www.ietdl.org Published in IET Communications Received on 21st May 2012 Revised on 2nd May 2013 Accepted on 6th June 2013 doi: 10.1049/iet-com.2012.0624
ISSN 1751-8628
Bayesian generalised likelihood ratio test-based multiple antenna spectrum sensing for cognitive radios Saeid Sedighi, Abbas Taherpour, Shaghayegh S.M. Monfared Department of Electrical Engineering, Imam Khomeini International University, Qazvin, Iran E-mail:
[email protected];
[email protected]
Abstract: In this study, the authors address the problem of multiple antenna spectrum sensing in cognitive radios by exploiting the prior information about unknown parameters. Specifically, under assumption that unknown parameters are random with the given proper distributions, the authors use a Bayesian generalised likelihood ratio test (B-GLRT) in order to derive the corresponding detectors for three different scenarios: (i) only the channel gains are unknown to the secondary user (SU), (ii) only the noise variance is unknown to the SU, (iii) both the channel gains and noise variance are unknown to the SU. For the first and third scenarios, the authors use the iterative expectation maximisation algorithm for estimation of unknown parameters and the authors derive their convergence rate. It is shown that the proposed B-GLRT detectors have low complexity and besides are optimal even under the finite number of samples. The simulation results demonstrate that the proposed B-GLRT detectors have an acceptable performance even under the finite number of samples and also outperform the related recently proposed detectors for multiple antenna spectrum sensing.
1
Introduction
The identification of spectrum holes by secondary users (SUs) constitutes a major requirement at the physical layer of cognitive radios (CRs) networks, where spectrum sensing techniques are sought to achieve a sufficiently reliable detection probability over the shortest possible sensing time. So far several different methods have been proposed for spectrum sensing [1, 2]. The energy detector (ED) is a popular method to detect an unknown signal in additive noise [3]. Unfortunately, the performance of the ED is susceptible to errors in the noise variance and it has been shown that to achieve a desired probability of detection under noise variance uncertainty, the signal-to-noise ratio (SNR) has to be above a certain threshold [4]. Using multiple antennas at the SU receiver is one the possible approaches to overcome the noise uncertainty problem and also to improve the performance of spectrum sensing by exploiting available observations in the spatial domain. Multiple antenna techniques have been used by different authors for spectrum sensing [5–12]. Shen et al. [5] considers a blind spectrum sensing approach where the empirical characteristic function of the multiple-antenna samples is used in formulation of the statistical test. In [6], the authors derive the optimal Neyman–Pearson (NP) and sub-optimal GLRT-based multiple antenna detectors of an orthogonal frequency division multiplexing (OFDM) signal with a cyclic prefix of known length. Multiple antenna spectrum sensing in frequency selective channels is IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
addressed in [7]. In [8–11], the authors derive the GLRT eigenvalue-based detectors for multiple antenna spectrum sensing, which are robust to noise variance uncertainty. The performance of the GLRT-based detector for detecting correlated primary user (PU) signals has been investigated in [12]. In the GLRT-based detectors, the parameters estimation and detection sub-problems are considered separately and the corresponding optimal solution for each of these sub-problems is obtained individually. Therefore although the GLRT-based detectors have acceptable performance, however, they are optimal only when the number of samples goes to infinity [13]. The motivation of this paper is to overcome this drawback by providing some new detectors which are optimal even under finite number of samples. Recently, in [14], a novel mathematical framework has been proposed for the GLRT in which the detection and parameters estimation sub-problems are considered jointly. In particular, under assumption that unknown parameters are random with the given prior distribution, by combining the NP lemma and the Bayesian criterion a B-GLRT has been proposed in [14], which involves the maximum a-posterior probability (MAP) estimation of unknown parameters and is optimal even under the finite number of samples. In this paper, by exploiting the proper prior information about unknown parameters, we obtain the corresponding Bayesian GLRT-based multiple antenna detectors for three different scenarios: (i) only the channel gains are unknown to the SU, (ii) only the noise variance is 2151 © The Institution of Engineering and Technology 2013
www.ietdl.org unknown to the SU, (iii) both of the channel gains and noise variance are unknown to the SU. The new proposed B-GLRT detectors are optimal even under the finite number of sample compared to the traditional GLRT-based detectors. In the first and third scenarios, as it will be shown, it is impossible to derive the MAP estimation of the unknown parameters in closed form and thus, we use the expectation maximisation (EM) algorithm [15] to obtain the MAP estimation of the unknown parameters in these scenarios. The EM algorithm is an iterative algorithm for computing the MAP estimation in which each iteration involves two steps, a expectation step (E-step) followed by a maximisation step (M-step) [16]. Moreover, owing to the significant impact of the convergence rate of the EM algorithm on the complexity and applicability of the proposed detectors, we discuss about the convergence rate of the EM algorithm in a separate section and show that it rapidly converges in our cases. The rest of the paper is organised as follows: In Section 2, we describe the system model and the basic assumptions about the PU signal, noise and channel gains. In Section 3, we derive the corresponding B-GLRT detectors for three mentioned scenarios. In Section 4, we derive the convergence rate of the EM algorithm for our cases. In Section 5, we represent the simulation results and evaluate the performance of the proposed B-GLRT detectors with simulation and compare them with some existing algorithms. Finally Section 6 concludes the paper. Notation: We use lightface letters for scalars, and boldface upper-case and lower-case letters for matrices and vectors, respectively.
2
System description and basic assumptions
We suppose that the SU is equipped with M receiving antennas and each antenna receives L samples. We denote the hypotheses of the presence and absence of the PU signal by H1 and H0 , respectively. The hypothesis testing problem for each antenna is given by yi =
ni hi s + ni
H0 , H1 ,
i = 1, 2, . . . , M
(1)
where hi [ C is the channel gain between the PU transmitter and ith antenna, s [ CL is the vector of the PU signal samples at ith antenna, and ni [ CL is the vector of additive noise samples at ith antenna which is assumed to be zero-mean circular complex Gaussian random variables with covariance matrix σ 2IL where σ 2 is the noise variance. We model the samples of the PU signal, that is, s, as independent identically zero-mean circular complex Gaussian random variables. If the PU system uses multicarrier modulation, like OFDM signal, with a sufficiently large number of sub-carriers, the Gaussian model for the PU signal is an accurate assumption based on the central limit theorem (CLT) [8, 9, 17]. Also, when the actual distribution of the PU signal is unknown, considering the Gaussian distribution for the PU signal, since noise also has the Gaussian distribution, is a customary assumption which makes the derivations simple and mathematically tractable [11, 18]. Without loss of generality, it can be assumed that the PU signal has unit power because any scaling of the PU signal can be incorporated into the channel gains. Under hypothesis H1 , we assume that the PU signal and noise are independent of each other. Let us 2152 © The Institution of Engineering and Technology 2013
define the observation matrix Y = [y1 , . . . , yM ]T [ CM ×L as the complex matrix containing the observed signals at M antennas. The observation matrix has a zero-mean Gaussian distribution under H 1 and H0 , that is, Y /H1 CN 0, R1 and Y /H0 CN 0, R0 , where R1 and R0 are the covariance matrix of Y under H1 and H0 , respectively, and defined as R0 = Cov vec(Y )|H0 = E vec(Y )vec(Y )H |H0 R1 = Cov vec(Y )|H1 = E vec(Y )vec(Y )H |H1
(2) (3)
and the vector vec(Y ) = y11 , y21 , . . . , yM 1 , y12 , . . . , yML ]T [ CML×1 is obtained by stacking the columns of ML samples on top of each other. Thus, the multiple antenna PU detection problem can be expressed as the following binary hypothesis test
H0 : Y CN 0, s2 I MN H1 : Y CN 0, I L ⊗ hhH + s2 I MN
(4)
where ⊗ denotes Kronecker’s product and the vector h [ CM denotes the channel gains vector between the PU and M antennas which is assumed as a constant parameter during each sensing time.
