ln(γ/e) and r = nR/C are the same, but there is an additional SNR offset factor e 2r(n−r) for r = R/ ln γ, which can be very significant for large n or r, as examples below demonstrate. Based on (29), (33) and (34), we remark that the SNR offset c γ is exponentially large in the diversity gain d(r) for various multiplexing gain definitions. While the ln(γ/e) term somewhat reduces the offset, it is a minor effect since ln(γ/e) increases very slowly with the SNR. ...
... [33] gives a detailed discussion of the importance of SNR offset in the capacity analysis of MIMO systems. Note that this offset is missing in(5).7 while, for most channels at finite SNR, d ′ γ and cγ are not SNRindependent constants but rather slowly-varying functions of the SNR, the DMT framework can still be used. ...
Diversity-multiplexing tradeoff (DMT) was characterized asymptotically (SNR-> infinity) for i.i.d. Rayleigh fading channel by Zheng and Tse [1]. The SNR-asymptotic DMT overestimates the finite-SNR one [2]. 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 criteria. 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.
... In (37), Y k is derived by applying [33,Corollary2] to the integral in the second line of (37), and its entries are defined as ...
... In (37), Y k is derived by applying [33,Corollary2] to the integral in the second line of (37), and its entries are defined as ...
... In general, the ergodic capacity of MIMO at high SNR, (i.e. when ρ → +∞) can be expressed by an affine expansion [37], as follows: ...
Reconfigurable intelligent surface (RIS) has emerged as a promising technology for enhancing the performance of wireless communication systems. However, the extent of this enhancement has yet to be defined in a simple and insightful manner, especially when RIS amplitude and phase responses are coupled. In this paper, we characterize the fundamental ergodic capacity limits of RIS-aided multiple-input multiple-output (MIMO), a.k.a. MIMO-RIS, when considering a practical amplitude response for the RIS, which is coupled to its phase shift response. By studying these fundamental limits, we provide insights into the performance of MIMO-RIS systems and inform the design and optimization of future wireless communications. Accordingly, we first derive a novel expression of MIMO-RIS ergodic capacity from a closed-form expression of the probability density function (pdf) of the cascaded channel eigenvalues. We then provide upper and lower bounds, alongside low SNR, high SNR, and large number of RIS element approximations to illustrate the dependence of the MIMO-RIS ergodic capacity on the amplitude and phase of RIS elements. These expressions helped us to define the maximum SNR gain of MIMO-RIS over MIMO systems. Next, simulations are used to validate the accuracy and correctness of our various capacity expressions. Furthermore, we investigate the impact of environmental factors, such as near-field or far-field path loss, on the MIMO-RIS ergodic capacity. Numerical results confirm the accuracy of our MIMO-RIS SNR gain expression and provide valuable insights into the performance of RIS-based systems in realistic scenarios. Consequently, this can contribute to the design of future wireless communications based on MIMO-RIS.
... The communication channel between the target and AP is modeled using the power loss model [28,29], where the power decreases with increasing distance. A node can communicate with the target node if both have the required signal-to-noise ratio (SNR), which is defined as the difference between the received power and the total noise received by the node (SNR = P r − Noise) [30]. The SNR is a measure of how much stronger the signal is than the noise, where a higher SNR means that the signal is more likely to be received correctly [31,32]. ...
... It started at position (80, 50, 3), and moved for 10, 000 steps while transmitting a spoofing signal at −32 dBm. The target and three APs were located at the coordinates (45, 35), (30,35), (60, 30), and (50, 55), respectively, as shown in Figure 11. The transmission power was set to −35 dBm for both the target and APs. ...
... It started at position (80, 50, 3), and moved for 10, 000 steps while transmitting a spoofing signal at −32 dBm. The target and three APs were located at the coordinates (45, 35), (30,35), (60, 30), and (50, 55), respectively, as shown in Figure 11. The transmission power was set to −35 dBm for both the target and APs. ...
In the context of the Internet of Things (IoT), location-based applications have introduced new challenges in terms of location spoofing. With an open and shared wireless medium, a malicious spoofer can impersonate active devices, gain access to the wireless channel, as well as emit or inject signals to mislead IoT nodes and compromise the detection of their location. To address the threat posed by malicious location spoofing attacks, we develop a neural network-based model with single access point (AP) detection capability. In this study, we propose a method for spoofing signal detection and localization by leveraging a feature extraction technique based on a single AP. A neural network model is used to detect the presence of a spoofed unmanned aerial vehicle (UAV) and estimate its time of arrival (ToA). We also introduce a centralized approach to data collection and localization. To evaluate the effectiveness of detection and ToA prediction, multi-layer perceptron (MLP) and long short-term memory (LSTM) neural network models are compared.
... where (a) holds because, for any X drawn from a unit-mean exponential distribution, -ln(X) follows a standard Gumbel distribution, whose mean equals C [86]. We then obtain (2.48) from (2.50) by deriving E r {ln r k } from the PDF of r k in (2.46). ...
... The second term, which does not depend on the transmit power, defines the rate-offset in the affine approximation, and it explains the improved performance of the ACC scheme. This type of affine approximations have been considered at high-SNR [86], but it will be shown later that, for this setting, it also offers an accurate characterization at practical SNR ranges. ...
... in (4.31), represents the high SNR power offset due to path-loss and fading [86]. ...
The thesis first addresses the worst-user bottleneck of wireless coded caching, which is known to severely diminish cache-aided multicasting gains. We present a novel scheme, called aggregated coded caching, which can fully recover the coded caching gains by capitalizing on the shared side information brought about by the effectively unavoidable file-size constraint. The thesis then transitions to scenarios with transmitters with multi-antenna arrays. In particular, we now consider the multi-antenna cache-aided multi-user scenario, where the multi-antenna transmitter delivers coded caching streams, thus being able to serve multiple users at a time, with a reduced radio frequency (RF) chains. By doing so, coded caching can assist a simple analog beamformer (only a single RF chain), thus incurring considerable power and hardware savings. Finally, after removing the RF-chain limitation, the thesis studies the performance of the vector coded caching technique, and reveals that this technique can achieve, under several realistic assumptions, a multiplicative sum-rate boost over the optimized cacheless multi-antenna counterpart. In particular, for a given downlink MIMO system already optimized to exploit both multiplexing and beamforming gains, our analysis answers a simple question: What is the multiplicative throughput boost obtained from introducing reasonably-sized receiver-side caches?
... The notable densification of the network results in a necessity of cooperation to avoid the congestion of the wireless medium. Multi-user cooperative networks and the extend of its theoretical capabilities have been thoroughly analyzed in the literature [27][28][29][30][31]. Network cooperation can take shape in many different forms. ...
... This measure has shown instrumental in several findings. In [29], Lozano et al. analyze the multiple-antenna point-to-point scenario, revealing that some system features which do not impact the DoF (as antenna correlation, fading...) do considerably impact the zero-order term, affecting the performance of the system at any possible SNR. In addition to expose the limitations of having only information about the DoF, [29] also reveals that the affine expansion offers appreciably tight approximations also at medium-to-low SNR. ...
... In [29], Lozano et al. analyze the multiple-antenna point-to-point scenario, revealing that some system features which do not impact the DoF (as antenna correlation, fading...) do considerably impact the zero-order term, affecting the performance of the system at any possible SNR. In addition to expose the limitations of having only information about the DoF, [29] also reveals that the affine expansion offers appreciably tight approximations also at medium-to-low SNR. This characterization has been also established for the Broadcast Channel (BC) with perfect CSIT using Dirty-paper coding and linear precoding [131], and for the BC with imperfect CSIT [132]. ...
