1
Multiple-Antenna Signal Detection in Cognitive Radio Networks with Multiple Primary User Signals
arXiv:1405.6408v1 [math.ST] 25 May 2014
David Morales-Jimenez, Raymond H. Y. Louie, Matthew R. McKay, and Yang Chen
Abstract—We consider multiple-antenna signal detection of primary user transmission signals by a secondary user receiver in cognitive radio networks. The optimal detector is analyzed for the scenario where the number of primary user signals is no less than the number of receive antennas at the secondary user. We first derive exact expressions for the moments of the generalized likelihood ratio test (GLRT) statistic, yielding approximations for the false alarm and detection probabilities. We then show that the normalized GLRT statistic converges in distribution to a Gaussian random variable when the number of antennas and observations grow large at the same rate. Further, using results from large random matrix theory, we derive expressions to compute the detection probability without explicit knowledge of the channel, and then particularize these expressions for two scenarios of practical interest: 1) a single primary user sending spatially multiplexed signals, and 2) multiple spatially distributed primary users. Our analytical results are finally used to obtain simple design rules for the signal detection threshold. Index Terms—Signal detection, cognitive radio, spectrum sensing, generalized maximum likelihood ratio test, sphericity test.
I. I NTRODUCTION Cognitive radio is a promising technology which can be used to improve the utilization efficiency of the radio spectrum [1], by allowing secondary user (SU) networks to co-exist with primary user (PU) networks through spectrum sharing. A key requirement is that SU transmission will not adversely affect the PUs’ performance. To achieve this, a common technique involves the SUs first detecting if at least one PU is transmitting, which is commonly referred to as “spectrum sensing”. If no signals are detected, the SUs are allowed to transmit. The importance of signal detection can be seen by its inclusion in the IEEE 802.22 standard; a standard built on cognitive radio techniques [2]. Signal detection has been extensively investigated over the past few decades (see [3]–[5] as examples of some seminal works), within different contexts of application (see, e.g., [6], The work of D. Morales-Jimenez and M. R. McKay was supported by the Hong Kong Research Grants Council under grant number 616713. R. H. Y. Louie was supported by HKUST research grant IGN13EG02. Y. Chen was supported by research grant FDCT 077/2012/A3. Part of this work will be presented at the IEEE International Conference on Communications (ICC), Sydney (Australia), June 2014. D. Morales-Jimenez, R. H. Y. Louie, and M. R. McKay are with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon (Honk Kong). (email:{eedmorales,eeraylouie,eemckay}@ust.hk) Y. Chen is with the Department of Mathematics, University of Macau, Av. Padre Tom´as Pereira, Taipa (Macau). (e-mail:
[email protected]) This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.
[7] in radar). Inspired by some of those seminal works, a number of signal detection tests have been proposed to detect PU transmission when there are multiple receive antennas at the SUs (see e.g., [8]–[12]). Optimality is often considered in the Neyman-Pearson sense, which involves comparing the generalized likelihood ratio (GLR) to a user-designed detection threshold. The GLR can be used to determine the false alarm and detection probabilities, which can then be subsequently used to design the threshold. The particular form of the GLR is dependent on the number of PU transmission signals, and whether noise and/or channel information is known at the SU receiver performing the signal detection. A reasonable scenario is to assume that nothing is known at the SU receiver, i.e., no noise and channel information are known. For this scenario, the false alarm and detection probability have been analyzed when there is only one PU signal (see e.g., [8], [9]). However, the simultaneous presence of multiple PU signals is a common occurrence in current and next generation systems. This may occur, for example, in systems where spatial multiplexing techniques are employed, or where different PUs transmit simultaneously. Furthermore, the number of PU signals is clearly expected to grow as evidenced by the forthcoming use of massive antenna arrays (either co-located or distributed) and by the trend towards more dense and heterogeneous networks [13]–[15]. As a result, we often encounter scenarios where the number of PU signals is no less than the number of receive antennas. For these scenarios, exact expressions for the false alarm probability and the detection probability were derived in [10] when there are two receive antennas. For more general scenarios with arbitrary number of receive antennas and observations, [11] conducted Monte Carlo simulations while [4] [12, pp. 230] derived infinite series expansions. However, the expressions in [4] involved complicated zonal polynomials or Meijer-G functions which are generally hard to compute, while the false alarm probability expression in [12, pp. 230] was not amenable to analysis. For the same general scenario, an approximation was considered in [10], however, the approximation therein was only justified for the false alarm probability, and only then for a very small number of antennas. In this paper, we derive accurate easy-to-compute approximations for the false alarm and detection probabilities of the GLR detector1 when there are an arbitrary number of receive antennas and observations. This is facilitated by an expression for the moments of the GLR test (GLRT) statistic which we 1 Note that the performance of the GLR detector has been previously shown to perform better than other detectors in many practical scenarios [10], and thus we do not consider such comparisons in this paper.
Then the GLR, used for determining H0 or H1 , admits max f x1 , . . . , xm N0 N0 ∈R+ L= max f x1 , . . . , xm N0 , H
derive. We also consider the scenario where the number of receive antennas and observations are large and of similar order. For this scenario, we first derive simple and accurate approximations for the moments and cumulants of the GLRT statistic, and then show that this statistic converges in distribution to a Gaussian random variable under the hypothesis of no PU signals being present. Further, we analyze the detection probability for a large number of PU signals being no less than the number of receive antennas at the SU. Using results from large random matrix theory, we show that the (instantaneous) detection probability can be accurately approximated without explicit knowledge of the channel for a practical number of antennas. Leveraging our analytical results, we then propose simple design rules to approximate the detection threshold that achieves a desired false alarm probability.
