# Comparison of Channel Estimation Protocols for Coherent AF Relaying Networks in the Presence of Additive Noise and LO Phase Noise

- Stefan Berger
^{1}Email author and - Armin Wittneben
^{1}

**2010**:861735

https://doi.org/10.1155/2010/861735

© S. Berger and A. Wittneben. 2010

**Received: **9 February 2010

**Accepted: **3 June 2010

**Published: **27 June 2010

## Abstract

Channel estimation protocols for wireless two-hop networks with amplify-and-forward (AF) relays are compared. We consider multiuser relaying networks, where the gain factors are chosen such that the signals from all relays add up coherently at the destinations. While the destinations require channel knowledge in order to decode, our focus lies on the channel estimates that are used to calculate the relay gains. Since knowledge of the compound two-hop channels is generally not sufficient to do this, the protocols considered here measure all single-hop coefficients in the network. We start from the observation that the direction in which the channels are measured determines (1) the number of channel uses required to estimate all coefficient and (2) the need for global carrier phase reference. Four protocols are identified that differ in the direction in which the first-hop and the second-hop channels are measured. We derive a sensible measure for the accuracy of the channel estimates in the presence of additive noise and phase noise and compare the protocols based on this measure. Finally, we provide a quantitative performance comparison for a simple single-user application example. It is important to note that the results can be used to compare the channel estimation protocols for any two-hop network configuration and gain allocation scheme.

## 1. Introduction

Cooperative networks offer diversity, multiplexing, and array gains as in MIMO systems but in a distributed fashion. The spatial diversity, that is inherently available, can be exploited by user cooperation to decrease the outage probability for a given rate, thus making the communication more robust against deep fades [1–5]. Furthermore, coherent beamforming allows for a distributed spatial multiplexing gain [6–9]. For interference networks comprising multiple source-destination pairs, this involves allowing the users to communicate concurrently on the same physical channel.

Note that the relays in these networks are usually not able to decode all data streams due to the large amount of inter-user interference. Instead, they assist the communication by simply forwarding scaled and rotated versions of their received signals, which corresponds to the multiplication with complex-valued gain factors. We refer to this type of forwarding protocol as multiuser AF relaying (e.g., [6]). The relay gains are chosen such that all signals add up coherently at the destination antennas. Global channel knowledge, that is, knowledge of all first-hop and second-hop channel coefficients, is usually required to calculate the gain factors accordingly. It is important to stress that information about the equivalent two-hop (source-relay-destination) channels (treated e.g., in [10–12]) is generally not enough to explicitly compute the relay gains. A gradient-based iterative scheme is required to find the gain factors in this case (e.g., [13]). Examples of papers discussing coherent cooperative gain allocation schemes, where the relay gains are computed from instantaneous, global CSI are [14–18].

Contribution

This work was triggered by the simple fact that the relay gains in coherent AF networks are computed from channel estimates. The quality of these estimates obviously has an impact on the accuracy with which the gain factors can be computed. This in turn determines the degree to which the signals from the relays combine coherently at the destinations and thus immediately affect the system performance. We furthermore observe that the direction in which the channels in a wireless network are measured (The source-relay (first-hop) and the relay-destination (second-hop) channels can be measured either in "forward direction", that is, from sources/relays to relays/destinations, or in "backward direction", that is, from relays/destinations to sources/relays.) (e.g., using training sequences [19]) determines the number of channel uses required to estimate all coefficients and the need for a global phase reference at a certain set of nodes [20]. In the presence of additive noise and LO phase noise, both factors have an impact on the quality of the channel estimates. In this work, we compare four channel estimation protocols that differ in the direction in which the single-hop channel matrices are measured. As a result, the accuracy of the channel estimates obtained by the protocols is different. For symmetry reasons we can constrain ourselves to the discussion of only two of the four protocols. We quantify the quality of the channel estimates and discuss which protocol delivers the most accurate channel estimates and thus allows for the best overall system performance. It turns out that there are situations where one protocol outperforms the other and vice versa.

