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.
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.
, 
, 
, 
, 
, 
, and 
denote transpose, complex conjugate, the ensemble average, diagonal matrix, Euclidean, the trace of the matrix 
, and maximum norm operations, respectively [
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.
and 
, and the relay 
as shown in Figure 
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).
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.
, 
, 
, 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.
= 
, can be expressed in the frequency domain as
(= 
), 
= 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.
and broadcasts 
. After GI removal and 
-point FFT, the signal received at the 
th user 
can be expressed as
th user 
first removes its self-information as
denotes the 
th user 
s data-modulated self-information vector. Then, one-tap zero forcing frequency domain equalization (ZF-FDE) is applied as
th user 
equalization weight matrix 
is chosen to satisfy the condition 
and it is given by
signifies the unitary complement operation (i.e., "NOT" operation) that performs logical negation of 
.
is replaced in (4) and (5) by the channel gain estimate matrix 
for 
.
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 
to the users as illustrated in Figure 
data frames as shown in Figure 
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 
-point FFT, the pilot signal received at the relay 
can be represented as
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 [
[
. 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
. Then, 
-point IFFT is applied to transform the estimated channel gain into the estimated CSI vector, 
, given by
and 
, respectively, denote the FFT matrix [
. The estimated CSI vector 
is illustrated in Figure 
is used to separate the 
th user's (
's) estimated CSI vector, 
, during the first slot represented as
, 
, 
, 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 
is shifted by 
samples as shown in Figure 
and 
between the relay and the first and second user during the first transmission stage). Estimated channel impulse response 
after filtering by (13)
after filtering by (13)
-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
is chosen to be larger than the channel number of paths 
.
broadcasts its pilot sequence, 
, to both users, 
and 
, during the second slot of the pilot frame as shown in Figure 
th user 
for 
as presented below.
th user 
can be expressed as
is obtained as [
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.
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.
have the equal number of paths 
. The MSE of the estimated channel gain 
during the 
th slot is defined by
denotes the estimated channel matrix between the relay 
and the 
th user at the 
th slot.
is an unbiased estimation of 
since 
. Using (13) and (16), the MSE of the estimated channel gain 
for 
during the first slot is given by
and 
. 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 
as shown by (17).
is an unbiased estimation of 
since 
. Using (15) and (16), the MSE of the estimated channel gain 
for 
during the second slot is given by
.
.
th user 
and define diagonal matrix 
. We assume that channel estimation error of the 
th user 
can be modeled as
denote the the channel estimation error matrix of the 
th user 
. It is assumed that elements of 
are modeled as 
, where elements of 
and 
are statistically independent.
denotes the channel estimation error matrix of 
.
. The interference power due to imperfect self-information removal is given by (Appendix B)
. Next, we present the decision variables in the case of perfect knowledge of CSI at the relay and destination terminals.
and (21) can be rewritten as
is given by (22) while the noise power is given by (29). Note that in this case 
.
, where 
denotes the complex operator). The 
th user 
instantaneous signal-to-interference plus noise ratio (SINR) 
of the 
th symbol in the transmitted data vector 
is given as
th data symbol for the given 
is given by
is the complementary error function [
th user 
transmitted data vector 
is numerically evaluated by averaging (32) over all possible realizations of 
for all 
as
is generated using (1) to obtain the channel gain matrix 
for 
, and, then, the equalization matrix 
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 
is computed using (32) for the given set of path gains 
. This is repeated a sufficient number of times to obtain the theoretical average BER given by (33).
-subcarriers, 
and ideal coherent QPSK modulation and demodulation with 
. The propagation channel is 
-path block Rayleigh fading channel, where 
are zero-mean independent complex variables with 
. We assume 
and that the 
th path time delay is 
, where Δ(≥1) is the time delay separation between the previous and following path. The maximum channel time delay is less than the GI length, and all channel paths are independent of each other. 
represents the normalized Doppler frequency, where 
is the transmission symbol rate (
corresponds to a mobile terminal speed of approximately 19 km/h for a transmission data rate of 100 Msymbols/s and a carrier frequency of 2 GHz). We assume error-free feedback channel from the relay 
to the users and no shadowing nor path loss. As a pilot, we use a Chu-sequence given by 
[
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.
with the channel estimation error variance 
as a parameter. The figure shows that the BER performance of broadband ANC scheme degrades as the channel estimation error variance 
increases from 10−7. 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 
is illustrated in Figure 
on average BER.
.
with 
as a parameter. The terms "Perfect CSI", "Effect of S-IR", and "Effect of CE" in Figure 
depending on the level of 
; for higher value of 
, 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 
becomes more sensitive to the CE error. This is because the small value of 
during the self-information removal still keeps a large portion of the desired signal at a high 
level, which significantly affects the BER performance.
with the number of data frames 
as a parameter. The power loss due to GI and pilot insertion is taken into consideration. The term "Pilot-assisted CE (w/o noise)" in Figure 
degradation is about 5, 4, 4.5, and 7 dB when 
= 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 
degradation, for BER = 10−3, is about 3.5, 3.7, 3.8, and 4 when 
= 3, 15, 47, and 79, respectively. The figure shows that the BER performance with proposed CE scheme, irrespective of 
, is the same for 
dB since the performance improvement is limited by the CE errors and the tracking errors.
on the BER performance with proposed CE scheme is discussed below. As 
increases, the tracking ability against the channel fading variations tends to be lost. Figure 
for 
= 15 dB, 30 dB, and 40 dB with 
as a parameter.
.
(which corresponds to about 2 km/h mobile terminal speed) almost the same BER performance is achieved irrespective of 
. In the case of 
(which corresponds to about 19 km/h mobile terminal speed), the BER performance slightly degrades irrespective of 
and 
. On the other hand, as 
increases to 
, which corresponds to about 190 km/h mobile terminal speed, the BER performance with 
severely degrades since the tracking ability of the channel estimator against the channel time selectivity is lost.
, we can expand 
into a polynomial sum (i.e., 
) given by
is a zero-mean Gaussian random variable, and, hence, it can be easily shown that the odd moments of (A.3) reduce to zero. Thus, (A.3) simplifies as
is given by
into a polynomial sum and consider only the first two terms. Thus, 
is given by
into polynomial sum and taking the expectation of its individual terms, we obtain (26).