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A Maximum-Likelihood Channel Estimator for Self-Interference Cancelation in Full-Duplex Systems Ahmed Masmoudi and Tho Le-Ngoc, Fellow, IEEE
Abstract—Operation of full-duplex systems requires efficient mitigation of the self-interference signal caused by the simultaneous transmission/reception. In this paper, we propose a maximumlikelihood (ML) approach to jointly estimate the self-interference and intended channels by exploiting its own known transmitted symbols and both the known pilot and unknown data symbols from the other intended transceiver. The ML solution is obtained by maximizing the ML function under the assumption of Gaussian received symbols. A closed-form solution is first derived, and subsequently, an iterative procedure is developed to further improve the estimation performance at moderate-to-high signal-to-noise ratios (SNRs). We establish the initial condition to guarantee the convergence of the iterative algorithm to the ML solution. In the presence of considerable phase noise from the oscillators, a phase noise estimation method is proposed and combined with the ML channel estimator to mitigate the effects of the phase noise. Illustrative results show that the proposed methods offer good cancelation performance close to the Cramer–Rao bound (CRB). Index Terms—Channel estimation, full-duplex communication, iterative method, maximum likelihood, multiple-input multipleoutput (MIMO), self-interference suppression.
I. I NTRODUCTION
F
ULL-DUPLEX transmission, by allowing simultaneous transmission/reception over the same channel, is emerging as an alternative to half-duplex communication to increase the transmission rate. The simultaneous transmission and reception creates large self-interference that needs to be properly canceled. Recent studies showed that, by using multiple cancelation stages, the self-interference can be sufficiently attenuated to detect the intended signal [1]–[3]. Self-interference cancelation is performed before low-noise amplifier (LNA) and analogto-digital conversion (ADC) to avoid overloading/saturation, and further self-interference suppression can be done after ADC at the baseband. The self-interference replica is created from the known transmitted signal and an estimate of the selfinterference channel, then subtracted from the received signal. In practice, it is not possible to completely cancel the selfinterference due to channel estimation error [4]. Therefore, Manuscript received April 21, 2014; revised September 9, 2014, January 11, 2015, April 5, 2015, and June 24, 2015; accepted July 18, 2015. Date of publication July 27, 2015; date of current version July 14, 2016. This work was supported in part by an R&D Contract from Huawei Technologies Canada and in part by a Grant from the Natural Sciences and Engineering Research Council of Canada. The review of this paper was coordinated by Dr. T. Jiang. The authors are with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 0E9, Canada (e-mail: ahmed.
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2015.2461006
channel estimation appears to be a critical issue in full-duplex systems. In [5], a least squares (LS)-based estimator was presented by dividing the received signal by the known transmitted signal in the frequency domain. Another approach to implement the LS criteria was proposed in [6] by iterating between channel estimation and intended signal detection. However, these two approaches treat the intended signal from the other transmitter as additive noise, which reduces the estimation performance. The LS estimate is also considered in [7] for full-duplex relay. Spatial domain cancelation could also reduce the selfinterference by precoding at the transmit chain and decoding at the receive chain [8]. In this paper, we jointly estimate the self-interference channel and the intended channel between the two transceivers using the maximum-likelihood (ML) criteria. We consider multipath multiple-input–multiple-output (MIMO) channels with large coherence time compared with the symbol period. While the self-interference channel can be estimated from the known transmitted symbols by the same transmitter, some pilot symbols are needed to estimate the intended channel. Since the received signal contains a mix of known and unknown data, the estimation process exploits these known data, and the second-order statistics of the unknown data from the intended transceiver toward the identification of the channels. The full use of the received signal reduces the number of pilot symbols needed, compared with training-based techniques. The unknown signal is approximated by a Gaussian process to formulate the likelihood function. Using some approximations, we give a closed-form solution to maximize the likelihood function. To further improve the estimation performance, we iteratively estimate the second-order statistics of the unknown signal and the channel coefficients. On the other hand, system nonlinearities are limited factors for full-duplex systems [9]. A detailed study of the effect of phase noise from the oscillators on cancelation is presented in [10] when independent oscillators are used in upconversion and downconversion. A shared oscillator can reduce the phase noise effects and improve the cancelation performance by 25 dB [11]. In this case, the difference between the phase noise at the transmit and receive chains depends on the delay that the self-interference signal experiences from the transmit to receive chain. A frequencydomain method to compensate such phase noise is proposed in [12], and a time-domain phase noise estimation technique is developed in [13]. The limiting factor of these methods is that they consider the intended signal as additive noise, which reduces the estimation accuracy. In the following, once an initial estimate of the channel coefficients is obtained, we propose an ML estimate of the phase noise affecting both the self-interference
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We suppose that each node is equipped with Nt transmitting antennas and Nr receiving antennas. For an orthogonal frequency-division multiplexing (OFDM) transmitted signal with N subcarriers, the tth received block, after removing the cyclic prefix, at antenna r is yr,t (n) =
Nt L
(s) h(i) r,q (l)zq,t (n−l)+hr,q (l)sq,t (n−l)+wr,t (n)
q=1 l=0
(1)
Fig. 1. Simplified block diagram of the full-duplex transceiver with RF and baseband self-interference cancelation stages.
and intended signals, which avoids the drawback of considering the intended signal as additive noise. The remainder of this paper is organized as follows. In Section II, we present the full-duplex communication system model under consideration. Analysis and development of the proposed ML channel estimation algorithm are presented in Section III, and the procedure to estimate the phase noise process is detailed in Section IV. Section V provides illustrative simulation results, and Section VI presents the conclusion. II. F ULL -D UPLEX T RANSCEIVER AND C ANCELATION S CHEME Consider a MIMO point-to-point transceivers operating in a full-duplex fashion by simultaneously transmitting and receiving in the same frequency band. In addition to the intended signal, each transceiver receives its own self-interference that needs to be canceled before demodulation. Fig. 1 shows a simplified block diagram of the different cancelation stages. The radio-frequency (RF) cancelation stage is done prior to the LNA/ADC to avoid saturation/overlapping. The baseband cancelation stage is performed after the ADC to reduce the residual self-interference. In the following, we assume that a first estimate of the self-interference channel is available to create the canceling signal in the RF cancelation stage1 and we take the received signal at the output of the RF cancelation. 1 The first estimate of the self-interference channel is obtained during a short initial half-duplex period. In the absence of the intended signal, the selfinterference channel can be obtained using any linear estimator, such as the classical LS estimator.
