# Fundamental Properties of Wireless Relays and Their Channel Estimation

**DOI:**https://doi.org/10.1007/978-3-319-32903-1_173-1

## Synonyms

## Definition

A wireless relay system involves at least three nodes: a source node, a relay node, and a destination node. The relay node assists the transmission of information from the source to the destination. The relay channels include all channels between these nodes.

## Historical Background

*S*intends to transmit messages to the destination node

*D*, the source node

*S*sends the signal in the first time slot and the signal is received by both the destination node

*D*and the relay node

*R*; then in the second time slot, the relay node forwards the received signal to the destination node

*D*. With an assistance of the relay node

*R*, two copies of signal transmitted from the source

*S*are received at the destination node

*D*where one from

*S*to

*D*and another from

*S*to

*R*then to

*D*. As the channels in those two links are generally independent, two copies of signals can be combined at the destination node

*D*to achieve a more reliable detection. From the communication theory point of view, the performance gain comes from an inherent spatial diversity. As here the wireless relay helps to forward one-way message delivery, i.e., from source to destination, we refer to this channel model as one-way relay channel.

*S*

_{1}desires to transmit signal

*x*

_{1}to source node

*S*

_{2}and source node

*S*

_{2}intends to transmit signal

*x*

_{2}to source node

*S*

_{1}, the whole transmission is completed in two time slots. In the first time slot, both source nodes transmit their signals to the relay node simultaneously. In this case, the relay node receives the combination of signals

*x*

_{1}and

*x*

_{2}. After performing certain processing, the combination is forwarded to nodes

*S*

_{1}and

*S*

_{2}in the second time slot. As two source nodes know their transmitted signals, the self-interference can be canceled from the received combination before decoding the desired signals. It is easy to observe that in two-way relay channel, two messages are delivered within two time slots. In contrast to the one-way relay channel, where two time slots are required to complete one message delivery, two-way relay channel significantly improves the spectrum efficiency.

Several relay processing strategies have been proposed in wireless relay channel. Basically, different strategies usually require different complexity and possess different performance. Two main strategies are amplify-and-forward (AF) strategy and decode-and-forward (DF) strategy. In AF strategy, the relay node simply amplifies the received signal and forwards it directly without decoding the messages. In DF strategy, the relay node decodes the messages from the received signals and regenerates new signals which are sent to the destination subsequently.

As the transmission protocol of the relay channel is quite different from the traditional point-to-point transmission, the corresponding physical layer techniques are greatly modified, especially the channel estimation which is used to obtain the channel state information required for physical layer designs including power allocation (Chen et al. 2017; Ma et al. 2014), precoding design (Cirik et al. 2014; Xu and Hua 2011; Yu and Hua 2010; Rong and Hua 2009; Rong et al. 2009), etc. In wireless relay channel, we generally need to estimate the channels of two-hop transmissions, i.e., from the source to the relay and from the relay to the destination. Lots of new and challenging problems are introduced. In this article, we aim to provide a brief review of the channel estimation in wireless relay channel.

## Foundations

According to transmission protocol, the two-hop channel estimation of wireless relay channel can be performed using the training signals received at the relay and the destination. If the relay node can perform the channel estimation and can transmit the training sequences, the two-hop channel estimation can be decoupled. For example, in the one-way relay channel, the first hop channel, i.e., the channel from the source *S* to the relay *R*, can be estimated at the relay node and the second hop channel, i.e., the channel from the relay *R* to the destination *D*, can be estimated at the destination node *D* separately. In this case, the overall channel estimation problem simply reduces to the two point-to-point channel estimations. In the two-way relay channel, we cannot directly decouple the two-hop channel estimation into two point-to-point channel estimation problems as it involves four independent channels. In this case, in the first hop, the channel can be considered as a multiple access channel (MAC) where received training signal at the relay node is used to estimate the channels from the source *S*_{1} to the relay node *R* and from the source *S*_{2} to the relay node *R*. While in the second hop, the channel can be treated as a broadcasting (BC) channel where the single training sequence sending from the relay node is received at the destination nodes and is then utilized to estimate the channels from the relay node *R* to two destination nodes *S*_{1} and *S*_{2}.