Network cooperation is known to bring multiplicative gains under certain ideal assumptions. However, current wireless settings cope with many challenging constraints, as tight delay constraints, fast-changing channels, or rate-limited backhaul links. The topic of analyzing how the non-fulfillment of the ideal hypotheses impacts the performance has generated great interest in the research community. Nevertheless, the main focus has been on settings in which the imperfect information is shared by all the nodes, which is not feasible in many scenarios. This thesis aims for shedding light on the performance of cooperative settings in which the information available at each node may be different. We focus on the distributed Network MIMO. This setting is characterized by two main aspects: The perfect sharing of the user's information data and the imperfect sharing of the channel information. We start by characterizing the Degrees-of-Freedom metric of the setting, which is an approximation of the capacity at high SNR. The contribution is twofold, as we provide both achievable schemes that considerably outperform the solutions in the literature and upper-bounds that illustrate up to which scale the distributed setting is harmed with respect to the perfect-sharing setting. The second perspective consists in restricting the transmission to the conventional paradigm of Zero-Forcing and analyzing the achievable rate at high SNR to understand whether the performance losses from decentralized information can be accurately calculated. We propose a novel zero-forcing scheme tailored to the decentralized configuration that asymptotically attains the centralized rate.
... In this paper, we study the rank improvement ability of an IRSaided single-user MIMO system while preserving the coherent phase alignment by optimizing the phase shifts. One of the classical bottlenecks of point-to-point MIMO communications is that the capacity gains provided by spatial multiplexing are only large at high SNR, and high SNR channels are mainly appearing in LoS scenarios where the channel matrix has low rank and therefore does not support spatial multiplexing [19,20]. Using a different setup than [18], we demonstrate how an IRS can be used and optimized in an LoS environment to increase the rank of the channel matrix, leading to substantial capacity gains. ...
... where Υ = (1 − cos (2π(Ω br + Ω bs ))) (1 − cos (2π(Ωru − Ωue))) , where Ωru = Ω ru 2,i − Ω ru 1,i = 1 dru (xud u H sin θr sin ϕr + yud u H sin θr cos ϕr+(zu−h)d u H cos θr) and Ω br = Ω br i,2 −Ω br (19), we note that the SNR scales with N 2 and the rate is a function of the BS, IRS and UE positions and their deployment angles. The arguments of the cosine functions in Υ are ...
... The locations of the BS and IRS are assumed fixed and the UE is moved from the point yu = −5 to yu = 2. By applying a linesearch algorithm, the maximum of the rate log 2 P 2 tot N 2 βcβ bu σ 4 Υ from (19) is obtained at yu = −0.94. Inside the logarithm, there are three terms that are functions of the position yu: the pathloss term βcβ bu , and cos (2π(Ω br + Ω bs )), cos (2π(Ωru − Ωue)). ...
An intelligent reflecting surface (IRS), consisting of reconfigurable metamaterials, can be used to partially control the radio environment and thereby bring new features to wireless communications. Previous works on IRS have particularly studied the range extension use case and under what circumstances the new technology can beat relays. In this paper, we study another use case that might have a larger impact on the channel capacity: rank improvement. One of the classical bottlenecks of point-to-point MIMO communications is that the capacity gains provided by spatial multiplexing are only large at high SNR, and high SNR channels are mainly appearing in line-of-sight (LoS) scenarios where the channel matrix has low rank and therefore does not support spatial multiplexing. We demonstrate how an IRS can be used and optimized in such scenarios to increase the rank of the channel matrix, leading to substantial capacity gains.
... The high-SNR regime provides important insights into the statistical performance of MIMO channels. In particular, it characterizes the minimum required transmit power, which is also referred to as the high-SNR power offset [13]. The considered high-SNR mutual information is directly related to the high-SNR power offset, where its mean values have been derived for different channel models in [13]. ...
... In particular, it characterizes the minimum required transmit power, which is also referred to as the high-SNR power offset [13]. The considered high-SNR mutual information is directly related to the high-SNR power offset, where its mean values have been derived for different channel models in [13]. As an application of our results, we may study as a possible future work the distribution of the power offset pertaining to the nonergodic channels, an open problem discussed in [13]. ...
... The considered high-SNR mutual information is directly related to the high-SNR power offset, where its mean values have been derived for different channel models in [13]. As an application of our results, we may study as a possible future work the distribution of the power offset pertaining to the nonergodic channels, an open problem discussed in [13]. This open problem has been partially addressed in [14] for the case of a product of two MIMO Rayleigh channels. ...
In this work, an analytical framework for deriving the exact moments of multiple-input- multiple-output (MIMO) mutual information in the high-signal-to-noise ratio (SNR) regime is proposed. The idea is to make efficient use of the matrix-variate densities of channel matrices instead of the eigenvalue densities as in the literature. The framework is applied to the study of the emerging models of MIMO Rayleigh product channels and Jacobi MIMO channels, which include several well-known channels models as special cases. The corresponding exact moments of any order for the high-SNR mutual information are derived. The explicit moment expressions are utilized to construct simple yet accurate approximations to the outage probability. Despite the high-SNR nature, simulation shows usefulness of the approximations with finite SNR values in the scenario of low outage probability relevant to MIMO communications.
... For this proposed secure transmission scheme, we derive new tight lower-bound expressions for the ergodic secrecy sum rate (ESSR) of the following three scenarios in the high signal-to-noise ratio (SNR) regime: 1) Without jamming (WoJ), where the jammer is not activated, 2) FJ, where the jamming signal is known a priori at the two sources, and 3) GNJ, where the jamming signal is unknown at the sources. We further characterize the high SNR slope and the high SNR power offset for the ESSR of the three WoJ, FJ, and GNJ scenarios, to explicitly determine the impact of network parameters on the ESSR [25]. Based on our analytical results, we further highlight the impact of several system design parameters including the EH time ratio, power allocation factor, transmit SNR, nodes distance, and path loss exponent on the ESSR performance. ...
... is the general asymptotic form of the ESSR performance [25]. For the ease of presentation, we assume that P S1 and P S2 grow large with P S1 = ξP S2 for some fixed ratio 0 < ξ < ∞. ...
In this paper, we propose a two-way secure communication scheme where two transceivers exchange confidential messages via a wireless powered untrusted amplify-and-forward (AF) relay in the presence of an external jammer. We take into account both friendly jamming (FJ) and Gaussian noise jamming (GNJ) scenarios. Based on the time switching (TS) architecture at the relay, the data transmission is done in three phases. In the first phase, both the energy-starved nodes, the untrustworthy relay and the jammer, are charged by non-information radio frequency (RF) signals from the sources. In the second phase, the two sources send their information signals and concurrently, the jammer transmits artificial noise to confuse the curious relay. Finally, the third phase is dedicated to forward a scaled version of the received signal from the relay to the sources. For the proposed secure transmission schemes, we derive new closed-form lower-bound expressions for the ergodic secrecy sum rate (ESSR) in the high signal-to-noise ratio (SNR) regime. We further analyze the asymptotic ESSR to determine the key parameters; the high SNR slope and the high SNR power offset of the jamming based scenarios. To highlight the performance advantage of the proposed FJ, we also examine the scenario of without jamming (WoJ). Finally, numerical examples and discussions are provided to acquire some engineering insights, and to demonstrate the impacts of different system parameters on the secrecy performance of the considered communication scenarios. The numerical results illustrate that the proposed FJ significantly outperforms the traditional one-way communication and the Constellation rotation approach, as well as our proposed benchmarks, the two-way WoJ and GNJ scenarios.