for n = 0, . . . , N −1, where zq,t (n) and sq,t (n), n = −Ncp, . . . , N − 1 are the self-interference (from its own transceiver) and intended (from the other intended transceiver) OFDM signals, (i) respectively; and hr,q (l), l = 0, . . . , L, is the L + 1-tap impulse response of the residual self-interference channel from antenna q to antenna r of the same transceiver after the RF cancelation stage, i.e., the difference between the actual self-interference channel and its first estimate by the RF cancelation stage. (s) hr,q (l), l = 0, . . . , L is the impulse response of the intended channel from antenna q to antenna r of the two different transceivers. N is the number of subcarriers, and Ncp is the length of the cyclic prefix. Note that L ≤ Ncp to avoid intersymbol interference, and the channels are zero-padded for channel order lower than L. In the following, we suppose that P subcarriers are dedicated to transmit pilot symbols. Let P = {p1 , . . . , pP } be the index set of the subcarrier reserved for pilots. The transmit signal sq,t (n) can be represented as the sum of the following two signals: spq,t (n) = sdq,t (n) =
P i=1
Sq,t (pi )ej2πpi n/N Sq,t (k)ej2πkn/N
(2)
k∈P
for n = 0, . . . , N − 1 where the first sequence spq,t (n) contains the pilot symbols Sq,t (pi ), pi ∈ P, and the second sequence sdq,t (n) contains the unknown transmit data symbols Sq,t (k), k ∈ P, during the tth OFDM block. Using (2), the received signal in (1) becomes yr,t (n) =
Nt L
p (s) h(i) r,q (l)zq,t (n − l) + hr,q (l)sq,t (n − l)
q=1 l=0 d + h(s) r,q (l)sq,t (n − l) + wr,t (n). (3)
For a more compact representation of (3), we define the set of N × (L + 1) circulant matrices Z q,cir,t , for q = 1, . . . , Nt , in which the first row is [zq,t (0), zq,t (N − 1), zq,t (N − 2), . . . , zq,t (N − L)] and the first column is [zq,t (0), zq,t (1), . . . , zq,t (N − 1)], and the N × Nt (L + 1) matrix Z t = [Z 1,cir,t , Z 2,cir,t , . . . , Z Nt ,cir,t ]. Matrix S pt is defined in the same way as Z t but using the sequence {spq,t (n)} instead of {zq,t (n)}. We also gather the channel coefficients from all the transmit antennas to the rth receive antenna as T (i) (i) (i) (i) h(i) r = hr,1 (0), . . . , hr,1 (L), . . . , hr,Nt (0), . . . , hr,Nt (L) T (s) (s) (s) (s) h(s) r = hr,1 (0), . . . , hr,1 (L), . . . , hr,Nt (0), . . . , hr,Nt (L) (s) (s) (s) (4) H (s) r = H r,1 , H r,2 , . . . , H r,Nt
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where the N × N ⎛ (s) hr,q (0) ⎜ (s) ⎜h (1) ⎜ r,q ⎜ . ⎜ . (s) ⎜ H r,q =⎜ . ⎜h(s) (L) ⎜ r,q ⎜ .. ⎝ .
circulant matrix H (s) r,q is defined as 0 .. .
··· 0
··· .. .
(s) hr,q (0)
.. (s)
0
(s)
hr,q (L)
hr,q (L)
. ···
··· .. .
(s)
⎞
hr,q (1) .. ⎟ . ⎟ ⎟ ⎟ (s) hr,q (L) ⎟ ⎟. ⎟ 0 ⎟ ⎟ .. ⎟ . ⎠ (s)
hr,q (0)
Using the previous notations, the received signal at antenna r can be reformulated in a vector form as p (s) (s) d y r,t = Z t h(i) r + S t hr + H r st + w r,t
(5)
where y r,t = [yr,t (0), . . . , yr,t (N − 1)]T is the received N × 1 vector after removing the cyclic prefix, and sdt = [sdq,t (0), . . . , sdq,t (N − 1)]T . By collecting the received vectors from the Nr receiving antennas, we can express (5) as y t = (I Nr ⊗Z t ) h(i) +(I Nr ⊗ S pt ) h(s) +H (s) sdt +wt (6) where ⊗ refers to the Kronecker product between two matrices, I Nr is the Nr × Nr identity matrix, and T (i) T (i) T (i) T h(i) = h1 , h2 , . . . , hNr T (s) T (s) T (s) T h(s) = h1 , h2 , . . . , hNr T (s) T (s) T (s) T (s) H = H 1 , H 2 , . . . , H Nr . (7) In the following, we assume that the noise and transmitted signals are independent, and the signal and noise variances are α2 and σ 2 , respectively. III. M AXIMUM L IKELIHOOD E STIMATOR To reduce the self-interference in (6), we need to estimate the residual self-interference channel h(i) from y t . Since the self-interference signal Z t is known, the straightforward way to estimate the corresponding channel is to resort to a linear estimator using the matrix Z t . However, this strategy gives poor performance since the intended signal is treated as noise. As an alternative, we propose a joint estimation of the selfinterference and intended channels, exploiting both the known pilot symbols and the statistic of the unknown part of the received signal. The use of the known and unknown transmit data in the estimation process is commonly referred as semiblind channel estimation [14], [15]. To that end, we introduce (i) T
(s) T
T
,h ] as the vector to be estimated and D t = h = [h [I Nr ⊗ Z t , I Nr ⊗ S pt ] as the matrix gathering the symbols sent by the same transceiver and the known pilot symbols sent by the other intended transceiver. For a Gaussian received data,2 y t is a Gaussian random vector with mean Dt h and covariance matrix H R = α2 H (s) H (s) + σ 2 I N Nr . A total of T OFDM symbols 2 The
Gaussian assumption is well justified for OFDM transmit signal [16].
