# On Channel Estimation for Analog Network Coding in a Frequency-Selective Fading Channel

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## Abstract

Recently, broadband analog network coding (ANC) was introduced for high-speed transmission over the wireless (frequency-selective fading) channel. However, ANC requires the knowledge of channel state information (CSI) for self-information removal and coherent signal detection. In ANC, the users' pilot signals interfere during the first slot, which renders the relay unable to estimate CSIs of different users, and, consequently, four time-slot pilot-assisted channel estimation (CE) is required to avoid interference. Naturally, this will reduce the capacity of ANC scheme. In this paper, we theoretically analyze the bit error rate (BER) performance of bi-directional broadband ANC communication based on orthogonal frequency division multiplexing (OFDM) radio access. We also theoretically analyze the performance of the channel estimator's mean square error (MSE). The analysis is based on the assumption of perfect timing and frequency synchronization. The achievable BER performance and the estimator's MSE for broadband ANC is evaluated by numerical and computer simulation. We discuss how, and by how much, the imperfect knowledge of CSI affects the BER performance of broadband ANC. It is shown that the CE scheme achieves a slightly higher BER in comparison with ideal CE case for a low and moderate mobile terminal speed in a frequency-selective fading channel.

### Keywords

Orthogonal Frequency Division Multiplex Channel Estimation Channel State Information Network Code Orthogonal Frequency Division Multiplex System## 1. Introduction

Future wireless communication networks are envisaged to provide multimedia broadband services to wireless users. The network capacity must be increased to accommodate these high bandwidth demanding services. Network coding [1] has been used in wired networks to increase the network capacity while its application in wireless relay-assisted networks [2, 3] can exploit the broadcast nature of the wireless medium and further increase the capacity. The relaying can be used to enable bidirectional communication between two users without a direct link between them. The conventional relaying requires four time slots to exchange information between the users, and, consequently, the spectrum efficiency is low.

Network coding at the physical layer (PNC) has been proposed to improve the spectrum efficiency (i.e., doubles the network capacity) of bidirectional relay-assisted communication over the conventional relaying in a flat (i.e., frequency-nonselective) fading channel [4, 5]. Henceforth, we refer to these schemes as narrowband PNC. The scheme enables the users to exchange the information within three time slots in comparison with conventional relaying. Two-slot narrowband analog network coding (ANC) introduced in [6] for bidirectional communication, where the user's signals are mixed in the wireless medium, is an extension of narrowband PNC. Recently [7], broadband ANC based on orthogonal frequency division multiplexing (OFDM) in a frequency-selective fading channel was introduced based on the assumption of perfect knowledge of channel state information (CSI). However, coherent detection and self-information removal in broadband ANC requires accurate amplitude and phase offset estimation (i.e., channel estimation (CE)).

In this paper, we present the performance of bidirectional broadband ANC communication with a pilot-assisted CE scheme based on OFDM in a frequency-selective fading channel. We theoretically analyze and discuss the performance of broadband ANC with imperfect CSI in terms of the estimator's mean square error (MSE) and bit error rate (BER). The theoretical analysis is based on the assumption that the guard interval (GI) is long enough to avoid the timing problem (i.e., perfect time synchronization) and perfect frequency synchronization. The performance of broadband ANC with imperfect knowledge of CSI is evaluated by Monte Carlo numerical computation method using the derived theoretical expressions and computer simulation. It is shown that the BER performance with imperfect self-information removal for the higher Open image in new window becomes more sensitive to the estimation errors. The CE error may degrade the BER performances of PNC and ANC differently (since PNC performs digital encoding at the packet level), but in this work, we only consider ANC since it is more spectrum efficient. Moreover, our interest lies on the physical layer techniques design, and the routing problem for ANC is out of the scope of this work.

The reminder of the paper is organized as follows. Section 3 gives an overview of the network model with pilot-assisted CE scheme for ANC in a frequency-selective fading channel. In Section 4, the theoretical BER performance analysis is presented. Numerical results and discussions are presented in Section 5. We summarize our findings in Section 6.

## 2. Related Work

The motivation for ANC is its higher spectrum efficiency in comparison with the conventional relaying and PNC since the bidirectional communication between two users is done via the relay within two time slots. Moreover, ANC has a lower computational complexity since there is no processing at the relay terminal.