... We note that the above expected natural logarithm of the determinant for q ≥ n s has been investigated in [34], where the derived expression is rather complicated, involving summations of determinants whose elements are in terms of the inverse of a certain Vandermonde matrix. We also note the q < n s and q = n s = s cases have been considered in [32,35]. ...
... . We can express (46) in the general form [34] ...
This paper presents an analytical characterization of the ergodic capacity of amplify-and-forward (AF) MIMO dual-hop relay channels, assuming that the channel state information is available at the destination terminal only. In contrast to prior results, our expressions apply for arbitrary numbers of antennas and arbitrary relay configurations. We derive an expression for the exact ergodic capacity, simplified closed-form expressions for the high SNR regime, and tight closed-form upper and lower bounds. These results are made possible to employing recent tools from finite-dimensional random matrix theory to derive new closed-form expressions for various statistical properties of the equivalent AF MIMO dual-hop relay channel, such as the distribution of an unordered eigenvalue and certain random determinant properties. Based on the analytical capacity expressions, we investigate the impact of the system and channel characteristics, such as the antenna configuration and the relay power gain. We also demonstrate a number of interesting relationships between the dual-hop AF MIMO relay channel and conventional point-to-point MIMO channels in various asymptotic regimes.
... It is worth noting that log 2 (p) in (146) can be rewritten as log 2 (p) = 10 log 10 (p) 10 log 10 2 = p| dB 3 dB . Hence, L ∞ is termed the high-SNR power offset in 3-dB units [101], and S ∞ is referred to as the high-SNR slope, the number of DoFs, the maximum multiplexing gain, or the pre-log factor in bits/s/Hz/(3 dB). The high-SNR slope S ∞ characterizes the ECC as a function of the transmit power, at high SNR, on a log scale. ...
... The evaluation of the ECC in (148) requires the application of random matrix theory; see [96]- [98], [102] for more details. Furthermore, the existing literature has shown that the asymptotic ECC for MIMO channels in the high-SNR regime can be also expressed in the standard form given in (146) (see, e.g., [101]). Besides the ECC and OP, the diversity-multiplexing tradeoff (DMT) is another important performance metric for statistical NFC MIMO channels. ...
Extremely large-scale antenna arrays, tremendously high frequencies, and new types of antennas are three clear trends in multi-antenna technology for supporting the sixth-generation (6G) networks. To properly account for the new characteristics introduced by these three trends in communication system design, the near-field spherical-wave propagation model needs to be used, which differs from the classical far-field planar-wave one. As such, near-field communication (NFC) will become essential in 6G networks. In this tutorial, we cover three key aspects of NFC. 1) Channel Modelling: We commence by reviewing near-field spherical-wave-based channel models for spatially-discrete (SPD) antennas. Then, uniform spherical wave (USW) and non-uniform spherical wave (NUSW) models are discussed. Subsequently, we introduce a general near-field channel model for SPD antennas and a Green's function-based channel model for continuous-aperture (CAP) antennas. 2) Beamfocusing and Antenna Architectures: We highlight the properties of near-field beamfocusing and discuss NFC antenna architectures for both SPD and CAP antennas. Moreover, the basic principles of near-field beam training are introduced. 3) Performance Analysis: Finally, we provide a comprehensive performance analysis framework for NFC. For near-field line-of-sight channels, the received signal-to-noise ratio and power-scaling law are derived. For statistical near-field multipath channels, a general analytical framework is proposed, based on which analytical expression for the outage probability, ergodic channel capacity, and ergodic mutual information are derived. Finally, for each aspect, the topics for future research are discussed.
... , β q }. According to [12], [13], the joint pdf of 0 ≤ β 1 ≤ · · · ≤ β q ≤ ∞ is given by ...
... In the general MIMO case, the ergodic capacity in the high SNR regime, i.e. when ρ → ∞, can be expressed through an affine expansion [12], as follows, ...
... The high-SNR slope is a key performance indicator for ASC at high SNRs and is given by [23,Eq. (10)] ...
... Following similar steps, we derive (17) by replacing each B and E with E and B, and the exact expression of T i,j is expressed as (23). ...
In this paper, we investigate the physical layer security over the Alternate Rician Shadowed fading channel, which is a novel channel model for body-centric wireless links and land mobile satellite. We derive exact closed-form expressions for the average secrecy capacity (ASC), secrecy outage probability (SOP), and probability of non-zero secrecy capacity (PNZ) for two cases: (i) m is a positive real number and (ii) m is a positive integer number, where m describes the level of fluctuation of the line-of-sight component. In the first case, SOP is derived in terms of the Meijer's G-function, while ASC and PNZ are derived in terms of the multivariate Fox's H-function. In the second case, ASC is derived in terms of the Meijer's G-function, while SOP and PNZ are derived in terms of elementary functions. In addition, we derive the asymptotic ASC, SOP and PNZ expressions which all match well the exact ones at high values of signal-to-noise ratio, respectively. The capacity slope of asymptotic ASC and the secrecy diversity order of asymptotic SOP have been derived for providing more physical insights. Finally, the accuracy of our derived expressions is validated by Monte-Carlo simulations.
... Proposition 1 clearly shows that the average secrecy rate with the proposed beamforming design is independent of the SI strength. Now, to obtain additional insights on the secrecy performance, based on (32), we derive two key performance indicators that determine the average secrecy rate at high SNR, namely the high SNR slope and the high SNR power offset [46]- [48]. The asymptotic average secrecy rate in (32) can be conveniently reexpressed ...
... Therefore, by substituting (46) and (47) into (45), the cdf of γ R L can be expressed as ...
We consider a secure overlay cognitive radio network with an eavesdropper wherein a multi-antenna secondary transmitter performs transmission in primary spectrum, on the condition that it helps primary system to perform secure and reliable transmission via cooperative relaying and jamming. To improve secrecy performance of primary network, we propose full-duplex jammer protocol and zero-forcing based beamforming design, which completely cancels the interferences at the primary and secondary users and simultaneously avoids the leakage of confidential information to eavesdropper. Moreover, we present new expressions for the average secrecy rate and a lower bound for the secrecy outage probability. Furthermore, an asymptotic analysis in the high signal-to-noise ratio regime is carried out to obtain closed-form average secrecy rate. Our analytical findings reveal that by exploiting beamforming and full-duplex at the secondary transmitter secrecy outage probability can be significantly reduced and a diversity order of min (N R −1, N T −2) can be achieved where N R and N T are the number of received and transmit antennas at secondary transmitter. Simulation results also demonstrate that as compared to the half-duplex scenario without jamming, the proposed cooperative FD overlay CR scheme with jamming can improve the average secrecy rate up to 224%. Index Terms Average secrecy rate, secrecy outage probability, jamming, full-duplex (FD), beamforming. Z. Mobini is with the Faculty
... An affine expansion of the capacity for high SNR values can be written as (see [29] for details) ...
... In our case, a typical assumption is to consider L < min{N R , N T } so that S ∞ = L. The high-SNR power offset, L ∞ , is given by [29,Formula (131)] ...
Random matrices are nowadays classical tools for modeling multiantenna wireless channels. Scattering phenomena typical of cellular frequencies and channel reciprocity features led to the adoption of matrices sampled either from the Gaussian Unitary Ensemble (GUE) or from more general Polynomial Ensembles (PE). Such matrices can be used to model the random impairments of the radio channel on the transmitted signal over a wireless link whose transmitter and receiver are both equipped with antenna
arrays. The exploitation of the millimeter-wave (mmWave) frequency band, planned for 5G and beyond mobile networks, prevents the use of GUE and PE elements as candidate models for channel matrices. This is mainly due to the lack of scattering richness compared to microwave-based transmissions. In this work, we propose to model mmWave Multi-Input–Multi-Output (MIMO) systems via products of random Vandermonde matrices.