3
B-GLRT detectors
In this section, we consider the spectrum sensing problem in (4) in the presence of unknown parameters. One useful solution for these composite hypothesis testing problems is the GLRT. However, it is known that the GLRT is asymptotically optimal only when the number of samples goes to infinity [13]. Recently, a joint solution for detection/estimation sub-problems in composite hypothesis testing has been proposed by combining the NP lemma and the Bayesian criterion and from there several B-GLRTs have been proposed which, as it is shown, are optimal even under the finite number of samples [14]. Let Θ1 and Θ0 be the set of unknown parameters under hypotheses H1 and H0 , with prior distributions f (Θ1) and f(Θ2), respectively. For a joint detection/estimation problem the combined detection/estimation structure can be defined as |Y, H ), f (Q |Y, H )}, (5) E = {d(H1 |Y), d(H0 |Y), f (Q 1 1 0 0 where of the two probabilities E is comprised d H1 |Y , d H0 |Y used to distinguish between the two hypotheses H1 and H0 and the two probability density |Y ) used to provide the |Y ), f (Q functions (PDFs) f (Q 1 0
parameter estimate. If C Q , Q denotes the cost of ji
j
i
deciding in favour of hypothesis Hj and providing the when the true hypothesis is H and the true estimate Q j i parameters is Θi, the conditional risk for each hypothesis as the average can be defined as
, Y dQ |Y A Q d H0 |Y f Q 0 0i 0 0
, Y dQ |Y A Q dY + d H1 |Y f Q 1 1i 1 1
J E|Hi =
where
(6)
,Y = C Q , Q f Y |H , Q f Q dQ . A ji Q j ij j i i i i i IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
www.ietdl.org By combining the NP lemma and the Bayesian criterion the following optimisation problem can be defined as an alternative to the classical NP approach
inf J E|H1 , E
subject to
J E|H0 ≤ a
(7)
where the level α constitutes the maximally allowable cost under hypothesis H0 . In [14], it has been shown that by selecting the following cost function
,Q =C Q , Q = 1; C01 Q 0 1 10 1 0
,Q =C Q ,Q C00 Q 0 0 11 1 1 2 ≤ D≪1 0 Qi − Q i = 1 otherwise
(9)
Q MAP1 = arg max f Y |H1 , Q1 f Q1
(10a)
Q MAP0 = arg max f Y |H0 , Q0 f Q0
(10b)
Q0
In this section, we derive the finite-sample optimal detectors based on this B-GLRT for the cases when only one or both of the channel gains vector and noise variance are unknown for the SU. 3.1 B-GLRT detector with unknown channel gains (B-GLRT1) In this part, we assume that the noise variance is known and the channel gains vector, that is, h, is unknown to the SU, but the SU has access to the prior statistical distribution of h which is assumed to be a circular complex Gaussian distribution with the covariance matrix Σh, that is, h CN 0, Sh . This assumption for h, for instance, is valid for a Rayleigh fading channel in which channel gains can generally be correlated. The logarithm of the PDF of the observation Y under H0 is obtained as [9]
ln f Y |H0 , s
2
tr YY H =− − ML ln p − ML ln s2 (11) s2
Under hypothesis H1 , we should first obtain the MAP estimation of the channel gains. With respect to (10a) the MAP estimation of h is given by hMAP = arg max ln f Y |H1 , h, s2 + ln f (h) h
IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
By using the matrix inversion lemma [19], we obtain −1
hhH s2 + h2
(14)
(M −1) det hhH + s2 I M = h2 + s2 s2
(15)
hhH + s2 I M
= s−2 I − s−2
and also it can be easily shown that (8)
where
Q1
ln f (Y |H1 , h, s2 ) + ln f (h) = −tr (hhH + s2 I M )−1 YY H − ML ln p − L ln det hhH + s2 I M (13) − M ln p − ln det Sh − hH S−1 h h
the optimisation problem in (7) lead to a B-GLRT which uses the MAP estimation of unknown parameters instead of maximum likelihood (ML) estimation and is optimal under the finite number of samples. This B-GLRT has the following form
f Y |H1 , Q f Q MAP1 MAP1 ,H 0
LR = h ≥H1 f Q f Y |H0 , Q MAP0 MAP0
Since h CN (0, Sh ) and by substituting ln f Y |H1 , h, s2 from [9], we will have
(12)
By substituting (14) and (15) in (13), we obtain ln f Y |H1 , h, s2 + ln f (h) tr YY H hH Y 2 − ML ln s2 − ML ln p + 2 =− 2 s s + h2 s2 h2 − L ln + 1 − M ln p − ln det Sh − hH S−1 h h 2 s (16) Thus, with respect to (12) and (16), the MAP estimation of the channel gains, that is, hMAP , is given by the solution of
hH Y 2 h2 H −1 arg max 2 − L ln + 1 − h Sh h s2 (s + h2 )s2 h (17) Since (17) is a M-dimensional, nonlinear optimisation problem, the MAP estimation of the channel gains vector cannot be found in closed-form under hypothesis H1 . Nevertheless, we can calculate the MAP estimation of the channel gains vector by using the EM algorithm. The key idea of the EM algorithm is rather simple. We can postulate that the available data is only a part of a larger data set, called the complete data, where the missing data, representing the difference between the complete and available data, is selected judiciously to ensure that the maximisation of the complete data likelihood function over the unknown parameters can be performed easily. Then the EM method is implemented as an iteration consisting of two phases, where in the E-phase, the conditional density of the missing data, given the actual observations, is estimated based on the current values of the unknown parameters and is used to evaluate the expected value of the complete log-likelihood function. Next, the M-phase finds the maximum of the estimated complete log-likelihood function with respect to the unknown system parameters, yielding new estimates of the parameters that can be used in the E-phase of the next iteration. In our problem, the EM algorithm applies the following iterations in order to calculate the MAP estimation of the channel gains when we select s as missing data 1. Initialisation: select an initial value of h0 2153 © The Institution of Engineering and Technology 2013
www.ietdl.org (22) to a threshold which results in
2. Expectation step (E-phase): Evaluate Q h, Qk = Es ln f Y , s|h, s2 + ln f (h)|Y ; hk
(18)
3. Maximisation step (M-phase): Evaluate hk+1 = arg max Q h, hk
⎛ ⎞ 2 H 2 h hMAP Y MAP ⎜ ⎟ + 1⎠ 2 − L ln⎝ s2 s2 + hMAP s2 −1 − hH MAP Sh hMAP
(19)
h
4. Looping decision: Until ||hk + 1 − hk||2 > ∈ return to step 2. Where 1 is a preset tolerance parameter and depends on the required or desired accuracy for the estimation. Lemma 1: After the convergence of the EM algorithm, the MAP estimation of the channel gains is obtained as hMAP =