The authors of [21] consider a very simple special case of this problem. They investigate the accuracy of a channel estimation protocol (corresponding to protocol B1 in this work) for a two-hop network with a single source-destination pair and multiple AF relays. The gain factors are to be computed from the channel estimates at the relays in a way that all signals combine coherently at the destination antenna. The authors neglect LO phase noise and implicitly assume a perfect carrier phase synchronization between all relays and the destination. In comparison to [21], this work compares four different channel estimation protocols, considers multiple source-destination pairs, takes LO phase noise into account, and drops the assumption of perfect phase synchronization.

Outline

The system model is presented in Section 2. We derive the input/output relation in Section 2.1 and discuss the impact of unknown and random LO phases on the signaling in Section 2.2. Section 3 then motivates the usage of the MSE of the estimated two-hop channels to judge the quality of the channel estimates. The four previously mentioned protocols are derived in Section 4. We will explain how the effort to estimate all channel coefficients in a distributed network depends on the direction in which the channel are measured. A scheme that can provide the relays with a global phase reference was originally presented in [22]. It is shortly revisited in Section 5. Section 6 then discusses the impact of additive noise and relay phase noise on the quality of the channel estimates delivered by the protocols. Finally, we compare the quality of the channel estimates produced by the protocols in Section 7.

Notation 1.

We use bold uppercase and lowercase letters to denote matrices and vectors, respectively. The operators , , and are the matrix transpose, hermitian transpose, and conjugate complex, respectively. We use to denote a convolution and is the expectation with respect to . is the identity matrix of size . The expression writes the elements of into a diagonal matrix. Finally, vectors with entries that are taken from a normal and a complex normal distribution with mean and variance are denoted by and , respectively.

## 2. System Model

*half-duplex constraint*; e.g., [4]). Consequently, a "transmission cycle" consists of two phases:

*phase one*comprises the "first-hop" transmission from the sources to all relays and

*phase two*the "second-hop" transmission from the relays to the destinations.

The relays shift their received signals to complex baseband, sample them, and store the samples until the end of the first phase. In the second phase, they retransmit scaled and rotated version of their received samples to the destinations. This corresponds to the multiplication of the samples with a complex-valued gain factor at each relay. As long as the sampling theorem is fulfilled, the analog transmit signal can be reconstructed perfectly from the stored samples.

Note that the direct link is not taken into account in this work because it is independent of LO phases of the relays. The quality of its estimates is therefore the same for all four channel estimation protocols. Without going further into details, we assume that the nodes are perfectly synchronized in time.

### 2.1. Input/Output Relation

All channels are assumed to be mutually independent and frequency flat. They are subject to *Rayleigh fading*, that is, the channel coefficients are zero-mean complex Gaussian random variables with variance
. The matrices
and
are called first-hop and second-hop channel matrix, respectively. The propagation environment is quasistatic, that is, the channels are constant during at least one transmission cycle while different channel realizations are temporally uncorrelated (block fading).

where is the channel coefficient from source to relay and the channel coefficient from relay to destination .

### 2.2. Local Oscillator Phase Offsets

They are reciprocal, that is, if A and B are phase synchronous, that is, (cf. [20]).

where is defined in (3). in (5) is independent of the LO phases of the relays because their impact on the signal during reception is compensated when the signal is retransmitted (If the relay phases change during the time between reception and retransmission (e.g., due to phase noise), they do not compensate. As a consequence, a phase error is introduced to the signal [24].).

Note that the way the signals from the relays add up at the destination antennas (constructively or destructively) is independent of both the LO phases of the sources and of the destinations. For this reason and do not have an impact on the accuracy of the gain factors that are computed from channel estimates. They can thus be chosen to be of any value without changing the result of the analysis. In order to keep the notation simple, we therefore set and to zero, that is, for all . This means that (see (5)).

