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Overhead and Spectral Efficiency of Pilot-Assisted Interference Alignment in Time-Selective Fading Channels
Abstract
The spectral efficiency achievable by interference alignment (IA) in a K-user multiple-input–multiple-output interference channel is studied in the face of time-selective continuous fading explicitly estimated through pilot-symbol observations. The robustness of IA in such operationally relevant conditions is assessed through a joint optimization of the pilot overhead and the IA update interval, which are characterized—in high-power conditions—as solutions of a fixed-point equation. Variations of the formulation are given for both frequency-division duplexing (FDD) and time-division duplexing, the former requiring explicit feedback of the fading estimates and the latter relying on fading reciprocity. For the FDD variation, analog feedback is considered. In addition to arbitrary numbers of users and antennas and arbitrary temporal fading correlation functions, the derivations accommodate forward and reverse links with asymmetric power levels.
... The analysis therein is based on a channel model with only transmit correlation, while the impact of receive correlation is omitted in consideration of its difficulty in analyzing. Other work on IA with imperfect CSI can be found in [17][18][19][20][21][22][23][24][25][26][27], including trade-off on overhead and spectrum efficiency, CSI sharing with limited feedback link and robust IA schemes conquering CSI imperfection, etc. ...
... Substitute the closed form analytical lower bound (19) ...
... Through the above analysis, we can perceive that the DoF of IA at high SNR regime can be achieved if the CSI accuracy improves along with the increase of SNR, which is true as shown by channel estimation theory [33,34]. However, it is worth noting that there exists a trade-off between the overhead paid for CSI improvement and effective data transmit [19]. ...
This paper studies the impact of channel error on the achievable rate of symmetrical K-user multiple-input multiple-output linear interference alignment (IA) networks. The upper and lower bounds of the achievable sum rate are derived analytically with the assumption of orthonormal transmit precoders and receive filters designed from imperfect channel state information (CSI) over both the uncorrelated and correlated channels. For uncorrelated channels, quite tight lower and upper bounds are obtained. The impact of channel error on the degrees of freedom (DoF) and the DoF persistence conditions are also investigated. Results show that the DoF of IA networks persists only if the channel error decreases in an order higher than the signal-to-noise ratio. For correlated channel, the lower and upper bounds for one realization of IA are derived. The derived upper bound can be used to characterize the achievable rate approximately. Simulation results indicate that the achievable rate of IA network is influenced significantly by CSI uncertainty. The obtained analytical bounds provide an intuitive way to show the impact of channel error on the achievable rate and thus can help practical systems deign.
... However, CSI is not a priori available at terminals and certain mechanisms should be designed to acquire CSI at terminals. In order to obtain CSI estimation at receivers, pilot-assisted channel training schemes have been developed for single-input multi-output (SIMO) point-to-point communication systems [4], single-input single-output (SISO) interference networks [5], and MIMO interference networks [6]. Also, digital/analog channel state feedback schemes have been proposed in the literature to acquire CSI estimation at transmitters (see e.g. ...
... Clearly, there is a trade-off for resource allocation between pilot transmission and data transmission: A more accurate channel estimation can be acquired by allocating more radio resources for channel training which implies that less radio resources are left for data transmission. The optimum length of channel training phase for MIMO systems has been investigated in [6], [10], and the optimum power allocation for SISO interference networks is addressed in [5]. ...
... The filters designed by helps of the equations (7) and (8) are independent from direct channel estimates and the estimation errors [14]. Therefore the achievable rate of user k is derived in [6] as ...
This paper addresses channel training and data communication over multi-input multi-input (MIMO) interference networks. We consider a pilot-assisted interference alignment scheme in which part of radio resources are allocated to channel training and the remaining resources are used for data transmission. A more accurate channel estimation can be obtained by increasing pilot transmission power. Since each transmitter has limited energy budget, this implies that less power is available for data transmission. Clearly, there is a trade off between the allocated power for channel training and the one for data communication. In order to investigate this trade off, first we compute an achievable sum-rate, and next we find the optimum power allocation to pilot transmission and data transmission. Finally, we verify these theoretical results with experimental measurements on USRP-based test-bed.
... The average CSI and precoder feedback bit rates were derived over timevariant MIMO interference channel in [9]. The robustness of IA in time-variant channels was assessed through a joint optimization of the pilot overhead and the IA update interval in [10]. However, the derivations only accommodate forward and reverse links with asymmetric power levels. ...
... For phase A, since we do channel estimation and feedback for each block, the effective average per-stream SINR keeps constant, and it is given by [10] 22 ...
... The "NP-NO" scheme is the conventional IA scheme that channel estimation and feedback is required at each time block with neither prediction nor optimization. While the "NP-O" scheme is an IA scheme of which the optimization objective is the interval to do channel estimation and feedback that achieves the maximum sumrate, but channel prediction is unconsidered [10]. The "PO" scheme is our proposed scheme in this paper. ...
Interference Alignment (IA) is a precoding technique that achieves the maximum multiplexing gain over an interference channel when perfect Channel State Information (CSI) is available at transmitters. Most of IA researches assume channels remain static for a period but vary independently from block to block, which neglects the temporal correlation of time-variant channels. In this paper, we propose a novel scheme that transmitters utilize a number of samples to predict CSI instead of obtaining CSI through feedback all the time. By making full use of the correlation of time-variant channels, our proposed scheme is able to reduce overhead and compensate for the feedback error due to low feedback Signal-Noise Ratio (SNR). Furthermore, we find an optimized prediction horizon achieving the maximum sum rate of our system, which is the best tradeoff between prediction error and overhead length. Simulation results verify that our scheme outperforms the traditional non-predictive feedback scheme.
... Hence, under fast fading scenario, to avoid CSI imperfection frequent CSI estimation procedures are required [10]. This results in an increased overhead and a subsequent decrease in the overall information rate [11][12][13]. Effect of CSI variation on IA is analyzed for block fading T channel [14] and continuous fading channel [12,13]. ...
... This results in an increased overhead and a subsequent decrease in the overall information rate [11][12][13]. Effect of CSI variation on IA is analyzed for block fading T channel [14] and continuous fading channel [12,13]. An analytical solution for estimating optimum pilot overhead for K-user interference channel is given in [12]. ...
Quality of service (QoS) and high data rate are two main parameters that drive the research in wireless communication. Perfect Channel State Information (CSI) at both transmitter and receiver sides is an important condition for achieving QoS and high data rate. But, achieving perfect CSI in mobile transmitters/receivers under fast fading is challenging. That is, frequent CSI updating is required to satisfy QoS requirements under fast fading scenario.
In this paper, we consider a multiple transmitter-receiver system, called X network, that has data flow in all the channels. Interference alignment (IA) is a practical solution to achieve maximum capacity in an X network. The present contribution explores the performance of the system with a mobile receiver (or transmitter) that employs CSI estimation using pilot symbols, and further optimize the performance so as to cater both bit error rate (BER) and data rate requirements. Using Monte-Carlo simulations, a novel algorithm is proposed to calculate optimal pilot overhead in M×2 X network or 2×M X network by setting an upper bound on BER. With simulation results we show that the optimal overhead varies with Doppler frequency and Signal-to-Noise Ratio (SNR).