L
2 pl I M + s2 S−1 sl + h
l=1
−1 ×
L
s∗l yl
,H0 h ≥H1 1
(23)
where h1 Wt + M ln p + ln det (Sh ). 2 H hMAP Y as
We
can
rewrite
L 2 H H H H hMAP hMAP YY hMAP = hMAP yl yl hMAP Y = = hH MAP
(20)
l=1
= hH MAP
(a)
where
L
yl yH l hMAP
l=1 L
l=1
yl s∗l
2 s + hMAP 2
l=1
(24)
hH MAP yl sl = 2 s + hMAP 2
(21a)
s2 s2 + hMAP 2
(21b)
pl =
where (a) follows from (21). By substituting (24) in (23), we obtain L hH hMAP 2 ∗ MAP +1 s y − L ln s2 l=1 l l s2
Denote, respectively, the conditional mean and conditional covariance of the PU signal evaluated with the MAP estimation of the channel gains. Proof: See Appendix 1
−1 − hH MAP Sh hMAP
,H0 h ≥H1 1
(25)
With respect to (20) we can easily show that
pl and hMAP , form a fixed Remark 1: After convergence, sl , point of the EM iteration in the sense that hMAP is used in the E-phase to calculate sl , pl , which are in turn used in the M-phase to obtain hMAP . Remark 2: A proper selection of the initial value of h0 is an important parameter for the convergence rate of the EM algorithm. The initial value could be set as h0 = 1M where 1M is a M × 1 vector whose elements are all 1’s. Now, we can use the MAP estimation of the channel gains to obtain the decision rule by employing ‘B-GLRT’ proposed in (9). By taking the logarithm from (9) and using (11) and (16), the logarithm of likelihood ratio (LLR) function is given by
f Y |H1 , hMAP , s2 f hMAP LLR = ln f Y |H0 , s2 ⎛ ⎞ 2 H 2 h hMAP Y ⎜ MAP ⎟ = + 1⎠ 2 − L ln⎝ s2 s2 + hMAP s2
−1 hH −M ln p − ln det Sh − MAP Sh hMAP
L s∗l yl = l=1
L 2 2 −1 sl + pl I M + s Sh hMAP
(26)
l=1
By substituting the above equation in (25) we can express the B-GLRT as 2 L
hMAP 2 + p s TB-GLRT1 (Y ) = l l s2 l=1 ⎛ ⎞ 2 ⎜hMAP ⎟ ,H0 − L ln⎝ + 1⎠ h s2 ≥H1 1
(27)
Remark 3: In the low-SNR regime where ( hMAP 2 /s2 ) ≪ 1, we will have
(22)
⎛ ⎞ 2 2 h h MAP MAP ⎜ ⎟ + 1⎠ ln⎝ 2 2 s s
(28)
For decision making, we should compare the LLR function in 2154 © The Institution of Engineering and Technology 2013
IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
www.ietdl.org By substituting (28) in (27), TB-GLRT1(Y) in the low-SNR case can be simplified as 2 L
hMAP 1 2 ,H0 h TB-GLRT1 (Y ) = pl − 1 sl + 2 L l=1 s ≥H1 (29) where hWh1 /L. Note that in the decision statistic (29), the term !L 2 (1/L) l=1 pl is the a posteriori average power of sl + the PU signal and the term 1 is the a priori average power of the PU signal. So, in fact, TB-GLRT1(Y) uses the increase in the a posteriori average power of the PU signal as the test statistic to decide between H1 and H0 . Remark 4: The computational complexity of the B-GLRT1 comes from two major operations: (1) computation of 2 hMAP , (2) computation of the decision statistic TB-GLRT1(Y). For the first part, in general NL(9M + 5) + N O M 2.4 multiplications and additions are required where N is the number of iterations needed until the EM algorithm converges. For the second part, L(7M + 5) + O(M ) multiplications and additions are needed. Thus the total computation complexity is as (M (9N + 7) + 5(N + 1))L + NO M 2.4
(30)
In practice the number of samples L is usually much larger than the number of antennas M and required iterations N, thus the dominant term in (31) is the first term. As can be seen, the number of required iterations for the convergence of the EM algorithm, or in other words the convergence rate of the EM algorithm, is an important parameter in the complexity of the B-GLRT1. For the sake of comparison, we consider the counterpart GLRT-based detector proposed in [9, Eq. (24)] which approximately needs 2M 2L multiplications and additions if L is assumed to be large enough [9]. Thus, the computational complexity of the ‘B-GLRT1’ is about ((M(9N + 7) + 5(N + 1))/2M 2) times that of its counterpart GLRT-based detector. Hence, if the number of required iterations for the convergence of the EM algorithm is not very large, or in other words the EM algorithm rapidly converges, the computation complexity of the ‘B-GLRT1’ is not much more than its counterpart GLRT-based detector. 3.2 B-GLRT detector with unknown noise variance (B-GLRT2) In this part, we assume the channel gains vector is known and the noise variance is unknown to the SU. We assume that the SU has the prior information on the distribution of the noise variance. In order to have a conjugate prior distribution for σ 2, we assume that σ 2 follows an Inverse-Gamma distribution with shape parameter α and scale parameter K [20, 21], that is, σ 2 ∼ Inv-Gamma(α, K). From (10), the MAP estimation of the noise variance under H0 is given by s2 MAP0 = arg max ln f Y |H0 , s2 + ln f s2 s2
IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
(31)
The logarithm of the joint PDF of the observations Y and the noise variance under H0 is as ln f Y |H0 , s2 + ln f s2 tr YY H − ML ln p − (ML + K + 1) ln s2 =− s2 a + K ln a − ln G(K) − 2 s
(32)
By maximising (32) with respect to σ 2, the MAP estimation of the noise variance under H0 is obtained as tr YY H + a 2 s MAP0 = ML + K + 1
(33)
From (10a), the MAP estimation of the noise variance under H1 is given by s2 MAP1 = arg max ln f Y |H1 , h, s2 + ln f s2
(34)
s2
The logarithm of the joint PDF of the observations Y and the noise variance under H1 is as ln f Y |H1 , h, s2 + ln f s2 tr YY H hH Y 2 − ML ln p + =− s2 s2 + h2 s2 h2 − L ln + 1 − (ML + K + 1) ln s2 + K ln a s2 a − ln G(K) − 2 s
(35)
By taking the derivative of (35) with respect to σ 2 and setting it to zero, we find As6 + Bs4 + C s2 + D = 0
(36)
which A=1
1 − tr YY H − a + 2h2 B= ML + K + 1
2hH Y 2 + h2 − 2h2 tr YY H − a C= + h4 ML + K + 1 h2 hH Y 2 − h4 tr YY H − a D= (37) ML + K + 1 Thus, the MAP estimation of σ 2 under H1 is the solution to third-order equation (36). 2155 © The Institution of Engineering and Technology 2013
www.ietdl.org After finding s2 MAP0 and s2 MAP1 , the LLR function can be calculated as
f Y |H1 , h, s2 MAP1 f s2 MAP1
LLR = ln f Y |H0 , s2 MAP0 f s2 MAP0 tr YY H + a =− − (ML + K + 1) s2 MAP1 ⎛ ⎞ s2 MAP0 ⎠ + (ML + K + 1) ln⎝ s2 MAP1
# $ Q Q1 , Q1k = Es ln f Y , s|Q1 + ln f Q1 |Y ; Q1k (43)
⎛
⎞
h Y h
− L ln⎝ + 1⎠ + 2 2 2 s2 MAP1 s MAP1 + h s MAP1 H