are used in the estimation process. Following the Gaussian model, the log-likelihood function is given by L(h(i) , h(s)) = −T log |R|−
T (y t −Dt h)H R−1 (y t −Dt h) t=1
(8) where | · | returns the determinant of a matrix. The ML estimates of h(i) and h(s) are obtained by maximizing the loglikelihood function L(., .). As the covariance matrix R depends on the unknown vector h(s) , maximizing the cost function with respect to h(s) appears to be computationally intractable since it involves a Nt Nr (L + 1)-dimensional grid search. To overcome this complexity, we ignore the relation between R and h(s) and maximize the log-likelihood function with respect to h = T T [h(i) , h(s) ]T and R. This separability is exploited to solve the problem in a low-complexity manner. In the following, we propose a closed-form solution and an iterative method to estimate the channels. A. Closed-Form Solution By considering separable variables h and R, the conditional approach to maximize the log-likelihood function can be used. In the conditional approach, the covariance matrix is modeled as deterministic and unknown. Therefore, the matrix R is substituted by the solution RML (h) that maximizes (8) for a fixed h. Hence, maximizing (8) with respect to R leads to [17] RML (h) =
T 1 (y − Dt h)(y t − D t h)H . T t=1 t
(9)
By substituting R by RML (h) in (8), we get the so-called compressed likelihood function [18], [19], i.e.,
T
1
H (y t − Dt h)(y t − D t h) (10) Lc (h) = −T log
T t=1 where the constant terms irrelevant to the maximization have been discarded. It follows that the ML channel estimate is given by hML = arg max Lc (h). h
(11)
To find a closed-form solution of (11), we first compute the LS estimate of the channel [20] T −1 T H hLS = D t Dt DH (12) t yt . t=1
t=1
= 1/T T dt dH . Then, we define dt = y t − D t hLS and R t t=1 Following these notations, the compressed likelihood function in (10) can be rewritten as
T
1
Lc (h) = −T log R Dt (h−hLS )(h−hLS )H DH + t
T t=1
H H H − Dt (h−hLS )dt −dt (h−hLS ) Dt . (13)
MASMOUDI AND LE-NGOC: ML CHANNEL ESTIMATOR FOR SELF-INTERFERENCE CANCELATION
Define ξ = h − hLS . As the block number T increases, the LS estimate hLS approaches the ML estimate hML . Therefore, the difference ξML = hML − hLS between the two estimates becomes small. Using the fact that for any matrix M satisfying3 M 1, we have |I + M | ≈ 1 + trace(M ), the loglikelihood function in (13) is rearranged to obtain the following: T 1 −1 log 1+ trace R Dt ξξH D H Lc (h) = −T log |R|−T t T t=1 H H − Dt ξdH t − dt ξ D t
(14)
where we substitute h − hLS by ξ. Since the log function is an increasing function, the maximization of Lc (h) is equivalent to ξML = arg max ξ
T
−1
ξH D H t R Dt ξ
t=1
−1
−1
H H − dH t R D t ξ − ξ D t R dt . (15)
By setting the first derivative with respect to ξ to zero, the solution to (15) is given by T −1 T H −1 −1 Dt R D t DH (16) ξ ML = t R dt . t=1
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The proposed approach iterates between (18) and (19). At the ith iteration, the estimate Ri−1 obtained at iteration i − 1 is used to find h as hi = hML (Ri−1 ). Then, the estimate of R is updated at iteration i as Ri = RML (hi ). The algorithm is stopped when there is no significant difference between two consecutive estimates. Like most of the iterative algorithms, initialization is a critical issue for convergence. In our case, setting R0 = I appears to be a reasonable starting point. At the first iteration, we obtain the LS estimate given in (12). As we iterate, the matrix Ri acts as a weighting matrix to improve the estimated channel. The proof of convergence to the global maximum of the log-likelihood function may not be straightforward because the function at hand is not verified to be convex. However, using the closed-form expression obtained earlier, it is possible to simply prove the convergence to the ML solution. In fact, when initializing the algorithm with R0 = I, the iterative algorithm returns, in the second iteration, the same channel estimate given in the closed-form solution in (17). That is, after two iterations, the algorithm operates close to the ML solution. Following the arguments in [21], we have L(hi , Ri ) = max L(hi , R) R
≥ L(hi , Ri−1 )
t=1
Using dt = y t − D t hLS and hML = hLS + ξML , the ML channel estimate is given by T −1 T H −1 −1 Dt R Dt DH (17) hML = t R yt . t=1
t=1
The ML estimate is different from the LS estimate because −1 . The ML and LS estimates are of the weighting matrix R equivalent in the presence of white Gaussian noise. In our case, the effective noise is composed of the thermal noise and unknown transmit signal, which is not a white noise. B. Iterative ML Estimator which is an The closed-form solution in (17) depends on R, estimate of the covariance matrix R. Therefore, a better estimate of R results in a better estimate of the channel vector h. On the other hand, matrix R depends on the unknown intended channel coefficients h(s) that we want to estimate. Exploiting again the separability of the log-likelihood function in h and R, a common approach in this situation is to resort to an iterative procedure. If the channel vector is given, the covariance matrix R that maximizes the log-likelihood function given h is RML (h) =
T 1 (y − Dt h)(y t − D t h)H . T t=1 t
(18)
Conversely, if R is available, the solution to the problem arg maxh L(h, R) can be computed as T −1 T H −1 −1 hML (R) = D t R Dt DH (19) t R yt . t=1
= max L(h, Ri−1 ) h
≥ L(hi−1 , Ri−1 ).
Therefore, the log-likelihood function is increased after each iteration, and for a good initialization, the convergence to the global maximum is rapid. Simulation results presented in Section V confirm that, when initializing the algorithm with R0 = I, the iterative algorithm converges to the ML solution after a reasonable number of iterations. The main complexity of the LS estimator comes from the computation of the inverse of ( Tt=1 DH t D t ), whereas the proposed ML algorithm involves at an additional matrix inversion of the N Nr × N Nr matrix R each iteration compared with the LS estimator. IV. P HASE N OISE S UPPRESSION Although the knowledge of the self-interference channel is essential to reconstruct the self-interference signal, some other RF components can affect the self-interference, and it is desired to consider their effects in self-interference cancelation. Phase noise, which is introduced by both transmitter and receiver oscillators, has been considered one of the main limiting factors in self-interference cancelation [10], [11], [13]. Considering phase noise in the received signal, (1) can be rewritten as N L t jφin−l,q h(i) yr,t (n) = r,q (l)zq,t (n − l)e q=1 l=0
t=1
+ 3 M
denotes the Frobenius norm of the matrix M .