The performance of an opportunistic network coding scheme that exploits interference (as in [6]) at the receivers by interpreting it as a form of a network code was investigated in [8]. In [9], the network coding for a multiuser communication problem has been presented and analyzed. The authors in [10] present and analyze an idea to decode the sum of the code-words at the relay followed by a broadcast phase which performs Slepian-Wolf coding with structured codes. The study on wireless multicasting in multiuser network (i.e., two sources to two destinations) with the assistance of a single half-duplex relay was investigated in [11], where the throughput and error performance of different analog and digital relay schemes has been presented. In [12], achievable rates for traditional multihop routing and network coding and various physical-layer network coding (PNC) approaches are considered, and a new method of PNC inspired by Tomlinson-Harashima precoding (THP), where a modulo operation is used to control the power at the relay, was introduced. The outage performance is investigated in [13], where the time-varying nature of the direct link is taken into consideration, and the outage regions of various PNC schemes are theoretically analyzed, and, then, the best combined strategies are derived in terms of the maximum goodput and robustness against the imperfect knowledge of CSI. However, coherent detection and self-information removal in broadband ANC requires accurate amplitude and phase offset estimation (i.e., channel estimation (CE)).

Unlike conventional (without relay) and cooperative (with relay) networks, where signals from different users are separated in time or frequency to avoid interference, in ANC the users' signals interfere in the same time slot. Hence, in the case of pilot transmission, the relay cannot estimate the CSIs of different users. To avoid this problem, a straightforward method is to allocate four time slots for pilot-assisted CE, which reduces the network throughput of ANC scheme. In [14], a complex maximum likelihood (ML) CE for narrowband ANC was presented based on a priori knowledge of the noise variance and the channel cross-correlation coefficients. However, the achievable estimator's MSE is high while the BER performance was not considered at all. Moreover, in [14], a simple symmetric Gaussian channel is assumed. However, in broadband wireless communications networks, the channel frequency selectivity is present due to many propagation paths having different time delays. In [15], the complex maximum likelihood channel estimation is presented for narrowband channels, where the channel gains for the both users are estimated at the relay, and, then, the power allocation algorithm is applied to allocate the power to different channel components so that the detection or CE at the user terminals is optimized. We note that the scheme in [15] requires the knowledge of the noise variance and the channel cross-correlation coefficients of the narrowband channels. In [16], tensor-based CE is presented to obtain the channel gains at both user terminals by solving a complex nonlinear least square problem in an iterative fashion based on a priori known identical and invariant channels over the two time-slots. Note that the design rules in [16] are derived without the effect of noise, which if taken into consideration must be available a priori similar to [14, 15], that may degrade the channel estimator's performance.

For broadband channels, in [17], a two-slot pilot-assisted CE scheme for ANC was presented. In the first slot, both users transmit their pilots to the relay, where one of the pilot signals is cyclically shifted [18] to allow the relay to separate and estimate the CSIs from both users. This stage is named multiple-input single-output channel estimation (MISO-CE) due to its analogy to multiple-input multiple-output (MIMO) OFDM systems [18]. During the second slot, the relay broadcast its pilot signal to the users, which estimate the corresponding CSIs. This stage is named single-input single-output channel estimation (SISO-CE). We note here that only BER performance has been evaluated by computer simulation in [17]. Therefore, in this work, we focus our attention to investigate and analyze the achievable performance of low-complexity pilot-assisted CE for broadband ANC in a frequency-selective fading channel.

## 3. Network Model

Throughout this paper, the following notations are used. Bold lowercase and uppercase letters are used to denote column vectors and matrices, respectively. Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window denote transpose, complex conjugate, the ensemble average, diagonal matrix, Euclidean, the trace of the matrix Open image in new window **,** and maximum norm operations, respectively [19]. A complex Gaussian distribution with mean Open image in new window and variance Open image in new window is denoted by Open image in new window . Open image in new window and Open image in new window denote the Open image in new window th element of the vector Open image in new window and element in the Open image in new window th row and Open image in new window th column of Open image in new window , respectively. Finally, Open image in new window and Open image in new window denote all zero entry and the Open image in new window identity matrix if otherwise not defined.