N0 ∈R+ ,H∈Cn×k
where
ˆ etr − mR N0 f x1 , . . . , xm N0 = (N0 π)mn
is the likelihood function of the observation matrix under hypothesis H0 and etr −mR ˆ HH† + In N0 −1 f x1 , . . . , xm N0 , H = m det (HH† + In N0 ) π mn
is the corresponding likelihood function under hypothesis H1 , with etr(·) = eTr(·) . The GLRT has been studied in the literature under different assumptions on the rank of HH† (see, e.g. [16] for rank-1 or [5], [11] for more general assumptions). We are concerned with the case of HH† having full rank. In this case, the GLRT yields the well-known sphericity test [3] and the GLRT statistic W to determine H0 or H1 admits3
II. P ROBLEM S TATEMENT Consider a wireless communications system where a SU receiver equipped with n antennas is tasked with determining if PU transmission signals are present from m independent and identically distributed (IID) observation sample vectors x1 , . . . , xm , where2 xℓ = CN n,1 (0n,1 , R) for ℓ = 1, . . . , m, and R is a n×n population covariance matrix. The ℓth sample vector xℓ for this hypothesis testing problem is modeled as H0 : H1 :
xℓ = nℓ xℓ = Hsℓ + nℓ
no signal present signals present
∆
W =
H1 :
R = In N0
signals present .
To evaluate the performance of the GLRT statistic (3), we consider the false alarm and the detection probability. The false alarm probability is ∆
PFA (η) = Pr (W0 > η) = 1 − FW0 (η)
∆
W0 =
Tr(XX† ) n 1 det(XX† ) n
,
X ∼ CN n,m (0n,m , In N0 )
and FW0 (η) denotes the cumulative distribution function (c.d.f.) of W0 . The threshold η is typically chosen to ensure the false alarm probability does not exceed a maximum value α0 ∈ (0, 1), i.e., η = P−1 FA (α0 ) .
The probability of detection is ∆
PD (η) = Pr (W1 > η) = 1 − FW1 (η)
(2)
(5)
where ∆
W1 =
m
X 1 ˆ = 1 xℓ x†ℓ = XX† . R m m ℓ=1
p,q
(4)
where
To proceed, it is convenient to introduce the observed data matrix X = [x1 , . . . , xm ] and the sample covariance matrix
20
(3)
A. False Alarm and Detection Probability
no signal present
R = HH† + In N0
1 ≷H H0 η,
where η is a user-specified detection threshold. Thus, at least one PU is deemed to be transmitting if W > η, while no PU transmission is deemed if W ≤ η.
(1)
where nℓ ∼ CN n,1 (0n,1 , In N0 ) denotes additive white Gausk sian h noise i with variance N0 , sℓ ∈ C is the signal vector with † E sℓ sℓ = Ik , H ∈ Cn×k is the channel matrix from the PUs to the SU detector, which is assumed to be constant during the m observation time periods, and k is the number of PU transmission signals. Both nℓ and sℓ are assumed IID over ℓ = 1, . . . , m, implying that the observation sample vectors xℓ are also IID. Note that, unless otherwise specified, the results in this paper do not assume a specific distribution for H. Thus, for example, our results can account for each PU transmission signal having different transmit power. We assume that H, k and N0 are unknown at the detector, and that HH† is positive-definite (full rank), i.e., k ≥ n. The latter condition can correspond to the scenario where there are at least n single-antenna transmitting PUs, or if there is at least one transmitting PU equipped with at least n antennas which are utilized for spatial multiplexing. The detection problem in (1) is equivalent to testing if the population covariance matrix R is one of two structures: H0 :
Tr(XX† ) n 1 det(XX† ) n
Tr(XX† ) n 1 det(XX† ) n
,
X ∼ CN n,m 0n,m , HH† + In N0
3 The GLRT statistic usually presented in literature is 1 , which is used W to form the sphericity test [3]. However, we work with W for mathematical convenience.
denotes the p × q matrix of all zeroes.
2
and FW1 (η) denotes the c.d.f. of W1 . Note that the probability of false alarm and detection are respectively obtained as the complementary c.d.f. of W0 and W1 . III. C. D . F.
OF
W0
AND
the form [19, eq. (45)] FX (x) ≈ Φ(˜ x) 2
j x ˜ s L x) Y 1 κX,ℓ+2 ℓ e− 2 X X Hes+2r (˜ − √ s+2r jℓ ! (ℓ + 2)! 2π s=1 {j} σX ℓ=1
W1 : N ON -A SYMPTOTIC A NALYSIS
, σX is the standard deviation of X, where x ˜ = x−E[X] σX r = j1 + j2 + . . . + js , and Φ(·) is the c.d.f. of a standardized Gaussian. The second summation enumerates all sets containing the nonnegative integer solutions of j1 +2j2 +. . .+sjs = s. Further, κX,p is the pth cumulant of X, related to the first p moments through
In this section, we derive expressions for the c.d.f. of W0 and W1 for arbitrary n and m. We first present closed-form expressions for the moments of W0 and W1 . A. Exact Moments
κX,1 = µX,1
Theorem 1. The pth (p ∈ Z+ ) moment of W0 and W1 , for p < n(m − n + 1), are respectively given by µW0 ,p =
κX,p = µX,p −
E [W0p ]
n−1 Y Γ m−n+1− p +j Γ (mn) n , = p n Γ (mn − p) j=0 Γ (m − n + 1 + j)
µW1 ,p = E [W1p ] Q − p n−1 p! ni=1 yi n Y Γ m − n + 1 − np + j = np Γ (m − n + 1 + j) j=0 n X Y Γ m − np + ki yiki , × Γ(ki + 1)Γ m − np k +...+k =p i=1 1
(8)
p−1 X ℓ=1
p−1 κX,ℓ µX,p−ℓ , ℓ−1
p ≥ 2,
(9)
with µX,p = E [X p ], and Heℓ (z) is the Chebyshev–Hermite polynomial [18, eq. (13)]
(6)
ℓ
⌊2⌋ X (−1)k z ℓ−2k Heℓ (z) = ℓ! , k!(ℓ − 2k)!2k k=0
where ⌊·⌋ denotes the floor function. In (8), a truncation limit L implies that κX,p , p = 3, 4, . . . , L+2, are involved in the corrected c.d.f. Particularizing (8) for L = 2 results in the following simple approximation for the c.d.f. of W0 and W1 : η − µWℓ ,1 (10) ; σWℓ , κWℓ ,3 , κWℓ ,4 , FWℓ (η) ≈ G σWℓ
(7)
n
where Γ(·) denotes the Gamma function, k1 , . . . , kn are nonnegative integers, and N0 < y1 ≤ y2 ≤ . . . ≤ yn < ∞ denote the eigenvalues of HH† + In N0 .