We illustrate the physical motivation of our model selection, discuss the meaning of the parameters and their impact on the spectral properties of the random matrix at hand, and provide both a list of results of immediate use for performance analysis of mmWave MIMO systems, as well as a list of open problems in the field.
... In the following, our attention is focused on deriving a closed-form expression for the asymptotic ESR of WFJ scenario with → ∞. The ESR at high SNR regime is calculated as [28] where S ∞ and L ∞ are the high SNR slope in bits/s/Hz/(3 dB) and the high SNR power offset in 3 dB units, respectively, which are defined as Note that at high SNR regime with → ∞ , we have ln (1 + ) ≈ ln ( ). Furthermore, under the assumption 12 ≪ 1r , r2 , and using Eqs. ...
... (28), the part I ′ D in Eq.(27) is calculated as follows where the approximation follows from the high SNR assumption. Due to the fact that ̂ has exponential distribution with the probability density function (pdf) of f̂(x) = 1 e − x , the part I d 1 in Eq. (29) can be evaluated as where ̄ is the average SNR of the estimated channel of the relay-to-D2, and c ≈ 0.577 denoted the Euler's constant. ...
In this paper, we investigate the impact of channel estimation errors on the physical layer security of an overlaying device-to-device (D2D) wireless network with an amplify-and-forward untrusted relay. An untrusted relay assists D2D communication while may capture the confidential data. Under the practical assumption of imperfect channel state information (ICSI) for the relay-to-receiver D2D link, we take into account optimal power allocation (OPA) problem to maximize the achievable secrecy rate of two different scenarios which are without jamming and with friendly jamming. Based on these OPA solutions, we study the secrecy performance of the two scenarios by driving closed-form expressions for the ergodic secrecy rate (ESR) in Rayleigh fading channel. We also calculate the high signal-to-noise ratio (SNR) slope and high SNR power offset of the optimized scenarios by finding the asymptotic ESR. Numerical results confirm the accuracy of our proposed theoretical analysis. The results also demonstrate that our proposed OPAs enhance the ESR performance compared with other power allocation techniques. Moreover, they show the effect of ICSI on the ESR such that as channel estimation error grows, the ESR performance reduction is occurred.
... For high SNR values, the capacity can be expressed as [16,33] where 3 dB= 10 log 10 (2) and the S ∞ and L ∞ are defined as ...
... Given (53), it was shown in [33] that S ∞ = 1 bit/s/Hz, regardless of N r and N s . The high SNR power offset L ∞ is derived by substituting (53) in (49) as [16] For the MRC scheme, by substituting (22) in (50) and calculating the resultant derivative with the aid of the identity dΓ(z)∕dz = (z)Γ(z) , we express L MRC ∞ as For the SC technique, L SC ∞ is given by ...
In this paper, we provide a performance analysis of communication systems over Rayleigh-product channels with two popular diversity combining techniques, namely maximal ratio combining (MRC) and selection combining (SC). We first derive new closed-form expressions for the exact cumulative distribution function (CDF) and probability density function (PDF) of the post-processing signal-to-noise ratio (SNR) for these two schemes. Secondly, we present the first-order asymptotic expansions for these CDF and PDF functions. Performance of MRC and SC techniques, in terms of outage probability, average symbol error rate (SER) and ergodic capacity, is derived using the exact expressions of CDF and PDF. Furthermore, we present new expressions for key metrics characterizing the system performance at the high and low SNR regimes. Thanks to the asymptotic CDF and PDF expressions, we compute the average SER in the high SNR regime and derive the diversity order and array gain parameters. In addition, we provide simple expressions for the ergodic capacity in the asymptotic low and high SNR regimes. Monte-Carlo simulations are conducted and their results agree well with the analytical results.
... To provide insight into the performance, the high-SNR slope is considered that is defined as [38]. To obtain it, the asymptotic expression for N's ER and the ceiling for F's ER are derived in the following proposition. ...
... In Fig. 5 corresponding analytical results that are derived from (27), (28), (38), and (39). It is also observed that the high-SNR approximations that are derived from (30), (31), (43), and (44) are accurate, and hence, it can be used to derive high-SNR slopes. ...
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 for the BS-IRS-UD link with Nakagami-m fading are investigated. For each scenario, closed-form expressions for the outage probability and the ergodic rate are derived. To gain further insight, the diversity order and the 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 elements and Nakagami-m 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.
... This measure has already proved instrumental in several findings. In [3], Lozano et al. analyzed the multiple-antenna point-to-point scenario, revealing that some system features that do not impact the DoF (as antenna correlation, fading...) do considerably impact the zero-order term, affecting the performance of the system at any possible SNR. In addition to exposing some limitations of the DoF metric, [3] also revealed that the affine expansion offers appreciably tight approximations also at medium-to-low SNR. ...
... In [3], Lozano et al. analyzed the multiple-antenna point-to-point scenario, revealing that some system features that do not impact the DoF (as antenna correlation, fading...) do considerably impact the zero-order term, affecting the performance of the system at any possible SNR. In addition to exposing some limitations of the DoF metric, [3] also revealed that the affine expansion offers appreciably tight approximations also at medium-to-low SNR. This characterization has been also established for the Broadcast Channel (BC) with perfect CSIT using Dirty-Paper Coding and linear precoding [38], and for the BC with imperfect CSIT [39]. ...
In this paper, we analyze the high-SNR regime of the M x K Network MISO channel in which each transmitter has access to a different channel estimation, possibly with different accuracy. It has been recently shown that, for some regimes, this setting attains the same Degrees-of-Freedom as the ideal centralized setting with perfect CSI sharing, in which all the transmitters are endowed with the best estimate available at any transmitter. This surprising result is restricted by the limitations of the Degrees-of-Freedom metric, as it only provides information about the slope of growth of the capacity as a function of the SNR, without any insight about the possible performance at a given SNR. In this work, we analyze the affine approximation of the rate on the high-SNR regime for this decentralized Network MISO setting, presenting the unexpected result that, for a regime of antenna configurations, it is possible to asymptotically attain the same achievable rate as in the ideal centralized scenario. Consequently, it is possible to achieve the beam-forming gain of the ideal perfect-CSIT-sharing setting even if only a subset of transmitters is endowed with accurate CSI. This outcome is a consequence of the synergistic compromise between CSIT accuracy at the transmitters and consistency between the locally-computed precoders. We propose a precoding scheme achieving the previous result, which is built on an uneven structure in which some transmitters reduce the accuracy of their own precoding vector for sake of using transmission parameters that can be more easily predicated by the other transmitters.
... We can further evaluate the asymptotic ESSR when ρ → ∞ by applying the general asymptotic form given by [12] R ...
In this paper, we examine the secrecy performance of two-way relaying between a multiple antenna base station (BS) and a single antenna mobile user (MU) in the presence of a multiple antenna friendly jammer (FJ). We consider the untrusted relaying scenario where an amplify-and-forward relay is both a necessary helper and a potential eavesdropper. To maximize the instantaneous secrecy sum rate, we derive new closed-form solutions for the optimal power allocation (OPA) between the BS and MU under the scenario of relaying with friendly jamming (WFJ). Based on the OPA solution, new closed-form expressions are derived for the ergodic secrecy sum rate (ESSR) with Rayleigh fading channel. Furthermore, we explicitly determine the high signal-to-noise ratio slope and power offset of the ESSR to highlight the benefits of friendly jamming. Numerical examples are provided to demonstrate the impact of the FJ's location and number of antennas on the secrecy performance.