... The theoretical design of CB schemes with multiple-antenna BSs and MTs has been lately the subject of many research papers [8]- [12]. Only very recently [13] was the spectral efficiency of the CB scheme known as interference alignment (IA) [8] studied under time-selective continuous fading channels that are explicitly estimated through pilot symbol observations. ...
... T and SNR = P/N 0 [17]. It is noted that, in [13], different forms of E b,k are presented modeling various realistic imperfections, such as CSI estimation errors and analog CSI feedback. Using the latter assumptions for the system model in (1), the power of the intended channel at each MT b and the power of the aggregate ICI (i.e., of the 2nd term in (1)) are related as ...
As network deployments become denser, interference arises as a dominant performance degradation factor. To confront with this trend, Long Term Evolution (LTE) incorporated features aiming at enabling cooperation among different base stations, a technique termed as Coordinated Multi Point (CoMP). Recent field trial results and theoretical studies of the performance of CoMP schemes revealed, however, that their gains are not as high as initially expected, despite their large coordination overhead. In this paper, we review recent advanced Coordinated Beamforming (CB) schemes, a special family of CoMP that reduces the coordination overhead through a joint choice of transmit and receive linear filters. We focus on assessing their resilience to uncoordinated interference and Channel State Information (CSI) imperfections, which both severely limit the performance gains of all CoMP schemes. We present a simple yet encompassing system model that aims at incorporating different parameters of interest in the relative interference power and CSI errors, and then utilize it for the performance evaluation of the state-of-the-art in CB schemes. It is shown that blindly applying CB in all system scenarios can indeed be counter-productive.
... Estimation algorithms are therefore required [12] that involve an exchange of information among transmitting and receiving nodes. This inevitably implies an increase of the associated overhead with a consequent decrease on the overall information rate [8]- [10]. Studies were conducted for IA in block fading channel [11] and continuous fading channel [8], [10]. ...
... This inevitably implies an increase of the associated overhead with a consequent decrease on the overall information rate [8]- [10]. Studies were conducted for IA in block fading channel [11] and continuous fading channel [8], [10]. The optimum pilot overhead and IA update interval for K-user interference channel in continuous fading have been obtained analytically in [8]. ...
... The studies in [13]- [15] consider block fading channels. The work of [16] extends [15] considering time-selective arXiv:1701.00376v1 [cs.IT] 2 Jan 2017 continuous fading in the payload part, while assuming constant fading for the training part. ...
In interference channels, channel state information (CSI) can be exploited to reduce the interference signal dimensions and thus achieve the optimal capacity scaling, i.e. degrees of freedom, promised by the interference alignment technique. However, imperfect CSI, due to channel estimation error, imperfect CSI feedback and time selectivity of the channel, lead to a performance loss. In this work, we propose a novel limited feedback algorithm for single-input single-output interference alignment in time-variant channels. The feedback algorithm encodes the channel evolution in a small number of subspace coefficients, which allow for reduced-rank channel prediction to compensate for the channel estimation error due to time selectivity of the fading process and feedback delay. An upper bound for the rate loss caused by feedback quantization and channel prediction is derived. Based on this bound, we develop a dimension switching algorithm for the reduced-rank predictor to find the best tradeoff between quantization- and prediction-error. Besides, we characterize the scaling of the required number of feedback bits in order to decouple the rate loss due to channel quantization from the transmit power. Simulation results show that a rate gain over the traditional non-predictive feedback strategy can be secured and a 60% higher rate is achieved at 20 dB signal-to-noise ratio with moderate mobility.
... Interference Alignment uses precoders and beamformers that are channel dependent. Naturally, channel estimation is critical to IA [19,20]. Both [17] and our previous works [18,21,22] either consider perfect channel state information or do not consider exact channel estimation error models. ...
This paper presents a high-throughput wireless access framework for future 6G networks. This framework, known as K-User MIMO, facilitates all-to-all communication between K access points and K mobile devices. For such a network, we illustrate the demodulation of K2 independent data streams through a new interference cancellation beamforming algorithm that improves spectral efficiency compared to massive MIMO. The paper derives a multi-user Shannon Capacity formula for K-User MIMO when K is greater than or equal to 3. We define an Orthogonal Frequency Division Multiplexing (OFDM) frame structure that demonstrates the allocation of time-frequency resources to pilot signals for channel estimation. The capacity formula is then refined to include realistic pilot overheads. We determine a practical upper-bound for MIMO array sizes that balances estimation overhead and throughput. Lastly, simulation results show the practical capacity in small cell geometries under Rayleigh Fading conditions, with both perfect and realistic channel estimation.
... The RMSE value 0.0002 (very close to 0) and R-square statistics value 0.9999 (very close to 1) signify goodness of the proposed approximation. The authors in [48] had proposed the following approximation in the same domain of interest: ...
5G wireless communication technologies aim at simultaneously achieving energy efficiency and spectral efficiency. 5G also demands high communication reliability. In this context, fine-grained temporal characterization of wireless channel can be used to enhance both. To this end, we propose a novel context-aware characterization of the temporally-varying wireless channel. Our characterization of temporal variation of the channel is based on the method of finite mixture of Gaussian distributions. However, unlike the classical Gaussian mixture model, the proposed characterization does not use an iterative algorithm for its parameter estimation; it depends on the current channel state and its statistics. Based on this characterization we estimate the quantity of data that can be transferred over the channel in a time interval without knowing the actual channel state in that duration. We propose an application context dependent upper bound on the time interval over which this estimation can be made. Our numerical results demonstrate that the present channel state plays a crucial role. When the proposed characterization is used in the context of channel adaptive communication, energy efficiency obtained is as high as 3.15 times over its nearest approach. A nontrivial trade-off between energy efficiency and precision of the proposed characterization is also investigated.
... The methods designed in [9] also exploit the advantages of CSI along with a quadratic function to improve alignment in small cells. This information carries the details of transmitters and concurrent users irrespective of the size of the network. ...
Communication between small and macro cells is cooperative to provide seamless access and to evade the issues caused due to open wireless interfaces. The major concern in these kinds of communication is the interference due to common channel exploitation. This article discusses a novel interference alignment (IA) method modeled using multi-objective least-square (MOLS) optimization. In this IA optimization, the received signal is classified using alignment vector boundary using least-square function. This helps to determine data and noise present in the signal. By deploying appropriate alignment vectors, using least-squares, the interference and data signal is classified at the receiver end. The integrated optimization method is efficient in classifying interference by mitigating the slope errors using alignment plots that help to reduce error rate. As the error is mitigated, the degree of freedom of the users is leveraged that improves the sum rate of the network.
... This results in ρ ∈ [1, 0.64]. In this region of interest, a suitable polynomial expansion of J 0 (·) is [50]: ...