Similar to Section 3.1 maximisation problem (42) have not any closed-form solution and the MAP estimation of h and σ 2 cannot be calculated directly. In this case as discussed before, we can use the EM algorithm to calculate the MAP estimation of h and σ 2. The EM algorithm applies the following iterations for calculating the MAP estimation of Θ1 when we select s as missing data: h0 1. Initialisation: select an initial value of Q10 = s20 2. Expectation step (E-phase): Evaluate
2
2
3. Maximisation step (M-phase): Evaluate
Q1k+1 = arg Q Q1 , Q1k
(38) Finally, by comparing the LLR function to a threshold, the B-GLRT detector is expressed as ⎛
⎞ s2 MAP s2 MAP0 0⎠ TB-GLRT2 (Y ) = −(ML + K + 1)⎝ − ln s2 s2 MAP1
MAP1
⎛
+
hH Y 2
s2 MAP1 + h2 s2 MAP1
− L ln⎝
⎞
h2 ,H0 + 1⎠ h 2 ≥H1 s
2 4. Looping decision: Until Q1k+1 − Q1k .[ return to step 2 (where ε is a preset tolerance parameter). Lemma 2: After the convergence of the EM algorithm, the MAP estimation of the channel gains and noise variance under H1 are obtained as
MAP1
(39)
hMAP
−1 L L −1 = pl I M + s2 MAP1 Sh × sl | 2 + sl ∗ yl l=1
3.3 B-GLRT detector with unknown channel gains and noise variance (B-GLRT3) In this part as the most general case, we assume that both the channel gains and noise variance are unknown to the SU. Also, we assume that the channel gains vector and noise variance follow circular complex Gaussian and Inverse-Gamma distributions, respectively. As shown in Section 3.2, the MAP estimation of the noise variance under H0 is as tr YY H + a 2 s MAP0 = ML + K + 1
(40)
(44)
Q1
s2 MAP1 =
l=1
(45a) 2 !L 2 h y − s + h pl + a l l MAP MAP l=1 ML + K + 1 (45b)
where sl =
hH MAP yl s2 MAP + hMAP 2
(46a)
1
Under hypothesis H1 , with respect to (10a), the MAP estimate of h is given by Q1 = s2 h MAP Q MAP1 = s2 MAP1 = arg max ln f Y |H1 , h, s2 + ln f (h) + ln f s2
pl =
(41)
h,s2
s2 MAP1 + hMAP 2
(46b)
denote, respectively, the conditional mean and conditional covariance of the PU signal evaluated with the MAP estimation of the channel gains and noise variance. Proof: See Appendix 2
Since h CN 0, Sh and σ 2 ∼ Inv-Gamma(α, K) and with respect to (16), the MAP estimation of the unknown parameters under H1 is solution to
s2 MAP1
h Y h2 − L ln + 1 arg max 2 s2 s + h2 s2 h,s2 a" 2 − hH S−1 h h − (ML + K + 1) ln s − 2 s H
2
2156 © The Institution of Engineering and Technology 2013
(42)
Note that as mentioned in Remark
1, after convergence, the pairs sl , pl and hMAP , s2 MAP1 form a fixed point of the EM iteration in the sense that hMAP and s2 MAP1 are used in the E-phase to evaluate sl , pl , which are in turn used in the M-phase to obtain hMAP and s2 MAP1 . The initial value could be set as Q10 = 1M +1 where 1M + 1 is a (M + 1) × 1 vector whose elements are all 1’s. IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
www.ietdl.org Remark 5: TB-GLRT3(Y) can be simplified in the low-SNR regime. In (50), it is observed that
The LLR function is as
hMAP f s2 MAP1 f Y |H1 , hMAP , s2 MAP1 f
LLR = ln f Y |H0 , s2 MAP0 f s2 MAP0
−1 s2 MAP0 hH MAP Sh hMAP =1+ s2 (ML + K + 1)s2 MAP1
s2 MAP0 s2 MAP0 − (ML + K + 1) ln = −(ML + K + 1) s2 s2 MAP1
MAP1
2 hH MAP Y
+ s2 MAP1 + hMAP 2 s2 MAP1 ⎛ ⎞ 2 h MAP − L ln⎝ + 1⎠ − M ln p s2
MAP1
in (50) are much smaller than 1, so we have (47) ln
For decision making, by comparing the LLR function to a threshold we obtain ⎛ ⎞ 2 2 hH Y h MAP
− L ln⎝ MAP + 1⎠ s2 MAP1 s2 MAP1 + hMAP 2 s2 MAP1 ⎛ ⎞ s2 MAP0 s2 MAP0 ⎠ − (ML + K + 1)⎝ − ln s2 s2
s2 MAP1
sl ∗ yl − L ln⎝
l=1
MAP1
| sl |2 + pl
⎞
≥H0 h ,H1 1
(49)
! sl ∗yl from (45a) in (49) we can express By substituting Ll=1 the B-GLRT detector as ⎛ ⎞ L 2 hMAP 2 h TB-GLRT3 (Y ) = | sl |2 + pl − Lln⎝ MAP + 1⎠ 2 s s2 l=1 MAP1
⎞ s2 MAP0 s2 MAP0 ,H0 ⎠ − ln h − (ML + K + 1)⎝ ≥H1 1 s2 s2 MAP1
(50)
IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
(53)
MAP1
By substituting (52) and (53) in (50) we will have the simplified TB-GLRT3 in the low-SNR regime as
(54)
MAP1
MAP1
(52)
MAP1
hMAP + 1⎠ s2
⎛
MAP1 l=1
2
MAP1
MAP1
L
L hMAP 2 1 ,H0 2 | sl | + pl − 1 TB-GLRT3 (Y ) = h 2 ≥H1 L s l=1
⎛ ⎞ s2 MAP0 s2 M A P0 ⎠ − ln − (ML + K + 1)⎝ sMAP1 s2 −1 − hH MAP Sh hMAP
hMAP 2 (ML + K + 1)s2
⎞ 2 h h 2 ln⎝ MAP + 1⎠ MAP s2 s2 (48)
⎛
L
+
⎛
MAP1
≥H0 h ,H1
−1 s2 MAP0 hH MAP Sh hMAP s2 MAP1 (ML + K + 1)s2 MAP1
and
With respect to (46a), (48) can be rewritten as
hH MAP
(51)
MAP1 l=1
hMAP 2 s2
MAP1
MAP1
L | sl | 2 + pl
In the low-SNR regime, the two last terms of (51) and term
−1 hH − ln det Sh − MAP Sh hMAP
−1 − hH MAP Sh hMAP
MAP1
hMAP 2 + (ML + K + 1)s2
where hWh1 + ML + K + 1. Note that the decision statistic in (54) is identical to the decision statistic derived in (29) for the case when only the channel gains are unknown except that σ 2 is now replaced by its estimate s2 MAP1 . Remark 6: Similar to the B-GLRT1 the computational complexity of the B-GLRT3 comes from two major operations: (1) Computation of Q MAP1 , (2) Computation of (Y). For the first part, in general the decision statistic T B-GLRT3 NL(18M + 8) + N O M 2.4 multiplications and additions are required where N is the number of iterations needed until the EM algorithm converges. For the second part, L(7M + 5) + O(M ) multiplications and additions are needed. Thus the total computation complexity is as (M (18N + 7) + 8N + 5)L + N O M 2.4
(55)
In practice the number of samples L is usually much larger than the number of antennas M and required iterations N,
2157 © The Institution of Engineering and Technology 2013
www.ietdl.org thus the dominant term in (55) is the first term. As can be seen, the number of required iterations for the convergence of the EM algorithm, or in other words the convergence rate of the EM algorithm, is an important parameter in the complexity of the ‘B-GLRT3’. For the sake of comparison, we consider the counterpart GLRT-based detectors proposed in [9, Eq. (39)] and [11, Eq.(14)] which approximately need 2M 2L multiplications and additions if L is assumed to be large enough [9]. Thus, the computational complexity of the B-GLRT3 is about ((M(18N + 7) + 8N + 5)/2M 2) times that of its counterpart GLRT-based detectors. Hence, if the number of required iterations for the convergence of the EM algorithm is not very large, or in other words the EM algorithm rapidly converges, the computation complexity of the B-GLRT3 is not much more than its counterpart GLRT-based detectors.
