Blind Channel Estimation in Full Duplex Systems: Identifiability Analysis, Bounds, and Estimators Abbas Koohian, Student Member, IEEE, Hani Mehrpouyan, Member, IEEE, Ali A. Nasir, Member, IEEE, Salman Durrani, Senior Member, IEEE, Mohammad Azarbad, Steven D. Blostein, Senior Member, IEEE.
arXiv:1511.04794v1 [cs.IT] 16 Nov 2015
Abstract We consider blind channel estimation in a single-input single-output full-duplex communication system, where both the self-interference and the communication channels need to be accurately estimated. In this context, blind estimators are attractive as they improve bandwidth efficiency but they suffer from the phase ambiguity problem. In this paper, we first formally define and analyse this ambiguity and then develop a general framework for testing and designing modulation sets for blind estimation of channel parameters. We mathematically show that simply shifting the mean of the M -PSK modulation resolves the ambiguity problem. We also show how this can be extended to more general modulation sets. Finally, we propose an expectation maximization (EM) iterative estimator and a closed form minimum mean square error (MMSE) estimator for use with the shifted modulation set. Since the non-data aided Cram´er-Rao lower bound (CRLB) or the Bayesian CRLB (BCRLB) are intractable, we derive the data-aided CRLB and the data-aided BCRLB to assess the performance of these estimators. Simulations show that both estimators reach the performance of their corresponding bounds. The EM estimator has considerably lower computational complexity compared to the MMSE estimator for a large number of observations. The MMSE estimator performs well for number of observations as low as N = 6, which is desirable for delay constrained systems. The simulation results also show the robustness of proposed estimators to increasing power of self-interference signal. Index Terms Blind channel estimation, identifiability analysis, full duplex communication, expectation maximization, minimum mean square error. A preliminary version of this work was presented at 2015 IEEE ICC in London, UK [1]. Abbas Koohian and Salman Durrani are with Research School of Engineering, Australian National University, Canberra, Australia (email: {abbas.koohian, salman.durrani}@anu.edu.au). Hani Mehrpouyan is with the Department of Electrical and Computer Engineering, Boise State University, Idaho, USA (email:
[email protected]). Ali A. Nasir is with School of Electrical Engineering and Computer Science, National University of Science and Technology, Islamabad, Pakistan (email:
[email protected]). Mohammad Azarbad is with the Department of Mathematics, Statistics, and Computer Sciences, Tehran University, Tehran, Iran (email:
[email protected]). Steven D. Blostein is with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, Ontario, Canada (email:
[email protected]).
1
I. I NTRODUCTION Background: Full-duplex (FD) communication, allowing devices to transmit and receive in the same frequency band at the same time, is a promising technology to double the spectral efficiency of future wireless communication systems [2]–[4]. Recent studies have shown the performance improvement achievable with FD communication both theoretically [5]–[8] and experimentally [9]–[12]. The key challenge in realizing the benefits of FD communication is the cancellation of the self-interference signal, which significantly corrupts the desired signal at the receiver [2]. There are two main approaches used in the literature to deal with this problem: (i) passive self-interference cancellation techniques, which attempt to isolate the transmit and receive antennas [13] and (ii) active self-interference cancellation techniques which use the knowledge of the self-interference signal to cancel the interference in the digital or the analog domain [9]. Although the power of self-interfering signal is reduced by passive techniques, the residual power can still distort the desired signal significantly. Hence, we consider active self-interference cancellation in this work. Motivation: Active cancelation requires more processing of the received signal compared to passive cancelation. For this reason and to have an effective active cancelation, accurate knowledge of the desired and the self-interference channels should be obtained [3], [9], [14]. The approach commonly used in the literature is to use pilots and to silence the transmitting node while estimating the selfinterference channel. Once the self-interference channel is estimated, pilots are transmitted for the estimation of the communication channel [9]–[11]. A semi-blind approach for channel estimation in full duplex communication systems is proposed in [15]. However, the approach in [15] requires few pilots to be transmitted and the unknown data