In this paper, we propose a novel system state aware multichannel protocol for energy harvesting cognitive radio networks (EH-CRNs). In contrast with the state-of-the-art approaches, the proposed scheme jointly accounts for the primary users traffic characteristics as well as the instantaneous channel state over a particular channel. Our analysis demonstrates that the channel state is a critical parameter for both transmission and harvesting. The proposed protocol, depending on the states of multiple channels and its data transmission versus energy harvesting priority, intelligently decides whether to enter transmission mode over a channel or harvesting mode over a potentially different channel, and the corresponding time duration. Further, exploiting the temporal correlation present in channel and also the PU traffic characteristics, the proposed protocol computes the inter-sensing interval instead of sensing the channel in every time slot, thereby enhancing the energy efficiency. Our numerical results, verified through extensive event-driven simulations, demonstrate that the proposed scheme offers an energy efficiency gain as high as 1.36 times over its nearest competitive approach.
... Papers [19,20] are examples of these works. Limited feedback channel and imperfect channel state information are the major concerns in some other papers [21,22] and blind algorithms are proposed to overcome this issue [23,24]. Although not specifically focused on femtocells, some results in [25,26] may also be used for this particular case as well. ...
In a conventional cellular network underlaid with femtocells, the authors employ interference alignment (IA) to restrict cross-layer co-channel interference (X-interference). In their scenario, locally obtained channel state information is allowed to be conveyed between heterogeneous nodes to construct a decentralised IA-based scheme. Furthermore, they provide closed-form formulas for the upper bounds of X-interference in both uplink and downlink directions and characterise how they asymptotically approach zero for big enough channel dimensions used by individual nodes. Simulation results demonstrate that X-interference is always bounded by the above-mentioned limits and prove accuracy of their analytical formulas.
... By contrast, the proposed algorithm for Fo-gRAN shows a throughput loss due to the distributed beamforming for high quality global CSI, but also a much higher robustness against CSI errors and even a close to optimal performance for σ 2 e = 1. Note that CSI errors in the order of σ 2 e = 0.1 correspond to realistic CSI qualities in usual cellular systems [16] [19]. Given the additional fronthaul delays, σ 2 e ≥ 0.1 hence constitutes a realistic region of interest for the considered CRAN framework. ...
By incorporating cloud computing capabilities to provide radio access functionalities, Cloud Radio Access Networks (CRANs) are considered to be a key enabling technology of future 5G communication systems. In CRANs, centralized radio resource allocation optimization is performed over a large number of small cells served by simple access points, the Remote Radio Heads (RRHs). However, the fronthaul links connecting each RRH to the cloud introduce delays and entail imperfect Channel State Information (CSI) knowledge at the cloud processors. In order to satisfy the stringent latency requirements envisioned for 5G applications, the concept of Fog Radio Access Networks (FogRANs) has recently emerged for providing cloud computing at the edge of the network. Although FogRAN may alleviate the latency and CSI quality issues of CRAN, its distributed nature degrades network interference mitigation and global system performance. Therefore, we investigate the design of tailored user pre-scheduling and beamforming for 5G FogRANs. In particular, we propose a hybrid algorithm that exploits both the centralized feature of the cloud for globally-optimized pre-scheduling using imperfect global CSIs, and the distributed nature of FogRAN for accurate beamforming with high quality local CSIs. The centralized phase enables the interference patterns over the global network to be considered, while the distributed phase allows for latency reduction, in line with the requirements of FogRAN applications. Simulation results show that our proposed algorithm outperforms the baseline algorithm under imperfect CSIs, jointly in terms of throughput, energy efficiency, as well as delay.
... Although all existing feedback topologies except the four-hop topology require the acquisition of two estimated forward channels at each Rx to calculate IA precoders, the initial forward channel training procedure (i.e., initial pilot sending) of these five topologies is omitted in [12] and [13]. For the proposed feedback topology, it requires initial pilot sending, and also needs pilot sending in time slot 2 and time slot 4. It is clear that pilot transmission incurs pilot overhead [26]. Ignoring the pilot overhead caused by the initial pilot sending, the transmission of training sequences in time slots 2 and 4 in the proposed feedback topology results in additional pilot overhead. ...
Since designing a feedback topology is a practical way to implement interference alignment at reduced cost of channel state information (CSI) feedback, six feedback topologies have been presented in prior works for a K-user multiple-input multiple-output interference channel. To fully reveal the potential benefits of the feedback topology in terms of the saving of CSI overhead, we propose a new feedback topology in this paper. By efficiently performing dimensionality-decreasing at the transmitter side and aligning interference signals at a subset of receivers, we show that the proposed feedback topology obtains substantial reduction of feedback cost over the existing six feedback designs under the same antenna configuration.
... The other is the maxsignal-to-interference-plus-noise ratio (SINR) algorithm, which intends to maximise the SINR of each stream. However, most existing IA algorithms ignored the optimal power allocation, where the equal power allocation between all data streams of transmitterreceiver pairs was assumed [11][12][13]. Therefore, energy-efficient power allocation in IA-based networks is of considerable importance. ...
As a promising interference management technique, interference alignment (IA) was proposed for improving system capacity and spectral efficiency by precoding and filtering design. However, the previous works assumed the equal power allocation among data streams in IA-based networks, and the sum-rate may fall short of the theoretical maximum without the energy-efficient optimisation. In this study, a novel approach (two-game theoretic) is presented, to solve the rate maximisation problem in the IA-based uplink multiple-input multiple-output cellular networks. First, a two-game analytical framework is presented, where the IA design and power allocation are modelled as two game processes, respectively, and the authors prove that both games have Nash equilibrium solutions. Second, based on the framework, two iterative algorithms of joint IA and power allocation that achieve the maximum sum-rate are proposed. In addition, the authors analyse the sum-rate performance loss under imperfect channel state information (CSI), which depends on the variance of CSI error. Simulation reveals that the sum-rate performances of the proposed iterative algorithms are higher than that of the previous schemes at low signal-to-noise ratio, whose computational complexities are acceptable.
... We use a system-level simulation to evaluation the performance of CoMP-MU-MIMO system, and assuming that use the same power to transmit all subcarrier. For detailed simulation parameters, refer to Table 1 [10]. In the simulation system, each user terminal has 2 receiving antennas, and each base node has 2 sending antennas, the number of CoMP-MU-MIMO is always 3, thus 3 cooperative transmission nodes and three user terminals can form virtual CoMP-MU-MIMO system for 6 × 6. Compare to the traditional pre-coding algorithm ZF and BD, this article proposed multiuser MSV-TZ pre-coding algorithm has an advantage on the same signal interference, shown as Figure 3. ...
... On one hand, a larger size of an IA network means a tighter constraint to align all interference, which may result in the failure to acquire a feasible precoding solution [3]. On the other hand, global channel state information (CSI) must be available at transmitters, and the amount of CSI overhead is a quadratic function of the number of small cells [4,5]. Hence, if the size of an IA network is too large, then the CSI overhead might be extremely high. ...