In another scenario, if the relay node cannot perform channel estimation or send training sequence, the channel estimation must be conducted at the destination nodes and corresponding channel estimation problem becomes relatively more complicated. Under this setup, besides estimating the individual channels of two-hop transmission, estimating the combined channels, i.e., the cascade of two-hop channels, is an efficient way to simplify the channel estimation problems. The following brief review of the channel estimation in wireless relay channel is given from four classifications: single-antenna single-carrier case, single-antenna multi-carrier case, multi-antenna single-carrier case, and multi-antenna multi-carrier case.

### Single-Antenna Single-Carrier Case

*h*

_{i}and

*g*

_{i}denotes the channel from source node

*S*to the

*i*-th relay node

*D*

_{i}and the channel from the

*i*-th relay node

*D*

_{i}to the destination node

*D*, respectively. Instead of estimating

*h*

_{i}and

*g*

_{i}individually, the authors considered to estimate the combined channel

*h*

_{i}

*g*

_{i}. The authors studied two estimation criterions, least square (L-S) and minimum mean-square-error (MMSE), by separately treating the unknown channels as deterministic variables and statistical random variables. The estimations of combined channel vector

**w**for LS and MMSE using the received signal at the destination node

**d**

_{2}can be expressed as

**A**

_{LS}and

**A**

_{MMSE}are related to the training signal and the beamforming matrix used at each relay node; furthermore, matrix

**A**

_{MMSE}is also related to the statistical information of the noise and channel. According to estimation theory, the authors derived the estimation error covariance matrix of two criterions. The training design was further investigated with an aim to minimize the estimation error, i.e., the trace of the estimation error covariance matrix. Different from the point-to-point channel estimation, the training design in this work included source training sequence design and relay beamforming matrices design. From the problem formulation, it is found that the optimization of source training sequence design and relay beamforming matrices design can be combined together, which implies that only one matrix variable

**C**needs to be equivalently optimized in the training design problem. For the LS estimation, the authors proved that the columns of

**C**should be orthogonal with each other at the optimal solution, and optimal training design can be found. For the MMSE estimation, the authors transformed the training design problem into a convex semi-definite programming (SDP) problem and the optimal solution can be efficiently solved by modern interior point method based on existing convex optimization software.

*S*

_{i}to the relay node by

*h*

_{i}, similarly to the one-way relay channel estimation in Gao et al. (2008), the authors proposed to estimate the variables

*|h*

_{i}

*|*

^{2}and

*|h*

_{1}

*h*

_{2}

*|*at terminal

*S*

_{i}instead of estimating

*h*

_{1}and

*h*

_{2}. Two channel estimation criterions, i.e., the maximum-likelihood (ML) and linear maximum average effective signal-to-noise ratio (LMSNR), were utilized to obtain the estimations. In specific, the ML estimation treats the unknown channels as two deterministic values and channel estimation is obtained by solving the following problem (take source

*S*

_{1}as an example)

*z*

_{1}is the received signal at source

*S*

_{1}and

*p*(

*z*

_{1}

*|h*

_{1}

*, h*

_{2}) is the probability density function (PDF) of

*z*

_{1}at given

*h*

_{1}and

*h*

_{2}. The LMSNR estimation treats the channels as unknown random values with known statistical information, and the channels are estimated by maximizing the effective SNR. Moreover, the training sequences at two sources were designed aiming to minimize the Cramer-Rao lower bound (CRLB) and maximize the average effective SNR. The results of optimizations showed that the optimal training sequences at two sources should be orthogonal with each other. In Xie et al. (2014), the authors proposed to use the Bayesian approach to estimate the cascaded source-relay-destination channel. The maximum a posteriori (MAP)-based estimation schemes were developed to estimate the cascaded channel and the amplitude of individual source-relay channels with apriori knowledge of channel distribution information. Additionally, to deal with some practical constraints, an iterative least square-MAP algorithm was developed when noise variance was unknown.