for ℓ = 0, 1, with
Proof: See Appendix A. Note that (6) has been derived previously in [10], while (7) is new. Condition p < n(m − n + 1): The condition on p, required for (6) and (7) to hold, becomes milder as the difference between the numbers of observations and antennas grow. The first few moments can be obtained for even a very small number of antennas and observations. For example, for p = 3, the condition is satisfied when n = 2 and m = 3.
r
x2
2 e− 2 κ3 (x2 − 1) π 12σ 3 κ4 (κ3 )2 4 2 2 + x(x − 3) + x x − 10x + 15 . 4σ 12σ 3
G(x; σ, κ3 , κ4 ) = Φ(x) −
(11)
More terms can be added (L > 2) with an expected increase in accuracy; however, with L = 2, i.e., involving up to the fourth cumulant, the c.d.f. of W0 and of W1 are already approximated with high accuracy. This can be observed in Figs. 1 and 2, which plot respectively the probability of false alarm PFA (η) and the probability of detection PD (η), both as a function of the detection threshold η. The ‘Analytical (Gaussian)’ curves correspond to a Gaussian approximation without any correction terms (L = 0), i.e., FW1 (η) ≈ Φ((η−µW1 ,1 )/σW1 ) and FW0 (η) ≈ Φ ((η − µW0 ,1 )/σW0 ), while the ‘Analytical (Correction)’ curves are plotted using (10). The ‘Analytical (Beta) [5]’ curves are plotted using the Beta function approximation introduced in [10]. These results are all compared with the true c.d.f., computed via Monte Carlo simulation4 . For the false alarm probability curves in Fig. 1, we observe that the Gaussian approximation deviates from Monte Carlo simulations, thus justifying the use of additional correction
B. C.d.f. Approximation: Edgeworth Expansion Armed with the derived expressions for the moments, we now aim at characterizing the c.d.f. of W0 and W1 in an easy-to-compute form, which can help in understanding the performance of the GLRT in order to design the detection threshold η. As observed in the conference version of this paper [17], the empirical distribution of W0 approaches a Gaussian as the number of antennas and observations grow large with fixed ratio. As also pointed out in [17], a similar phenomenon is observed for W1 . This convergence motivates us to consider a Gaussian approximation for the c.d.f. of W0 and W1 , corrected with additional terms obtained by the Edgeworth expansion [18], [19]. Specifically, the c.d.f. of an arbitrary random variable X in L-truncated Edgeworth expansion takes
4 To simulate the detection probability, we generate H as ∼ CN n,k 0n,k , In , which is held constant for m observation periods. This corresponds to a scenario where a single PU transmits k spatially multiplexed signals and where the channel undergoes Rayleigh fading.
3
are thus motivated in this section to look into more convenient asymptotic expressions for the moments and cumulants, which allow in turn for an efficient computation of the c.d.f. with (moderately) large n and m.
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NO
A. Moments and Cumulants of W0 We aim to obtain simple expressions for the cumulants of W0 which, plugged into (8), allow for an efficient computation of the false alarm probability. Letting n and m be large but finite, we first provide an asymptotic expansion for the moments.
2 1 , 0
/ . , +
Proposition 1. The pth (p ∈ Z+ ) moment of W0 , with p < n n(m − n + 1) and c = m ∈ (0, 1), admits the expansion
?@A@BACDE FGH@IGDJKL M
µW0 ,p =
Probability of false alarm vs. detection threshold, with k = n.
Fig. 1.
∞ X βp,j (c) j=0
n2j
with βp,0 (c) = eAp,0 (c) , and for j > 0,
P
YQZU
βp,j (c) = eAp,0 (c)
YQZ
X
i1 +2i2 +...+jij =j l k j i b h g ] _ f _ d e d ` c b a ` _ ] ^ \ ] \
YQXU
YQX
|}PU~ T
YQWU
YQW
YQVU
YQV
YQUU
PQP
PQR
PQS
PQT
PQU
(13)
and, for q > 1,
YQU P
j Y Ap,r (c)ir , ir ! r=1
where the sum is taken over the non-negative integer solutions to i1 + 2i2 + . . . + jij = j, and the coefficients Ap,q (c) are 1 Ap,0 (c) = p − p 1 − log(1 − c), (14) c cp(12 − 11c + 6(1 − p)(c − 1)) −p2 log(1 − c) + , Ap,1 (c) = 2 12(c − 1) (15)
|}RY~
[
PQV
PQW
PQX
(cp)q c + 12p − 12cp − cq 2 Ap,q (c) = 12cq (−1 + q 2 ) (1 − c)q (cp)q −12p + c −1 + q 2 cq + − Hp,−q 12cq (−1 + q 2 ) q q−1 X B2j+2 c2j (2j)q−j (pc)q−j + 1 − (1 − c)−j−q , 4j (j + 1) (q − j)! j=1
mnonpoqrs tuvnwurxyz {
Fig. 2. k = n.
(12)
Probability of detection vs. detection threshold, with N0 = 5 and
terms. With these terms, the ‘Analytical (Correction)’ curve closely matches the simulations with improved accuracy as n and m increase. Finally, the ‘Analytical (Beta) [5]’ curve shows a satisfactory agreement for {n = 4, m = 15}, but it starts deviating for larger number of antennas and observations. For the detection probability curves in Fig. 2, we again observe a significant deviation of the Gaussian approximation from the Monte Carlo simulations, especially for the case {n = 4, m = 15}. Moreover, for {n = 10, m = 20}, the ‘Analytical (Beta) [5]’ curves are inaccurate for most detection probabilities as opposed to our ‘Analytical (Correction)’ curves, which closely match the simulations for both configurations.
