1.1.1 On the Capacity of Distributed AF MIMO Relay Network
Relays have been considered a useful means for coverage extension and capacity enhance-ment of wireless systems [33]. Among all conceivable relaying strategies, two have received the most attention: amplify-and-forward (AF) [30] and decode-and-forward (DF) [31]. In AF systems the relays amplify or beamform the received signals without further process-ing, while in DF systems they decode (or demodulate if there is no channel coding) the received signals and transmit the re-encoded (or remodulated) signals to the destination.
Besides the forwarding strategies, an important subject in relay system design is the over-all wireless system architecture. In this, due to the capacity advantage of multi-input multi-output (MIMO) transmission over single-input single-output (SISO) transmission, many have sought to incorporate some MIMO concepts one way or another. The present work is concerned with AF-based distributed relay networks, whose architecture will be described further later.
The simplest relay-aided transmission system consists of three nodes: source, relay (cooperator) and destination [44]. To facilitate MIMO transmission, an intuitive approach is to install multiple antennas on one or more of the nodes. For simplicity, consider the situation where the source and the destination have an equal number of antennas. A case with a single-antenna source (SAS) and a single relay (SR) equipped with multiple antennas (MAR) is considered in [48]. A natural extension to have a multiple-antenna source (MAS) and an SR-MAR to enable spatial multiplexing [25, 6]. In studies of MAS-SR-MAR systems, the multiple antennas on each terminal are usually assumed to be fully connected and may have arbitrary interconnection weights. In this case, known matrix theory can be used to decompose a MIMO transmission channel into parallel SISO links (via, for example, the singular value decomposition (SVD) or the QR decomposition).
Each spatially multiplexed signal stream can then be transported over one parallel link, and the matrix decomposition can be viewed as simultaneous beamforming for these streams. Typical performance measures, such as the signal-to-interference-plus-noise ratio (SINR) or the mean-square error (MSE) in received signal values, can be expressed in terms of the parameters of the decomposed channel. System optimization may then become essentially a problem of power allocation among the individual streams [25, 6].
On the other hand, use of multiple, parallel relays (PR) has also been considered by many researchers and shown to be potentially beneficial in various aspects [5, 21, 40, 22, 13, 9, 12, 24, 26, 11, 15, 47, 20, 8, 3]. For example, it is found that an increased number of relays can benefit the system capacity [5]. In fact, the parallel relays can function as virtual transmitter antennas and effect transmitter diversity either in the form of distributed space-time coding [21, 40] or in the form of distributed beamforming [22, 13, 9]. The corresponding diversity order has been examined in [22] and [12], respectively. Moreover, parallel relaying has also been studied in the contexts of sensor networks [24], two-way relaying [26], and secrecy communication [11]. However, despite the potential benefits, the fact that the relays are not connected but stand in parallel raises a cooperation problem which, if not dealt with, could severely limit the realizable benefit.
To see why, let L be the number of parallel relays and Ni (1 ≤ i ≤ L) the number of antennas on relay i. Let M denote the number of antennas on the source terminal as well as that on the destination terminal. Consider first the simplest case where each terminal has only one antenna, i.e., SAS-PR-SAR where M = Ni = 1 ∀i [13, 9]. In this case, the relays effectively constitute a distributed beamformer for the single signal stream. Applying the same design philosophy to an MAS-PR-MAR system with M > 1 and Ni > 1 ∀i, there can be MS = min{M, Ni ∀i} concurrent signal streams. The beamforming techniques used in MAS-SR-MAR systems can be extended to this scenario with a twist [15, 47]. That is, the available antennas on the relays can be used to provide MS parallel subchannels between the source and the destination. Systems operating in the above ways have been considered in some works [48, 13, 9, 15, 47, 25, 6]. In terms of capacity, however, such systems suffer from two consequences. First, the number of supported subchannels (i.e., the number of concurrent spatially multiplexed streams) does not grow with the relay number L, but is upper-bounded by MS. Secondly, to increase the number of streams we need to ensure that all relays are equipped with sufficient antennas.
Designs that can obviate the above limits are of interest and importance.
In this work we consider the design of MAS-PR-SAR systems (where Ni = 1 and P Ni = L) to support multiple signal streams. More specifically, we consider the design of AF relay forwarding gains for maximization of system capacity. Previously, Jin et al.
[20] considered the case where the relays had equal gain and analyzed the statistics of the resulting ergodic capacity. Chen et al. [8] considered the minimization of transmission power subject to per-stream SINR targets. The problem is related to system capacity, but somewhat indirectly. Bae and Lee [3] proposed algorithms for capacity optimization under the condition that the product of the source-to-relay and the relay-to-destination
channel matrices was asymptotically diagonal in the limit of a large number of relays.
But in sum, there is as yet no extensive work on the design of distributed parallel relay networks for capacity maximization. Actually, the relationship between number of relay terminals and system capacity also needs to be further clarified. The present work is motivated by these observations.