## 3. Performance Measure

which will be used as a figure of merit.

## 4. Channel Estimation Protocols

- (1)
*Number of required channel uses:*the effort required to estimate all first-hop and second-hop channel coefficients depends on the direction in which they are measured. In the following, we assume that it takes one channel use to estimate one channel coefficient. - (2)
*Need for global phase reference:*it turns out that for two of the protocols, the gain factors can only be computed correctly if the relays possess a global phase reference.

The channel coefficients in the two-hop network shown in Figure 1 can be measured (e.g., using training sequences or pilot symbols) either in *forward direction*, that is, from sources/relays to relays/destinations, or in *backward direction*, that is, from relays/destinations to sources/relays. In order to highlight the impact of the LO phases of the relays, estimation noise is omitted in this section. Measuring the first-hop and second-hop channels in forward direction consequently yields knowledge of the coefficients
and
. In contrast to that, estimating the channels in backward direction yields knowledge of
and
(see (4)). There are altogether four combinations of directions in which the first-hop and second-hop channel matrices can be measured. The four corresponding protocols are as follows.

Protocol A1

Protocol A2

which is the same as for protocol A1.

Protocol B1

The phase that enters may be random and unknown. As long as it is the same for all relays (due to a global phase reference), it has no impact on the way the signals add up at the destination antennas. Since , (12) implies that the anticipated SINR at destination (which is based on ) is equal to the actual one.

Protocol B2

Direction of measurement and required number of channel uses to estimate all first-hop and second-hop channel coefficients.

Apart from the effort to measure all channel coefficients, the four protocols differ in the quality of the channel estimates they deliver in the presence of noise. In Section 6, we will discuss impact of additive noise and relay phase noise on the quality of the channel estimates. Since the anticipated equivalent two-hop channels are the same for protocols A1 and A2, (see (9) and (10)), it suffices to consider only one of them. Furthermore, (12) and (14) reveal that . Consequently, the MSE of the anticipated equivalent two-hop channels is the same for protocols B1 and B2. In the following, we will thus confine ourselves to the discussion of protocols A1 and B1. The results then also hold for A2 and B2.

It is important to realize that in a distributed network, each node can only estimate the channels to itself. For example, using protocol B1, relay can only estimate the th row of the first-hop channel matrix and the th column of the second-hop channel matrix. We call this kind of channel knowledge "local CSI". In contrast to that, "global CSI" refers to the knowledge of all channel coefficients. In the two-hop network shown in Figure 1, this means knowledge of the complete first-hop and second-hop channel matrices, that is, and .

There exists no channel estimation protocol that yields global CSI at an individual node in a distributed network. In order to obtain global CSI at the relays in Figure 1 (so that they can compute their gain factors locally), all locally estimated channel coefficients have to be disseminated. Since the number of channel coefficients that have to be disseminated is identical for all protocols, the effort is the same in all cases. It has thus no impact on the comparison presented in this work and is omitted in the following considerations.

## 5. Distributed Phase Synchronization Scheme

In the previous section, we have seen that the gain factors can only be computed correctly from channel estimates obtained with protocols B1 or B2 if the relays are phase synchronous. Two approaches to provide the relays with a global phase reference have been presented in [22] and [25, 26]. The scheme presented in [22] will be used for channel estimation protocol B1 in Section 6 and is therefore shortly revisited in this section. Please refer to [22] for a more detailed description and a comparison to the scheme presented [25, 26]. We again focus on LO phase offsets and omit estimation noise in this section. Furthermore, the LO phases of all relays are assumed to be constant during a transmission cycle.

It has the same form as (12), where , and is independent of the LO phases of the relays. Note that knowledge of is used to compensate the phase error introduced to by the channel estimates. This means that the phase synchronization scheme only has to be performed when the channel estimates are updated (and has become outdated due to phase noise).