### Single-Antenna Multi-Carrier Case

**h**from the source node to the relay node and the channel

**g**from the relay node to the destination node in the time domain. In Gao et al. (2011), to allow the destination node to estimate

**h**and

**g**, besides sending the training signals at the source node, the relay superimposes its own training signal to the received training signal and sends them to the destination node. In this case, the transmit signal

**r**

_{t}at the relay can be represented as

**r**

_{r}is the received signal at the relay node,

*α*is the scaling factor used at the relay to satisfy the power constraint, and

**t**

_{r}is the new training signal superimposed at the relay node. It was shown that the closed-form expressions of estimated channel are generally not available in OFDM setup. To obtain an accurate estimation, an iterative channel estimation was proposed where

**h**and

**g**were separately updated during the iteration. Different from Gao et al. (2011), the channel was estimated jointly with the unknown carrier frequency offset (CFO) and phase noise (PN) in Wang et al. (2016) under the OFDM framework. To enable the joint estimation of MIMO individual channel, CFO and PN at the destination, the training sequences are assumed to be transmitted at both the source and the relay. Let

**y**

^{[r]}denote the received signal at the destination when relay transmits the training signal and

**y**

^{[s]}denote the received signal at the destination when source transmits the training signal. According to the MAP criterion, the estimations of individual channel, CFO, and PN can be obtained by solving the following optimization problem

*ϕ*

^{[s − d]}and

*η*

^{[s − d]}denote the combined CFO from source to destination and the combined PN from source to destination; \( \left\{{\hat{\phi}}^{\left[\mathrm{s}-\mathrm{d}\right]},{\hat{\eta}}^{\left[\mathrm{s}-\mathrm{d}\right]},\hat{\mathbf{h}},\hat{\mathbf{g}}\right\} \) denotes the estimated version of \( \left\{{{\phi}}^{\left[\mathrm{s}-\mathrm{d}\right]},{{\eta}}^{\left[\mathrm{s}-\mathrm{d}\right]},{\mathbf{h}},{\mathbf{g}}\right\};\,p\left({\phi}^{\left[\mathrm{s}-\mathrm{d}\right]},{\eta}^{\left[\mathrm{s}-\mathrm{d}\right]},\right. \)\( \left.\mathbf{h},\mathbf{g}|{\mathbf{y}}^{\left[\mathrm{s}\right]},{\mathbf{y}}^{\left[\mathrm{r}\right]}\right) \) denotes the posterior distribution of the parameters of interests given

**y**

^{[s]}and

**y**

^{[r]}. The ambiguities among the estimated PN, CFO, and channels were analyzed and showed that the estimated source-relay channel suffers from a phase ambiguity. Based on this analysis, a hybrid Cramer-Rao lower bound (HCRLB) for analyzing the performance was derived, which can effectively avoid the estimation ambiguities.

Regarding the single-antenna and multi-carrier two-way relay channel, the channel estimation was studied in the framework of OFDM modulation in Gao et al. (2009a). Different from Gao et al. (2011), the authors assumed that the training sequences were only transmitted from two sources while no training sequence was superposed at the relay node. Two estimation schemes, i.e., block-based and pilot-tone-based, were proposed to estimate the cascaded source-relay-destination channel and the individual channels, respectively. The block-based estimation scheme uses all carriers in one or more OFDM blocks for channel estimation and generally applies to a scenario where the training sequence is long enough. The pilot-tone estimation scheme uses several pilots residing in one OFDM block to estimate the channel and applies to the scenario with a length-limited training sequence. The estimation ambiguities of two schemes were further analyzed. In specific, the authors showed that when the length of training sequence is larger than a threshold, only the sign ambiguity can be introduced and it does not affect the finally data decoding.