(20)
h(s) r,q (l)sq,t (n
− l)e
jφsn−l,q
e−jφn,r + wr,t (n) (21)
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where φin,q is the phase noise of its own qth oscillator of the transmitter side affecting the nth self-interference sample, φsn,q is the phase noise of the qth intended transmitter oscillator affecting the nth intended received sample, and φn,r is the oscillator phase noise at the rth receive antenna affecting both the self-interference and intended signals. For generality, we use different notations for the phase noise process from different antennas. This later allows us to apply the proposed method for independent oscillators at the different antennas or a common shared oscillator. The phase noise processes in (21) change from one OFDM symbol to another and should be indexed by time t, but this notation is ignored for clarity while we keep in mind that the phase noise changes from one OFDM symbol to another. Since the transmitted symbols multiplied by different phase noise realizations are further convoluted by the multipath channel impulse response, the received sample n is affected by L + 1 different realizations of phase noise. However, phase noise due to oscillator imperfection is typically a very narrowband process and, hence, changes slowly over time. As a result, the difference in phases during these L + 1 consecutive symbols can be assumed negligible to simplify the development of the algorithm, i.e., φin−l,q = φin,q and φsn−l,q = φsn,q for l = 0, . . . , L. Therefore, the received signal in (21) becomes
yr,t (n) =
Nt L
i
jφn,r,q h(i) + h(s) r,q (l)zq,t (n − l)e r,q (l)
q=1 l=0
×
spq,t (n
− l) +
sdq,t (n
s − l) ejφn,r,q + wr,t (n) (22)
is an underdetermined problem and may have many different solutions. Thus, exploiting the fact that the phase noise is slowly varying over time, we consider that it remains constant over lp consecutive samples, which divides the number of unknown by i s lp . Let Φr,q and Φr,q denote the reduced version of Φir,q and i
Φsr,q whose elements are defined as Φr,q (n) = Φir,q (nlp ) and s Φr,q (n) = Φsr,q (nlp ), respectively, and let ⎛ ⎞ zr,q,t (0) 0 ··· 0 .. .. ⎜ ⎟ ⎜ ⎟ . . ⎜ ⎟ ⎜zr,q,t (lp −1) ⎟ 0 ⎜ ⎟ ⎜ ⎟ .. ⎜ ⎟ 0 z (l ) . r,q,t p ⎜ ⎟ ⎟ r,q,t =⎜ . . . Z .. .. .. ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 0 zr,q,t (2lp −1) ⎜ ⎟ ⎜ zr,q,t (N −lp )⎟ ⎜ ⎟ ⎜ ⎟ . . . . . . ⎝ ⎠ . . . zr,q,t (N −1)
0
p d where and S r,q,t and S r,q,t are defined in the same way as r,q,t using the pilot part sp (n) and the unknown part sd (n) Z q,t q,t r,1,t , . . . , Z r,N ,t ] r,t = [Z of the intended signal. Denoting Z t p = [S p , . . . , S p and S ], the received vector in (23) can r,t r,1,t r,Nt ,t be approximated by y r,t ≈
Nt
r,q,t Φi + S p Φs + S d Φs + wr,t Z r,q r,q,t r,q r,q,t r,q
q=1
where φin,q,r = φin,q − φn,r and φsn,q,r = φsn,q − φn,r are the combined transmit and receive phase noise processes affecting the self-interference and the intended signals, respectively. It has to be noticed that we adopt this assumption only during the development of the algorithms. However, simulations are performed using the actual phase noise process. (i) pr,q,t (n) = Denotingz zr,q,t (n) = L l=0 hr,q (l)zq,t (n − l), s L (s) (s) p d dr,q,t (n) = L l=0 hr,q (l)sq,t (n − l), and s l=0 hr,q (l)sq,t (n − l), the received N received samples of the tth OFDM block can be expressed as
r,t Φr + S r,t Φr + S r,t Φr + wr,t =Z i
i
p
d
s
iT
iT
T
s
s
(25)
sT
sT
T
with Φr = [Φr,1 , . . . , Φr,Nt ] and Φr = [Φr,1 , . . . , Φr,Nt ] . Finally, by collecting the phase noise processes as iT i iT iT T Φ = Φ1 , Φ2 . . . , ΦN r sT s sT sT T Φ = Φ1 , Φ2 . . . , ΦN r
(26)
with block t and S and defining the block diagonal matrices Z t p diagonal elements Z r,t and S r,t , for r = 1, . . . , Nr , respecp
y r,t =
Nt
diag ( zr,q,t (n)) Φir,q
T
q=1
+ diag spr,q,t (n) + diag sdr,q,t (n) Φsr,q + wr,t
where diag( zr,q,t (n)) is a diagonal matrix with diagonal elements given by { zr,q,t (0), . . . , zr,q,t (N − 1)}, and T i i i Φir,q = ejφ0,q,r , ejφ1,r,q , . . . , ejφN −1,r,q s T s s Φsr,q = ejφ0,r,q , ejφ1,q,r , . . . , ejφN −1,q,r .