(16)
where Ha,b and Bk are, respectively, the Harmonic and Bernoulli numbers. Proof: See Appendix B. We may now obtain corresponding expansions for the cumulants. Proposition 2. The pth (p ∈ Z+ ) cumulant of W0 , with p < n n(m − n + 1) and c = m ∈ (0, 1), admits the expansion κW0 ,p where
IV. A SYMPTOTIC A NALYSIS
∞ X αp,j (c) = , 2(p−1+j) n j=0
(17)
αp,j (c) = βp,p−1+j (c) p−r+j p−1 X X p−1 · − αr,ℓ (c) βp−r, p−r+j−ℓ (c) , r−1 r=1 ℓ=0 (18)
The false alarm and detection probabilities can be calculated for arbitrary number of antennas n and observations m using the moment expressions (6) and (7). However, the computation of such expressions, involving n-products of Gamma functions, gets rather involved when n and m are large. We 4
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when c = 1 is found to be
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κW0 ,1 ≈ e 1 κW0 ,2 ≈ 2 e2 (C + log n) n 1 κW0 ,p ≈ p ep (p − 1)! ζ(p − 1), n
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Fig. 3.
(26)
We now show the asymptotic convergence of W0 to a Gaussian distribution.
Probability of false alarm vs. detection threshold, with k = n.
Theorem 2. Let n → ∞ with with βp,j (c) given by (13).
n m
→ c¯ ∈ (0, 1). Then5
d
n (W0 − µ ¯) → N 0, σ ¯2
Proof: Follows by substituting (12) into (9) and rearranging terms in the resultant series. n ∈ (0, 1), the For large (but finite) n and m, and c = m leading-order term αp,0 (c) dominates the pth cumulant, giving κW0 ,p ≈
p > 2,
B. Gaussian Convergence of W0
¡
(25)
where ζ(·) is the Riemann-Zeta function. This result, albeit practically less meaningful than for the case c 6= 1, may still be of theoretical interest in other contexts.
« ª © ¨ § ¦
(24)
αp,0 (c) . n2(p−1)
where µ ¯ = e(1 − c¯)
,
(27)
(28)
and
(19)
2
Using (18) to obtain αp,0 (c) for the first four cumulants yields κW0 ,1 ≈ a−d e −1 κW0 ,2 ≈ 2 a−2d e2 [c + b] n −1 −3d−1 3 2 κW0 ,3 ≈ 4 a e c (2c − 3) − 6cab − 3ab2 n −1 −4d−2 4 3 κW0 ,4 ≈ 6 a e c (16 + c(6c − 23)) n 2 −12ac (3c − 4)b + 48ca2 b2 + 16a2 b3 ,
1−¯ c c ¯
2
σ ¯ = e (1 − c¯)
(20)
2(1−¯ c) c ¯
1 ln − c¯ . 1 − c¯
(29)
Proof: See Appendix D.
(21) (22)
C. Asymptotic Analysis of W1 Here, we first aim to obtain simple expressions for the cumulants of W1 which, plugged into (8), allow for an efficient computation of the detection probability for large (but finite) n and m. Recall that W1 is the GLRT statistic under the hypothesis of transmitted PU signals being present. In this case, as shown by (7), the moments (and therefore the cumulants) involve the eigenvalues of HH† + In N0 . Hence, it is convenient to first define certain functions of these eigenvalues.
(23)
where a = 1 − c, b = ln a, and d = 1 − 1c . Note that the leading-order cumulant approximations in (20)-(23) are much simpler to compute than the exact cumulants, especially when n is large. The false alarm probability can thus be efficiently computed via the approximated cumulants by plugging (20)-(23) into (10). To see the accuracy of such approximation, we compare it with the same c.d.f. expression (10) computed with the exact cumulants. This comparison is shown in Fig. 3, where we observe that the difference between the “asymptotic cumulants” and the “exact cumulants” curves is indistinguishable even for n as low as 4.
Definition 1. Let {y1 , y2 , . . . , yn } denote the eigenvalues of HH† + In N0 , with HH† positive definite, then Ψ1 ,
n Y
1 −n
yi
(30)
i=1
Ψℓ+1 ,
Remark 1 (The case c = 1). Note that the expansions for the moments and the cumulants, as given in Propositions 1 and 2, are not valid when c = 1. For this particular case, different expansions involving all powers of n−1 are obtained in Appendix C and provide analytic continuation to the aforementioned propositions. Remarkably, the leadingorder approximation to the pth cumulant, p < n(m − n + 1),
n
1X ℓ y , n i=1 i
ℓ ≥ 1,
with Ψ1 ∈ 0, N0−1 , Ψℓ+1 ∈ (N0 , ∞) . d 5x →
5
y implies x converges in distribution to y.
(31)
n ∈ Proposition 3. For large (but finite) n and m, and c = m (0, 1), the first three cumulants of W1 are given by 1−c 1 c (32) κW1 ,1 = e(1 − c) Ψ1 Ψ2 + O n2 1−c 1 κW1 ,2 = 2 Ψ21 e2 (1 − c)2 c n 1 1 − 2c + cΨ3 + O (33) × Ψ22 ln 1−c n4 1−c 1 κW1 ,3 = 4 Ψ31 e3 (1 − c)3 c n 2 c (−10 + 9c) 3 + 3 ln(1 − c)(4c + ln(1 − c)) × Ψ2 −1 + c 1 2 − Ψ2 Ψ3 3c(3c + 2 ln(1 − c)) + Ψ4 2c + O , n6 (34)
Let us first connect Ψ1 with the mutual information of the PU-SU channel through 1 1 1 , (37) exp − IH Ψ1 = N0 n N0 where IH
Proof: See Appendix E. Although only the first three cumulants are given in Proposition 3, higher order cumulants can be derived by following the same approach. However, such derivations become more tedious as the cumulant order increases. Nevertheless, as shown later in this paper, the first three cumulants are enough for an accurate computation of the detection probability. Plugged into (8), these cumulants yield the approximate detection probability for a given PU-SU channel realization H, which determines Ψℓ in Proposition 3. In practice, H is a random communication channel, and thus the detection probability can be seen as a random function of H. It turns out, however, that as n and k grow large, this function converges to a deterministic value, which depends only on some statistical properties of H, but not on its specific distribution. Deterministic Equivalents for Ψℓ : Note from (30)-(31) that 1 Ψ1 = det− n HH† + In N0 , ℓ 1 Ψℓ+1 = Tr HH† + In N0 , ℓ ≥ 1, (35) n ¯ ℓ such that6 for which we aim to find the convergence value Ψ as n → ∞ with
a.s.