### Multi-Antenna Single-Carrier Case

When considering multiple antennas at each node, the channel estimation of single-carrier one-way relay channel was investigated in Rong et al. (2012) and Kong and Hua (2011). The challenge of estimating the multi-input multi-output (MIMO) channels lies in the fact that the estimation variables become unknown matrices, while not the unknown values as in the single-antenna case. In Rong et al. (2012), the MIMO channels in source-relay-destination link and the MIMO channel in direct link were estimated without knowledge of the channel statistical information. In particular, the MIMO channel in direct link was estimated using LS criterion. Regarding the source-relay-destination link, according to the parallel factor (PARAFAC) analysis, the bilinear alternating least-squares (BALS) algorithm was proposed to obtain individual MIMO channels for source-relay link and relay-destination link. It was shown that with a mild length of training sequence, the MIMO channel matrices of two hops can be estimated up to permutation and scaling ambiguities. Moreover, the authors proposed to exploit the knowledge of the relay factors to remove the permutation ambiguity. In Kong and Hua (2011), the authors assumed that the statistical channel information was known in prior, and then the linear MMSE (LMMSE) estimation method was proposed to estimate the MIMO channels. To estimate the individual MIMO channels in each hop of the source-relay-destination link, the authors proposed a two-step estimation strategy where in the first step, the MIMO relay-destination channel is estimated assuming that the relay node is able to transmit training sequences. With the estimated MIMO relay-destination channel, the MIMO source-relay channel is then estimated at the destination node utilizing the training sequence sent from the source. For the first step, the optimal structure of relay training sequence matrix was derived according to the statistical information of the relay-destination channel. While for the second step, an algorithm was developed to compute the optimal training sequence matrix used at the source and the optimal precoding matrix used at the relay.

Later on, the channel estimation of the MIMO single-carrier two-way relay channel was studied in Wang et al. (2015). Similar to Kong and Hua (2011), the authors assumed that the statistical channel information was known in prior under a Kronecker-correlation model. Additionally, the MIMO channels were estimated in a colored noise environment by considering the impact of the antenna correlation and the interference from neighboring users. To estimate each individual MIMO channel, the authors proposed to decompose the bidirectional transmission of two-way relay channel into two phases, i.e., MAC phase and the BC phase. The optimal LMMSE estimators were derived for each phase. Two iterative training design algorithms were further proposed to obtain the training sequences for the general conditions and they were verified to produce training sequences achieving near optimal channel estimation performance. For certain specific practical scenarios where the covariance matrices of the channel or disturbances are of particular structures, the optimal training sequence design guidelines were provided. To assess the estimation performance, the relationship between the estimation performance and the length of training sequences were established, which showed that when the training sequence length is shorter than the threshold, a lower bound of estimation performance exists no matter how to increase the powers.

### Multi-Antenna Multi-Carrier Case

The MIMO channel estimation was extended to multi-carrier case for two-way relay channel in Kang et al. (2017). Instead of estimating the individual channels, the authors in this study proposed to estimate the convolution of two MIMO individual channels using the self-interfering link and information-bearing link under the LMMSE criterion. The training sequences were optimized with an aim to minimize the total MSE under the power constraints at the sources and the relay. To obtain the optimal training sequences, the authors derived optimal structure which then converted the training design optimization problem into a tractable convex form.

## Key Applications

Wireless relay is one of the fundamental techniques in cellular wireless communications. It can be efficiently used to extend the wireless coverage and enhance the network throughput in harsh environments with low economy cost.