(24)
Assuming an estimate of the channel coefficients is available, from the proposed estimator in Section III, the joint estimation of the phase noise vectors Φir,q and Φsr,q , for q = 1, . . . , Nt , involves recovering 2N Nt parameters from N equations. This
tively, the received vector y t = [y T1,t , . . . , y TNr ,t ] over the Nr antennas can be written in the following compact form: tΦ + S Φ +S Φ + wt . yt = Z t t i
p
s
d
s
(27)
Similar to Section III, we gather the parameters to be estimated iT sT in one vector Φ = [Φ , Φ ]T and the known transmitted t, S p ]. Thus, by adopting the t = [Z signals in one matrix as D t Gaussian model to the received vector y t , the log-likelihood function to estimate the phase noise, knowing the channel coefficients, is expressed as i s t Φ)H R−1 (y t − D t Φ) Lpn (Φ , Φ ) = −log |Rpn | − (y t − D pn (28)
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where Rpn is the covariance matrix of y t given by 2 dH dt Φs ΦsH S Rpn = E(S t ) + σ I N Nr . Obviously, maximizi s ing Lpn (., .) with respect to Φ and Φ leads to the ML estimate of the phase noise processes. Compared with the problem of channel estimation in Section III, the covariance matrix Rpn depends on statistic properties of the phase noise process and not on its actual realization of the process. Therefore, the prob t Φ)H R−1 (y t − D t Φ) lem is reduced to minimizing (y t − D pn with respect to Φ. By setting the first derivative to zero, it can be shown that the ML estimate of the phase noise is given by −1 H H t R−1 y t . R−1 D D ΦML = D pn pn t t
(29)
The closed-form expression of Rpn depends on the oscillator type. In Appendix A, the expression of Rpn is given for a phase-locked loop (PLL)-based oscillator and a free-running oscillator. The main complexity of the phase noise estimation procedure comes from the inversion of the 2N Nt Nr /lp × 2N Nt Nr /lp t . Note that the phase noise estimator needs H R−1 D matrix D t
pn
also the inverse of the covariance matrix R−1 pn , which is computed only one time since it depends only on the characteristics of the oscillators and not on the transmitted signal. H depends on the self-interference and inThe matrix D t tended channels. An iterative technique is used to jointly estimate the channel coefficients and the phase noise required to suppress the self-interference signal. First, an initial estimate of the channel is obtained using the proposed ML algorithm by ignoring the presence of the phase noise. Then, the estimated channels are used to obtain an estimate of the phase noise vector Φ from (29). Next, we use the estimate of the phase noise to shift the transmitted self-interference signal and intended signal and estimate the channel coefficients again from the shifted reference signal. We iterate this procedure until the algorithm converges. As any iterative method, the convergence to the actual solution should be discussed. The proposed method may converge to a stationary point that is different from the actual channel (i) coefficients and the phase noise processes. For example, if hr,q i (i) and ejφn,r,p are solutions to our problem, then hr,q ejβ and j(φin,r,p −β) are also solutions. The phase noise process at time e i i , where δn,r,q is the n is written as φin,r,q = φin−1,r,q + δn,r,q innovation process at time n. Thus, the phase noise process φin,r,q can be expressed as the sum of a constant term φi0,r,q i and a variable term Δin,r,q = nk=1 δk,r,q . When combined with the propagation channel, it is not possible to separate the phase of the channel coefficients and φi0,r,q . Therefore, the (i)
i
channel estimation algorithm returns an estimate of hr,q ejφ0,r,q s (s) and hr,q ejφ0,r,q while leaving the variable part of the phase noise Δin,r,q and Δsn,r,q to be estimated during the second part of the procedure. The iterative algorithm is more likely i (i) to converge to the points (hr,q ejφ0,r,q , Δin,r,q ) than the actual (i)
points (hr,q , φin,r,q ). A more detailed study on convergence is presented in Appendix B.
Fig. 2. Self-interference channel estimation MSE versus SNR with Nt = Nr = 2.
V. S IMULATION R ESULTS Here, we provide some simulation results to illustrate the performance of the proposed algorithm in terms of estimation error and self-interference cancelation capability in different scenarios. The wireless channels are represented as a frequencyselective Rayleigh fading channel with nine equal-variance resolvable paths (L = 8). The signal-to-noise ratio (SNR) is the average intended-signal-to-thermal-noise power ratio and the intended signal-to-interference power ratio SIRin at the input of the RF cancelation stage is assumed to be −50 dB. A first estimate of the self-interference channel is obtained during an initial half-duplex period as −1 (I Nr ⊗ Z t )H y t (30) hRF = (I Nr ⊗ Z t )H (I Nr ⊗ Z t ) and the input to the proposed algorithm is the output of the RF cancelation stage. The data are drawn from 4-quadrature amplitude modulation constellation and then passed through an OFDM modulator. Unless otherwise specified, the number of observed OFDM blocks is set to T = 60. The pilot symbols are inserted periodically in some subcarriers before the OFDM modulator. A. Performance in the Absence of Phase Noise We first evaluate the performance of the proposed channel estimator in the absence of phase noise. Figs. 2 and 3 depict the MSE of the proposed method when estimating the residual selfinterference h(i) and the intended channel h(s) , respectively. The pilot symbols represent 20% of the total transmitted data. To assess properly the performance of the proposed channel estimator, the MSE of the algorithm is compared with the Cramer–Rao bound (CRB), as a benchmark for the performance evaluation of the estimator. The expression of the CRB is given in Appendix C. We also compare the ML estimator with the LS estimator. The iterative ML algorithm is initialized with R0 = I since it guarantees convergence to the ML solution. As mentioned earlier, the LS estimate is obtained in the first
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Fig. 3. Intended channel estimation MSE versus SNR with Nt = Nr = 2.
Fig. 5. Amount of baseband cancelation versus SIRin .
Fig. 4. SINRout after self-interference cancelation versus SNR.
Fig. 6. Self-interference channel (in solid lines) and signal channel (in dashed lines) estimation MSE versus percentage of pilots.
iteration of the proposed algorithm. From the simulation results, the performances of the closed-form ML and LS are close to the CRB for moderate SNR, and the LS saturates at high SNR. This saturation is due to the presence of the unknown received signal from the intended user, which acts as noise floor as the SNR increases. However, at low SNR, the thermal noise is dominant as compared with the unknown transmitted signal, and the estimation performances are mostly affected by the thermal noise. The closed-form ML presents also a noise floor at high SNR because of the different approximations adopted to obtain the closed-form expression. As we iterate, this saturation is reduced since we have a better estimate of the covariance matrix R. At low SNR, convergence is obtained after three or four iterations, whereas more iterations are needed at high SNR. Fig. 4 represents the relation between the input SNR and output SINR SINRout after self-interference cancelation. It can be seen that the proposed iterative algorithm outperforms the
closed-form ML solution at moderate and high SNRs. The MSE saturation of the closed-form ML is reflected in SINRout since the cancelation performance is directly related to the accuracy of the estimated self-interference channel. Fig. 5 shows the amount of baseband cancelation αBB , defined as the selfinterference power after the RF stage divided by the remaining self-interference power after the baseband cancelation. Both iterative and closed-form solutions are compared with the LS estimator. Clearly, the proposed algorithm outperforms the LS estimator. Note that the performance of the LS estimator saturates at high interference power as a consequence of the channel estimation MSE saturation at high SNR shown in the previous simulations. The estimation of the intended channel needs some pilot symbols from the other transceiver. In Fig. 6, we evaluate the performance of the proposed algorithms when varying the
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TABLE I N UMBER OF I TERATIONS V ERSUS N UMBER OF A NTENNAS FOR SNR = 20 dB
Number of antennas Number of iterations
1 4
2 6
3 7
4 9
5 10
Fig. 7. Relationship between the cancelation gains of RF and baseband cancelation stages.