(n)
(38)
ℓ X ℓ N0ℓ−r Jr , r r=1
(39)
where Jr is the rth moment: Jr = n→∞ lim k n →β
r 1 Tr HH† . n
(40)
Proof: Follows straightforwardly from (35) by using the binomial expansion. Both the mutual information and Jr have been broadly studied for n, k → ∞ in the context of information theory and large random matrix theory under different assumptions on the statistics of H. The rich body of existing results allows ¯ ℓ for a broad number of for computing the limiting values Ψ scenarios. Among these results, we focus on two particular cases, especially relevant in our detection problem: 1) IID case: the entries of H are IID with zero mean and 1 n variance. This is the case for multiple PU signals being colocated as, e.g., when a single PU sends spatially multiplexed signals with equal transmit powers. In this case, the asymptotic value of (38) is given by [22, Eq. (105)] (see also, e.g., [23, Eq. (95)], [21, Eq. (1.14)]), which yields −β ¯1 = 1 1 + 1 − 1F 1 , β Ψ N0 N0 4 N0 −1 β 1 1 1 N0 × 1+ − F ,β ,β F exp N0 4 N0 4 N0 (41)
(36)
k n
k n
→ β, and 2 q q √ √ x(1 + z)2 + 1 − x(1 − z)2 + 1 . F (x, z) =
as n → ∞ with → β, and Fβ (x) commonly referred to as the asymptotic spectrum with density denoted by fβ (x). Under the above assumption, commonly adopted in large random matrix theory (see e.g. [20], [21]), the limiting quan¯ ℓ are closely related to some asymptotic results which tities Ψ we invoke next. 6 x a.s. →
1 = log det In + HH† N0
¯ ℓ+1 = N ℓ + Ψ 0
k → β > 1, n
∀x ∈ R − {0}
¯ ℓ+1 , ℓ ≥ 1, are given by Lemma 1. The convergence values Ψ
¯ ℓ does not depend on the PU-SU where the limiting value Ψ channel realization H, but only on its statistical properties. To that end, we first state the following assumption. Assumption 1: The empirical distribution of the eigenvalues (n) of P an n × n Hermitian matrix HH† , denoted by FHH† (x) = n 1 i=1 1{yi ≤ x}, satisfies n FHH† (x) → Fβ (x) ,
1 N0
is the mutual information of H with average SNR 1/N0 . The asymptotics of this quantity have been extensively studied in information theory under different assumptions on the statistics of H (see, e.g., [21]–[25]). In view of (37), the existing asymptotic results for (38) can be directly translated into the ¯ 1. corresponding limiting value7 Ψ Let us now turn our attention to Ψℓ+1 , ℓ ≥ 1, which can be related to the so-called “moments” of HH† as shown in the following lemma.
with Ψℓ given in Definition 1.
a.s. ¯ Ψℓ → Ψ ℓ
with
(42)
7 Note that the limiting value of (38) yields the limiting value of (37) due to smoothness of the exponential function.
y implies that x converges almost surely to y.
6
Furthermore, Jr is obtained as [21, Eq. (2.102)] r r 1X r βi, Jr(iid) = r i=1 i i−1
ú íö
ÿ ÿ é ÿ ÿ é
(43)
þÿ ÿ
$ # " !
¯ ℓ+1 , ℓ ≥ 1. We can now which plugged into (39) yield Ψ ¯ ¯ ¯ compute the limiting Ψ2 , Ψ3 , Ψ4 , required for the cumulants in Proposition 3. Thus, setting ℓ = {1, 2, 3} in (39) gives
÷ù íö
¯ 2 = β + N0 , Ψ (44) ¯ 3 = β 2 + β (1 + 2N0 ) + N02 , Ψ (45) 3 3 2 2 ¯ Ψ4 = β + 3β (1 + N0 ) + β 1 + 3N0 + 3N0 + N0 . (46)
æçûé êçëì
2) Unequal variances: H = Hiid Σ1/2 where Hiid has IID entries with zero mean and n1 variance, whilst Σ is a diagonal matrix with non-negative entries σ12 , . . . , σk2 uniformly bounded. This accommodates multiple signals from spatially distributed PUs having different transmit powers. For this case we invoke [25, Thm. 1] to obtain − n1 2 Y k β ¯ 1 = β δ exp β δ δ˜ 1+ δσi2 , (47) Ψ N0 N0 N0 i=1 with
k n
→ β,
−1 1 β ˜ δ= 1+ δ , β N0
í
i=1
β ki
íîí
íîï
íîð
íîñ
íîò
íîó
íîô
íîõ
%&'&(')*+ ,-.&/-*012 3
Detection probability vs. detection threshold, with k = n, N0 = 6.
Fig. 4.
QON
E NOM D C B ; A @ 6 8 ? 8 = NOK > = 9 < ;
(48)
: 9 8 6 7 5 6 5
NOH F G HI JI KI LI M
4
(49)
klm nopqr NOP
For Jr , according to [26, Thm. 4], Jr(unequal) = r i X
æçëüé êçýü
÷ø íö
and δ˜ being the unique positive solution to k 1 X σi2 (β δ˜ + N0 ) . δ˜ = k i=1 σi2 + N0 + β δ˜
æçèé êçëì
fg QN
fh QN
fj QN
i QN
RSTUV WTSXY ZX[\S\]T]^_` Zab cde
X
r1 +...+ri =r
ξ(r1 , . . . , ri ) · Tr (Σr1 ) · · · Tr (Σri ) ,
Fig. 5. Analytic ROC curve (detection probability vs. false alarm probability) with N0 = 6, m = ⌈5n⌉ and k = n.