The emergence of small cells provides a cost-effective way to satisfy users’ explosive traffic requirements. The massive deployment of small cells, nevertheless, causes severe inter-cell interference in Orthogonal Frequency Division Multiple-Access -based cellular networks. As such, conventional interference management strategies may be inefficient and interference alignment (IA) has been proposed as a promising technology to cope with inter-cell interference. To perfectly align all interference in a reduced-dimensional subspace, IA transmitters generally call for global channel state information (CSI) across small cell networks through receivers’ feedback. However, the number of total feedback bits scales as the square of the number of small cells. Hence, IA achieves a greater multiplexing gain at the cost of substantial overhead. To enable a tradeoff between multiplexing gain and overhead reduction, in this paper we present a new metric termed average effective degrees of freedom (AEDoF), which embodies the average degrees of freedom of small cell networks with CSI overhead considered. Furthermore, for reducing the computational complexity, we propose a graph-based clustering algorithm to solve the formulated AEDoF maximization problem. Simulation results verify that our proposed algorithm is of low complexity and achieves the maximum spectrum efficiency among several existing clustering methods.
... The studies in [13]- [15] consider block fading channels. The work of [16] extends [15] considering time-selective arXiv:1701.00376v1 [cs.IT] 2 Jan 2017 continuous fading in the payload part, while assuming constant fading for the training part. ...
In interference channels, channel state information (CSI) can be exploited to reduce the interference signal dimensions and thus achieve the optimal capacity scaling, i.e. degrees of freedom, promised by the interference alignment technique. However, imperfect CSI, due to channel estimation error, imperfect CSI feedback and time selectivity of the channel, lead to a performance loss. In this work, we propose a novel limited feedback algorithm for single-input single-output interference alignment in time-variant channels. The feedback algorithm encodes the channel evolution in a small number of subspace coefficients, which allow for reduced-rank channel prediction to compensate for the channel estimation error due to time selectivity of the fading process and feedback delay. An upper bound for the rate loss caused by feedback quantization and channel prediction is derived. Based on this bound, we develop a dimension switching algorithm for the reduced-rank predictor to find the best tradeoff between quantization- and prediction-error. Besides, we characterize the scaling of the required number of feedback bits in order to decouple the rate loss due to channel quantization from the transmit power. Simulation results show that a rate gain over the traditional non-predictive feedback strategy can be secured and a 60% higher rate is achieved at 20 dB signal-to-noise ratio with moderate mobility.
... Lemma 1: In the high-SNR regime, can be approximated by (19) where and are given by (20) and (21). ...
This paper proposes a framework for the artificial noise assisted secure transmission in multiple-input, multiple-output, multiple antenna eavesdropper (MIMOME) wiretap channels in frequency-division duplexed (FDD) systems. We focus on a practical scenario that only the eavesdroppers' channel distribution information (CDI) is available and the imperfect channel state information (CSI) of the legitimate receiver is acquired through training and analog feedback. By taking explicitly into account the signaling overhead and training power overhead incurred by channel estimation and feedback, we define the achievable effective ergodic secrecy rate (ESR), and investigate a joint power allocation and training overhead optimization problem for the maximization of effective ESR. We first derive a deterministic approximation for the achievable effective ESR which facilitates the joint optimization. Then, efficient iterative algorithms are proposed to solve the considered nonconvex optimization problem. In particular, in the high-SNR regime, a block coordinate descent method (BCDM) is proposed to handle the joint optimization. In the low-SNR regime, we transform the problem into a sequence of geometric programmings (GPs) and locate its Karush-Kuhn-Tucker (KKT) solution using the successive convex approximation (SCA) method. For the general case of SNR, we maximize the lower bound of the achievable effective ESR. Simulation results corroborate the theoretical analysis and illustrate the secrecy performance of the proposed secure transmission scheme.
... With the degree of CSI required for IA, a superior MU-MIMO baseline could be implemented. Overheads associated with precoder computation [36], [37] have also been disregarded. ...
Capitalizing on the analytical potency of stochastic geometry and on some new ideas to model intercell interference, this paper presents analytical expressions that enable quantifying the spectral efficiency of interference alignment (IA) in cellular networks without the need for simulation. From these expressions, the benefits of IA are characterized. Even under favorable assumptions, IA is found to be beneficial only in very specific and relatively infrequent network situations, and a blanket utilization of IA is found to be altogether detrimental. Applied only in the appropriate situations, IA does bring about benefits that are significant for the users involved but relatively small in terms of average spectral efficiency for the entire system.
We present centralized based iterative algorithms that jointly determine the optimal transceiver filters as well as the optimum power allocation for a ����-user multiple-input multiple-output (MIMO) interference channel (IC). The optimality criterion is developed on the basis of the average per user multiplexing gain and the achievable sum-rate in the MIMO IC. By allowing channel state information (CSI) exchanged between base stations (BSs) and a central unit (CU), we design a feedback topology where CU collects local CSIs from all BSs, computes all transceiver filters and sends them to corresponding user-BS pairs. Note that the local CSIs at BSs are obtained from the estimation of the channel states during the so-called uplink-training phase. At the CU, using the alternating optimization strategy, various iterative algorithms are proposed to design the filters. In other related studies, the equal transmit power policy for all user-BS pairs is chosen ignoring the essential need to search for the optimal power allocation policy; they do not take the full advantage of the system’s total power. Thus, another key aspect of this paper is to make optimal power allocation decisions for all the user-BS pairs based on the sum-rate maximization strategy and under a sum power constraint.
Recent studies have demonstrated that millimeter-wave cellular networks may operate either in the noise-or in the interference-limited regime, depending on several parameters, which include the density of base stations, the density and size of obstacles/blockages, the antenna beamwidth, and the transmission bandwidth. The objective of the present paper is to exploit tools from stochastic geometry for obtaining a mathematically tractable framework that allows us to identify the operating conditions under which millimeter-wave cellular networks operate in the inference-limited regime and to assess the potential advantages of interference alignment, by taking into account the overhead cost of base station cooperation due to the estimation of channel state information.
This paper investigates an interference alignment (IA) scheme suitable for a K-user multiple-input multiple-output (MIMO) wireless X network. The K-user MIMO X-network is a communication architecture where each transmitter, equipped with multiple antennas, has independent messages for each of the receivers, also equipped with multiple antennas. Earlier only 2 × N or M × 2 X networks were considered to be achievable. In this paper we remove this restriction by employing time division multiple access scheme. The proposed IA scheme allows each user to achieve a degree of freedom of KA/2K−1 a particular time slot, where A is the number of antennas at each transmitter and receiver. The capacity and bit error-rate (BER) performance of the proposed scheme is compared to that of another scheme recently proposed by Park and Ko. Simulation results are reported to show the BER performance for K-user (for a toy example, K is set equal to 3) and two-user X channel IA in case of transmission over flat-fading Rayleigh channels.