number of pilots with SNR = 20 dB and seven iterations for iterative ML algorithm. Clearly, the MSE for self-interference channel estimation (in solid lines) is less affected by the number of pilots than the intended channel estimation (in dashed lines). For very small number of pilots, the self-interference channel is estimated by using the known self-signal and the statistic of the intended signal. For more pilots, the intended channel can be estimated, and a more accurate estimate of the statistic of the intended signal can be obtained. To investigate the effect of the RF cancelation on the performance of the proposed scheme, we investigate the relation between the two following parameters: 1) the RF cancelation gain αRF defined as the ratio between the self-interference input power and output power of the RF cancelation stage; and 2) the baseband cancelation gain αBB . Clearly, these two parameters present the performance of the RF and BB self-interference cancelation stages. The results plotted in Fig. 7 show that, as αRF increases, αBB decreases. In fact, as αRF increases, the amount of self-interference left for the baseband cancelation stage is reduced, leaving not much interference to be canceled in the next stage. Increasing number of transmit antennas Nt and receive antennas Nr results in more channel coefficients between the different antennas; thus, more parameters to estimate from a larger number of observations coming from the receive antennas. Thus, as the number of antennas increases, the size of the matrices involved in the closed form, and iterative solutions also involved in the closed-form soluincrease. In fact, the matrix R tion has size of N N r × N Nr , whose complexity increases with the number of iterations. On the other hand, one may intuitively think that, as the number of parameters increases, convergence requires more iterations. Table I summarizes the number of iterations required for convergence to estimate the self-interference and intended channels for different values of Nt and Nr (taking Nt = Nr ) obtained from extensive simulations.
Fig. 8. SINRout after self-interference cancelation versus SNR in the presence of phase noise from a shared oscillator for f3dB = 100 Hz.
B. Performance in the Presence of Phase Noise In the following, we evaluate the performance of the proposed algorithm in the presence of phase noise. The estimate of the self-interference channel used in the RF cancelation stage is still obtained during the initial half-duplex period in the presence of all noise components. Figs. 8 and 9 show the SINRout for different values of SNR in the presence of phase noise with independent and shared PLL-based oscillators at the transmitters and receivers, respectively. The quality of the oscillator is often measured by its 3-dB bandwidth f3dB . Higher 3-dB bandwidth level results in a fast varying process, making its estimation more difficult. In these figures, we set f3dB = 100 Hz. It is observed that the presence of phase noise reduces the SINRout after cancelation, and the shared-oscillator case offers higher SINRout than the separate-oscillator case. The effects of phase noise can be reduced by using the same common oscillator in the upconversion and downconversion of the same transceiver. In this case, the difference between the phase noise in the transmitter and the receiver depends on the delay that the self-interference experiences from the transmitter to the receiver. While the proposed method improves the cancelation performance by estimating the phase noise and mitigating its effects, a noise floor appears at high SNR. This noise floor comes from the approximation of the same phase noise over a set of samples used when developing the phase noise estimation algorithm. We also mention that a shared oscillator is possible in practice when the transmit and receive chains are located at the same transceiver.
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and the intended channels. An iterative procedure was also proposed to avoid the performance saturation of the closedform solution as high SNR. The iterative procedure incorporates the statistic of the unknown received signal to improve the estimation performance. In the presence of significant phase noise, a method that exploits the previous channel estimate was proposed to mitigate the effects of the phase noise. A PPRENDIX A pn C OVARIANCE M ATRIX R
Fig. 9. SINRout after self-interference cancelation versus SNR in the presence of phase noise from separate oscillators for f3dB = 100 Hz.
pn Here, we give the expression of the covariance matrix R used to estimate the phase noise in (29). This is done for the cases of using PLL or free-running oscillators. First, for separate oscillators in both the transmitter and receiver, the s s covariance of ejφn,r,q and ejφn ,r ,q is given by s s E ejφn,r,q e−jφn ,r ,q s s = E ej (φn,r −φn ,r +φn ,q −φn,q ) ⎧ s j (φn ,q −φsn,q ) j(φn,r −φn ,r ) ⎪ E(e , for q = q , r = r
)E e ⎪ ⎪ ⎪ s ⎪ r ⎨ j (φn ,q −φsn,q ) −σpn Ee , for q = q , r = r
e = t ⎪e−σpn E(ej(φn,r −φn ,r ) ), ⎪ for q = q , r = r
⎪ ⎪ ⎪ t r ⎩e−(σpn +σpn ), for q = q , r = r
(31) t r and σpn are the phase noise variances at the transwhere σpn mitter and receiver oscillators, respectively. For a free-running oscillator, the phase noise process is modeled by a Brownian motion process with the difference between two realizations of the phase noise at time n and n following a normal-distributed random variable with zero mean and variance 4πf3dB Ts |n − n |, and f3dB is a parameter describing the oscillator quality [22]. Then, we have
4πf3dB Ts |n−n | s s 2 E ej (φn ,q −φn,q ) = e− Fig. 10. SINRout after self-interference cancelation versus 3-dB bandwidth of phase noise f3dB .
The residual self-interference depends obviously on the quality of the oscillators f3dB . The resulting SINRout as a function of f3dB is given in Fig. 10 for the common and separate oscillator cases. The SNR is fixed at 35 dB. In the common oscillator case, the proposed algorithm can keep the SINRout constant for f3dB lower than 300 Hz. In the case with separate oscillators, one can see that the phase noise causes the SINRout to increase, starting from f3dB = 100 Hz.