(50)
where {r1 , . . . , ri } are the strictly positive integer solutions to r1 + . . . + ri = r satisfying r1 ≤ . . . ≤ ri , and ξ(r1 , . . . , ri ) =
r! , (r − i + 1)!f1 ! · · · fr !
ℓ = 1, . . . , 4, given respectively by either (41) and (44)-(46), or (47) and (52)-(54). Plugging these cumulants in (8), we can compute the detection probability in two typical scenarios of interest: 1) detection of a single PU which transmits k spatially multiplexed signals with equal power, or 2) multiple spatially distributed PUs with different transmit powers. As an example, Fig. 4 shows the detection probability vs. the detection threshold corresponding to the first scenario (IID case), with the same number of antennas at the PU and the SU, i.e., k = n. The solid curve is obtained with the asymptotic ¯ ℓ , while the dots are computed via the exact Ψℓ for a values Ψ particular channel realization H. On the one hand, we see that the correction curve provides a satisfying match with Monte Carlo simulations. On the other hand, the correction based on ¯ ℓ is almost indistinguishable from the one based on Ψℓ for Ψ a not-so-large n = 8, which shows the quick convergence of Ψℓ . This is encouraging since, even for a moderate number of antennas and observations, the detection probability can be computed without explicitly knowing H.
(51)
with fj being the number of entries in {r1 , . . . , ri } equal to j. ¯ ℓ+1 , ℓ ≥ 1, for the unequal variances We can now compute Ψ case by plugging (50) in (39), which results in ¯ 2 = N0 + β Tr(Σ), (52) Ψ k 2 ¯ 3 = N 2 + 2N0 β Tr(Σ) + β Tr(Σ2 ) + β Tr2 (Σ), (53) Ψ 0 k k k2 2 β β 2 3 2β 2 ¯ Ψ4 = N0 + 3N0 Tr(Σ) + 3N0 Tr(Σ ) + 2 Tr (Σ) k k k 2 3 β β β (54) + Tr(Σ3 ) + 3 2 Tr(Σ)Tr(Σ2 ) + 3 Tr3 (Σ). k k k ¯ ℓ is attained As we will see shortly, the convergence Ψℓ →Ψ with high accuracy for not-so large n. Thus, under either the IID or the unequal variances assumption on H, the cumulants ¯ ℓ, in Proposition 3 can be accurately approximated using Ψ 7
|
V. T HRESHOLD D ESIGN
z{
Having derived expressions to compute the probabilities of detection and false alarm, we now put these expressions to work in order to design the detection threshold. As previously discussed, the PU-SU channel is typically unknown in practice, and therefore, the design usually relies on a false alarm probability requirement. Once the threshold is set, we can, under some statistical assumptions on H, e.g., IID or with unequal variances, compute the corresponding detection probability for different PU-SU scenarios such as, e.g., a single transmitting PU with k antennas, or k spatially distributed PUs. From (10), the false alarm probability is approximated by
z{ ¬ « z{ ª £ © ¨ z{ § ¥ ¦ ¥ z{ ¡ ¤ £ ¢ z{ ¡ z{~
~
z{}
z{|
z
Â
uv
st
uw st
ux
y
st
st
®¯°±² ³°¯´µ ¶´·¸¯¸¹°¹º»¼ ¶½¾¿ÀÁ
PFA (η) = 1 − FW0 (η) η − µW0 ,1 , σW0 , κW0 ,3 , κW0 ,4 , ≈1−G σW0
Fig. 6. ROC curve (detection probability vs. false alarm probability) with N0 = 6, m = ⌈5n⌉ and k = n. The vertical dotted line at a false alarm probability of 0.01 is shown for convenience.
(55)
2 where µW0 ,1 , σW , κW0 ,3 , and κW0 ,4 are given by (20)-(23), 0 and thus the minimum threshold which can satisfy a false alarm probability requirement of α0 can be approximated as
η0 ≈ P−1 FA (α0 ).
Ì
ÊËÔ
(56)
ÊËÓ
This threshold can be computed numerically from (55), and then used to obtain the corresponding detection probability PD (η0 ) = 1 − FW1 (η0 ) r ≈ 1 − Φ(x) +
2
− x2
2 e 2 3 κW1 ,3 (x − 1), π 12σW 1
(57)
η −µ
2 , κW1 ,3 given by (32)with x = 0 σWW1 ,1 and µW1 ,1 , σW 1 1 (34). The pair of values {PFA (η), PD (η)} defines the receive operating characteristics (ROC) curve, which is plotted in Fig. 5 for the IID scenario (a single transmitting PU with k antennas), with N0 = 6, k = n, and m = 5n. We see that, with an average SNR of −10 log10 N0 = −7.78 dB, a low false alarm probability and high detection probability can be simultaneously achieved with n > 7 antennas and m > 35 observation samples. The above design and ROC analysis is based on accurate representations of PFA (η) and PD (η), which have been numerically validated earlier in this paper. However, the design threshold needs to be computed by numerical search. In order to simplify the threshold design, we focus on our asymptotic results next. For large n and m, Theorem 2 suggests that the false alarm probability can be approximated as n (η − µ ¯) 1 √ , (58) 1 − erf PFA (η) = 1 − FW0 (η) ≈ 2 2¯ σ2
ÊËÒ
ÊËÑ
ÊËÐ Õ Ö Î× Ï× Ð× Ñ× Ò
ÊËÏ
ÊËÎ
ØÙÚ ÛÜÝÞß
ÊËÍ
àáÝßâáãäå æç èéêë ÊËÌ
Ê
àáÝßâáãäå æç èéìë
ÅÆ
ÃÄ
ÅÇ ÃÄ
ÅÈ
É
ÃÄ
ÃÄ
íîïðñ òïîóô õóö÷î÷øïøùúû õüýþÿ
Fig. 7. ROC curve (detection probability vs. false alarm probability) with N0 = 6, m = ⌈7n⌉ and k = n. The vertical dotted line at a false alarm probability of 0.01 is shown for convenience.