Channel state information (CSI) at the transmitter and receiver is an essential requirement for interference alignment (IA) schemes. For moving users the channel coefficients vary with time and, therefore, it is required to update CSI both at the transmitter and receiver at regular intervals. Meanwhile it is important to note that frequent updates of CSI will reduce data rate and delayed updates will cause a large variation in CSI. In this context we explore the error performance of IA in two-user multiple-input multiple-output (MIMO) X channel where the channel suffers continuous time-varying fading. The bit error rate (BER) performance of MIMO two-user X channel is evaluated for different Doppler frequencies. We also propose a method for calculating optimal pilot overhead for time-varying channels by setting an upper bound on BER.
In this paper the performance of interference alignment is evaluated for the downlink of 3GPP UMTS LTE via the Vienna LTE Link Level simulator. Interference alignment is compared to closed-loop spatial multiplexing, a non-cooperative communication scheme. In order to reduce the computational complexity of solving the interference align-ment problem for each subcarrier separately, the same pre-coding and interference suppression matrices are used for disjoint subsets of subcarriers. The performance impair-ment in terms of average throughput reduction of this ap-proach is analyzed numerically. Furthermore, the perfor-mance of interference alignment is investigated for realistic fast-fading channels employing outdated precoding and in-terference suppression matrices.
Interference alignment (IA) is a cooperative transmission strategy that,
under some conditions, achieves the interference channel's maximum number of
degrees of freedom. Realizing IA gains, however, is contingent upon providing
transmitters with sufficiently accurate channel knowledge. In this paper, we
study the performance of IA in multiple-input multiple-output systems where
channel knowledge is acquired through training and analog feedback. We design
the training and feedback system to maximize IA's effective sum-rate: a
non-asymptotic performance metric that accounts for estimation error, training
and feedback overhead, and channel selectivity. We characterize effective
sum-rate with overhead in relation to various parameters such as
signal-to-noise ratio, Doppler spread, and feedback channel quality. A main
insight from our analysis is that, by properly designing the CSI acquisition
process, IA can provide good sum-rate performance in a very wide range of
fading scenarios. Another observation from our work is that such overhead-aware
analysis can help solve a number of practical network design problems. To
demonstrate the concept of overhead-aware network design, we consider the
example problem of finding the optimal number of cooperative IA users based on
signal power and mobility.
We examine the capacity of beamforming over a multi-input/single-output block Rayleigh fading channel with finite training for channel estimation and limited feedback. A fixed-length packet is assumed, which is spanned by T training symbols, B feedback bits, and the data symbols. The training symbols are used to obtain a minimum mean squared error (MMSE) estimate of the channel vector. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing 2B i.i.d. random vectors, and relays the corresponding B bits back to the transmitter. We derive bounds on the capacity and show that for a large number of transmit antennas Nt , the optimal T and B, which maximize the bounds, are approximately equal and both increase as Nt/logNt. We conclude that with limited training and feedback, the optimal number of antennas to activate also increases as Nt/logNt
Interference alignment (IA), given uncorrelated channel components and perfect channel state information, obtains the maximum degrees of freedom in an interference channel. Little is known, however, about how the sum rate of IA behaves at finite transmit power, with imperfect channel state information, or antenna correlation. This paper provides an approximate closed-form signal-to-interference-plus-noise-ratio (SINR) expression for IA over multiple-input-multiple-output (MIMO) channels with imperfect channel state information and transmit antenna correlation. Assuming linear processing at the transmitters and zero-forcing receivers, random matrix theory tools are utilized to derive an approximation for the postprocessing SINR distribution of each stream for each user. Perfect channel knowledge and i.i.d. channel coefficients constitute special cases. This SINR distribution not only allows easy calculation of useful performance metrics like sum rate and symbol error rate, but also permits a realistic comparison of IA with other transmission techniques. More specifically, IA is compared with spatial multiplexing and beamforming and it is shown that IA may not be optimal for some performance criteria.
We examine the capacity of beamforming over a single-user, multiantenna link taking into account the overhead due to channel estimation and limited feedback of channel state information. Multi-input-single-output (MISO) and multi-input-multi-output (MIMO) channels are considered subject to block Rayleigh fading. Each coherence block contains L symbols, and is spanned by T training symbols, B feedback bits, and the data symbols. The training symbols are used to obtain a minimum mean squared error estimate of the channel matrix. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing 2B i.i.d. random vectors, and sends the corresponding B bits back to the transmitter. We derive bounds on the beamforming capacity for MISO and MIMO channels and characterize the optimal (rate-maximizing) training and feedback overhead (T and B) as L and the number of transmit antennas Nt both become large. The optimal Nt is limited by the coherence time, and increases as L/logL. For the MISO channel the optimal T/L and B/L (fractional overhead due to training and feedback) are asymptotically the same, and tend to zero at the rate 1/log Nt. For the MIMO channel the optimal feedback overhead B/L tends to zero faster (as 1/log2 Nt).
While interference alignment schemes have been employed to realize the full multiplexing gain of K-user interference channels, the analyses performed so far have predominantly focused on the case when global channel knowledge is available at each node of the network. This paper considers the problem where each receiver knows its channels from all the transmitters and feeds back this information using a limited number of bits to all other terminals. In particular, channel quantization over the composite Grassmann manifold is proposed and analyzed. It is shown, for K-user multiple-input, multiple-output (MIMO) interference channels, that when the transmitters use an interference alignment strategy as if the quantized channel estimates obtained via this limited feedback are perfect, the full sum degrees of freedom of the interference channel can be achieved as long as the feedback bit rate scales sufficiently fast with the signal-to-noise ratio. Moreover, this is only one extreme point of a continuous tradeoff between the achievable degrees of freedom region and user feedback rate scalings which are allowed to be non-identical. It is seen that a slower scaling of feedback rate for any one user leads to commensurately fewer degrees of freedom for that user alone.
Cooperation is viewed as a key ingredient for interference management in
wireless systems. This paper shows that cooperation has fundamental
limitations. The main result is that even full cooperation between transmitters
cannot in general change an interference-limited network to a noise-limited
network. The key idea is that there exists a spectral efficiency upper bound
that is independent of the transmit power. First, a spectral efficiency upper
bound is established for systems that rely on pilot-assisted channel
estimation; in this framework, cooperation is shown to be possible only within
clusters of limited size, which are subject to out-of-cluster interference
whose power scales with that of the in-cluster signals. Second, an upper bound
is also shown to exist when cooperation is through noncoherent communication;
thus, the spectral efficiency limitation is not a by-product of the reliance on
pilot-assisted channel estimation. Consequently, existing literature that
routinely assumes the high-power spectral efficiency scales with the log of the
transmit power provides only a partial characterization. The complete
characterization proposed in this paper subdivides the high-power regime into a
degrees-of-freedom regime, where the scaling with the log of the transmit power
holds approximately, and a saturation regime, where the spectral efficiency
hits a ceiling that is independent of the power. Using a cellular system as an
example, it is demonstrated that the spectral efficiency saturates at power
levels of operational relevance.
