Spectrum Sensing of MIMO SC-FDMA Signals in Cognitive Radio Networks: UMP-Invariant Test

Transcription

1 Spectrum Sensing of MIMO SC-FDMA Signals in Cognitive Radio Networks: UMP-Invariant Test Amir Zaimbashi Department of Electrical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran. ABSTRACT Key role in cognitive radio (CR) process play the spectrum sensing techniques which are employed by the unlicensed secondary users in order to detect the spectrum holes. It is, therefore, very important to do this with an optimal method. This inspire us to consider a Uniformly Most Powerful Invariant (UMPI) test as an optimum invariant test for the spectrum sensing problem. To this end, we develop UMPI -based spectrum sensing scheme for multiple-input multipleoutput (MIMO) single carrier frequency division multiple access (SC-FDMA) cognitive radio networks. In [1], authors have solved this detection problem via the widely known generalized likelihood ratio test (GLRT) with no optimality property for a finite number of observations, whereas the derived UMPI test is the optimum invariant test. Interestingly, we find that the UMPI test is equivalent to the GLRT obtained in [1]. Hence, we show that the GLRT is an optimal test for a finite number of observations in the above spectrum sensing problem. Keywords: Cognitive Radio, Spectrum Sensing, UMPI, GLRT, SC-FDMA, MIMO. 1. INTRODUCTION Cognitive Radio(CR) is the enabling technology for dynamic spectrum access allowing unlicensed secondary users (SUs) to utilize a frequency band when it is vacant of the licensed primary users (PUs) [1]-[3]. Hence, the important step of the CR process is to reliably detect spectrum areas that are temporarily unused by licensed PUs. Authors of [1] developed a generalized likelihood ratio test (GLRT) based spectrum sensing scheme for multiple-input multiple-output (MIMO) single-carrier frequency-division multiple access networks 1. As we all know, for small sample size, the GLRT shares no known optimality properties. In this paper, we aim to develop a Uniformly Most Powerful Invariant (UMPI) test as an optimum invariant test for this problem. Generally, for the small size sample problems, the purpose of considering invariant principle for detection is that in situations where no Uniformly Most Powerful (UMP) test exists, it is sometimes possible to find an optimal test in the smaller class of invariant tests. Based on this, we consider the spectrum sensing problem introduced in [1], and derive UMPI test. The results presented in this paper show that the UMPI and GLR detectors are identical. Therefore, the GLRT test derived in [1] is also optimum invariant test even with small size sample. It is worth noting that the derivation of the UMPI test is not as straightforward as the GLRT one and it is usually a skill that is developed more by experience than by deep mathematical insight. Based on this fact, the rest of paper is organized as follows. In section, we formulate the spectrum sensing problem in a MIMO SC-FDMA network as a composite hypothesis test problem. Section 3 is devoted to derive UMPI test as an invariant domain UMP test. Section 4 concludes this paper. Notation: Throughout the paper, scalars are denoted by nonboldface type, vectors by boldface lowercase letters, and matrices by boldface uppercase letters. Superscripts(. ) T, and (. ) H denote transpose and complex conjugate transpose, respectively. C N P denotes the set of complex N P matrices, I N denotes the Identity matrix of size N N and denotes the Kronecker matrix product.. MIMO SC-FDMA COGNITIVE RADIO SYSTEM MODEL Consider a MIMO SC-FDMA cognitive radio system described in [1] with N primary users for which M subcarriers are allocated and a single secondary user. Let N t and N r denote the number of transmit antennas at the primary system, and the number of receive antennas at the secondary system, respectively. The received signal vector over N r receive antennas of the secondary user, when the n th primary user transmit pilot symbols, can be formulated as y = (I Nr X)h + w (1) In the above expression, y C MKN r 1 is a concatenated vector of received signal vectors at the output of the M- point Inverse Discrete Fourier Transform (IDFT) post-precessing block over N r receive antennas of primary user; X C MK MN t is the pilot symbol matrix in which K being the total number of pilot symbol vectors transmitted (K > N t [1]), and h C MN rn t 1 and w C MKN r 1 are the concatenated channel and noise vectors. The interested reader is refereed to 1 SC-FDMA is a blockwise transmission scheme that efficiently exploits fast Fourier transform (FFT) to create a multiple-access technique with reasonable complexity, good performance, and low peak-to-average-power ratio (PAPR)[]-[3]. 1