and 7 for c = 1/5 and c = 1/7 respectively. The ‘ROC curve’ is plotted using Monte Carlo simulations. The threshold computed via the Gaussian approximation in (59) for a target false alarm probability of α0 = 0.01 is shown. For comparison, the threshold computed with the c.d.f. approximation function (through higher order cumulants) in (56) is also shown. We observe that the approximate threshold (59) yields a false alarm probability slightly above the requirement α0 = 0.01, whilst this requirement is successfully met with the threshold in (56). As expected, this loss in accuracy diminishes as n (and consequently m) increases, which is in agreement with Theorem 2. Further, for satisfactory detection probabilities above 80%, the threshold in (56) results in a false alarm probability that tightly meets the requirement α0 = 0.01. Moreover, we observe in Figs. 6 and 7 that decreasing c for the same n results in a higher detection probability, as more observations are utilized for detection.
and thus the minimum threshold which can satisfy a false alarm probability requirement of α0 can be approximated as √ 2¯ σ 2 −1 η0 ≈ µ ¯+ erf (1 − 2α0 ) . (59) n A natural question then is whether this approximation is accurate enough for practical numbers of antennas and observations. To investigate this, we plot the ROC curve in Figs. 6 8
Leibniz integral rule, and b follows from [28, Eq. (50)]. From [29, Eq. (8.3.10)], we rewrite (62) as Qn n(n−1) −m (−1) 2 ℓ=1 yℓ µW,p = p Qn Q −1 −1 n ℓ=1 (m − ℓ)! i 2.
∞ k k−1 X c a k=1
(98)
n2k
with m = n/c and Proposition 1 we arrive at 1 µW1 ,2 = Ψ21 eA2,0 (c) Ψ22 + 2 Ψ22 (A2,1 (c) − c) + Ψ3 c n ! 2 A (c) 1 2,1 + A2,2 (c) + 4 Ψ22 n 2 1 +(Ψ3 − Ψ22 )(A2,1 (c) + c2 ) + O . (99) n6
(91) 2
a nm
=
(92) (93)
D. Proof of Theorem 2
2 Now, the variance σW = µW1 ,2 − µ2W1 ,1 is obtained using 1 (95) and (99) as
Through the invariance and homogeneity property of cumulants, the pth cumulant of n(W0 − µ ¯), for p ≥ 2, can be written as np κW0 ,p . From (17), we thus observe that for p ≥ 3, limn→∞ np κW0 ,p = 0, and thus limn→∞ n(W0 − µ ¯) follows a Gaussian distribution, with zero mean and variance given by limn→∞ n2 κW0 ,2 which is obtained from κW0 ,2 given by (21).
2 = σW 1
1 2 A2,0 (c) Ψ e n2 1
Ψ22 (A2,1 (c)
11
− c) + Ψ3 c −
2Ψ22 A1,1 (c)
+O
1 , n4 (100)
which yields (33) upon substituting A2,0 (c), A2,1 (c), and A1,1 (c) with their respective values given in Proposition 1. For the third cumulant, we first compute the third moment from (7). For p = 3 we have 2 Q 6 ni=1 yin µW0 ,3 µW1 ,3 = n3 m − n1 m − n2 X n 1 3 1 3 m− +2 m− +1 3 6 n n y i=1 i X 1 1 3 3 + m− +1 m− 2 2 n n i,j=1,...,n yi yj
is obtained using (95), (99), and (105), resulting in κW1 ,3 =
2 3 + m− n
X
i,j,k=1,...,n i6=j6=k
1 . yi yj yk
i,j,k=1,...,n i6=j6=k
i6=j
R EFERENCES [1] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE J. Select. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [2] C. Cordeiro, K. Challapali, and N. S. Shankar, “IEEE 802.22: the first worldwide wireless standard based on cognitive radios,” in 1st IEEE International Symposium on New Frontiers in Dynamic Spectrum Access Networks (DySPAN), Baltimore (USA), Nov. 2005, pp. 328–337. [3] J. W. Mauchly, “Significance test for sphericity of a normal n-variate distribution,” Annals of Mathematical Statistics, vol. 11, no. 2, pp. 204– 209, Jun. 1940. [4] K. Pillai, “On the distribution of the sphericity test criterion in classical and complex normal populations having unknown covariance matrices,” Annals of Mathematical Statistics, vol. 42, no. 2, pp. 764–767, Apr. 1971. [5] M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 33, no. 2, pp. 387–392, Apr. 1985. [6] D. Cochran, H. Gish, and D. Sinno, “A geometric approach to multiplechannel signal detection,” IEEE Trans. Signal Processing, vol. 43, no. 9, pp. 2049–2057, Sep. 1995. [7] S. Sirianunpiboon, S. Howard, and D. Cochran, “Multiple-channel detection of signals having known rank,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), May 2013, pp. 6536–6540. [8] A. Taherpour, M. Nasiri-Kenari, and S. Gazor, “Multiple antenna spectrum sensing in cognitive radios,” IEEE Trans. Wireless Commun., vol. 9, no. 2, pp. 814–823, Feb. 2010. [9] P. Wang, J. Fang, and N. Han, “Multiantenna-assisted spectrum sensing for cognitive radio,” IEEE Trans. Veh. Commun., vol. 59, no. 4, pp. 1791–1800, May 2010. [10] L. Wei and O. Tirkkonen, “Spectrum sensing in the presence of multiple primary users,” IEEE Trans. Commun., vol. 60, no. 5, pp. 1268–1277, May 2012. [11] D. Ram´ırez, G. Vazquez-Vilar, R. L´opez-Valcarce, J. V´ıa, and I. Santamar´ıa, “Detection of rank-P signals in cognitive radio networks with uncalibrated multiple antennas,” IEEE Trans. Signal Processing, vol. 59, no. 8, pp. 3764–3774, Aug. 2011. [12] Y. Fujikoshi, V. V. Ulyanov, and R. Shimizu, Multivariate Statistics: High-Dimensional and Large-Sample Approximations. New Jersey: John Wiley, 2010. [13] T. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Transactions on Wireless Communications, vol. 9, no. 11, pp. 3590–3600, Nov. 2010. [14] F. Rusek, D. Persson, B. K. Lau, E. Larsson, T. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities and challenges with very large arrays,” IEEE Signal Processing Mag., vol. 30, no. 1, pp. 40–60, Jan. 2013. [15] H. Li, J. Hajipour, A. Attar, and V. Leung, “Efficient HetNet implementation using broadband wireless access with fiber-connected massively distributed antennas architecture,” IEEE Wireless Commun. Mag., vol. 18, no. 3, pp. 72–78, Jun. 2011. [16] P. Bianchi, M. Debbah, M. Maida, and J. Najim, “Performance of statistical tests for single-source detection using random matrix theory,” IEEE Trans. Inform. Theory, vol. 57, no. 4, pp. 2400–2419, Apr. 2011.