Transmitter channel state information (CSIT) is crucial for the multiplexing
gains offered by advanced interference management techniques such as multiuser
MIMO and interference alignment. Such CSIT is usually obtained by feedback from
the receivers, but the feedback is subject to delays. The usual approach is to
use the fed back information to predict the current channel state and then
apply a scheme designed assuming perfect CSIT. When the feedback delay is large
compared to the channel coherence time, such a prediction approach completely
fails to achieve any multiplexing gain. In this paper, we show that even in
this case, the completely stale CSI is still very useful. More concretely, we
show that in a MIMO broadcast channel with K transmit antennas and K
receivers each with 1 receive antenna, degrees of freedom is achievable even when the fed back channel state is
completely independent of the current channel state. Moreover, we establish
that if all receivers have independent and identically distributed channels,
then this is the optimal number of degrees of freedom achievable. In the
optimal scheme, the transmitter uses the fed back CSI to learn the side
information that the receivers receive from previous transmissions rather than
to predict the current channel state. Our result can be viewed as the first
example of feedback providing a degree-of-freedom gain in memoryless channels.
We consider a MIMO fading broadcast channel where the fading channel coefficients are constant over time-frequency blocks that span a coherent time a coherence bandwidth. In closed-loop systems, channel state information at transmitter (CSIT) is acquired by the downlink training sent by the base station and an explicit feedback from each user terminal. In open-loop systems, CSIT is obtained by exploiting uplink training and channel reciprocity. We use a tight closed-form lower bound on the ergodic achievable rate in the presence of CSIT errors in order to optimize the overall system throughput, by taking explicitly into account the overhead due to channel estimation and channel state feedback. Based on three time-frequency block models inspired by actual systems, we provide some useful guidelines for the overall system optimization. In particular, digital (quantized) feedback is found to offer a substantial advantage over analog (unquantized) feedback.
While interference alignment schemes have been employed to realize the full multiplexing gain of K-user interference channels, the analyses performed so far have predominantly focused on the case when global channel knowledge is available at each node of the network. This paper considers the problem where each receiver knows its channels from all the transmitters and feeds back this information using a limited number of bits to all other terminals. In particular, channel quantization over the composite Grassmann manifold is proposed and analyzed. It is shown, for K-user multiple-input, multiple-output (MIMO) interference channels, that when the transmitters use an interference alignment strategy as if the quantized channel estimates obtained via this limited feedback are perfect, the full sum degrees of freedom of the interference channel can be achieved as long as the feedback bit rate scales sufficiently fast with the signal-to-noise ratio. Moreover, this is only one extreme point of a continuous tradeoff between achievable degrees of freedom region and user feedback rate scalings which are allowed to be non-identical. It is seen that a slower scaling of feedback rate for any one user leads to commensurately fewer degrees of freedom for that user alone.
In this paper, we propose a simple framework for evaluating the required repetition rate of channel estimation for multiple-input multiple-output (MIMO) systems in continuous slowly fading radio channels. An analytical formula for the interference due to the temporal variation of the channel coefficients is given and verified by link level simulations based on synthetic and measured impulse responses. The proposed interference model makes it possible to optimize the lengths of the training and data phases under different performance criteria. As an example, we investigate the bit-error rate performance of MIMO systems with zero-forcing detection and determine the time interval, after which the channel has to be estimated again in order to keep the error probability below a desired threshold. In the second part, the lengths of the training and data phases is optimized based on the information-theoretical capacity. It is shown that these lengths depend not only on the Doppler spread of the channel, but also on the antenna configuration and noise power.
Knowledge of accurate and timely channel state information (CSI) at the transmitter is becoming increasingly important in wireless communication systems. While it is often assumed that the receiver (whether base station or mobile) needs to know the channel for accurate power control, scheduling, and data demodulation, it is now known that the transmitter (especially the base station) can also benefit greatly from this information. For example, recent results in multiantenna multiuser systems show that large throughput gains are possible when the base station uses multiple antennas and a known channel to transmit distinct messages simultaneously and selectively to many single-antenna users. In time-division duplex systems, where the base station and mobiles share the same frequency band for transmission, the base station can exploit reciprocity to obtain the forward channel from pilots received over the reverse channel. Frequency-division duplex systems are more difficult because the base station transmits and receives on different frequencies and therefore cannot use the received pilot to infer anything about the multiantenna transmit channel. Nevertheless, we show that the time occupied in frequency-duplex CSI transfer is generally less than one might expect and falls as the number of antennas increases. Thus, although the total amount of channel information increases with the number of antennas at the base station, the burden of learning this information at the base station paradoxically decreases. Thus, the advantages of having more antennas at the base station extend from having network gains to learning the channel information. We quantify our gains using linear analog modulation which avoids digitizing and coding the CSI and therefore can convey information very rapidly and can be readily analyzed. The old paradigm that it is not worth the effort to learn channel information at the transmitter should be revisited since the effort decreases and the gain increases with the number of antennas.
Multiple-antenna wireless communication links promise very high data rates with low error probabilities, especially when the wireless channel response is known at the receiver. In practice, knowledge of the channel is often obtained by sending known training symbols to the receiver. We show how training affects the capacity of a fading channel-too little training and the channel is improperly learned, too much training and there is no time left for data transmission before the channel changes. We compute a lower bound on the capacity of a channel that is learned by training, and maximize the bound as a function of the received signal-to-noise ratio (SNR), fading coherence time, and number of transmitter antennas. When the training and data powers are allowed to vary, we show that the optimal number of training symbols is equal to the number of transmit antennas-this number is also the smallest training interval length that guarantees meaningful estimates of the channel matrix. When the training and data powers are instead required to be equal, the optimal number of symbols may be larger than the number of antennas. We show that training-based schemes can be optimal at high SNR, but suboptimal at low SNR.
We present a model for time-varying communication single-access
and multiple-access channels without feedback. We consider the
difference between mutual information when the receiver knows the
channel perfectly and mutual information when the receiver only has an
estimate of the channel. We relate the variance of the channel
measurement error at the receiver to upper and lower bounds for this
difference in mutual information. We illustrate the use of our bounds on
a channel modeled by a Gauss-Markov process, measured by a pilot tone.
We relate the rate of time variation of the channel to the loss in
mutual information due to imperfect knowledge of the measured channel
We consider a MIMO fading broadcast channel and compute achievable ergodic rates when channel state information is acquired at the receivers via downlink training and it is provided to the transmitter by channel state feedback. Unquantized (analog) and quantized (digital) channel state feedback schemes are analyzed and compared under various assumptions. Digital feedback is shown to be potentially superior when the feedback channel uses per channel state coefficient is larger than 1. Also, we show that by proper design of the digital feedback link, errors in the feedback have a minor effect even if simple uncoded modulation is used on the feedback channel. We discuss first the case of an unfaded AWGN feedback channel with orthogonal access and then the case of fading MIMO multi-access (MIMO-MAC). We show that by exploiting the MIMO-MAC nature of the uplink channel, a much better scaling of the feedback channel resource with the number of base station antennas can be achieved. Finally, for the case of delayed feedback, we show that in the realistic case where the fading process has (normalized) maximum Doppler frequency shift 0 < F < 1/2, a fraction 1 - 2F of the optimal multiplexing gain is achievable. The general conclusion of this work is that very significant downlink throughput is achievable with simple and efficient channel state feedback, provided that the feedback link is properly designed. Comment: Revised for IEEE Trans. Information Theory, May 2009. (Original submission: Nov. 2007)
The spectral efficiency achievable by IA (interference alignment) in a K-user interference channel is studied in the face of time-selective fading explicitly estimated through pilot-symbol observations. The robustness of IA is assessed through a joint optimization of the pilot overhead and the IA update interval, which are characterized-in the high-power conditions where IA is relevant-as solutions of a fixed-point equation.