### 3.1. Transmission Signal Representation

First Slot

The Open image in new window th user Open image in new window data-modulated symbol vector is represented by Open image in new window for Open image in new window . The Open image in new window th user Open image in new window symbol vector is fed to an Open image in new window -point inverse fast Fourier transform (IFFT) to generate the OFDM signal waveform Open image in new window . An Open image in new window -sample guard interval (GI) is inserted, and, then, the signals from the users are transmitted over a frequency-selective fading channel.

where Open image in new window , Open image in new window , Open image in new window , and Open image in new window denote the channel number of paths, the Open image in new window th path gain between the Open image in new window th user Open image in new window and the relay Open image in new window during the Open image in new window th slot, the Open image in new window th path time delay normalized by the sampling period of IFFT (i.e., Open image in new window ), and the delta function, respectively.

where Open image in new window (= Open image in new window ), Open image in new window = diag[ Open image in new window , Open image in new window , Open image in new window ] and Open image in new window = Open image in new window , respectively, denote the transmit signal power, the channel gain matrix between the Open image in new window user Open image in new window and the relay Open image in new window at the Open image in new window th slot with Open image in new window and the noise vector which elements are modeled as Open image in new window with Open image in new window . Open image in new window and Open image in new window denote the data-modulated symbol energy and the single-sided noise power spectral density. Note that the frequency domain signal representation at the relay is used for the sake of consistency with the following analysis.

Second Slot

where the bar over Open image in new window signifies the unitary complement operation (i.e., "NOT" operation) that performs logical negation of Open image in new window .

Estimates of the channel gains are required to perform self-information removal and equalization given by (4) and (5), respectively. The channel gain matrix Open image in new window is replaced in (4) and (5) by the channel gain estimate matrix Open image in new window for Open image in new window .

### 3.2. Channel Estimation [17]

This section is devoted, in part, to the problem statement of conventional CE for ANC scheme, and, then, the proposed pilot-assisted CE scheme and its MSE performance analysis are presented.

#### 3.2.1. Problem Statement

- (i)
In both conventional (without relaying) and cooperative (relay-assisted) networks, different users' pilot signals are separated by orthogonal frequencies or different time slots to avoid interference as shown in [20, 21]. However, in ANC, the users transmit simultaneously during the first time slot, and, as a result of this, the pilot signals interfere with each other. Consequently, the relay cannot estimate the CSIs of different users.

- (ii)
To estimate all CSIs in bidirectional ANC scheme, a conventional method is to allocate four time slots to separate different users' pilot signals. However, this significantly reduces the network throughput since additional two slots must be added for pilot-assisted CE.

- (iii)
The channels between the users and the relay are estimated at the relay during the first slot. These estimated CSIs have to be fed back to the user terminals, which additionally reduces the throughput. The problem of CSI feedback is not considered in this work, and it is left as an interesting future work. We note that we theoretically analyze the channel estimator's performance in terms of MSE and BER (see Sections 4.1 and 4), and, thus, the assumption of an ideal feedback channel simplifies the analysis (if a nonideal feedback channel is assumed, the theoretical analysis may become very difficult if not impossible).

To address the above-mentioned problems (1) and (2) the proposed CE is presented below.

#### 3.2.2. Two-Slot CE for Broadband ANC [17]

Unlike the conventional approach, where four time slots must be allocated to support bidirectional communication, we present a two-slot pilot-assisted CE scheme for bidirectional ANC with two users assisted by a relay.

MISO-CE

The users, Open image in new window and Open image in new window , transmit their pilots to the relay Open image in new window over a frequency-selective fading channel during the first slot of a pilot frame as shown in Figure 3.

where Open image in new window with Open image in new window . To avoid overlapping of the CSIs from different users during the first slot, the pilot signal Open image in new window of Open image in new window is cyclicly shifted by Open image in new window samples relative to the pilot signal Open image in new window of Open image in new window ; Open image in new window as used in MIMO-OFDM systems [18].

**,**and Open image in new window , respectively, denote all zero entry, the matrix represented by the rows from Open image in new window until Open image in new window of Open image in new window , the matrix represented by the last Open image in new window rows of Open image in new window , and the matrix represented by the first Open image in new window rows of Open image in new window . The Open image in new window th user Open image in new window filtered CSI vector estimate Open image in new window for Open image in new window is illustrated in Figures 4(b) and 4(c), respectively. After filtering, the estimated CSI vector Open image in new window is shifted by Open image in new window samples as shown in Figure 4(c).

Note that (13) holds as long as Open image in new window is chosen to be larger than the channel number of paths Open image in new window .