In the following, we shortly assess the effort required to perform this phase synchronization scheme. Assume to this end that all relays transmit on orthogonal channels to the master node, which again transmits on orthogonal channels back to the relays. This results in a total of orthogonal channel uses if none of the relay nodes acts as a master node (If a relay acts as master, the number of orthogonal channel uses reduces to . In the following, we will, however, assume that no relay acts as master node.). It yields the most accurate phase synchronization results (because there is no interference) but also requires the biggest effort. If the transmissions from relays to master node and back are orthogonalized in time, this corresponds to a total of timeslots. For a wideband system, orthogonality can instead be achieved in frequency domain, which then only requires a total of timeslots. In the following, we will denote the number of channel uses required to perform the phase synchronization scheme by .

## 6. Impact of Noise

Up to now, phase noise and additive noise perturbing the channel estimates have been neglected. Both will, however, degrade the quality of the channel estimates and therefore the performance of any coherent gain allocation scheme. While the impact of estimation noise on all protocols of Section 4 is the same, the impact of phase noise is not. In this section, the impact of relay phase noise and estimation noise on the quality of the channel estimates produced by protocols A1 and B1 is investigated. The result allows for a comparison that states which protocol delivers better channel estimates under which circumstances.

All relays are assumed to employ free running LO. Wiener phase noise is in this case an appropriate model that describes the LO phase fluctuations as sampled Wiener process (e.g., [27]). The severity of the unknown and random phase changes is then a linear function of time. Consequently, the protocols requiring more channel uses to estimate all coefficients suffer more from phase noise than those requiring less channel uses. In order to assess the impact of relay phase noise on the quality of the channel estimates, the notion of "block phase noise" is introduced: the LO phases of the relays stay constant for a single channel use and change randomly afterwards (similar to a block fading channel model). In the Wiener phase noise model, the phase changes are mutually independent, zero-mean Gaussian random variables. Their variance is in the following denoted by . It is assumed to be the same for all relays.

In the following, we derive expressions for the perturbed single-hop channel estimates obtained by protocols A1 or B1. These are then used as basis for the subsequent performance comparison of both protocols.

### 6.1. Single-Hop Channel Estimates: Protocol A1

*First-Hop Channels*. Let denote the phase offset of relay in timeslot 1. Furthermore, the phase change between timeslots and is denoted by , . Consequently, the phase offset of relay in timeslot , that is, while source is transmitting its training sequence, is given by

where is given in (19) and is AWGN (cf. (18)).

*Second-Hop Channels*. From timeslot until timeslot , the relays transmit training sequences to the destinations. Let be defined as in (21) for . Then the estimated channel coefficients are

The phase changes are zero-mean Gaussian with variance . Furthermore, the scaling factor is assumed to be the same as for the estimation of the first-hop channel coefficients because the channel coefficients and noise samples have the same statistics.

### 6.2. Single-Hop Channel Estimates: Protocol B1

*First-Hop Channels:* the estimated first-hop channel coefficients are the same as for protocol A1. They are given in (22).

*Phase Synchronization:*at timeslot , the relays start to transmit their training symbols on orthogonal channels to the master node M. The phase offset of relay at this time is denoted by

is due to phase noise and is due to the additive noise components in (30). In [29] it was shown that for large SNR, is approximately Gaussian. For the following considerations, this assumption is made and we have and .

*Second-Hop Channels:*for the estimation of the second-hop channel coefficients, the relay phases stay constant for a single channel use and change independently afterwards. In contrast to protocol A1, the second-hop channels are now estimated in backward direction. This means that the channel coefficients are measured at the relays. Their estimates are given by

where the phase changes and are given in (26) and (29), respectively. Furthermore, and for . The variance of is larger than the variance of for because it took the master timeslots to transmit to all relays during the phase synchronization procedure.