This monograph introduces to the reader the idea of interference alignment, traces its origins, reviews a variety of interference alignment schemes, summarizes the diverse settings where the idea of interference alignment is applicable and highlights the common principles that cut across these diverse applications. The focus is on theoretical aspects.
The performance of any transmission scheme is coupled with the receive strategy. Herein the behavior of transmissions based on interference alignment scheme is investigated under different receive strategies. Moreover, interference alignment is compared with different state-of-art transmission schemes under the assumption of intrabase station and inter-base station coordination. The performance of the above-mentioned techniques is assessed with a fully 3GPP compliant downlink LTE simulator, which gives a very realistic picture of the behavior in real systems. The results have shown that interference alignment is underperforming with respect to the baseline using base station coordination, when a realistic number of base stations is considered. Moreover, the gains of interference alignment are also highly dependent on the receiver type, whereas the baselines show relatively low differences with different receivers.
Cooperation in a large wireless network with pilotassisted coherent communication is shown to have certain fundamental limitations, namely that even perfect cooperation cannot in general change an interference-limited network to a noise-limited one. Specifically, we demonstrate the existence of a spectral efficiency upper bound that does not grow with the transmit power, when channels are estimated via pilot signals. This is because pilot-assisted channel estimation is only possible within finite cooperation clusters, resulting in out-of-cluster interference that scales with the transmit power. Making the clusters excessively large can actually worsen this effect, nor does sidestepping the pilot-assisted channel estimation via noncoherent demodulation provide an escape. Using a cellular system as an example, it is demonstrated that the spectral efficiency saturates at power levels of operational relevance, indicating that the lackluster gains from cooperation observed in practice may be based on fundamental information-theoretic limitations, rather than current technology imperfections.
A simple limited feedback scheme is proposed for interference alignment on the K-user Multiple-Input-Multiple-Output Interference Channel (MIMO-IC). The scaling of the number of feedback bits with the transmit power required to preserve the multiplexing gain that can be achieved using perfect channel state information (CSI) is derived. This result is obtained through a reformulation of the interference alignment problem in order to exploit the benefits of quantization on the Grassmann manifold, which is well investigated in the single-user MIMO channel. Furthermore, through simulations we show that the proposed scheme outperforms the naive feedback scheme consisting in independently quantizing the channel matrices, in the sense that it yields a better sum rate performance for the same number of feedback bits.
Distributed cooperation schemes such as Interference Alignment hold the promise of an increased number of spatial degrees of freedom and, with that, of substantially higher spectral efficiencies. Most results available to date, however, have been obtained in simplified settings featuring a small number of transmitters and receivers in isolation. While such controlled settings are excellent platforms to develop ideas and build intuition, they also conceal important aspects that are inherent to actual wireless systems. Chief among these is the fact that any small set of cooperating transmitters and receivers is bound to be embedded within a large system featuring many other transmitters and receivers. This paper studies the fundamental performance of IA, in the context of a large cellular network, and contrasts it with that of non-cooperative MIMO.
Interference alignment (IA) is a multiplexing gain optimal transmission strategy for the interference channel. While the achieved sum rate with IA is much higher than previously thought possible, the improvement comes at the cost of requiring network channel state information at the transmitters. This can be achieved by explicit feedback, a flexible yet potentially costly approach that incurs large overhead. In this paper we propose analog feedback as an alternative to limited feedback or reciprocity based alignment. We show that the full multiplexing gain observed with perfect channel knowledge is preserved by analog feedback and that the mean loss in sum rate is bounded by a constant when signal-to-noise ratio is comparable in both forward and feedback channels. When signal-to-noise ratios are not quite symmetric, a fraction of the multiplexing gain is achieved. We consider the overhead of training and feedback and use this framework to numerically optimize the system's effective throughput. We present simulation results to demonstrate the performance of IA with analog feedback, verify our theoretical analysis, and extend our conclusions on optimal training and feedback length.
Cooperation is viewed as a key ingredient for interference management in wireless networks. This paper shows that cooperation has fundamental limitations. First, it is established that in systems that rely on pilot-assisted channel estimation, the spectral efficiency is upper-bounded by a quantity that does not depend on the transmit powers; in this framework, cooperation is possible only within clusters of limited size, which are subject to out-of-cluster interference whose power scales with that of the in-cluster signals. Second, an upper bound is also shown to exist if the cooperation extends to an entire (large) system operating as a single cluster; here, pilot-assisted transmission is necessarily transcended. Altogether, it is concluded that cooperation cannot in general change an interference-limited network to a noise-limited one. Consequently, the existing literature that routinely assumes that the high-power spectral efficiency scales with the log-scale transmit power provides only a partial characterization. The complete characterization proposed in this paper subdivides the high-power regime into a degree-of-freedom regime, where the scaling with the log-scale transmit power holds approximately, and a saturation regime, where the spectral efficiency hits a ceiling that is independent of the power. Using a cellular system as an example, it is demonstrated that the spectral efficiency saturates at power levels of operational relevance.
Interference alignment (IA) has attracted great attention in the last few
years for its breakthrough performance in interference networks. However,
despite the numerous works dedicated to IA, the feasibility conditions of IA
remains unclear for most network topologies. The IA feasibility analysis is
challenging as the IA constraints are sets of high-degree polynomials, for
which no systematic tool to analyze the solvability conditions exists. In this
work, by developing a new mathematical framework that maps the solvability of
sets of polynomial equations to the linear independence of their first-order
terms, we propose a sufficient condition that applies to MIMO interference
networks with general configurations. We have further proved that this
sufficient condition matches with the necessary conditions under a wide range
of configurations. These results further consolidate the theoretical basis of
IA.
Interference alignment is evaluated as a technique to mitigate inter-cell interference in the downlink of a cellular network using OFDMA. The sum mutual information achieved by interference alignment together with a zero-forcing receiver is considered, and upper and lower bounds are derived for the case of imperfect channel knowledge. The sum mutual information achieved by interference alignment when the base stations share their information about the channels is shown to compare favorably to the achievable sum-rate of methods where the base stations do not cooperate, even under moderately accurate knowledge of the channel state.