symbols are assumed Gaussian. Note that this assumption leads to significant performance degradation as in reality transmitted symbols follow a discrete distribution [16]. Since FD communication requires two channels to be estimated, blind channel estimators are more attractive as they can significantly improve bandwidth efficiency. Blind channel estimation for conventional half-duplex (HD) communication systems has been thoroughly investigated in the literature. It is well-known that blind estimators can only estimate the channel up to a scaling factor and cannot recover the channel phases. The degree of this ambiguity can be determined using identifiability analysis and depends directly on the prior information available about the transmitted symbols [16]–[18]. In this regard, a parameter is said to be identifiable if it can be estimated without any ambiguity [19]. Recently, blind estimators have been proposed for new emerging systems like amplify and forward two way relay networks (AF-TWRN) [20], [21] and interference limited networks [22], [23]. A shortcoming of recent research into blind estimators is a lack of parameter
2
identifiability analysis. This has prevented achieving ambiguity-free blind channel estimation. This motivates a more thorough analysis of identifiability problem in emerging communication systems. Blind channel estimation in FD communication also requires thorough analysis including appropriate performance bounds and estimators, on top of identifiability analysis. Paper contributions: In this paper, we investigate the blind channel estimation problem in a singleinput single-output (SISO) FD communication system. Unlike previous works in the literature, we analyze the identifiability problem for blind channel estimation in FD communication system. The novel contributions of this work are as follows: •
Identifiability analysis: We formulate and derive the mathematical condition for identifiability of the channel parameters in FD communication systems (c.f. Theorem 1). Theorem 1 can be applied as a general framework for testing and designing modulation sets for communication systems where two channels are to be blindly estimated with no piloting (e.g., FD communication, TWRN, and interference limited systems). If the condition of the theorem is met by a given modulation set then ambiguity-free blind channel estimation becomes possible and channels can be estimated without any piloting. Considering M -PSK modulation and using Theorem 1, we (i) mathematically prove that the zero-mean M -PSK modulation suffers from an ambiguity problem, (ii) show that one solution would be to use a shifted non-zero-mean M -PSK modulation set, and (iii) how these ideas can be extended to more general modulation sets.
•
Proposed estimators: We present two estimators for simultaneous estimation of both interfering and communication channels using shifted modulation set: (i) a computationally efficient expectation maximization (EM) estimator to numerically obtain the maximum likelihood (ML) solution of the channel parameters and (ii) a closed form minimum mean square error (MMSE) estimator to take advantage of channel statistics. The choice of MMSE estimator is motivated as follows: (i) prior information about the channels haa and hba may be available [3], (ii) Bayesian estimators are known to minimize the Bayes risk, i.e., average MSE, which is an important performance metric for assessing the performance of estimators [24], and (iii) the MMSE estimator is closed form, which in turn means it does not suffer from convergence issues and does not require initialization as opposed to the EM estimator. Since the non-data aided Cram´er-Rao lower bound (CRLB) or the Bayesian CRLB (BCRLB) are intractable, we derive the data-aided CRLB and the data-aided BCRLB to assess the performance of these estimators.
•
Results: Simulations show that both the EM and the MMSE blind estimators reach the performance of corresponding data-aided bounds. This performance is achieved at the cost of higher energy
3
which is needed for shifting the modulation set mean. The EM estimator has considerably lower computational complexity compared to the MMSE estimator for a large number of observations. On the other hand, the MMSE estimator performs well for number of observations as low as N = 6. This property of the MMSE estimator is very desirable for delay constrained systems. Simulation results also show the robustness of proposed estimators to increasing self-interference power. Notations: The following notation is used: bold face lower case letters, e.g., x, are used for vectors. Bold face upper case capital letters, e.g., X, are used for matrices. IN represents the N × N identity √ matrix. j , −1, and the real and imaginary parts of a complex quantity are represented by
