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 two-way 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 amplify-and-forward protocol (AF-P).
First Slot
The
th user
data-modulated 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 frequency-selective fading channel.
The propagation channel can be expressed by the discrete-time 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 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
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 self-information as
where
denotes the
th user
s data-modulated self-information vector. Then, one-tap zero forcing frequency domain equalization (ZF-FDE) 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 self-information 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 pilot-assisted 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 (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.
The channel estimation scheme for broadband ANC network is illustrated in Figure 2. The figure shows that ANC scheme can be seen as a multiple-input single-output (MISO) system during the first slot and two independent single-input single-output (SISO) systems during the second slot. Consequently, the first slot of the CE process, illustrated in Figure 2(a), is based on the MISO-CE 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 SISO-CE 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 SISO-CE rather than a single-input multiple-output- (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 MISO-CE.
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 MISO-CE, 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 SISO-CE 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.
MISO-CE
The users,
and
, transmit their pilots to the relay
over a frequency-selective 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 MIMO-OFDM systems [18].
Using the time-shifting 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
.
SISO-CE
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 MISO-CE 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 self-information and detect the signal from the partner as described in Section 3.