(32)
with Ts being the sample period. For the PLL oscillator, the output phase noise is modeled as the Ornstein–Uhlenbeck process [23]. For shared oscillators at thes transmitter and the receiver,4 the s covariance of ejφn,r,q and ejφn ,r ,q reduces to s s s s E ejφn,r,q e−jφn ,r ,q = E(ej(φn,r −φn ,r ) )E ej (φn ,q −φn,q ) (33) which can be evaluated given the nature of the oscillator. For an independent transmit signal and phase noise process, it can be shown that d d s sH dH Eφ (Φs ΦsH )S dH Φ Φ S = Es S (34) E S t t t t
VI. C ONCLUSION In this paper, ML channel estimation in full-duplex MIMO transceivers has been investigated. A closed-form expression was obtained to jointly estimate the residual self-interference
4 One shared oscillator is used between the antennas of one transceiver, but two different oscillators at the transmitter and the receiver are needed since they are located a two different transceivers.
MASMOUDI AND LE-NGOC: ML CHANNEL ESTIMATOR FOR SELF-INTERFERENCE CANCELATION
where Es (·) and Eφ (·) denote the expectation with respect to the unknown intended data and the phase noise, respectively. d Eφ (Φs ΦsH )S dH is d, S Considering the block structure of S t t t written as ⎛ d S ⎜ 1. ⎜ .. ⎝
d S Nt
⎞⎛ R1,1 ⎟⎜ . ⎟⎝ . . ⎠ RNr ,1
··· ···
⎛
dH d R1,1 S S 1 1 ⎜ .. =⎜ . ⎝ d dH S Nr RNr ,1 S 1 s sH E(Φm Φn )
where Rm,n = can be shown that
dH = d Rm,n S S m n
R1,Nr .. .
⎞⎛ dH S 1 ⎟⎜ .. ⎜ ⎠⎝ .
RNr ,Nr ··· ···
⎞
dH S Nt
⎟ ⎟ ⎠
⎞ dH d R1,N S S r 1 Nr ⎟ .. ⎟ . ⎠ d dH S Nr RNr ,Nr S Nr
is a N Nt × N Nt matrix. Then, it
Nt Nt
d Rqq S dH S m,q m,n m,q
(35)
q=1 q =1
s
sH
with Rqq m,n = E(Φm,q Φn,q ). By developing element (i, j) of dH dm,q Rqq the matrix S m,n S m,q , we have L L d dH m,q Rqq S [i, j] = S hsm,q (l1 )sdq (i − l1 )Rqq m,n m,q m,n l1 =0 l2 =0 d∗ × [i, j]hs∗ n,q (l2 )sq (j − l2 ).
(36)
For intended signals independently transmitted over the differ
ent antennas (i.e., E(sdq (i)sd∗ q (j)) = 0 if q = q , it is easy to d Rqq S dH ][i, j]) = 0 for q = q . For q = q
verify that E([S m,n m,q E(sdq (i)sd∗ q (j))
m,q
and noting that = 0 if i = j, the element (i, j) in (36) can therefore be calculated as E
dH Rqq S dH S m,q m,n m,q [i, j]
min(L,L+i−j)
= α2
s∗ hsm,q (l)Rqq m,n [i, j]hn,q (l). (37)
l=max(0,i−j)
pn is Thus, combining (35) and (37), the expression of R obtained with (31) for separate oscillators and (33) for shared oscillators. A PPRENDIX B P ROOF OF C ONVERGENCE We prove here that the proposed iterative method converges i s (i) (s) to (hr,q ejφ0,r,q , Δin,r,q ) and (hr,q ejφ0,r,q , Δsn,r,q ). Following the notations in Section IV, each phase noise process φin,r,q is represented as the sum of a constant term φi0,r,q and a variable term Δin,r,q . Small values of Δin,r,q and Δsn,r,q satisfy i ejΔn,r,q ≈ 1 + jΔin,r,q . Therefore, we have Φ = Φ0 + jΔt ,
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and the log-likelihood function (8) when considering the presence of phase noise can be rewritten as L(h(i) , h(s) ) = −T log |R| T (y t − D t h)H R−1 (y t − Dt h) + F (Δt , h) (38) − t=1 (i)
i
where the elements of h are written as hr,q ejφ0,r,q and s (s) hr,q ejφ0,r,q , and F (Δt , h) is a function of the channel coefficients and the variable part of the phase noise. The first iteration of the joint channel and phase noise estimation procedure is performed by arg max = −T log |R|− h
T (y t −Dt h)HR−1(y t −Dt h) (39) t=1
and ignores the other terms containing the variable part of the phase noise. Thus, the problem in (39) returns an estimate of h, which will be considered an estimate of the channel, leaving the variable terms Δt to be estimated in the second step. As we iterate, the reference matrix Dt used to estimate the channel is rotated by the phase noise coefficients obtained in the previous iteration, which has the effect of reducing the contribution of F (Δt , h) in the log-likelihood function (38). Therefore, a better estimate of h can be obtained, which results in a better estimate of Δt during the second step of the iteration. The iterative procedure guarantees a monotonic increase of the log-likelihood functions through the set of reestimation transformations. As a conclusion, the proposed algorithm converges to the point (h, Δt ) instead of (h, Φ). A PPRENDIX C S TOCHASTIC C RAMER -R AO B OUND The CRB is defined as the inverse of the Fisher information matrix (FIM) [20]. Following the derivations in [14], the real FIM can be formulated as: (J hh ) − (J hh ) (J hh∗ ) − (J hh∗ ) +2 J R (h) = 2
(J hh )
(J hh∗ ) (J hh ) (J hh∗ ) (40) where T H −1 J hh (i, j) = Dt R Dt (i, j) t=1 −1 ∂R −1 ∂R + trace R R ∂h∗ (i) ∂h∗ (j) ∂R ∂R J hh∗ (i, j) = trace R−1 ∗ R−1 ∗ . (41) ∂h (i) ∂h (j) The first derivative of R with respect to h∗ (i) is 0, for i = 1, . . . , Nt Nr (L + 1) ∂R = ∂H (s) ∂h∗ (i) α2 H (s) ∂h , otherwise. ∗ (i) (42) The expression of the CRB depends on the specific realization of the channel. Therefore, we average the obtained CRB over a set of independent realizations of the channel coefficients. Note that in (42), we keep the dependence of the covariance matrix R on h(s) .