We consider two approaches to maximizing the capacity of a distributed relay network with presence of perfect channel state information (CSI). The first is algorithmic, as so far no closed-form solution to the problem exists. However, although algorithmic optimization can yield good results, it provides little insight into the analytical properties of the solutions. We thus also attempt an analytical approach. Because no closed-form solution can be obtained for the general situation, we consider two asymptotic situations which are more amenable to analysis. In one of them the relay noise dominates the overall noise in the received signal at the destination and in the other the destination terminal noise dominates. Alternatively, these two situations can also be viewed as providing two upper bounds to the system capacity.
Given the simplification lead by noise-dominant models, it is still a challenging task to optimize the relay network performance. Instead of considering all relays simultaneously, we discover that closed form optimization is possible if only partial of relays are active.
In consequence, we develop algorithms for capacity improvement which work with relay selection. In addition, we then observe the proposed selection-based algorithms could be modeled and analyzed with equivalent single-hop MIMO antenna selection systems [16, 17]. Given the results and analysis of existing works in the area of MIMO antenna selection, we gain more in depth understanding about the application of proposed algo-rithms for distributed relay networks.
As the number of relays increases, it is found that both capacity and outage diversity order increases. However, the proposed algorithms based on relay selection may become intractable as the number of relay terminals is large. To counteract the situation, we design further simplifications by modified criterion and shrunken solution space. Though being suboptimal in nature, the proposed algorithms avoid computational intensive oper-ation and could fully utilize the whole relay network. Simuloper-ations confirm the proposed approaches could result in slightly inferior or better performance than selection-based schemes, with considerably smaller computation cost as number of relays is large.
1.1.2 On the OFDM Channel Estimation and Applications in Relay Networks
Coherent demodulation of orthogonal frequency-division multiplexing (OFDM) signals critically depends on proper channel estimation. Since OFDM systems usually reserves some subcarriers as pilots, most channel estimation methods are pilot-aided. A com-mon approach is to estimate the channel frequency response at pilot locations first, and
then “extend” the estimate to other subcarrier locations. One frequently considered way of “extension” is low-order polynomial interpolation, which can take the form of one-dimensional interpolation in the frequency domain (within the boundary of one OFDM symbol) or two-dimensional interpolation over frequency and time (across several OFDM symbols) [23], [7]. The performance of this sort of methods is limited by the pilot density and the channel characteristics. For example, if the channel has small coherence band-width (i.e., long delay spread) and low coherence time (e.g., due to fast motion) and the pilots are widely spaced in frequency, then they would have difficulty obtaining accurate channel estimates.
Another way of “extension” is based on exploiting the time-domain characteristics of the channel [29]. Since, in many cases, only a few multipaths contribute significantly to the channel response (in other words, the channel is “sparse”), the unknowns in time-domain estimation (which consist of the path coefficients of the significant multipaths if their delays are known) are usually much fewer than that in frequency-domain-based interpolation (which consist of the frequency response at all subcarriers). Hence the few pilots can be put to better use and result in more accurate channel estimates. This is especially the case when the pilots are very few and very widely spaced (as, for example, in the case of the IEEE 802.16-2004 OFDMA uplink [19]).
Evidently, a fundamental issue in time-domain channel estimation is to find the delays of the significant multipaths. In [46], an effective delay acquisition technique is developed, but the pilots need to be equally spaced. In [32], the MUSIC algorithm widely used for spectrum analysis is employed for channel estimation, but again assuming equally spaced pilots. The algorithm can be easily extended to deal with irregular pilot spacings, but the pilot locations in the multiple OFDM symbols used in channel estimation should be identical. To the best of our knowledge, there is no time-domain channel estimation technique proposed so far that makes use of arbitrarily organized pilots in multiple OFDM symbols in the presence of channel fading.
In this work, we propose an effective technique for time-domain sparse channel esti-mation based on the matching pursuit (MP) approach. MP algorithms have been used in audio and video signal processing to select bases of subspaces [27], [1]. We extend the prior MP method for multipath delay estimation by jointly considering a group of OFDM symbols; thus we term the proposed algorithm a group MP (GMP) algorithm. And we design the algorithm such that it can deal with arbitrary pilot assignment that may vary from OFDM symbol to OFDM symbol. In [43], [34] the similar MP based processing for multiple measurement vectors are discussed. Note that the dictionary, or the range of basis searching, in these works is unique. In the scenario of this contribution, however, there are multiple reference dictionaries due to arbitrary pilot assignment.
The proposed subspace-based approaches for OFDM channel estimation is effective to highly dispersive frequency-selective channels with limited resource of pilots, which is also an potential threat to distributed relays with OFDM transmission. For most works on relay systems, flat-fading channels are the presumed channel models. However in
practice frequency selective channels would pose substantial effects to relay networks We demonstrate that for AF distributed relays, the end-to-end OFDM transmission could be modeled as equivalent single-hop OFDM with a delay profile composed of summation of multiple convolution, and consequently exhibits severe channel dispersion. We would apply the proposed schemes for the relay transmission and examine the performance.