4
Convergence rate of the EM algorithm
In the following
parts, we
use Lemma 3 to calculate I c QMAP1 and I m QMAP1 and then the asymptotic convergence rate of the EM algorithm for the ‘B-GLRT1’ and ‘B-GLRT3’. 1. B-GLRT1 (Θ1 = h): The matrix Ic(h) is evaluated in Appendix 3. By using the results of Appendix 3 we have
hMAP = Ic
+
MAP1
1k
MAP1
converges monotonically to zero with a linear rate, in other words 2 2 Q1k+1 − Q MAP1 = r(W )Q1k − QMAP1
(56)
where ρ(W) < 1 is the spectral radius of P × P matrix W and W is defined as W =
I −1 c
Q MAP1
I m QMAP1
# 2 I c Q1 W Es −∇Q ln f Y , s|h, s2 1 $ 2 − ∇Q |Y , Q ln f Q 1 1 1
(58)
# $ 2 2 ln f s|Y , h, s I m Q1 W Es −∇Q |Y , Q 1 1
(59)
2 ∇Q G 1
∂ 2 G Q1 Q1 = ∂Q1 ∂QH 1
which shows the second-order vector derivative of G(Θ1) with respect to vector Θ1. Proof: See [22] 2158 © The Institution of Engineering and Technology 2013
L hH 1 MAP yl I + S−1 h 2 2 s l=1 s2 + hMAP
(60)
hMAP = Im +
L I s2 + hMAP 2
L
hH 1 MAP yl I + B h MAP s2 l=1 s2 + hMAP 2
(61)
where B(h) is defined in Appendix 4. The asymptotic convergence rate of the EM algorithm
for the ‘B-GLRT1’
hMAP and can be obtained by using I c hMAP and I m calculating the matrix W. The derivations (60) and (61) can be also used for monitoring the convergence rate of the iterative EM algorithm. As a matter of fact, around the proximity of convergence point, the matrix W can be approximated as W k = I −1 c hk I m hk
(62)
(57)
where Ic(Θ1) and Im(Θ1) are, respectively, the information matrices associated to the complete and missing data and are defined as
with
L I + hMAP 2
hMAP can be obtained with respect to Also, the matrix I m Appendix 4 as
As shown the convergence rate of the EM algorithm is an important parameter in the complexity of the ‘B-GLRT1’ and ‘B-GLRT3’ so that the complexity is reduced by increasing the convergence rate of the EM algorithm. In this section, we investigate the convergence rate of the EM algorithm for the ‘B-GLRT1’ and ‘B-GLRT3’. The following lemma shows the asymptotic convergence rate of the EM algorithm. P Lemma 3: Let Q MAP1 [ C be the exact MAP estimation of P unknown vector Θ1 ∈ C and Q1k [ C P be the estimation of Θ1 obtained from kth iteration of the EM algorithm. In 2 the norm Q − Q a neighbourhood of Q
s2
which the spectral radius of Wk provides a measure of the convergence rate of the EM algorithm as it approaches the convergence point. This information can be used to decide when to stop the EM algorithm iterations, that is, when the spectral radius of Wk tends to a constant value, it implies that hk is in the proximity of hMAP. 1. B-GLRT3 (Θ1 = [h σ 2]T): Similar to the previous part and according to the results of Appendix 2 and (58) we can easily calculate Ic(Θ1), which we do not bring it here because of lack of space. Then, by substituting Q MAP1 in Ic(Θ1) and doing some manipulation, we obtain
I c ij Q MAP1 ⎧ 0 ⎪ ⎨
hiMAP = ⎪ ⎩ 2 Sh ii s MAP1
∀i , j,
i = 1, . . . , M − 1
∀i , j,
i = 1, . . . , M ,
j =M +1 (63)
IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
www.ietdl.org and (see (64))
∗ and also we have I c ij Q MAP1 = I c ji QMAP1 .
Also, the matrix I m Q MAP1 is obtained as (see Appendix 5)
Im Q MAP1 = I c QMAP1 + Q + C QMAP1
(65)
where the matrices Q and C(Θ1) are defined in Appendix 5. The asymptotic convergence rate of the EM algorithm for the B-GLRT3 can be obtained by calculating W by using (63)–(65) for
the derivations
I c QMAP1 and I m QMAP1 . Similar to the previous part, we can approximate W by Wk and use it to monitor the convergence rate of the EM algorithm as it gets close to the convergence point. This information can be used to predict the number of the EM algorithm iterations needed until it converge.
5
Fig. 1 Probability of missed detection of different detectors against average SNR for Pfa = 10−2, M = 4 and L = 16
Simulation results
In this section, we provide the simulation results to evaluate the performance of the proposed detectors. Also, we compare the proposed detectors with the optimal NP detector [9, Eq. (10)] which has perfect knowledge on the channel gains and noise variance, the GLRD1 [9, Eq. (24)], the ED, the GLRD3 [9, Eq. (39)] and the arithmetic to geometric mean (AGM) method [11, Eq. (14)]. Note that the GLRD1 and ED need to know the exact value of the noise variance and the GLRD3 and AGM are blind detectors. For all of the provided simulations, the channel gains and noise variance were drawn from the assumed prior distributions in Section 3 and we set the parameter values of the prior distributions to α = 3, K = 2 and 2 w w2 w2 Sh = diag , , ..., M M M where we adjust the channel gains variance j 2 to achieve an average SNR, defined as Ave SNR W Eh,s2
h2 aw2 = 2 s K
(66)
In Fig. 1, we illustrate the probability of missed detection Pm of the proposed detectors and the five detectors mentioned above against average SNR at a false alarm rate of Pfa = 10−2, L = 16 and M = 4. As can be observed, the performance of all detectors improves with increasing the average SNR. Fig. 1 shows that the optimal detector performs slightly better than the ‘B-GLRT2’ and the ‘B-GLRT2’ outperforms ‘B-GLRT1’. The ‘B-GLRT1’ outperforms the ‘GLRD1’ and the ED, for instant, at Pm = 10 − 4, the ‘B-GLRT1’ outperforms the ‘GLRD1’ by 1 dB and the ED by 1.4 dB. Also, the
‘B-GLRT3’ performs better than the ‘GLRD3’ and the AGM, for instant, at Pm = 10− 4, the ‘B-GLRT3’ outperforms the ‘GLRD3’ by 1.6 dB and the AGM by 2.46 dB. Note that if there is a mismatch between the real data and our model parameters, the performance of our proposed detectors might be degraded and hence the appropriate choice of prior distributions becomes important. In fact, because of this importance, besides the mathematical tractability, the mentioned distributions have been chosen so that to enable us with high accuracy to capture and model the broad category of the cases which may arise in practice by fitting the available parameters (degrees of freedom) of the prior distribution to the practical data and measurements. In Fig. 2, we depict the probability of missed detection Pm of the different detectors against number of samples, that is, L, at a false alarm rate of Pfa = 10−2, M = 4 and the average SNR of 2 dB. Note that at M = 2 the ‘GLRD3’ and AGM are equal. Fig. 2 shows that the logarithm of the probability of missed detection of the proposed detectors decreases approximately with a linear rate by increasing the number of samples, that is, L. Thus, in a fixed false alarm rate, number of antennas and SNR, there is an exponentially relationship between the probability of missed detection, that is, Pm, and the number of samples, that is, L. In other words, we have Pm ∝ exp (−αL), that α is a constant parameter that is a function of Pfa, M and average SNR. Also, it can be observed that the performance of the proposed detectors improves with a higher rate than the performance of ‘GLRD3’ and AGM by increasing L. Fig. 3 depicts the probability of missed detection Pm of the proposed detector against the number of antennas, that is, M, at a false alarm rate of Pfa = 10−2, L = 12 and the average SNR of 2 dB. From Fig. 3, we can see that the probability of missed detection Pm of the proposed detectors has also an exponentially relationship with the