... The rate offset of log 2 b (per user) can easily be translated into a power offset, which is a more useful metric from the design perspective. Since a multiplexing gain of M is achieved with zero-forcing, the [32]. Thus, b = 2 corresponds to a 3 dB offset, and the resulting scaling of bits takes on a particularly simple form when a 3 dB offset is desired: ...
Multiple transmit antennas in a downlink channel can provide tremendous capacity (i.e. multiplexing) gains, even when receivers have only single antennas. However, receiver and transmitter channel state information is generally required. In this paper, a system where each receiver has perfect channel knowledge, but the transmitter only receives quantized information regarding the channel instantiation is analyzed. The well known zero forcing transmission technique is considered, and simple expressions for the throughput degradation due to finite rate feedback are derived. A key finding is that the feedback rate per mobile must be increased linearly with the SNR (in dB) in order to achieve the full multiplexing gain, which is in sharp contrast to point-to-point MIMO systems in which it is not necessary to increase the feedback rate as a function of the SNR.
... On that account, the study of the low power cornerstone, i.e., the low-SNR regime, is of great significance. There is no reason to consider the high-SNR regime, because in this regime an important metric such as the high-SNR slope S ∞ = lim pr→0 R lk (pr,α) log 2 pr [31] is zero due to the finite rate, as shown in (17). ...
In this paper, we analyze the performance of the uplink communication of massive multi-cell multiple-input multiple-output (MIMO) systems under the effects of pilot contamination and delayed channels because of terminal mobility. The base stations (BSs) estimate the channels through the uplink training, and then use zero-forcing processing to decode the transmit signals from the users. The probability density function (PDF) of the signal-to-interference-plus-noise ratio is derived for any finite number of antennas. From this PDF, we derive an achievable ergodic rate with a finite number of BS antennas in closed form. Insights of the impact of the Doppler shift (due to terminal mobility) at the low signal-to-noise ratio regimes are exposed. In addition, the effects on the outage probability are investigated. Furthermore, the power scaling law and the asymptotic performance result by infinitely increasing the numbers of antennas and terminals (while their ratio is fixed) are provided. The numerical results demonstrate the performance loss for various Doppler shifts. Among the interesting observations revealed is that massive MIMO is favorable even in channel aging conditions.
... When the channel propagation conditions are somewhat favorable (e.g., distinct LOS directions as in scenario 1, or NLOS propagation as in scenario 2), maximizing the channel beamforming gain is the better strategy. The relative performance of D-GOB and D-SUB depends in general on ρ: For all N ,c D-SUB (ρ, N ) goes to 1 in the limit ρ → ∞, with the difference between C TDD (ρ) and C D-SUB (ρ, N ) constant [42], [43]. Meanwhile, for interferencelimited D-GOB we have thatc D-GOB (ρ, N ) must go to 0 as ρ → ∞, if N < 128, and to 1, if N = 128. ...
Downlink beamforming in Massive MIMO either relies on uplink pilot measurements - exploiting reciprocity and TDD operation, or on the use of a predetermined grid of beams with user equipments reporting their preferred beams, mostly in FDD operation. Massive MIMO in its originally conceived form uses the first strategy, with uplink pilots, whereas there is currently significant commercial interest in the second, grid-of-beams. It has been analytically shown that in isotropic scattering (independent Rayleigh fading) the first approach outperforms the second. Nevertheless there remains controversy regarding their relative performance in practice. In this contribution, the performances of these two strategies are compared using measured channel data at 2.6 GHz.
... As an alternative to deriving exact analytical results, some works focus on extracting parameters that characterize the channel capacity under extreme SNR scenarios (see [3] - [4] for more details on the extreme SNR characterization). The low-SNR regime is characterized through the minimum transmit E b /N 0 that enables reliable communications, i.e., E b /N 0 min , and the low-SNR spectral efficiency slope S 0 . ...
The open problem of calculating the limiting spectrum (or its Shannon transform) of increasingly large random Hermitian finite-band matrices is described. In general, these matrices include a finite number of non-zero diagonals around their main diagonal regardless of their size. Two different communication setups which may be modeled using such matrices are presented: a simple cellular uplink channel, and a time varying inter-symbol interference channel. Selected recent information-theoretic works dealing directly with such channels are reviewed. Finally, several characteristics of the still unknown limiting spectrum of such matrices are listed, and some reflections are touched upon.
... In the work [ 18], the authors have studied the downlink and uplink intelligent reflective surface support networks: NOMA and OMA. In [ 19], the authors surveyed the IRS-NOMA Integrated Communication Network. ...
The intelligent reflecting surface (IRS) with the ability to flexibly control the reflection direction as well as the electromagnetic wave phase shift (EM) is considered as a potential solution to improve radio quality in the modern age. 5G and 6G legacy. Non-orthogonal multiple access (NOMA) network is considered the optimal technique in 5G network, because NOMA network has high spectral efficiency and high energy efficiency. In this study, we study a NOMA network incorporating IRS connected to two users at the border. Here, the IRS is used as a bridge that supports the user equipment (UE) at the cell edge to communicate with the base station (BS) without having a direct link to the base station. We analyze and evaluate NOMA network performance in combination with IRS and survey and analyze BS-IRS-UE channel statistics with Nakagami-m fading distribution. We construct closed-form expressions for stopping probability, ergodic rate and approximation of stopping probability, ergodic ratio of UEs to signal-to-noise ratio (SINR) with low and high value. Finally, we analyze, simulate and verify by Monte Carlo simulation using Matlab software.KeywordsIntelligent reflecting surfaceNon-orthogonal multiple accessOutage probabilityErgodic rate
... 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 (15) and (29), we can evaluate S ∞ and L ∞ in as follows. Theorem 5. When N r ≥ N t , the high-SNR slope and high-SNR power offset of the RIS-aided MIMO channel are, respectively, given by ...
... With an affine expansion [20], and following the same process as [5] (omitted for lack of space), the MIMO-RIS ergodic capacity in high-SNR regime can be found as follows: ...
Metamaterial-based antenna designs, such as Reconfigurable Intelligent Surface (RIS), are expected to play a significant role in next generation communication networks (i.e. 6G) because of their ability to improve wireless communication environments. This paper investigates the ergodic capacity of RIS-aided multiple input multiple output (MIMO), a.k.a. MIMO-RIS, systems over Rayleigh-Rician fading channels. We consider that the transmitter-RIS and receiver-RIS links experience Rayleigh and Rician fading, respectively. An exact analytical expression of the ergodic capacity is derived based on closed-form expressions of the probability density function (pdf) of the cascaded channel. Moreover, a high SNR expression and a large RIS approximation are provided to unveil further system insights. Simulations result validate the correctness of our expressions and show the impact of the Rician fading and the number of RIS elements on the capacity.
... The curve in Fig. 1.14 grows as K log 2 (SNR) + constant at high SNR, where K = 5 is the number of multiplexed users. This is a classical scaling behavior for multi-antenna systems [34] and demonstrates that the K channel vectors span a K-dimensional vector space. The factor K is called the spatial degrees-of-freedom (DoF) and manifests the maximum capacity that the system can achieve. ...
The number of users that can be spatially multiplexed by a wireless access point depends on the aperture of its antenna array. When the aperture increases and wavelength shrinks, "new" electromagnetic phenomena can be utilized to further enhance network capacity. In this chapter, we describe how extremely large aperture arrays (ELAA) can extend the radiative near-field region to kilometer distances. We demonstrate how this affects the propagation models in line-of-sight (LoS) scenarios and enables finite-depth beamforming. In particular, it becomes possible to simultaneously serve users that are located in the same direction but at different distances.