2 [1] for more detail about received signal model. Based on this, the problem of detecting the n th primary user in the MIMO SC-FDMA based cognitive radio system can be formulated as a binary composite as follows { H 0: y = w, () H 1 : y = Ah + w, or equivalently, { H 0: h = 0, (3) H 1 : h 0 where A = (I Nr X), and H 0 and H 1 are hypothesis corresponding to the absence and presence of the primary user, respectively. In the following, it is assumed that the additive Gaussian noise w to be independent of the received signal Ah and has the covariance matrix R ww = σ I with unknown σ. In addition, A is known since it is constructed based on a prior known pilot symbol matrix X typically broadcasted for channel estimation, while the channel vector h is not known. Hence, the parameter space, Θ say, partitioned as follows Θ = {0, σ } {h, σ } (4) It is known that no UMP test exist for the detection problem at hand [4]. In what follows, we attempt to use the invariance principle to reduce observation data and parameter space to find a UMP test in invariant domain, which is known as UMPI test. Since matrix A is known, we can rewrite (1) into an equivalent form by left-multiplying (1) with MKN r MKN r matrix P, defined as P = [ (AH A) 1 A H ] (5) U H where U H A = 0 Here, A is MN t N r -dimensional signal subspace, and U is its corresponding MN r (K N t )-dimensional orthogonal subspace. We can then write the transformed vector as z 1 (A H A) 1 A H y z Py = [ U H y ] [ z ] (6) 3. UMPI-BASED SPECTRUM SENSING RECEIVER In detection problems in which no UMP test exists, it makes sense to focus on the class of detectors invariant to some transformations. To do so, we resort to the invariance principle to find a statistic called the maximal invariant, which maximally condenses the observations while preserves the necessary information from the observations to perform signal activity detection [4]-[9]. This means that a test obtained in an invariant domain depends on the observations only through a maximal invariant. In general, as we all know, the GLR test is invariant with respect to transformations for which the hypothesis test is itself invariant, but it is not any guarantee that the GLR statistic captures all information which are necessary to signal activity detection, while maximal statistic are expected to do it. Before proceeding further, let us first introduce the concepts of invariance and maximal invariant statistic that will be used in the sequel. Definition 1: A detector is said to be invariant to the group of transformations G if [4] d(g(z)) = d(z), g G (7) where d(. ) is the test function and z are the observations. Definition : A statistic M(z) is a maximal invariant if it satisfies [4] M(g(z)) = M(z) (8) for all g G and if M(z) = M(z ) (9) implies z = g(z) for some g G. Therefore every invariant test may be written as a function of a maximal invariant statistic, i.e., d(z) = d(m(z)) (10) Definition 3: The test d(z) = d(m(z)) is UMP-invariant of size α for testing H 0 : θ Θ 0 against H 1 : θ Θ 1 in model admit probability density function (pdf) z: f(z; θ) if, for the test d(z) and for every competing test ρ(z) that is also Ginvariant [4], 1. (size) sup θ Θ0 E θ {d(m(z))} = α; sup θ Θ0 E θ {ρ(z)} α. (power) E θ {d(m(z))} E θ {ρ(z)} for all θ Θ 1 where the subscript θ indicates that the expectation is a function of θ. This means that a UMPI test is a UMP test in an invariant domain, so it has uniformly higher power over the parameter set Θ 1 (higher detection probability) than any other invariant tests including GLR test.

3 To find a UMPI test, it is first necessary to identify the problem invariance with respect to some transformation groups. In the following, we show that the detection problem at hands is invariant under the transformation groups G defined as follows: V 1 0 G = {g: z γvz, V = [ 0 V ]} (11) where V 1 C MN tn r MN t N r and V C MN r(k N t ) MN r (K N t ) are both unitary matrices, and γ is a complex scalar. To show that the transformation group G is invariant to our decision problem, in the following, we show that the distribution of the observation and the parameter spaces remain unchanged under each hypothesis. In fact, under H 1, we have z 1 ~CN((A H A) 1 h, σ I MNt N r ) and z ~CN(0, σ I MNr (K N t )) and so, for the transformed data we get γv 1 z 1 ~CN(γV 1 (A H A) 1 h, γ σ I MNt N r ) and γv z ~CN(0, γ σ I MNr (K N t )). Since h and σ both are unknown, the distribution family of the transformed data does not change. UnderH 0, the proof is similar, except that h = 0. Thus, the detection problem is invariant under the set of transformation in G. It is easy to show that the GLR statistic obtained in [1] is also invariant to the transformation group G. In general, however, it is not any guarantee that the GLR statistic captures all information which are necessary to signal activity detection, while maximal statistic are expected to do it. More precisely, from the invariant property of