⎧ L ⎪ h∗iMAP yli L 1 1 ⎪ ⎪ + + ⎪ ⎪ 2 2 ⎪ 2 2 2 S
⎨ s MAP1 + hMAP s MAP1 l=1 s MAP1 + hMAP h ii 2 I c ii Q ! M A P1 = L ⎪ ⎪ hMAP + a ML + K + 1 sl ⎪ l=1 yl − ⎪ ⎪ − ⎪ ⎩ s2 MAP1 s2 MAP1 IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
∀i = 1, . . . , M (64) ∀i = M + 1
2159 © The Institution of Engineering and Technology 2013
www.ietdl.org
Fig. 2 Probability of missed detection of different detectors against L for Pfa = 10−2, average SNR = 2 dB and M = 2
number of antennas, that is, M, In other words, we also have Pm ∝ exp(−βM). Moreover, it can be seen that the performance improvement reduces by increasing the number of antennas. For example, Pm decreases about 0.12, 0.126 and 0.21, respectively, for ‘B-GLRD1’, ‘B-GLRD2’ and ‘B-GLRD3’ by increasing M from 1 to 2, whereas Pm decreases about 0.013, 0.011 and 0.068, respectively, for ‘B-GLRD1’, ‘B-GLRD2’ and ‘B-GLRD3’ by increasing M from 2 to 3. On the other hand, the computational and implementation complexities grow by increasing the number of antennas. Thus, in practice, we cannot increase the number of antennas arbitrarily. In addition, as expected, the simulation results indicate that increasing the number of antennas, that is, M, compared to increasing the number of samples, that is, L, has more substantial effect on the performance improvement of the proposed detectors. Fig. 4 shows the complementary receiver operating characteristics (ROC) or the probability of missed detection, that is, Pm, against the probability of the false alarm, that is, Pfa, for the different detectors in the absence of the noise variance mismatch for M = 4, L = 8, and the average SNR of 1 dB. Also, Fig. 5 shows these curves for the same values
Fig. 3 Probability of missed detection of the proposed detectors against M for Pa = 10−2, average SNR = 2 dB and L = 12 2160 © The Institution of Engineering and Technology 2013
Fig. 4 Complementary ROC of different detectors for average SNR = 1 dB, M = 4 and L = 8
Fig. 5 Effect of noise variance mismatch on the performance of different detectors, for αdB = 1, average SNR = 1 dB, L = 8 and M=4
Fig. 6 Effect of noise variance mismatch on the performance of different detectors, for αdB = 2.5, average SNR = 1 dB, L = 8 and M=4 IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
www.ietdl.org of M, L and average SNR under the noise variance mismatch of 1 dB. In practice, the noise uncertainty factor, that is, αdB, in receiver is normally 1–3 dB which is because of the existence of interference can be much higher [4, 23]. It is seen that the ‘B-GLRT2’, ‘B-GLRT3’, ‘GLRD3’ and AGM are robust to the noise variance uncertainty but the performance of the ‘B-GLRD1’, ED and optimal detector degrades significantly in the presence of the noise variance mismatch. In fact, under the noise variance mismatch of 1 dB, the ‘B-GLRT2’ outperforms the optimal detector and the optimal detector performs better than all the other detectors. Also, the ‘B-GLRT1’ performs similar to the ‘B-GLRT3’ and the ‘B-GLRT3’ outperforms the ED, ‘GLRD3’ and AGM. Fig. 6 shows the same curves shown in Fig. 5 for the same values of M, L and average SNR under the noise variance mismatch of 2.5 dB. Fig. 6 shows that under a greater noise uncertainty factor, the performance of the ED, ‘B-GLRT1’ and optimal detector
Fig. 7
degrades more substantially and the ‘B-GLRT3’ presents a better performance than the optimal detector. From these figures, we conclude that the optimal detector can outperform the proposed detectors provided that the optimal detector knows the noise variance accurately enough. However, in practice, there is uncertainty about the noise variance which under these unavoidable circumstances, the ‘B-GLRT2’ and ‘B-GLRT3’ outperform the optimal detector. Figs. 7a and 8a, b show the values of ||hk||2 and σk 2 as a function of iterations k at M = 4, L = 30 and the average SNR of 1 dB for the cases when only the channel gains are unknown, and both of the channel gains and noise variance are unknown, respectively. The exact value of the channel gains vector norm and noise variance are also shown in these figures. It is seen that, as expected, after the convergence of the EM algorithm, there is a percentage of error between the estimation value and the exact value of the unknown parameters. Also, Figs. 7b and 8c show the
Values of ||hk|| 2 and P(W)k as a function of iterations k, when channel gains are unknown at average SNR = 1 dB, L = 30 and M = 4
a Iterates ||hk||2 b The spectral radius of Wk, ρ(W)k
Fig. 8 Values of ||hk|| 2, σ2k and ρ(Wk) as a function of iterations k when channel gains and noise variance are unknown at average SNR = 1 dB, L = 30 and M = 4 a Iterates ||hk||2 b Iterates s2k c Spectral radius of Wk, ρ(W)k IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
2161 © The Institution of Engineering and Technology 2013
www.ietdl.org spectral radius of Wk, that is, ρ(Wk), against the number of iterations at M = 4, L = 30 and the average SNR of 1 dB based on the results obtained in Section 4 for the the cases when only the channel gains are unknown, and both of the channel gains and noise variance are unknown, respectively. It is seen that by increasing k and approaching the convergence point, the spectral radius of Wk tends to a constant value. As it can be seen in these figures, the EM algorithm rapidly converges in about 6 and 10 iterations for cases when only the channel gains are unknown, and both of the channel gains and noise variance are unknown, respectively. Thus, computing the MAP estimation of unknown parameters via the EM algorithm in the ‘B-GLRT1’ and ‘B-GLRT3’ has not high computational complexity.
6
Conclusion
In this paper, we considered the spectrum sensing problem in CRs by using multiple antennas at the SU receiver. Under assumption that unknown parameters are random with some known prior distributions, we used the B-GLRT, which is optimal under the finite number of samples, to derive the finite-sample optimal detectors for the cases when only one or both of the channel gains vector and noise variance are unknown to the SU. We also derived the convergence rate of the EM algorithm in terms of the available parameters for the cases when we used the EM algorithm for calculating the MAP estimation of unknown parameters, and then we discussed about it in the simulation results section and showed that it rapidly converges in our cases. The simulation results revealed that the proposed B-GLRT detectors are optimal under the finite number of samples, outperform their counterpart existing detectors and therefore can be used whenever the additional statistical side information of the system parameters are available.