... In addition to simplifying the performance analysis, we can observe several insights from the approximate expressions. For example, similar to the diversity order [53], the high-SNR slope is a key performance indicator that can explicitly capture the impact of channel parameters on OP performance, which is defined as [54] ...
TeraHertz (THz) communications can satisfy the high data rate demand with massive bandwidth. However, severe path attenuation and hardware imperfection greatly alleviate its performance. Therefore, we utilize the reconfigurable intelligent surface (RIS) technology and investigate the RIS-aided THz communications. We first prove that the small-scale amplitude fading of THz signals can be accurately modeled by the fluctuating two-ray distribution based on two THz signal measurement experiments conducted in a variety of different scenarios. To optimize the phase-shifts at the RIS elements, we propose a novel swarm intelligence-based method that does not require full channel estimation. We then derive exact statistical characterizations of end-to-end signal-to-noise plus distortion ratio (SNDR) and signal-to-noise ratio (SNR). Moreover, we present asymptotic analysis to obtain more insights when the SNDR or the number of RIS’s elements is high. Finally, we derive analytical expressions for the outage probability and ergodic capacity. The tight upper bounds of ergodic capacity for both ideal and non-ideal radio frequency chains are obtained. It is interesting to find that increasing the number of RIS’s elements can significantly improve the THz communications system performance. For example, the ergodic capacity can increase up to 25% when the number of elements increases from 40 to 80, which incurs only insignificant costs to the system.
... In this paper, we present closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral χ 2 -distributed RV with even or odd degrees of freedom. Note that while the probability density function (PDF), the moment-generating function (MGF), and the moments of a noncentral χ 2 -distributed RV are well-known, the expected logarithm and the negative integer moments have only been derived relatively recently for even degrees of freedom [1][2][3][4][5][6], but-to the best of our knowledge-for odd degrees of freedom they have been completely unknown so far. These expectations have many interesting applications. ...
Closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral χ 2 -distributed random variable are presented in the cases of both even and odd degrees of freedom. Moreover, some basic properties of these expectations are derived and tight upper and lower bounds on them are proposed.
... We now provide the asymptotic analysis of the ASR, i.e., the ASR performance when transmit SNR ρ = P N 0 goes to infinity. To this aim, according to the definition in [35], the high SNR slope S ∞ and the high SNR power offset L ∞ are obtained, respectively, as ...
This paper proposes a novel cooperative secure unmanned aerial vehicle (UAV) aided transmission protocol, where a source sends confidential information to a destination via an energy-constrained UAV-mounted amplify-and-forward relay in the presence of a ground eavesdropper. We adopt destination-assisted cooperative jamming as well as simultaneous wireless information and power transfer at the UAV-mounted relay to enhance physical-layer security and transmission reliability. Assuming a low-altitude UAV, we derive connection probability, secrecy outage probability, instantaneous secrecy rate, and average secrecy rate of the proposed protocol over Air-Ground channels, which are modeled as Rician fading with elevation-angel dependent parameters. Further, we analyze the asymptotic average secrecy rate performance of the proposed UAV-relaying scheme and derive high signal-to-noise ratio measures of the average secrecy rate to highlight the effect of various channel features on the system performance. By simulations, we verify our novel theoretical exact and approximate results and demonstrate significant performance improvement of our protocol, when compared to conventional transmission protocol with ground relaying and UAV-based transmission protocol without exploiting destination jamming. Finally, we evaluate the impacts of various system parameters, specifically, find the optimal UAV placement on the proposed protocol in terms of the aforementioned secrecy metrics.
Non-orthogonal multiple access (NOMA) can be a desirable method since it can utilize resources efficiently to provide larger capacity. Similarly, optical transmission is also becoming popular due to the availability of licence-free large bandwidth and low-cost transceiver implementation. Although optical NOMA system can be promising for long range wireless systems, investigation of its reliability performance is essential for different channel models. In this study, outage probability, channel capacity, symbol error rate expressions, diversity and coding gains for arbitrary number of users are obtained for downlink-NOMA over free space optical links modeled with Málaga distribution. Theoretical results are verified by numerical examples for various scenarios providing useful background for future analyses and designs.
The number of users that can be spatially multiplexed by a wireless access point depends on the aperture of its antenna array. When the aperture increases and wavelength shrinks, “new” electromagnetic phenomena can be utilized to further enhance network capacity. In this chapter, we describe how extremely large aperture arrays (ELAA) can extend the radiative near-field region to kilometer distances. We demonstrate how this affects the propagation models in line-of-sight (LoS) scenarios and enables finite-depth beamforming. In particular, it becomes possible to simultaneously serve users that are located in the same direction but at different distances.
The intelligent reflecting surface (IRS) has emerged as a promising solution to enhance the quality of radio transmission by allowing flexible control over the direction of reflection and phase shift of electromagnetic waves. In this study, we present a downlink wireless network considering non-orthogonal multiple access (NOMA) and two IRSs to improve performance for a group of user equipment (UE) at the cell edge, when the UEs cannot reach out direct links from the base station (BS). We start by discussing the framework to design a wireless system relying on both IRS and NOMA, and then calculate the link channel statistics between the BS, IRS, and UEs with Rayleigh fading distributions using the probability density function (PDF) and cumulative distribution function (CDF). Based on these distributions, we derive closed form expressions for the outage probability (OP) for evaluating the performance of the user pair. Additionally, we compute the bit error rate (BER), the ergodic rates (ER) of the UE as functions of the signal-to-interference-plus-noise ratio (SINR) using the Chebyshev-Gauss quadrature method, and hence the complete the overall evaluation of the system’s performance can be achieved. Finally, we verify the mathematical analysis through comparing with Monte-Carlo simulations and indicate that the number of IRS meta-surfaces elements as the main limiting factor to achieve better OP, ER and BER performance.
The reconfigurable intelligent surface (RIS) is useful to effectively improve the coverage and data rate of end-to-end communications. In contrast to the well-studied coverage-extension use case, in this paper, multiple RIS panels are introduced, aiming to enhance the data rate of multi-input multi-output (MIMO) channels in presence of insufficient scattering. Specifically, via the operator-valued free probability theory, the asymptotic mutual information of the large-dimensional RIS-assisted MIMO channel is obtained under the Rician fading with Weichselberger’s correlation structure, in presence of both the direct and the reflected links. Although the mutual information of Rician MIMO channels scales linearly as the number of antennas and the signal-to-noise ratio (SNR) in decibels, numerical results show that it requires sufficiently large SNR, proportional to the Rician factor, in order to obtain the theoretically guaranteed linear improvement. This paper shows that the proposed multi-RIS deployment is especially effective to improve the mutual information of MIMO channels under the large Rician factor conditions. When the reflected links have similar arriving and departing angles across the RIS panels, a small number of RIS panels are sufficient to harness the spatial degree of freedom of the multi-RIS assisted MIMO channels.