the GLRT derived in [1], it has not been clear whether or not the GLRT has any optimality properties, so the question is, "can the GLRT be improved upon?". In the following, we try to answer this important question by obtaining the maximal statistic, which preserves all necessary information for the purpose of signal activity detection. The following lemma provides the expression of maximum invariant(mi) statistic under the group G. Lemma 1: The one-dimensional MI statistic under the group G is given by M(z) = z 1 (1) z where z k is the Euclidean norm of vector z k. Based on the original observation y, the MI given by t = yh Π A y y H (13) Π A y where Π A A(A H A) 1 A H is the projection matrix that projects a vector onto the columns of A and where Π A = I H Π A U U is the orthogonal projection matrix that projects a vector onto the space orthogonal to that spanned by the columns of A( see Appendix ). In the following, we show that the one-dimensional data t, which contains all the information of y necessary for the design of receiver, gives us better chances of finding a UMP test in invariant domain. To see this, it is thus necessary to obtain the pdf of the devised MI under both H 0 and H 1 hypotheses. In (1), y is distributed as y~cn(0, σ I MKNr ) under the null hypothesis, and y~cn(ah, σ I MKNr ) under the alternate hypothesis. This gives rise to the following distribution of t = MN rn t t as follows [9] MN r (K N t ) t ~ { F MN r N t,mn r (K N t ) under H 0 F MNr N t,mn r (K N t )(δ) under H 1 (14) where F R,L denotes a central complex F distribution with R numerator complex degree of freedom and L denominator complex degree of freedom, and F R,L (δ) denotes a noncentral complex F distribution with R numerator complex degree of freedom and L denominator complex degree of freedom and noncentrality parameter δ, given by δ = Ah (15) σ This shows that the probability density f(t ; δ) of MI only depends on the pair of unknown parameters θ = [h, σ ] T though a scalar parameter δ, which is the induced maximal invariant under induced group actions G, G say. As a result, the principle of invariance reduces the two-sided problem (1) to one-sided one of { H 0: δ = 0, (16) H 1 : δ > 0 with the reduced parameter space, Θ (r) say, partitioned as follows Θ (r) = {0} {δ > 0} (17) only with the unknown parameter δ. Now, according to the Karlin-Rubin theorem, a UMP test exist when f(t ; δ) has Monotone Likelihood Ratio(MLR) property. Fortunately, it is not difficult to show that noncentral complex F distribution has MLR in its noncentrality parameter[4]. Hence the UMP test is to reject H 0 if t > η t, or equivalently t = yh Π A y y H Π A y > η t (18) where threshold η t is set to satisfy the false alarm probability (P fa ) requirement in the test t > η t. It is worth to note that under H 0 the distribution of statistic t does not depends on δ, in which δ = 0, and therefore test based on t will Note that maximal invariants t and t contain the same information since they have a one-to-one relationship. 3

4 automatically be constant False alarm rate (CFAR). In this case, the probability of false alarm and the probability of detection for the UMPI detector based on the statistic t can be computed with the aid of (14), and they are given respectively by P fa = Q FMN rn t,mnr(k N t ) (η t ) (19) P d = Q FMN rn t,mnr(k N t )(δ)(η t ) (0) where Q FMN rn t,mnr(k N t ) and Q F MN rn t,mnr(k N t )(δ) are the right-tail probability of central and noncentral complex F distribution, respectively. By comparing the GLR test obtained in [1] with that of (18), it can be found that the the proposed UMPI test is equivalent to the GLR test. Based on the results of this paper, we show that the GLR test obtained in [1] is a UMP test in the invariant domain for a finite number of observations. Despite the difficulty of the UMPI test derivation as compare to that of GLR test, the payoff for the extra effort in the spectrum sensing applications can be high since we found that this is the best we can do when UMP test does not exist in the original domain. 4. CONCLUSIONS In this paper, it is shown that it is possible to find a uniformly most powerful test within a smaller invariant class, named UMPI test, even though no general UMP test may exist. Indeed, we see that invariant principle acts as a data reduction technique leading to a reduced observation space with significantly lower dimensionality than the original one. Furthermore, this reduction results in a scalar induced maximal invariant, which enable us to find UMP test in invariant domain. Generally speaking, a UMPI test has weaker optimality property than a UMP one but it may achieve a better performance as compared with the conventional GLR one, which does not possess any optimality property. In this case, we see that the proposed UMPI test is equivalent to that of GLR test obtained in [1]. As a result, we show that the GLRT is an optimal invariant test for a finite number of observations in the detection problem at hand. More precisely, we show that the GLR statistic captures all information which are necessary to signal activity detection for multiple-input multipleoutput (MIMO) single carrier frequency division multiple access (SC-FDMA) cognitive radio networks. APPENDIX: PROOF OF LEMMA 1 In this appendix, the maximal invariant (MI) statistic for the transformation group G is derived. From (11), it is seen that transformation group G is generated by two subgroups G V and G γ defined as follows: V 1 0 G V = {g V : z Vz, V = [ 0 V ]} (1) and G γ = {g γ : z γz, } () The following lemma shows that how the process of determining a MI in steps can be carried out for problems in which G generated by two subgroups G V and G γ. Lemma : Let G be a group of transformations, and let G V and G γ be two subgroups generating G. Suppose that p = M V (z) is maximal invariant with respect to G V, and for any g γ G γ, any z and any z, we have M V (z ) = M V (z) M V (g γ (z )) = M V (g γ (z)) (3) If t = M H (p) is MI under the group H of transformations g H defined by g H (p) M V (g γ (z)), then t = M H (M V (z)) is MI with respect to G. (see, e.g., [4], pp. 17, Ch. 6, Lemma 6..). According to above lemma, it can be shown that an MI for G V is a reduced-dimension function of the data z = [z 1 T, z T ] T, given by p = M V (z) = [ z 1, z ] T (4) This statistic is invariant, since for all g V G V we have M V (g V (z)) = [ V 1 z 1, V z ] T (5) Since V 1 and V are both unitary matrices, it can be concluded that M V (g V (z)) = M V (z). This show that the statistic p is invariant. To see the maximally of p, it is requires showing that given any pairs of data vectors z and z such that M V (z ) = M V (z), then there exist a transformation g V (. ) of the form (1), such that z = g V (z). Suppose that for given z and z, we have M V (z ) = M V (z), it follows from (4) that { z 1 = z 1 z = z (6) Consider two unitary matrices V 1 and V, with dimension MN r N t MN r N t and MN r (K N t ) MN ( K N t ) respectively, constructed by given vectors z = [z 1T, z T ] T and z = [z 1 T, z T ] T as V i = [ z i z i, E i] [ z i z i, E i] T (7) 4

5 for i = 1,. Here, matrices E i and E i must be constructed such that E ih z i = 0 and E H i z i = 0 for i = 1,. From (6), we can conclude that V i z i = z i for i = 1,. Substituting V 1 and V into (1), we are able to construct transformation g V (. ) such that z = g V (z). To obtain a MI for second group, we first verify the condition in (3), which leads to M V (g γ (z )) = M V (γz ) = γ M V (z ). From the assumption M V (z ) = M V (z), we obtain M V (g γ (z )) = M V (g γ (z)). Hence, the theorem condition of (3) is satisfied. Now, we should first find a group G H that acts on p and then find a MI under that group. According to Lemma, the transanimation group G H is given by G H = {g γ : p γ p} (8) The MI for the group G H is given by M H (p) = M H ([p 1, p ] T ) = p 1 in which p p 1 = z 1 and p = z. This statistic is invariant, since M H (g H (p)) = M H (p). Furthermore, from M H (p ) = M H (p) for given p and p, we obtain p 1 = p 1 and p p so p 1 = p γ. Therefore, [p 1, p 1] = γ [p p 1 p 1, p ] or p = g H (p) for some g H G H, as was to be shown. Hence, M H (p) is the MI under group G H. The MI under the composite group G is given by M(z) = M H (M V (z)) = z 1 z. REFERENCES [1] A. Kumar, S. Dwivedi, and A.K. Jagannatham, GLRT based Spectrum Sensing for MIMO SC-FDMA Cognitive Radio Networks, IEEE Wireless Communication Letter, DOI /LWC [] R. Dinis, D. Falconer, C. T. Lam, and M. Sabbaghian, A multiple access scheme for the uplink of broadband wireless systems, in Proc. IEEE GLOBECOM, Nov./Dec. 004, vol. 6, pp [3] A. Mirdamadi, and M. Sabbaghian, Spectrum Sensing of Interleaved SC-FDMA Signals in Cognitive Radio Networks, Vehicular Technology, IEEE Transactions on. vol.64, no.4, pp , 015. [4] E.L., Lehman, J.P.: Romano,Testing statistical hypothesis, (New York: Springer Verlag, 005). [5] H. S. Kim, Adaptive target detection in radar imaging, Commun. Signal Process. Lab., Univ. Michigan, Ann Arbor, MI, Tech. Rep. 34, Apr [6] A.Zaimbashi, Invariant Target Detection in Distributed MIMO Radar: Geometry Gain Helps Improving Moving Target Detection, IET Radar, Sonar and Navigation, pp.1-1, 016. [7] De Maio, S. Kay, and A. Farina, On the Invariance, Coincidence, and Statistical Equivalence of the GLRT, Rao Test, and Wald test,signal Processing, IEEE Transactions on. vol.58, no.4, pp , 010. [8] D. Ramirez, J. V, I. Santamara and L. L. Scharf, Locally most powerful invariant tests for correlation and sphericity of Gaussian vectors, Information Theory, IEEE Transactions on, vol.59, no.4, pp , 013. [9] L. Scharf, and B. Friedlander,"Matched Subspace Detectors," IEEE Trans. Signal Process., 4, 8 (1994),