## Cross-References

## References

- Chen L, Meng W, Hua Y (2017) Optimal power allocation for a full-duplex multicarrier decode- forward relay system with or without direct link. Signal Process 137:177–191CrossRefGoogle Scholar
- Cirik AC, Rong Y, Ma Y, Hua Y (2014) On MAC-BC duality of multihop MIMO relay channel with imperfect channel knowledge. IEEE Trans Wirel Commun 13(10):5839–5854CrossRefGoogle Scholar
- Gao F, Cui T, Nallanathan A (2008) On channel estimation and optimal training design for amplify and forward relay networks. IEEE Trans Wirel Commun 7(5):1907–1916CrossRefGoogle Scholar
- Gao F, Zhang R, Liang YC (2009a) Channel estimation for OFDM modulated two-way relay networks. IEEE Trans Signal Process 57(11):4443–4455MathSciNetCrossRefGoogle Scholar
- Gao F, Zhang R, Liang YC (2009b) Optimal channel estimation and training design for two-way relay networks. IEEE Trans Commun 57(10):3024–3033CrossRefGoogle Scholar
- Gao F, Jiang B, Gao X, Zhang XD (2011) Superimposed training based channel estimation for OFDM modulated amplify-and-forward relay networks. IEEE Trans Commun 59(7):2029–2039CrossRefGoogle Scholar
- Guthery S (1997) Wireless relay networks. IEEE Netw 11(6):46–51CrossRefGoogle Scholar
- Hong YW, Huang WJ, Chiu FH, Kuo CCJ (2007) Cooperative communications in resource-constrained wireless networks. IEEE Signal Process Mag 24(3):47–57CrossRefGoogle Scholar
- Hua Y, Bliss DW, Gazor S, Rong Y, Sung Y (2012) Guest editorial theories and methods for advanced wireless relays; issue i. IEEE J Sel Areas Commun 30(8):1297–1303. https://doi.org/10.1109/JSAC.2012.120901 CrossRefGoogle Scholar
- Hua Y, Bliss DW, Gazor S, Rong Y, Sung Y (2013) Guest editorial: Theories and methods for advanced wireless relays; issue ii. IEEE J Sel Areas Commun 31(8):1361–1367. https://doi.org/10.1109/JSAC.2013.130801 CrossRefGoogle Scholar
- Kang JM, Kim IM, Kim HM (2017) Optimal training design for MIMO-OFDM two-way relay networks. IEEE Trans Commun 65(9):3675–3690CrossRefGoogle Scholar
- Kong T, Hua Y (2011) Optimal design of source and relay pilots for MIMO relay channel estimation. IEEE Trans Signal Process 59(9):4438–4446MathSciNetCrossRefGoogle Scholar
- Ma Y, Liu A, Hua Y (2014) A dual-phase power allocation scheme for multicarrier relay system with direct link. IEEE Trans Signal Process 62(1):5–16MathSciNetCrossRefGoogle Scholar
- Nosratinia A, Hunter TE, Hedayat A (2004) Cooperative communication in wireless networks. IEEE Commun Mag 42(10):74–80CrossRefGoogle Scholar
- Rong Y, Hua Y (2009) Optimality of diagonalization of multi-hop MIMO relays. IEEE Trans Wirel Commun 8(12):6068–6077CrossRefGoogle Scholar
- Rong Y, Tang X, Hua Y (2009) A unified framework for optimizing linear nonregenerative multicarrier MIMO relay communication systems. IEEE Trans Signal Process 57(12):4837–4851MathSciNetCrossRefGoogle Scholar
- Rong Y, Khandaker MR, Xiang Y (2012) Channel estimation of dual-hop MIMO relay system via parallel factor analysis. IEEE Trans Wirel Commun 11(6):2224–2233CrossRefGoogle Scholar
- Sheng Z, Leung KK, Ding Z (2011) Cooperative wireless networks: from radio to network protocol designs. IEEE Commun Mag 49(5):64–69CrossRefGoogle Scholar
- Soldani D, Dixit S (2008) Wireless relays for broadband access [radio communications series]. IEEE Commun Mag 46(3):58–66CrossRefGoogle Scholar
- Wang R, Tao M (2012) Joint source and relay precoding designs for MIMO two-way relaying based on MSE criterion. IEEE Trans Signal Process 60(3):1352–1365MathSciNetCrossRefGoogle Scholar
- Wang R, Tao M, Mehrpouyan H, Hua Y (2015) Channel estimation and optimal training design for correlated MIMO two-way relay systems in colored environment. IEEE Trans Wirel Commun 14(5):2684–2699CrossRefGoogle Scholar
- Wang R, Mehrpouyan H, Tao M, Hua Y (2016) Channel estimation, carrier recovery, and data detection in the presence of phase noise in OFDM relay systems. IEEE Trans Wirel Commun 15(2):1186–1205CrossRefGoogle Scholar
- Xie X, Peng M, Zhao B, Wang W, Hua Y (2014) Maximum a posteriori based channel estimation strategy for two-way relaying channels. IEEE Trans Wirel Commun 13(1):450–463CrossRefGoogle Scholar
- Xu S, Hua Y (2011) Optimal design of spatial source-and-relay matrices for a non-regenerative two-way MIMO relay system. IEEE Trans Wirel Commun 10(5):1645–1655MathSciNetCrossRefGoogle Scholar
- Yu Y, Hua Y (2010) Power allocation for a MIMO relay system with multiple-antenna users. IEEE Trans Signal Process 58(5):2823–2835MathSciNetCrossRefGoogle Scholar