This work evaluates the secrecy rate of transmit antenna selection (TAS) in the amplify and forward (AF)-relay system with multi-antenna relay and user. This paper considers that the receiver of relays, users, and an eavesdropper have non-identical channel estimation error. To assess the secrecy rate, this work derives new analytical expressions for the cumulative distribution function (CDF) and probability density function (PDF) of the received signal to noise ratio (SNR) using infinite series for maximal-ratio combining (MRC) and selection combining (SC) receivers. Using the CDF for the received SNR, we derive new exact analytical expressions for the secrecy rate with non-identical channel estimation error for four scenarios. Further, using the PDF for the received SNR, we derive the asymptotic analytical expressions for the secrecy rate and quantify the high SNR slope and the high SNR power offset for four scenarios. From the numerical results, we observe that the relay and user selection has the high SNR power offset gain compared to the non-selection. The secrecy rate of the proposed system can be improved according to legitimate receiver's parameters such as the number of receive antennas and channel correlation coefficient and can be degraded according to eavesdropper's parameters. The superiority of the secrecy rate of the MRC receiver and the SC receiver depends on the channel estimation accuracy of the MRC receiver.
Despite the numerous results in the literature about the eigenvalue distributions of Wishart matrices, the existing closed-form probability density function (pdf) expressions do not allow for efficient sampling schemes from such densities. In this letter, we present a stochastic representation for the eigenvalues of
complex central uncorrelated Wishart matrices with an arbitrary number of degrees of freedom (referred to as dual Wishart matrices). The draws from the joint pdf of the eigenvalues are generated by means of a simple transformation of a chi-squared random variable and an independent beta random variable. Moreover, this stochastic representation allows a simple derivation, alternative to those already existing in the literature, of some eigenvalue function distributions such as the condition number or the ratio of the maximum eigenvalue to the trace of the matrix. The proposed sampling scheme may be of interest in wireless communications and multivariate statistical analysis, where Wishart matrices play a central role.
Millimeter wave (mmWave) is a promising future wireless communication technology. To overcome the high path loss in mmWave band, distributed antenna systems (DASs) with each base station (BS) connecting to multiple remote radio units (RRUs) is a candidate solution. To better characterize the small-scale fading of mmWave channels, the fluctuating two-ray (FTR) was recently proposed. In this letter, we study the performance of RRU selection in mmWave DASs over the FTR fading. First, simplified closed-form expressions are, respectively, provided for the probability density function (PDF) and cumulative distribution function (CDF) of signal-to-noise ratio (SNR). Then, we derive the exact closed-form expressions and their high-SNR approximations for the outage probability and ergodic rate, respectively. The effect of RRU number and FTR-fading parameters on the performance are further revealed from these high-SNR approximations. Simulations validate our analytical results.
In this paper, we analyze the high-SNR regime of the
Network MISO channel in which each transmitter has access to a different channel estimate, possibly with different precision. It has been recently shown that, for some regimes, this setting attains the same Degrees-of-Freedom as the ideal centralized setting with perfect Channel State Information (CSI) sharing, in which all the transmitters are endowed with the best estimate available at any transmitter. This result is restricted by the limitations of the Degrees-of-Freedom metric, as it only provides information about the slope of growth of the capacity as a function of the SNR, without any insight about the possible performance at a given SNR. In order to overcome this limitation, we analyze the affine approximation of the rate on the high-SNR regime for this decentralized Network MISO setting for the antenna configurations in which it achieves the Degrees-of-Freedom of the centralized setting. We show that, for a regime of antenna configurations, it is possible to asymptotically attain the same achievable rate as in the ideal centralized scenario. Consequently, it is possible to achieve the beamforming gain of the ideal perfect-CSI-sharing setting even if only a subset of transmitters is endowed with precise CSI, which can be exploited in scenarios such as distributed massive MIMO where the number of transmit antennas is much bigger than the number of served users. This outcome is a consequence of the synergistic compromise between CSI precision at the transmitters and consistency between the locally-computed precoders, which is an inherent trade-off of decentralized settings that does not exist in the centralized CSI configuration. We propose a precoding scheme achieving the previous result, which is built on an uneven structure in which some transmitters reduce the precision of their own precoding vector for the sake of using transmission parameters that can be more easily predicted by the other transmitters.
In this paper, novel ergodic capacity (EC) performance evaluation results of a power beacon (PB)-assisted multi-input multi-output (MIMO) wireless powered communication network are presented. In the considered system, the energy harvesting node harvests energy from the radio-frequency signals sent by the dedicated PB and uses this energy to communicate with the destination node. To accurately model the combined effect of multi-path fading and shadowing, it is assumed that the energy transfer link is subject to
-
shadowed fading. Performance evaluation results are presented for two cases, depending upon the availability of channel state information (CSI) at the PB, namely,
no
CSI and
full
CSI. In the former case, equal power allocation is assumed, whereas, in the later case, energy beamforming is employed to increase energy transfer efficiency. For the performance evaluation of EC under
full
CSI, a closed-form approximation for the probability density function of the maximum eigenvalue of a
-
shadowed distributed random matrix is derived. For both
no
CSI and
full
CSI cases, lower and upper bounds on the achievable EC are derived in closed-form. Moreover, in order to obtain further insights on the impact of key parameters on the system performance, asymptotic EC expressions which become very tight at low- and high-signal-to-noise ratio regimes, are obtained. Using the proposed EC lower bound as well as these asymptotic results, simple closed-form expressions for the optimal time split that maximize the achievable EC are derived. Numerically evaluated results accompanied with Monte-Carlo simulations are further presented to corroborate the theoretical analysis.
Due to the increasing demand for higher capacity in mobile communication devices, non-orthogonal multiple access (NOMA) has become a promising technique. Compared to existing orthogonal multiple access methods, NOMA can serve multiple users with better spectral efficiency by allowing co-channel interference, thus its reliability performance over various channel models needs further research. In this paper, closed-form outage probability, average channel capacity and symbol error rate are derived for a downlink-NOMA system with an arbitrary number of users over generalized-K fading channels. Finally, theoretical findings are validated via Monte Carlo simulations for several channel parameters.
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.
We study memoryless, discrete time, matrix channels with additive white Gaussian noise and input power constraints of the form , where , and are complex, i=1..m, j=1..n, and H is a complex matrix with some degree of randomness in its entries. The additive Gaussian noise vector is assumed to have uncorrelated entries. Let H be a full matrix (non-sparse) with pairwise correlations between matrix entries of the form , where C,D are positive definite Hermitian matrices. Simplicities arise in the limit of large matrix sizes (the so called large-N limit) which allow us to obtain several exact expressions relating to the channel capacity. We study the probability distribution of the quantity . S is non-negative definite and hermitian, with . Note that the expectation E[f(H)], maximised over S, gives the capacity of the above channel with an input power constraint in the case H is known at the receiver but not at the transmitter. For arbitrary C,D exact expressions are obtained for the expectation and variance of f(H) in the large matrix size limit. For C=D=I, where I is the identity matrix, expressions are in addition obtained for the full moment generating function for arbitrary (finite) matrix size in the large signal to noise limit. Finally, we obtain the channel capacity where the channel matrix is partly known and partly unknown and of the form , being known constants and entries of H i.i.d. Gaussian with variance 1/n. Channels of the form described above are of interest for wireless transmission with multiple antennae and receivers.
In this article we propose matrix variate Kummer-Gamma distribution which is an extension of matrix variate Gamma distribution. Several properties of this distribution have been studied. Distributional results on randorn quadratic forms involving Kummer-Gamma matrix have also been derived.
Previous studies have shown that single-user systems employing
n-element antenna arrays at both the transmitter and the receiver can
achieve a capacity proportional to n, assuming independent Rayleigh
fading between antenna pairs. We explore the capacity of
dual-antenna-array systems under correlated fading via theoretical
analysis and ray-tracing simulations. 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.
In both cases, the theoretic capacity growth is linear in n but the
growth rate is 10-20% smaller in the presence of correlated fading. We
analyze our assumption of separable transmit/receive correlations via
simulations based on a ray-tracing propagation model. Results show that
empirical capacities converge to the limit capacity predicted from our
asymptotic theory even at moderate n = 16. We present results for both
the cases when the transmitter does and does not know the channel
realization
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 re- ceiving 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 for exponential correlation model. Finally, we derive an exact expression for the mean value of the capacity for arbitrary correlation matrices.
Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing and small-world networks. This article provides a tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained. Furthermore, the application of random matrix theory to the fundamental limits of wireless communication channels is described in depth.
Information theoretic properties of flat fading channels with multiple antennas are investigated. Perfect channel knowledge at the receiver is assumed. Expressions for maximum information rates and outage probabilities are derived. The advantages of transmitter channel knowledge are determined and a critical threshold is found beyond which such channel knowledge gains very little. Asymptotic expressions for the error exponent are found. For the case of transmit diversity closed form expressions for the error exponent and cutoff rate are given. The use of orthogonal modulating signals is shown to be asymptotically optimal in terms of information rate.
Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing and small-world networks. Random Matrix Theory and Wireless Communications is the first tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained. Furthermore, the application of random matrix theory to the fundamental limits of wireless communication channels is described in depth. The authors have created a uniquely comprehensive work that provides the reader with a full understanding of the foundations of random matrix theory and demonstrates the trends of their applications, particularly in wireless communications. Random Matrix Theory and Wireless Communications is a valuable resource for all students and researchers working on the cutting edge of wireless communications.
The past decade has seen many advances in physical layer wireless communication theory and their implementation in wireless systems. This textbook takes a unified view of the fundamentals of wireless communication and explains the web of concepts underpinning these advances at a level accessible to an audience with a basic background in probability and digital communication. Topics covered include MIMO (multi-input, multi-output) communication, space-time coding, opportunistic communication, OFDM and CDMA. The concepts are illustrated using many examples from real wireless systems such as GSM, IS-95 (CDMA), IS-856 (1 x EV-DO), Flash OFDM and UWB (ultra-wideband). Particular emphasis is placed on the interplay between concepts and their implementation in real systems. An abundant supply of exercises and figures reinforce the material in the text. This book is intended for use on graduate courses in electrical and computer engineering and will also be of great interest to practising engineers.
Jacobians, Exterior Products, and Related Topics Kronecker Products
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Abstract ● ● ● Transmit Antenna Array,, ● ● Receive
Over the last decade, the increases in capacity promised by multiantenna communication techniques have spurred many information-theoretic analyses. Furthermore, information theory has been used as a design tool to optimize the signals fed to the transmit array and to motivate signal processing strategies at the receiver. In this chapter, we catalog a number of misconceptions that have arisen in the multiantenna literature. The focus is on information-theoretic results and their interpretations, rather than on the validity of the various modeling assumptions. After some introductory material, we address several misconceptions about optimum signaling and about the impact of model features, such as antenna correlation, line-of-sight components, and intercell interference. Particular attention is given to the low- and high-power regions. We also briefly touch upon the relationship of the capacity results with some practical transmit and receive architectures. The chapter deals mostly, but not exclusively, with coherent communication. Definitions Denoting by the number of transmit and receive antennas, respectively, we shall abide by the frequency-flat complex vector model where is a deterministic scalar that represents the average channel gain, while the random matrix H is normalized to satisfy While the distribution of H is known to both transmitter and receiver, we shall specify when its realization is known to either.
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.
From the Publisher:
IEEE Press is pleased to bring back into print this definitive text and reference covering all aspects of microwave mobile systems design. Encompassing ten years of advanced research in the field, this invaluable resource reviews basic microwave theory, explains how cellular systems work, and presents useful techniques for effective systems development. The return of this classic volume should be welcomed by all those seeking the original authoritative and complete source of information on this emerging technology. An in-depth and practical guide, Microwave Mobile Communications will provide you with a solid understanding of the microwave propagation techniques essential to the design of effective cellular systems.
We study memoryless, discrete time, matrix channels with additive white Gaussian noise and input power constraints of the
form Y
i = ∑
j
H
ij
X
j
+ Z
i
, where Y
i
, X
j
and Z
i
are complex, i = 1… m, j = 1… n, and H is a complex m× n matrix with some degree of randomness in its entries. The additive Gaussian noise vector is assumed to have uncorrelated
entries. Let H be a full matrix (non-sparse) with pairwise correlations between matrix entries of the form E[H
ik
H
*
jl] = 1/n
C
ij
D
kl, where C, D are positive definite Hermitian matrices. Simplicities arise in the limit of large matrix sizes (the so called large-n limit) which allow us to obtain several exact expressions relating to the channel capacity. We study the probability distribution
of the quantity f(H) = log (1+PH
†
SH) . S is non-negative definite and hermitian, with TrS = n and P being the signal power per input channel. Note that the expectation E[f(H)], maximised over S, gives the capacity of the above channel with an input power constraint in the case H is known at the receiver but not at the transmitter. For arbitrary C, D exact expressions are obtained for the expectation and variance of f(H) in the large matrix size limit. For C = D = I, where I is the identity matrix, expressions are in addition obtained for the full moment generating function for arbitrary (finite)
matrix size in the large signal to noise limit. Finally, we obtain the channel capacity where the channel matrix is partly
known and partly unknown and of the form α; I+ β H, α,β being known constants and entries of H i.i.d. Gaussian with variance 1/n. Channels of the form described above are of interest for wireless transmission with multiple antennae and receivers.
The past decade has seen many advances in physical-layer wireless communication theory and their implementation in wireless systems. This textbook takes a unified view of the fundamentals of wireless communication and explains the web of concepts underpinning these advances at a level accessible to an audience with a basic background in probability and digital communication. Topics covered include MIMO (multiple input multiple output) communication, space-time coding, opportunistic communication, OFDM and CDMA. The concepts are illustrated using many examples from wireless systems such as GSM, IS-95 (CDMA), IS-856 (1 × EV-DO), Flash OFDM and ArrayComm SDMA systems. Particular emphasis is placed on the interplay between concepts and their implementation in systems. An abundant supply of exercises and figures reinforce the material in the text. This book is intended for use on
This paper characterizes the capacity-achieving input covariance for multi-antenna channels with (zero-mean) arbitrary distribution. The solution accom-modates a wide range of correlation structures, not necessarily separate transmit-receive. Our characterization of the covariance encompasses both its eigenvectors, which are found explicitly, and its eigenvalues, for which we present necessary and sufficient conditions as well as an iterative algorithm. In addition, we iden-tify the correlation structures for which an isotropic input achieves capacity.
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 independenceassumptions for the fades and noises at different receiving antennas.
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.
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.
Wireless systems using multi-element antenna arrays simultaneously at the 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-antenna-array systems via theoretical analysis and simulation experiments. We present expressions for the asymptotic growth rate of capacity with n for both independent and correlated fading cases; the latter is derived under some assumptions about 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 predictions of our asymptotic theory to the capacities of channels simulated using ray tracing, and find good agreement even for moderate n, i.e., 1⩽n⩽16. Our results address both the cases when the transmitter does and does not know the channel realization
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 characterize the capacity-achieving input covariance for multi-antenna channels known instantaneously at the receiver and in distribution at the transmitter. Our characterization, valid for arbitrary numbers of antennas, encompasses both the eigenvectors and the eigenvalues. The eigenvectors are found for zero-mean channels with arbitrary fading profiles and a wide range of correlation and keyhole structures. For the eigenvalues, in turn, we present necessary and sufficient conditions as well as an iterative algorithm that exhibits remarkable properties: universal applicability, robustness and rapid convergence. In addition, we identify channel structures for which an isotropic input achieves capacity.
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.