*Disseminated Channel Coefficients:*after the first-hop and second-hop channel coefficients have been measured, the estimates have to be disseminated to all relays. The disseminated first-hop and second-hop channel estimates are as given in (22) and

respectively (cf. (16)). The phase correction term is the result of the phase synchronization scheme. It is given in (31).

### 6.3. Channel Estimation Error: Equivalent Two-Hop Channels

A sensible performance measure for the channel estimation schemes was found to be how well the anticipated equivalent two-hop channels match the actual ones. In this section, we derive defined in (8) for protocols A1 and B1, respectively. The main results are (41) and (48).

*Protocol A1:*for channel estimation protocol A1, the estimates of the first-hop and second-hop channel coefficients are given in (22) and (23), respectively. The anticipated and the actual equivalent two-hop channel coefficients between source and destination are in this case

where is defined in (36). The proof is included in [30] but is omitted in this work due to space limitation. The gradient of the MSE with respect to the gain factors is , where is the vector comprising all . It can easily be derived from (41) and is useful for gradient-based gain allocations that optimize the relay gains for robustness against channel estimation errors.

*Protocol B1:*for channel estimation protocol B1, the estimates of the first-hop and second-hop channel coefficients are given in (22) and (35), respectively. They can be written as

where is defined in (36). The gradient can be easily computed from (48).

### 6.4. Channel Estimation Error: Single-Hop Channels

Instead of averaging over all channel and noise realizations, the MSEs in the previous section have been computed for fixed channel estimates. It is not clear how well the actual quality of the estimates is reflected in this measure. In this section, we investigate an alternative measure that is very simple. Since both protocols deliver the same estimates for the first-hop channels, we compare them based on the quality of the second-hop channel estimates.

## 7. Performance Comparison

The scaling factor ensures that an average transmit power constraint is met. Since the gain factors are explicit functions of the channel estimates, we can furthermore assess the accuracy with which the approximations in Sections 6.3 and 6.4 judge the performance of the protocols: averaging the squared estimation error over the perturbations (estimation noise and phase noise) delivers reference MSE of the anticipated equivalent two-hop channels in closed-form. They are denoted by and for protocols A1 and B1, respectively.

We compare the quality of the channel estimates by computing the ratio of MSE. The reference will be denoted by "Two-hop MSE (reference)". A value larger than one means that the estimates produced by B1 are more accurate than those produced by A1, a value smaller than one means that B1 delivers more accurate estimates than A1. Note that the number of source-destination pairs and relays in the network has an impact on the quality of the channel estimates. While the estimated first-hop channels are equal for protocols A1 and B1, the MSEs of the second-hop estimates are not. Their MSEs (and thus the quality of their estimates) are equal if (cf. (49) and (52)). Although being independent of , this point is a function of the destination index . Increasing the while keeping constant is therefore in favor of protocol A1. If the number of relays increases, the relation between and determines which protocol delivers the better estimates of the second-hop channel coefficients.

- (1)
*Section 6.3*: in order to compare the quality of the estimates produced by A1 and B1 based on (41) and (48), we average and over all channel estimates in for the case that the gain factors are given in (53). The ratio is then denoted by "Fixed estimate MSE". - (2)
*Section 6.4*: since (49) depends on the order in which the relays transmit their training sequences, we perform an averaging over all relays and defineThe ratio is then denoted by "Second-hop MSE", where and are given in (49) and (52), respectively.

Comparing the curves to the respective references ("Two-hop MSE (reference)") shows that the measure in Section 6.3 is very accurate for high-estimation SNR (from about ). Furthermore, the measure in Section 6.4 is very accurate in medium estimation SNR ( ) and low-phase noise ( ). In the respective range of parameters, both measures are able to judge the performance of both channel estimation protocols very well.