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R EFERENCES [1] J. I. Choi, M. Jain, K. Srinivasan, P. Levis, and S. Katti, “Achieving single channel, full duplex wireless communication,” in Proc. ACM MobiCom, New York, NY, USA, 2010, pp. 1–12. [2] M. Duarte and A. Sabharwal, “Full-duplex wireless communications using off-the-shelf radios: Feasibility and first results,” in Proc. ASILOMAR Signals, Syst., Comput., 2010, pp. 1558–1562. [3] M. Duarte et al., “Design and characterization of a full-duplex multiantenna system for WiFi networks,” IEEE Trans. Veh. Technol., vol. 63, no. 3, pp. 1160–1177, Mar. 2014. [4] D. Kim, H. Ju, S. Park, and D. Hong, “Effects of channel estimation error on full-duplex two-way networks,” IEEE Trans. Veh. Technol., vol. 62, no. 9, pp. 4666–4672, Nov. 2013. [5] M. Duarte, C. Dick, and A. Sabharwal, “Experiment-driven characterization of full-duplex wireless systems,” IEEE Trans. Wireless Commun., vol. 11, no. 12, pp. 4296–4307, Dec. 2012. [6] S. Li and R. D. Murch, “Full-duplex wireless communication using transmitter output based echo cancellation,” in Proc. IEEE Global Telecommun. Conf., 2011, pp. 1–5. [7] J. Ma, G. Y. Li, J. Zhang, T. Kuze, and H. Iura, “A new coupling channel estimator for cross-talk cancellation at wireless relay stations,” in Proc. IEEE Global Telecommun. Conf., 2009, pp. 1–6. [8] T. Riihonen, S. Werner, and R. Wichman, “Mitigation of loopback selfinterference in full-duplex MIMO relays,” IEEE Trans. Signal Process., vol. 59, no. 12, pp. 5983–5993, Dec. 2011. [9] E. Ahmed, A. M. Eltawil, and A. Sabharwal, “Self-interference cancellation with nonlinear distortion suppression for full-duplex systems,” in Proc. ASILOMAR Signals, Syst., Comput., 2013, pp. 1199–1203. [10] A. Sahai, G. Patel, C. Dick, and A. Sabharwal, “On the impact of phase noise on active cancelation in wireless full-duplex,” IEEE Trans. Veh. Technol., vol. 62, no. 9, pp. 4494–4510, Nov. 2013. [11] V. Syrjala, M. Valkama, L. Anttila, T. Riihonen, and D. Korpi, “Analysis of oscillator phase-noise effects on self-interference cancellation in full-duplex OFDM radio transceivers,” IEEE Trans. Wireless Commun., vol. 13, no. 6, pp. 2977–2990, Jun. 2014. [12] E. Ahmed, A. Eltawil, and A. Sabharwal, “Self-interference cancellation with phase noise induced ICI suppression for full-duplex systems,” in Proc. IEEE Global Telecommun. Conf., 2013, pp. 3384–3388. [13] E. Ahmed and A. M. Eltawil, “On phase noise suppression in full-duplex systems,” IEEE Trans. Wireless Commun., vol. 14, no. 3, pp. 1237–1251, Mar. 2015. [14] E. De Carvalho and D. T. Slock, “Cramer-Rao bounds for semi-blind, blind and training sequence based channel estimation,” in Proc. 1st IEEE Workshop SPAWC, 1997, pp. 129–132. [15] F. Wan, W.-P. Zhu, and M. Swamy, “A semiblind channel estimation approach for MIMO-OFDM systems,” IEEE Trans. Signal Process., vol. 56, no. 7, pp. 2821–2834, Jul. 2008. [16] H. Ochiai and H. Imai, “Performance analysis of deliberately clipped OFDM signals,” IEEE Trans. Commun., vol. 50, no. 1, pp. 89–101, Jan. 2002. [17] L. L. Scharf, Statistical Signal Processing, vol. 98. Reading, MA, USA: Addison-Wesley, 1991. [18] P. Stoica and A. Nehorai, “Performance study of conditional and unconditional direction-of-arrival estimation,” IEEE Trans. Acoust., Speech Signal Process., vol. 38, no. 10, pp. 1783–1795, Oct. 1990.
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Ahmed Masmoudi was born in Ariana, Tunisia, on February 10, 1987. He received the Diplôme d’Ingénieur degree in telecommunication from the Higher School of Communication of Tunis, Tunis, Tunisia, in 2010 and the M.Sc. degree from Institut National de la Recherche Scientifique, Montréal, QC, Canada, in 2012. He is currently working toward the Ph.D. degree with the Department of Electrical and Computer Engineering, McGill University, Montréal. His research interests include signal processing and estimation of parameters for wireless communications.
Tho Le-Ngoc (F’97) received the B.Eng. degree (with distinction) in electrical engineering and the M.Eng. degree from McGill University, Montréal, QC, Canada, in 1976 and 1978, respectively, and the Ph.D. degree in digital communications from the University of Ottawa, Ottawa, ON, Canada, in 1983. From 1977 to 1982, he was with Spar Aerospace Limited, where he was involved in the development and design of satellite communications systems. From 1982 to 1985, he was an Engineering Manager of the Radio Group with the Department of Development Engineering, SRTelecom Inc., where he developed the new pointto-multipoint demand-assigned time-division multiple-access/time-division multiplexing Subscriber Radio System SR500. From 1985 to 2000, he was a Professor with the Department of Electrical and Computer Engineering, Concordia University, Montréal. Since 2000, he has been with the Department of Electrical and Computer Engineering, McGill University. His research interests include broadband digital communications. Dr. Le-Ngoc holds a Canada Research Chair (Tier I) on Broadband Access Communications. He received the Canadian Award in Telecommunications Research in 2004 and the IEEE Canada Fessenden Award in 2005. He is a Fellow of Engineering Institute of Canada, the Canadian Academy of Engineering, and the Royal Society of Canada.
