Throughout this paper, the following notations are used. Bold lowercase and uppercase letters are used to denote column vectors and matrices, respectively. , , , , , , and denote transpose, complex conjugate, the ensemble average, diagonal matrix, Euclidean, the trace of the matrix , and maximum norm operations, respectively [19]. A complex Gaussian distribution with mean and variance is denoted by . and denote the th element of the vector and element in the th row and th column of , respectively. Finally, and denote all zero entry and the identity matrix if otherwise not defined.
3.1. Transmission Signal Representation
We consider a twoway relay network with the users, and , and the relay as shown in Figure 1. The users and the relay communicate using time division duplex (TDD) in two slots; (1) and transmit their respective signals to the relay, and (2) the relay broadcasts the received signal to the users using an amplifyandforward protocol (AFP).
First Slot
The th user datamodulated symbol vector is represented by for . The th user symbol vector is fed to an point inverse fast Fourier transform (IFFT) to generate the OFDM signal waveform . An sample guard interval (GI) is inserted, and, then, the signals from the users are transmitted over a frequencyselective fading channel.
The propagation channel can be expressed by the discretetime channel impulse response given by
where , , , and denote the channel number of paths, the th path gain between the th user and the relay during the th slot, the th path time delay normalized by the sampling period of IFFT (i.e., ), and the delta function, respectively.
The signal received at the relay, = , can be expressed in the frequency domain as
where (= ), = diag[,,] and = , respectively, denote the transmit signal power, the channel gain matrix between the user and the relay at the th slot with and the noise vector which elements are modeled as with . and denote the datamodulated symbol energy and the singlesided 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
The relay amplifies the received signal by and broadcasts . After GI removal and point FFT, the signal received at the th user can be expressed as
The th user first removes its selfinformation as
where denotes the th user s datamodulated selfinformation vector. Then, onetap zero forcing frequency domain equalization (ZFFDE) is applied as
The th user equalization weight matrix is chosen to satisfy the condition and it is given by
where the bar over signifies the unitary complement operation (i.e., "NOT" operation) that performs logical negation of .
Estimates of the channel gains are required to perform selfinformation removal and equalization given by (4) and (5), respectively. The channel gain matrix is replaced in (4) and (5) by the channel gain estimate matrix for .
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 pilotassisted CE scheme and its MSE performance analysis are presented.
3.2.1. Problem Statement
If a classical CE approach is to be used with ANC scheme the following problems arise.

(i)
In both conventional (without relaying) and cooperative (relayassisted) 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 pilotassisted 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 abovementioned problems (1) and (2) the proposed CE is presented below.
3.2.2. TwoSlot CE for Broadband ANC [17]
Unlike the conventional approach, where four time slots must be allocated to support bidirectional communication, we present a twoslot pilotassisted CE scheme for bidirectional ANC with two users assisted by a relay.
The channel estimation scheme for broadband ANC network is illustrated in Figure 2. The figure shows that ANC scheme can be seen as a multipleinput singleoutput (MISO) system during the first slot and two independent singleinput singleoutput (SISO) systems during the second slot. Consequently, the first slot of the CE process, illustrated in Figure 2(a), is based on the MISOCE principle since the signals from two users' antennas are received by a single antenna at the relay. The second slot of the CE process, illustrated in Figure 2(b), is based on two independent SISOCE schemes. It is assumed that the users are out of each others coverage area, and, consequently, the signals received by each user's antenna are independent. Thus, we refer to the second CE stage as SISOCE rather than a singleinput multipleoutput (SIMO) CE. Note that in the first CE stage, both user signals arrive at the relay's antenna at the same time, and, thus, we refer to this stage as MISOCE.
Figure 3 illustrates the pilot and data transmission frame structure of the two users, and , and the relay . The pilot and data frames are divided in two time slots, where each slot has a length of samples, where and , respectively, denote the number of subcarriers and the GI length. The first slot of the pilot frame corresponds to the MISOCE, which is used to transmit the pilot signals, and , from and , respectively, as illustrated in Figure 2(a). The second slot is used by the relay during the SISOCE to broadcast its pilot signal to the users as illustrated in Figure 2(b). The pilot frame transmission is followed by data frames as shown in Figure 3.
MISOCE
The users, and , transmit their pilots to the relay over a frequencyselective fading channel during the first slot of a pilot frame as shown in Figure 3.
After GI removal and point FFT, the pilot signal received at the relay can be represented as
where with . To avoid overlapping of the CSIs from different users during the first slot, the pilot signal of is cyclicly shifted by samples relative to the pilot signal of ; as used in MIMOOFDM systems [18].
Using the timeshifting property of Fourier transform applied to [22] in (7) we obtain
where . To avoid overlapping of different users CSIs the time shift is chosen to be larger than the GI length since we assume that the channel number of paths does not exceeds the GI. The estimate of the channel gain is obtained by reverse modulation as
where . Then, point IFFT is applied to transform the estimated channel gain into the estimated CSI vector, , given by
where and , respectively, denote the FFT matrix [19] and the time domain shifted CSI vector of . The estimated CSI vector is illustrated in Figure 4(a). A filter is used to separate the th user's ('s) estimated CSI vector, , during the first slot represented as
with
where , , , and , respectively, denote all zero entry, the matrix represented by the rows from until of , the matrix represented by the last rows of , and the matrix represented by the first rows of . The th user filtered CSI vector estimate for is illustrated in Figures 4(b) and 4(c), respectively. After filtering, the estimated CSI vector is shifted by samples as shown in Figure 4(c).
Finally, an point FFT is applied to to obtain the estimate of the channel gain between the relay and the th user during the first slot given by
Note that (13) holds as long as is chosen to be larger than the channel number of paths .
SISOCE
The relay broadcasts its pilot sequence, , to both users, and , during the second slot of the pilot frame as shown in Figure 3. Without loss of generality, we focus on the processing of the th user for as presented below.
The pilot signal received at the th user can be expressed as
The estimate of the channel gain matrix is obtained as [20]
where and . Then, point IFFT is applied to , to obtain the estimated CSI vector between the relay and the th user . The estimated CSI vector is filtered by for as . Finally, the filtered signal is fed to point FFT to obtain the channel gain matrix estimate between the relay and the th user during the second slot.
The estimates of CSI during the MISOCE stage are required at the users terminals. In this paper, we assume that the channel gains, for , are sent from the relay to the users by an ideal feedback channel. The channel gain matrix for is used by the users and to remove selfinformation and detect the signal from the partner as described in Section 3.