Expressions relating spectral efficiency, power, and Doppler spectrum, are derived for Rayleigh-faded wireless channels with Gaussian signal transmission. No side information on the state of the channel is assumed at the receiver. Rather, periodic reference signals are postulated in accordance with the functioning of most wireless systems. The analysis relies on a well-established lower bound, generally tight and asymptotically exact at low SNR. In contrast with most previous studies, which relied on block-fading channel models, a continuous-fading model is adopted. This embeds the Doppler spectrum directly in the derived expressions, imbuing them with practical significance. Closed-form relationships are obtained for the popular Clarke-Jakes spectrum and informative expansions, valid for arbitrary spectra, are found for the low- and high-power regimes.While the paper focuses on scalar channels, the extension to multiantenna settings is also discussed.
Recent results establish the optimality of interference alignment to approach the Shannon capacity of interference networks at high SNR. However, the extent to which interference can be aligned over a finite number of signalling dimensions remains unknown. Another important concern for interference alignment schemes is the requirement of global channel knowledge. In this work, we provide examples of iterative algorithms that utilize the reciprocity of wireless networks to achieve interference alignment with only local channel knowledge at each node. These algorithms also provide numerical insights into the feasibility of interference alignment that are not yet available in theory.
The optimization of the pilot overhead in single-user wireless fading channels is investigated, and the dependence of this overhead on various system parameters of interest (e.g., fading rate, signal-to-noise ratio) is quantified. The achievable pilot-based spectral efficiency is expanded with respect to the fading rate about the no-fading point, which leads to an accurate order expansion for the pilot overhead. This expansion identifies that the pilot overhead, as well as the spectral efficiency penalty with respect to a reference system with genie-aided CSI (channel state information) at the receiver, depend on the square root of the normalized Doppler frequency. It is also shown that the widely-used block fading model is a special case of more accurate continuous fading models in terms of the achievable pilot-based spectral efficiency. Furthermore, it is established that the overhead optimization for multiantenna systems is effectively the same as for single-antenna systems with the normalized Doppler frequency multiplied by the number of transmit antennas.
We consider the \emph{effective} degrees of freedom (DoF) achieved by interference alignment when channel state information (CSI) is acquired by training and feedback. For a flat block-fading interference channel with power P per transmitter and M antennas per node, we show that interference alignment achieves higher DoF than orthogonal transmission (e.g., TDMA) only if the channel coherence time is large and the capacity of feedback link is at least as . Under this condition, to maximize the effective DoF, each receiver needs to feed back CSI via bits per coherence interval; smaller growth rate of feedback bits will decrease the effective DoF. We also show that in the presence of training and feedback cost, K=3 achieves the optimal DoF for a broad range of channel characteristics; with larger number of user pairs, the DoF falls short of its optimum and beyond a certain point becomes a decreasing function of K.
Using interference alignment, it has been shown that the number of degrees of freedom in the interference chan- nel scales linearly with the number of users. Unfortu- nately, closed-form solutions for interference alignment over constant-coefficient channels with more than 3 users are difficult to derive. This paper proposes an algorithm for inter- ference alignment in the MIMO interference channel with an arbitrary number of users, antennas, or spatial streams. The algorithm is an alternating minimization over the precoding matrices at the transmitters and the interference subspaces at the receivers, and is proven to converge. Numerical results show how the algorithm is useful for simulation and can give insight into the possible limitations of interference alignment.
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)
While the best known outerbound for the K user interference channel states that there cannot be more than K/2 degrees of freedom, it has been conjectured that in general the constant interference channel with any number of users has only one degree of freedom. In this paper, we explore the spatial degrees of freedom per orthogonal time and frequency dimension for the K user wireless interference channel where the channel coefficients take distinct values across frequency slots but are fixed in time. We answer five closely related questions. First, we show that K/2 degrees of freedom can be achieved by channel design, i.e. if the nodes are allowed to choose the best constant, finite and nonzero channel coefficient values. Second, we show that if channel coefficients can not be controlled by the nodes but are selected by nature, i.e., randomly drawn from a continuous distribution, the total number of spatial degrees of freedom for the K user interference channel is almost surely K/2 per orthogonal time and frequency dimension. Thus, only half the spatial degrees of freedom are lost due to distributed processing of transmitted and received signals on the interference channel. Third, we show that interference alignment and zero forcing suffice to achieve all the degrees of freedom in all cases. Fourth, we show that the degrees of freedom D directly lead to an capacity characterization of the form for the multiple access channel, the broadcast channel, the 2 user interference channel, the 2 user MIMO X channel and the 3 user interference channel with M>1 antennas at each node. Fifth, we characterize the degree of freedom benefits from cognitive sharing of messages on the 3 user interference channel.
We consider single-antenna interference networks where M sources, each with an average transmit power of P/M, communicate with M destinations over frequency-selective channels (with L taps each) and each destination has perfect knowledge of its channels from each of the sources. Assuming that there exist error-free non-interfering broadcast feedback links from each destination to all the nodes (i.e., sources and destinations) in the network, we show that naive interference alignment, in conjunction with vector quantization of the impulse response coefficients according to the scheme proposed in Mukkavilli et al., IEEE Trans. IT, 2003, achieves full spatial multiplexing gain of M/2, provided that the number of feedback bits broadcast by each destination is at least M(L-1) log P. Comment: 5 pages, 0 figures
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
Multiple-input-multiple-output (MIMO) systems can provide high data rate wireless services in a rich scattering environment. We study one of the proposals for MIMO systems, the Bell Labs Layered Space-Time (BLAST) architecture. Channel estimation using training sequences is required for coherent detection in BLAST. We apply the maximum-likelihood channel estimator and the optimal training sequences for block flat fading channels to continuous flat fading channels and analyze the estimation error. The optimal training length and training interval that maximize the throughput for a given target bit error-rate are found by computer simulations as functions of the Doppler frequency and the number of antennas.
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 correspondence, 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 signal-to-noise ratio (SNR) (in decibels) in order to achieve the full multiplexing gain. This is in sharp contrast to point-to-point multiple-input multiple-output (MIMO) systems, in which it is not necessary to increase the feedback rate as a function of the SNR
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.
BLAST (Bell Labs Layered Space-Time) is a multiple-antenna communication scheme whose outage capacity in a Rayleigh flat fading environment grows linearly with the minimum of the number of transmit and receive antennas, with no increase in bandwidth or transmitted power. Based on its knowledge of the matrix of propagation coefficients, the receiver performs two critical operations: nulling and cancellation, that in effect create independent virtual subchannels. Assume that the receiver estimates the propagation matrix from a known set of transmitted training signals, and then uses the estimate as though it were correct for nulling and cancellation. How much training is needed for satisfactory operation? The optimal training signals are orthogonal with respect to time among the transmit antennas, and each transmit antenna is fed equal energy. Errors in estimating the propagation matrix manifest themselves as crosstalk among the virtual subchannels. If its magnitude is too large, the cro...
1536-1276 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permissionhtml for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication
- M Medard
M. Medard, " The effect upon channel capacity in wireless communica-tions of perfect and imperfect knowledge of the channel, " IEEE Trans. on Inform. Theory, vol. 46, no. 3, pp. 933–946, 2000. 1536-1276 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TWC.2014.2327217, IEEE Transactions on Wireless Communications IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2014