SISO-CE

The relay Open image in new window broadcasts its pilot sequence, Open image in new window , to both users, Open image in new window and Open image in new window , during the second slot of the pilot frame as shown in Figure 3. Without loss of generality, we focus on the processing of the Open image in new window th user Open image in new window for Open image in new window as presented below.

where Open image in new window and Open image in new window . Then, Open image in new window -point IFFT is applied to Open image in new window , to obtain the estimated CSI vector Open image in new window between the relay Open image in new window and the Open image in new window th user Open image in new window . The estimated CSI vector is filtered by Open image in new window for Open image in new window as Open image in new window . Finally, the filtered signal is fed to Open image in new window -point FFT to obtain the channel gain matrix estimate Open image in new window between the relay Open image in new window and the Open image in new window th user Open image in new window during the second slot.

The estimates of CSI during the MISO-CE stage are required at the users terminals. In this paper, we assume that the channel gains, Open image in new window for Open image in new window , are sent from the relay to the users by an ideal feedback channel. The channel gain matrix Open image in new window for Open image in new window is used by the users Open image in new window and Open image in new window to remove self-information and detect the signal from the partner as described in Section 3.

## 4. Performance Analysis

This section is devoted to theoretical analysis of broadband ANC with imperfect knowledge of CSI. We first derive the channel estimators MSE, and, then, the expressions for decision variables with the conditional BER expressions are presented. Note that the analysis is based on the assumption of perfect timing and frequency synchronization, and their impacts on the performance of bidirectional broadband ANC are left as an interesting future work.

### 4.1. Channel Estimator's MSE

where Open image in new window denotes the estimated channel matrix between the relay Open image in new window and the Open image in new window th user at the Open image in new window th slot.

MIMO-CE

where we assumed Open image in new window and Open image in new window . Although different CE schemes (i.e., MISO-CE and SISO-CE) are applied for different users, the MSE is not a function of the user parameter Open image in new window as shown by (17).

SISO-CE

This confirms that the MSE of the channel estimator for MISO-CE and SISO-CE are the same. Finally, the average MSE is given by Open image in new window .

### 4.2. Decision Variables

We begin by developing a general mathematical model for decision variables in three cases; (1) effect of imperfect knowledge of CSI, (2) effect of self-information removal due to imperfect knowledge of CSI, and (3) perfect knowledge of CSI and self-information removal. We note here that imperfect self-information removal may be caused by different factors such as imperfect synchronization, carrier frequency offset, imperfect knowledge of CSI, and so forth. In this paper, we only consider the impact of imperfect knowledge of CSI on the self-information removal.

where Open image in new window denote the the channel estimation error matrix of the Open image in new window th user Open image in new window . It is assumed that elements of Open image in new window are modeled as Open image in new window , where elements of Open image in new window and Open image in new window are statistically independent.

#### 4.2.1. Effect of Imperfect Knowledge of CSI

where Open image in new window denotes the channel estimation error matrix of Open image in new window .

where the first term denotes the desired signal; the second term denotes the effect of imperfect knowledge of CSI on the desired signal; the third term denotes the interference due to imperfect self-information removal; the last term denotes the noise. Next, we derive the signal and interference powers due to channel estimation errors.

Next, we present the decision variables in the case of imperfect self-information removal due to imperfect knowledge of CSI at the relay and destination terminals.

#### 4.2.2. Effect of Self-Information Removal due to Imperfect Knowledge of CSI

Note that in this case Open image in new window . Next, we present the decision variables in the case of perfect knowledge of CSI at the relay and destination terminals.

#### 4.2.3. Perfect Knowledge of CSI

where the first and second terms denote the desired signal and the noise, respectively. The desired signal power Open image in new window is given by (22) while the noise power is given by (29). Note that in this case Open image in new window .

### 4.3. BER

The evaluation of the theoretical average BER is done by Monte-Carlo numerical computation method based on the analysis presented in Section 4 as follows. A set of path gains Open image in new window is generated using (1) to obtain the channel gain matrix Open image in new window for Open image in new window , and, then, the equalization matrix Open image in new window is computed using (6) for each source terminal. The conditional BER as a function of the average signal energy per symbol-to-AWGN power spectrum density ratio Open image in new window is computed using (32) for the given set of path gains Open image in new window . This is repeated a sufficient number of times to obtain the theoretical average BER given by (33).

## 5. Numerical Results and Discussions

Simulation parameters.