## 8. Conclusions

In this work, we investigated different channel estimation protocols for two-hop AF relaying networks (single-user and multiuser) in the presence of additive estimation noise and relay phase noise. They differ in the direction in which the single-hop links are measured and thus the required effort to estimate all channel coefficients in the network. We used the MSE of the channel estimates as an indicator for the performance of the protocols. This is a sensible measure because computing the gain factors from more accurate channel estimates will on the average lead to better system performance. It was possible to draw conclusions independently of the gain allocation by comparing the MSE of the second-hop estimates only. Finally, we compared the protocols quantitatively for a single-user application example. It is important to note that the results can as well be used to assess the channel estimation protocols for any two-hop network configuration and gain allocation.

## Authors’ Affiliations

## References

- Laneman JN, Wornell GW: Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks.
*IEEE Transactions on Information Theory*2003, 49(10):2415-2425. 10.1109/TIT.2003.817829MathSciNetView ArticleMATHGoogle Scholar - Sendonaris A, Erkip E, Aazhang B: User cooperation diversity—part I: system description.
*IEEE Transactions on Communications*2003, 51(11):1927-1938. 10.1109/TCOMM.2003.818096View ArticleGoogle Scholar - Sendonaris A, Erkip E, Aazhang B: User cooperation diversity—part II: implementation aspects and performance analysis.
*IEEE Transactions on Communications*2003, 51(11):1939-1948. 10.1109/TCOMM.2003.819238View ArticleGoogle Scholar - Laneman JN, Tse DNC, Wornell GW: Cooperative diversity in wireless networks: efficient protocols and outage behavior.
*IEEE Transactions on Information Theory*2004, 50(12):3062-3080. 10.1109/TIT.2004.838089MathSciNetView ArticleMATHGoogle Scholar - Muhaidat H, Uysal M: Cooperative diversity with multiple-antenna nodes in fading relay channels.
*IEEE Transactions on Wireless Communications*2008, 7(8):3036-3046.View ArticleGoogle Scholar - Wittneben A, Rankov B: Distributed antenna systems and linear relaying for gigabit MIMO wireless.
*Proceedings of the 60th IEEE Vehicular Technology Conference (VTC '04), September 2004, Los Angeles, Calif, USA*3624-3630.Google Scholar - Morgenshtern VI, Boelcskei H, Nabar RU: Distributed orthogonalization in large interference relay networks.
*Proceedings of IEEE International Symposium on Information Theory (ISIT '05), September 2005*1211-1215.Google Scholar - Bölcskei H, Nabar RU, Oyman Ö, Paulraj AJ: Capacity scaling laws in MIMO relay networks.
*IEEE Transactions on Wireless Communications*2006, 5(6):1433-1444.View ArticleGoogle Scholar - Dana AF, Hassibi B: On the power efficiency of sensory and ad hoc wireless networks.
*IEEE Transactions on Information Theory*2006, 52(7):2890-2914.MathSciNetView ArticleMATHGoogle Scholar - Patel CS, Stüber GL: Channel estimation for amplify and forward relay based cooperation diversity systems.
*IEEE Transactions on Wireless Communications*2007, 6(6):2348-2355.View ArticleGoogle Scholar - Cui T, Gao F, Nallanathan A: Optimal training design for channel estimation in amplify and forward relay networks.
*Proceedings of the 50th Annual IEEE Global Telecommunications Conference (GLOBECOM '07), November 2007*4015-4019.Google Scholar - Gao F, Cui T, Nallanathan A: On channel estimation and optimal training design for amplify and forward relay networks.
*IEEE Transactions on Wireless Communications*2008, 7(5):1907-1916.View ArticleGoogle Scholar - Eşli C, Wagner J, Wittneben A: Distributed gradient based gain allocation for coherent multiuser AF relaying networks.