1 , yi yj yk
and
k=1
k=1
1
X
y2y i,j=1,...,n i j i6=j
,
(103)
we can rewrite (101) as µW0 ,3 µW1 ,3 = Ψ31 m − n1 m − n2 ! 2 3 3 2 3 3 m− + Ψ2 m − + Ψ4 2 Ψ2 Ψ3 n n n n 1 2 nm 1 − nm 1 + 3Ψ2 Ψ3 − 4Ψ32 1 nm 1 − nm
= Ψ31 µW0 ,3 Ψ32 + Ψ32
+ (2Ψ4 − 3Ψ2 Ψ3 )
1
n 2 m2 1 −
1 nm
1−
2 nm
!
.
(104)
Then, using (98) together with the expansion for µW0 ,3 given in Proposition 1, we arrive at µW1 ,3 1 Ψ32 + 2 Ψ32 A3,1 (c) + 3Ψ2 Ψ3 c − 3Ψ32 c = n ! 2 A (c) 1 3,1 3 + A3,2 (c) + 3Ψ2 Ψ3 cA3,1 (c) + 4 Ψ2 n 2 1 3 2 3 2 . (105) +O −2Ψ2 c − 3Ψ2 cA3,1 (c) + 2Ψ4 c n6 Ψ31 eA3,0 (c)
Finally, the third cumulant,
κW1 ,3 = µW1 ,3 − 3µW1 ,2 µW1 ,1 + 2µ3W1 ,1 ,
A22,1 (c) − 3A2,2 (c) 2
which finally yields (34) upon substituting Ap,q (c) with the expressions given by Proposition 1, and further simplifications.
(102) n n n X X 1 X 1 1 = 2 y yk yk3 i=1 i
A23,1 (c) + A3,2 (c) − 3cA3,1 (c) 2
15 2 A (c) + 3A1,2 (c) 2 1,1 +Ψ2 Ψ3 3c(A3,1 (c) − A1,1 (c) − A2,1 (c) − c) + Ψ4 2c2 , (107) +3A2,1 (c)c + c2 +
(101)
Noting that !3 n n X X X X 1 1 1 +6 +3 = 3 3 2 y y y y i=1 i i,j=1,...,n i j i=1 i
Ψ32
−3A1,1 (c)(A2,1 (c) − c) − 3
i6=j
1 A3,0 (c) 3 e Ψ1 n4
(106) 12
[17] R. H. Y. Louie, M. R. McKay, and Y. Chen, “Multiple-antenna signal detection in cognitive radio networks with multiple primary user signals,” in IEEE International Conference on Communications (ICC), Sydney (Australia), Jun. 2014. [18] S. Blinnikov and R. Moessner, “Expansions for nearly Gaussian distributions,” Astron. Astrophysics Suppl. Ser, vol. 130, pp. 193–205, May 1998. [19] S. Li, M. R. McKay, and Y. Chen, “On the distribution of MIMO mutual information: An in-depth Painlev´e-based characterization,” IEEE Trans. Inform. Theory, vol. 59, no. 9, pp. 5271–5296, Sep. 2013. [20] O. Ledoit and M. Wolf, “Spectrum estimation: A unified framework for covariance matrix estimation and PCA in large dimensions,” 2013. [Online]. Available: http://ssrn.com/abstract=2198287 [21] A. Tulino and S. Verd´u, Random Matrix Theory and Wireless Communications, ser. Foundations and Trends in Communications and Information Theory. Hanover (USA): NOW Publishers, 2004. [22] A. Moustakas, S. Simon, and A. Sengupta, “MIMO capacity through correlated channels in the presence of correlated interferers and noise: a (not so) large n analysis,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2545–2561, Oct. 2003. [23] Y. Chen and M. McKay, “Coulumb fluid, Painlev´e transcendents, and the information theory of MIMO systems,” IEEE Trans. Inform. Theory, vol. 58, no. 7, pp. 4594–4634, Jul. 2012. [24] P. Kazakopoulos, P. Mertikopoulos, A. Moustakas, and G. Caire, “Living at the edge: A large deviations approach to the outage MIMO capacity,” IEEE Trans. Inform. Theory, vol. 57, no. 4, pp. 1984–2007, Apr. 2011. [25] W. Hachem, O. Khorunzhiy, P. Loubaton, J. Najim, and L. Pastur, “A new approach for mutual information analysis of large dimensional multi-antenna channels,” IEEE Trans. Inform. Theory, vol. 54, no. 9, pp. 3987–4004, Sep. 2008. [26] L. Li, A. Tulino, and S. Verd´u, “Design of reduced-rank MMSE multiuser detectors using random matrix methods,” IEEE Trans. Inform. Theory, vol. 50, no. 6, pp. 986–1008, Jun. 2004. [27] A. T. James, “Distributions of matrix variates and latent roots derived from normal samples,” Annals of Math. Statistics, pp. 475–501, 1964. [28] M. Chiani, M. Z. Win, and A. Zanella, “On the capacity of spatially correlated MIMO Rayleigh-fading channels,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2363–2371, Oct. 2003. [29] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. San Diego, CA: Academic, 1965. [30] A. Voros, “Spectral functions, special functions and the Selberg Zeta function,” Communications in Mathematical Physics, pp. 439–465, 1987. [31] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. New York: Dover Publications, 1970.
13