Transmitter Open image in new window | Block size | |
---|---|---|

GI | ||

Data modulation | QPSK | |

Channel | Open image in new window -path block Rayleigh fading with Open image in new window | |

Relay | Protocol | Amplify-and-forward |

Feedback | Perfect | |

Receiver Open image in new window | FDE | ZF |

Channel estimation | Pilot-assisted |

### 5.1. Effect of Imperfect Knowledge of CSI and Self-Information Removal

First, we discuss the overall impact of channel estimation error Open image in new window on the achievable BER performance. Later we investigate the effect of self-information removal, when the effect of imperfect CSI on the desired signal is not taken into consideration.

^{−7}. On the other hand, to evaluate the impact of self-information removal due to imperfect knowledge of CSI, we plot the BER performance as a function of the channel estimation error variance Open image in new window is illustrated in Figure 7.

The figure illustrates the achievable BER performance as a function of the channel estimation error variance Open image in new window with Open image in new window as a parameter. The terms "Perfect CSI", "Effect of S-IR", and "Effect of CE" in Figure 7, respectively, denote the BER performance with perfect knowledge of CSI at all terminals given by (30), the effect of self-information removal due to imperfect knowledge of CSI given by (28) and the overall performance degradation due to imperfect knowledge of CSI given by (21). The figure shows that the BER performance with imperfect knowledge of CSI at the destination and relay terminals is sensitive on the channel estimation error variance Open image in new window depending on the level of Open image in new window ; for higher value of Open image in new window , the system becomes more sensitive to the channel estimation error. The figure confirms that the BER performance with imperfect self-information removal for the higher Open image in new window becomes more sensitive to the CE error. This is because the small value of Open image in new window during the self-information removal still keeps a large portion of the desired signal at a high Open image in new window level, which significantly affects the BER performance.

### 5.2. BER Performance

It can be seen from the figure that the two-slot pilot-assisted CE scheme achieves a satisfactory performance while allocating only two slots for the proposed CE, which maintains a higher transmission data-rate in comparison with four-slot pilot-assisted CE schemes. The BER performance of the proposed channel estimator for broadband ANC degrades in comparison with the perfect CSI case; for BER = 10^{−3} the Open image in new window degradation is about 5, 4, 4.5, and 7 dB when Open image in new window = 3, 15, 47, and 79, respectively. This performance degradation is due to three factors: (*i*) the CE errors, (*ii*) the tracking errors, and (*iii*) imperfect self-information removal due to CE errors. In the case of CE without noise (long-dotted lines in Figure 8), where only the propagation errors due to the channel time selectivity are considered, the Open image in new window degradation, for BER = 10^{−3}, is about 3.5, 3.7, 3.8, and 4 when Open image in new window = 3, 15, 47, and 79, respectively. The figure shows that the BER performance with proposed CE scheme, irrespective of Open image in new window , is the same for Open image in new window dB since the performance improvement is limited by the CE errors and the tracking errors.

### 5.3. Impact of Channel Time-Selectivity

It can be seen from the figure that for Open image in new window (which corresponds to about 2 km/h mobile terminal speed) almost the same BER performance is achieved irrespective of Open image in new window . In the case of Open image in new window (which corresponds to about 19 km/h mobile terminal speed), the BER performance slightly degrades irrespective of Open image in new window and Open image in new window . On the other hand, as Open image in new window increases to Open image in new window , which corresponds to about 190 km/h mobile terminal speed, the BER performance with Open image in new window severely degrades since the tracking ability of the channel estimator against the channel time selectivity is lost.

## 6. Conclusion

In this paper, we theoretically analyzed the performance of bidirectional broadband ANC communication based on OFDM radio access in terms of the channel estimator's MSE and BER. Assuming perfect timing and frequency synchronization, the achievable BER performance and the channel estimator's MSE for broadband ANC were evaluated by Monte-Carlo numerical and computer simulation. We discussed how much imperfect knowledge of CSI affects the BER performance of broadband ANC. Our results show that the BER performance of broadband ANC with practical CE gives a satisfactory performance for a low and moderate mobile terminal speed in a frequency-selective fading channel.

The feedback of the estimated CSIs at the relay terminal was not considered in this paper. Time and frequency synchronization problems are a major design challenge for relay-assisted networks due to simultaneous signal reception from different users. Moreover, to provide bidirectional communication for more than two users either time division multiple access (TDMA), frequency division multiple access (FDMA), or code division multiple access (CDMA) can be used. These are left as an interesting future work. The performance analysis and comparison among digital network coding (i.e., PNC) and ANC with pilot-assisted CE are also left as an interesting future work.

## Notes

### Acknowledgment

This paper was supported, in part, by 2010 KDDI Foundation Research Grant Program.

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