*Proceedings of the IEEE International Conference on Communications (ICC '09), June 2009*Google Scholar - Berger S, Wittneben A: Cooperative distributed multiuser MSE relaying in wireless ad-hoc networks.
*Proceedings of the 39th Asilomar Conference on Signals, Systems and Computers, October-November 2005, Pacific Grove, Calif, USA*1072-1076.Google Scholar - Khajehnouri N, Sayed AH: A distributed MMSE relay strategy for wireless sensor networks.
*Proceedings of the 6th Workshop on Signal Processing Advances in Wireless Communications (SPAWC '05), June 2005*796-800.Google Scholar - Shi H, Abe T, Asai T, Yoshino H: Analyses of achievable gains in half duplex MIMO relaying schemes using multiple relay nodes.
*IEICE Transactions on Communications*2007, E90-B(9):2541-2551. 10.1093/ietcom/e90-b.9.2541View ArticleGoogle Scholar - Guan W, Luo H: Joint MMSE transceiver design in non-regenerative MIMO relay systems.
*IEEE Communications Letters*2008, 12(7):517-519.View ArticleGoogle Scholar - Ding Z, Chin WH, Leung KK: Distributed beamforming and power allocation for cooperative networks.
*IEEE Transactions on Wireless Communications*2008, 7(5):1817-1822.View ArticleGoogle Scholar - Meyr H, Moeneclaey M, Fechtel SA:
*Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing, Wiley Series in Telecommunications and Signal Processing*. John Wiley & Sons, New York, NY, USA; 1997.Google Scholar - Berger S, Wittneben A: When do non-regenerative two-hop relaying networks require a global phase reference?
*Proceedings of IEEE Global Telecommunications Conference (GLOBECOM '09), November-December 2009, Honolulu, Hawaii, USA*Google Scholar - Behbahani AS, Eltawil A: On channel estimation and capacity for amplify and forward relay networks.
*Proceedings of IEEE Global Telecommunications Conference (GLOBECOM '08), December 2008*3685-3689.Google Scholar - Berger S, Wittneben A: Carrier phase synchronization of multiple distributed nodes in a wireless network.
*Proceedings of the 8th IEEE Signal Processing Advances in Wireless Communications (SPAWC '07), June 2007, Helsinki, Finland*Google Scholar - Proakis JG:
*Digital Communications, McGraw-Hill Series in Electrical and Computer Engineering*. McGraw-Hill, New York, NY, USA; 2001.Google Scholar - Berger S, Wittneben A: Impact of local-oscillator imperfections on nonregenerative TDD and FDD relaying.
*Proceedings of Vehicular Technology Conference (VTC '10), May 2010, Taipei, Taiwan*Google Scholar - Tu Y-S, Pottie GJ: Coherent cooperative transmission from multiple adjacent antennas to a distant stationary antenna through AWGN channels.
*Proceedings of the 55th Vehicular Technology Conference (VTC '02), 2002, Birmingham, Ala, USA*1: 130-134.Google Scholar - Barriac G, Mudumbai R, Madhow U: Distributed beamforming for information transfer in sensor networks.
*Proceedings of the 3rd International Symposium on Information Processing in Sensor Networks (IPSN '04), April 2004*81-88.Google Scholar - Demir A, Mehrotra A, Roychowdhury J: Phase noise in oscillators: a unifying theory and numerical methods for characterization.
*IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications*2000, 47(5):655-674. 10.1109/81.847872View ArticleGoogle Scholar - Yoo T, Goldsmith A: Capacity and power allocation for fading MIMO channels with channel estimation error.
*IEEE Transactions on Information Theory*2006, 52(5):2203-2214.MathSciNetView ArticleMATHGoogle Scholar - Berger S, Wittneben A: Impact of noisy carrier phase synchronization on linear amplify- and-forward relaying.
*Proceedings of the 50th Annual IEEE Global Telecommunications Conference (GLOBECOM '07), November 2007*795-800.Google Scholar - Berger S:
*Coherent cooperative relaying in low-mobility multiuser networks, Ph.D. dissertation*. ETH, Zurich, Switzerland; 2010. Appendix A1, http://www.nari.ee.ethz.ch/~berger/Appendix_A1.pdfGoogle Scholar

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