7
References
1 Yucek, T., Arslan, H.: ‘A survey of spectrum sensing algorithms for cognitive radio applications’, IEEE Commun. Surv. Tutor., 2009, 11, (1), pp. 116–130 2 Axell, E., Geert, L., Larsson, E.G., Poor, H.V.: ‘Spectrum sensing for cognitive radio: State-of-the-art and recent advances’, IEEE Signal Process. Mag., 2012, 29, (3), pp. 101–116 3 Urkowitz, H.: ‘Energy detection of unknown deterministic signals’, Proc. IEEE, 1967, 55, (4), pp. 523–531 4 Tandra, R., Sahai, A.: ‘SNR walls for signal detection’, IEEE J. Sel. Top. Signal Process., 2008, 2, (1), pp. 4–17 5 Shen, L., Wang, H., Zhang, W., Zhao, Z.: ‘Multiple antennas assisted blind spectrum sensing in cognitive radio channels’, IEEE Commun. Lett., 2012, 16, (1), pp. 92–94 6 Axell, E., Larsson, E.: ‘Optimal and sub-optimal spectrum sensing of OFDM signals in known and unknown noise variance’, IEEE J. Sel. Areas Commun., 2011, 29, (2), pp. 290–304 7 Nguyen, H., De Carvalho, E., Prasad, R.: ‘Spectrum sensing for cognitive radio based on multiple antennas’. IEEE 75th Vehicular Technology Conf. (VTC), May 2012, pp. 1–5 8 López-Valcarce, R., Vazquez-Vilar, G., Sala, J.: ‘Multiantenna spectrum sensing for cognitive radio: overcoming noise uncertainty’. Second Int. Workshop on Cognitive Inform. Process. (CIP), June 2010, pp. 310–315 9 Taherpour, A., Nasiri-Kenari, M., Gazor, S.: ‘Multiple antenna spectrum sensing in cognitive radios’, IEEE Trans. Wirel. Commun., 2010, 9, (2), pp. 814–823 10 Bianchi, P., Debbah, M., Mada, M., Najim, J.: ‘Performance of statistical tests for single-source detection using random matrix theory’, IEEE Trans. Inf. Theory, 2011, 57, (4), pp. 2400–2419 11 Zhang, R., Lim, T., Liang, Y., Zeng, Y.: ‘Multi-antenna based spectrum sensing for cognitive radios: a glrt approach’, IEEE Trans. Commun., 2010, 58, (1), pp. 84–88 2162 © The Institution of Engineering and Technology 2013
12 Yang, X., Peng, S., Zhu, P., Chen, H., Cao, X.: ‘Effect of correlations on the performance of GLRT detector in cognitive radios’, IEICE Trans. Commun., 2011, 94, (4), pp. 1089–1093 13 Wilks, S.S.: ‘Mathematical statistics’ (Lightning Source Inc., 2008) 14 Moustakides, G.: ‘Finite sample size optimality of GLR tests’. Arxiv preprint arXiv:0903.3795, 2009 15 Ng, S., Krishnan, T., McLachlan, G.: ‘The EM algorithm’, in Gentle, J., Hardle, W., Mori, Y. (Eds.): ‘Handbook of computational statistics’ (Springer, Berlin Heidelberg, 2012), pp. 139–172 16 Dempster, A., Laird, N., Rubin, D.: ‘Maximum likelihood from incomplete data via the EM algorithm’, J. R. Stat. Soc. Series B (Methodological), 1977, 39, (1), pp. 1–38 17 Ramrez, D., Vazquez-Vilar, G., López-Valcarce, R., Va, J., Santamara, I.: ‘Detection of rank-p signals in cognitive radio networks with uncalibrated multiple antennas’, IEEE Trans. Signal Process., 2011, 59, (8), pp. 3764–3774 18 Wang, P., Fang, J., Han, N., Li, H.: ‘Multiantenna-assisted spectrum sensing for cognitive radio’, IEEE Trans. Veh. Technol., 2010, 59, (4), pp. 1791–1800 19 Strang, G.: ‘Introduction to linear algebra’ (Cambridge Publication, 2003) 20 Gelman, A., Carlin, J., Stern, H., Rubin, D.: ‘Bayesian data analysis’ (Chapman & Hall/CRC, 2003) 21 Box, G., Tiao, G.: ‘Bayesian inference in statistical analysis’ (Wiley Classics Library, 1992) 22 Hero, A., Fessler, J.: ‘Convergence in norm for alternating expectation-maximization (em) type algorithms’, Stat. Sin., 1995, 5, (1), pp. 41–54 23 Tandra, R., Sahai, A.: ‘Fundamental limits on detection in low SNR under noise uncertainty’. Int. Conf. on Wireless Networks, Communications and Mobile Computing, June 2005, pp. 464–469
8
Appendix 1: proof of Lemma 1
By substituting (18) inside (19), we obtain hk+1 = arg max Es ln f Y , s|h, s2 + ln f (h)|Y ; hk h
= arg max Es ln f Y |s, h, s2 + ln f s|h, s2 h
+ ln f (h)|Y ; hk
(67)
The random vector s is statistically independent from h and σ 2, thus ln f (s|h, σ 2) = ln f(s). Since ln f(s) is constant with respect to h, the above equation is simplified as hk+1 = arg maxEs ln f Y |s, h, s2 + ln f (h)|Y ; hk (68) h
Owing to the Gaussian assumption, it is straightforward to show that Y |s, h, s2 CN hs ⊗ I L , s2 I ML , thus we have ln f Y |s, h, s2 = −ML ln p − ML ln s2 −
L 1 y − s h 2 l l s2 l=1
(69)
Since h CN 0, Sh and by substituting (69) in (68), we find Es ln f Y |s, h, s2 + ln f (h)|Y ; hk = −ML ln p − ML ln s2 −
L # $ 1 y − s h2 |Y ; h E s l l k s2 l=1 l
−M ln p − ln det Sh − hH S−1 h h
(70) IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
www.ietdl.org Similar to Appendix 1 we can show that
We can show that # $ 2 Esl yl − sl h |Y ; hk = yl 2 − E sl |Y ; hk yH l h ∗ H 2 − E sl |Y ; hk h yl + E |sl | |Y ; hk h2 = yl 2 − slk yH slk ∗ hH yl + | slk |2 + plk h2 l h − = yl − slk h2 + plk h2
(71)
# $ # $ 2 Esl yl − sl h |Y ; Q1k = yl 2 − E sl |Y ; Q1k }yH h l # $ # $ − E sl ∗|Y ; Q1k hH yl + E |sl |2 |Y ; Q1k h2
= yl 2 − slk yH slk ∗hH yl + | sl k | 2 + plk h2 l h − = yl − slk h2 + plk h2
where
slk W E sl |Y ; hk
(77) hH y = 2 k l 2 s + hk
plk W Cov sl |Y ; hk =
(72a)
s2 s2 + hk 2
where slk W E{sl |Y ; Q1k } =
(72b)
By substituting (71) in (70), we obtain
plk W Cov(sl |Y ; Q1k ) =
Es ln f Y |s, h, s2 + ln f (h)|Y ; hk = −ML ln p − ML ln s2 −
L 2 1 slk h hl − 2 s l=1
L h − 2 p − M ln p − ln det Sh − hH S−1 h h (73) s l=1 lk
L L 2 1 1 h2 slk h − 2 plk yl − 2 s l=1 s l=1 − ML ln p − M ln p − ln det Sh − hH S−1 h h a − (K + 1) ln s2 + K ln a − ln G(K) − 2 s (79)
l=1
Equations (74), (72a) and (72b) are iterated for finding the MAP estimation of the channel gains vector until the EM algorithm converges.
9
By taking the derivative from (79) with respect to Θ1 and setting to zero, the next iteration of the channel gains and noise variance can be obtained as
Appendix 2: proof of Lemma 2
By substituting (43) inside (44) and simplifications like Appendix 1, we obtain Q1k+1
doing
some
hk+1 =
= arg max Es h,s2
L
| slk | + plk I M + s2 S−1 h
−1
2
l=1
×
L slk ∗yl l=1
(80a)
# $ = arg max Es ln f Y |s, Q1 + ln f (Q1 )|Y ; Q1k Q1
(78b)
= −ML ln s2 −
−1 L 2 L 2 −1 plk I + s Sh × slk + slk ∗ yl (74) l=1
s2k s2k + hk 2
# $ Es ln f Y |s, h, s2 + ln f (h) + ln f (s2 )|Y ; Q1k
By taking the derivative of (73) with respect to h and setting to zero, we can obtain the next iteration of the channel gains as hk+1 =
(78a)
By substituting (77) in (76), we obtain
2
hH k yl , s2k + hk 2
2 2 y − s h + h p l lk k k lk + a l=1
!L
s2k+1 =
ln f Y |s, h, s2 .
ML + K + 1
(80b)
Equations (80a), (80b), (78a) and (78b) are iterated for finding the MAP estimation of h and σ 2 until the EM algorithm converges.
+ ln f (h) + ln f (s )|Y ; Q1k 2
(75)
10
Appendix 3: derivation of Ic(h)
With respect to (69), we can easily show that According to (58), we have Es { ln f (Y |s, h, s ) + ln f (h) + ln f (s )|Y ; Q1k } 2
2
L # $ 1 y − s h2 |Y ; Q E l l 1k s2 l=1 − ML ln p − M ln p − ln det Sh − hH S−1 h h a − (K + 1) ln s2 + K ln a − ln G(K) − 2 s
I c (h) = Es −∇h2 ln f Y , s|h, s2 − ∇h2 ln f (h)|Y , h (81)
= −ML ln s2 −
IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
(76)
From (67) in Appendix 1, the above equation is simplified as I c (h) = Es −∇h2 ln f Y |s, h, s2 − ∇h2 ln f (h)|Y , h (82) Since h CN 0, Sh and with respect to (69) in Appendix 1, 2163 © The Institution of Engineering and Technology 2013
www.ietdl.org and
it is straightforward to show that L 1 |s |2 I ∇h2 ln f Y |s, h, s2 = − 2 s l=1 l
(83)
∇h2 ln f (h) = −S−1 h
(84)
∀ i = 1, 2, . . . , M : ∂2 ln f Y |h, s2 Bii (h) = ∂hi ∂h∗i ! hH Y 2 + hi Mj=1 h∗j yH i yj =− 2 2 2 2 s + h s
and
yi 2 |hi |2 hH Y 2 + 2 + 2 s + h2 s2 s2 + h2 s2
By substituting (83) and (84) inside (82) we obtain
I c (h) = =
1 s2
L
−
E |sl |2 |Y , h I + S−1 h
L L|hi |2 + 2 s2 + h2 s2 + h2
l=1
L L 1 hH yl I + I + S−1 h s2 l=1 s2 + h2 s2 + h2
(85)
and since B(h) is a Hermitian matrix, Bij (h) = B∗ji (h). According to (89) and (90), B(h) is only a function of Y, h and σ 2 which result in Es B(h)|Y , h = B(h)
Appendix 4: derivation of Im(h)
11
(90)
In order to calculate Im(h), after some manipulation, we can rewrite (59) as I m (h) = I c (h) − Es −∇h2 ln f (h)|Y , h + Es ∇h2 ln f Y |h, s2 |Y , h
(91)
Finally, with respect to (87) and (91), we obtain
I m (h) = (86)
L L 1 hH y l I + I + B(h) s2 l=1 s2 + h2 s2 + h2
(92)
From (84) and (85), we can easily show that I c (h) − Es −∇h2 ln f (h)|Y , h =
L L 1 hH yl I + I s2 l=1 s2 + h2 s2 + h2
12 (87)
For calculating Im(Θ1), similar to Appendix 4, after some manipulation, we can rewrite (59) as # $ 2 I m Q1 = I c Q1 − Es −∇Q ln f (h)|Y , Q 1 1 # $ 2 2 −Es −∇Q1 ln f s |Y , Q1 # $ 2 ln f Y |Q |Y , Q + E s ∇Q 1 1 1
Let define the matrix B(h) as B(h) W ∇h2 ln f Y |h, s2
(88)
where
Appendix 5: derivation of Im(Θ1)
(93)
If we define the matrix C(Θ1) as ∀ i , j,
i = 1, 2, . . . , M :
Bij (h) =
∂2 ln f Y |h, s ∂hi ∂h∗j
2 C Q 1 W ∇Q ln f Y |Q 1 1
2
we can derive different elements of C(Θ1) as !M
hj j=1 h∗j yH yH i yj i yj = 2 − 2 2 2 s + h s s2 + h2 s2 ! H h∗i M h∗i hj hH Y 2 i=1 hj yi yj − + 3 2 s2 + h2 s2 s2 + h2 s2 +
h∗i hj
s2 + h2
(94)
2
2164 © The Institution of Engineering and Technology 2013
∀ i ≤ j, i = 1, 2, . . . , M :
C ij Q1
∂2 ln f Y |Q1 = = Bij (h) ∂hi ∂h∗j
(95)
(89) and IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
www.ietdl.org Also we can define the matrix Q as
∀i = 1, 2, . . . , M , j = M + 1: ∂2 ln f Y |Q1 C ij (Q1 ) = ∂hi ∂s2 !M ∗ H !M ∗ H j=1 hj yi yj j=1 hj yi yj = − − 2 2 s2 + h2 s2 s2 + h2 s4 h∗j hH Y 2 h∗j hH Y 2 + + 3 3 s2 + h2 s2 s2 + h2 s4
# $ 2 Q = −Es −∇Q ln f (h)|Y , Q 1 1 # $ 2 2 |Y , Q ln f s − Es −∇Q 1 1 (96)
Lh∗j
s2 h2 + − 2 2 2 s + h s2 + h2 and also C ji Q1 = C ∗ij Q1 . Similar to Appendix 4 we can see that all the elements of C (Θ1) depend only on Y and Θ1, therefore (97) Es C(Q1 )|Y , Q1 = C Q1
IET Commun., 2013, Vol. 7, Iss. 18, pp. 2151–2165 doi: 10.1049/iet-com.2012.0624
=
2 ∇Q 1
ln f (h) +
2 ∇Q 1
ln f s
2
⎛
=⎝
−S−1 h 01×M
⎞ 0M ×1 K+1 a ⎠ − 6 s4 s (98)
Finally, with respect to (93), (97) and (98), Im(Θ1) is obtained as I m Q 1 = I c Q 1 + Q + C Q1
(99)
2165 © The Institution of Engineering and Technology 2013
