Subspace-based Cooperative Spectrum Sensing for ... - CiteSeerX

Subspace-based Cooperative Spectrum Sensing for Cognitive Radio Raghavendra Rao, Qi Cheng ECEN, Oklahoma State University Stillwater, OK 74078

Priyadip Ray EECS, Syracuse University Syracuse, NY 13244

Abstract— Cognitive radio with its capability to sense the radio environment and dynamically access the vacant spectrum opportunistically, is an important technology to improve the efficiency of current spectrum utilization. In this paper, a cooperative wideband spectrum sensing approach based on subspace methods is proposed to estimate the number of primary user signals present in the band of interest and their carrier locations. Specifically, each secondary user implements eigen-decomposition of its sample covariance matrix to provide a local estimate. The fusion center generates a global estimate through a weighted sum of these local estimates. Experimental results demonstrate the efficiency of the proposed algorithm in detecting the correct number of primary users and estimating their carrier frequencies.

I. I NTRODUCTION The ever increasing number of wireless services has resulted in making the electromagnetic radio spectrum one of the scarcest natural resources. Recent studies have shown that many portions of the spectrum are used sporadically [1] and the efficient utilization of the spectrum can provide a solution to the problem of the scarcity of the spectrum. This necessitates a shift from the fixed spectrum assignment policy of the past to an intelligent and dynamic assignment policy. Cognitive radio with its capability to sense its radio environment is a perfect tool to exploit this intermittent spectrum access and opportunistically find unused frequency bands which is best suited for the user’s communication requirements [2]. The first and key operation in the cognitive radio cycle is spectrum sensing [2], i.e., the ability to sense the spectrum and dynamically detect the presence of primary users. An important requirement of a sensing algorithm is that it can be executed in real-time while maintaining the desired rates of false alarms and missed detections. Existing spectrum sensing approaches can be broadly divided into two categories: narrowband and wideband approaches. For narrowband spectrum sensing, one particular frequency band of interest is examined at a time to determine whether it is occupied by a primary user. Depending on the amount of prior knowledge we have of the signal to be detected, different approaches can be adopted. A matched filter approach requires prior information of the signal structure and an energy detector can be adopted when we do not have that information. However, the energy detector is vulnerable to the noise floor uncertainty and needs to have an accurate information of it [3]. In [4], features such as cyclostationarity in signals have been explored for primary user detection.

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However, the aforementioned methods are limited to the detection of signals on a single prefixed frequency band. For more efficient spectrum access, the awareness of spectrum utilization over a wide frequency band is required and this calls for wideband spectrum sensing techniques. An early approach employs a bank of narrowband bandpass filters for wideband sensing [5]. Instead of considering one band at a time, [6] proposed a multiband detection framework in which multiple narrowband energy detectors are jointly optimized. However, in these approaches, the location and bandwidth of multiple bands are prefixed, which may lead to performance degradation in realistic scenarios when signals straddle the boundaries of frequency bins. A wavelet approach was proposed in [7] [8] to identify the boundary locations of non-overlapping spectrum bands, but it is sensitive to noise, causing spurious local extrema. Subspace methods, which use eigen-decomposition to separate the socalled structured (carrier signals) and unstructured (noise) components, require little knowledge of the number and type of signals and have been applied in [9] to gain knowledge of the quality and usage of the spectrum. Spectrum sensing by a single cognitive terminal using the methods discussed in [5]–[9] provides some information of the spectrum usage, but the performance of a single terminal is severely impaired due to the presence of weak signals due to channel fading. This is often the cause of missed detections, leading to potential interference to licensed primary users. To reduce this, if we lower the detection threshold, then more false alarms are generated and this reduces the efficiency of spectrum utilization. Spatial diversity has been explored extensively in wireless communication to combat fading. It can be similarly applied in the context of spectrum sensing and such cooperation among several secondary users provides significant advantages in alleviating the effect of destructive channel fading [10] [11]. While most of the existing cooperative spectrum sensing schemes are built on narrowband sensing approaches such as energy detection, there has not been much focus on cooperative schemes for wideband sensing. In this paper, a wideband sensing scheme based on the subspace method is explored as the building block for the proposed cooperative spectrum sensing scheme. As mentioned earlier, subspace-based methods require little or no knowledge of the number and type of signals present and are free from the boundary effects. In this approach,

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Asilomar 2008

the eigen-decomposition of the sample covariance matrix is conducted at individual secondary users. The estimation results regarding the number of primary signals and their corresponding carrier frequencies within the band of interest (BOI) are combined at a fusion center, which maybe one of the secondary users or the access point. It is shown that the effects of fading can be effectively suppressed through the collaboration among secondary users using the proposed method, leading to improved detection performance and more accurate carrier frequency estimation. The remainder of the paper is organized as follows. The problem under consideration is formulated in Section II. The proposed subspace-based cooperative spectrum sensing scheme is provided in Section III. In Section IV, a simulated example is given to demonstrate the effectiveness of the algorithm. Conclusions and future work are provided in the last section.

III. S UBSPACE - BASED C OOPERATIVE SPECTRUM SENSING ALGORITHM

A. Local processing at secondary users Let Z(t) be a vector of m samples at SUd ,1 Z(t) = [yd (t), yd (t + 1), · · · , yd (t + m − 1)]

yd (t) =

n

jωk t

hkd sk e

+ wd (t)

(1)

k=1

where t = 1, 2, . . . , T and d = 1, 2, . . . , D. The n primary signals, each has a carrier frequency fk . ωk is defined as ωk = 2πfk ΔT rad/sample, where ΔT is the sampling interval. hkd is the channel gain between the kth PU and the dth SU. Here, we consider Rayleigh fading channels, i.e., hkd follows a complex Gaussian distribution with zero mean and unit variance. sk is the complex symbol (e.g., QPSK) of the kth PU signal. We assume that T samples are taken within the smallest symbol duration among all the n signals, i.e., T · ΔT < min1≤k≤n Tsk , where Tsk is the symbol duration for the kth signal. For most communication systems, transmitted signals generally experience slow fading. Thus, we can assume that the channel gains remain constant for T samples. {wd (t)}Tt=1 is a sequence of complex Gaussian noise with variance σ 2 and is assumed to be independent across secondary users. The primary objective is to collaboratively determine the following, 1) How many primary signals (n) are present in the BOI? 2) What are their carrier frequencies (fk ’s)? After extracting the carrier information, the subsequent task is to obtain more detailed information regarding each PU signal, such as bandwidth occupied etc. The secondary users can then try to access the unoccupied portions of spectrum oppurtunistically for data transmission. This task is beyond the scope of this paper and we focus only on the solution for the sensing task.

(2)

It can be easily shown that Z(t) can be rewritten in a more compact form, Z(t) = AS(t)H + W (t)

(3)

where A is an m × n matrix A = [A1 , A2 , · · · , An ] with T column Ak = 1, ejωk , · · · , ejωk (m−1) for k = 1, . . . , n. S(t) is an n × n diagonal matrix, the kth diagonal element T of which is signal sk ejωk t . H = [h1d , h2d , · · · , hnd ] and T W (t) = [wd (t), wd (t + 1), · · · , wd (t + m − 1)] . Then the covariance matrix of Z(t) is given by

II. P ROBLEM F ORMULATION An unknown number (n) of primary users (PUs) are transmitting signals over a wide BOI. They may differ in their carrier frequencies and occupied bandwidth. There are D secondary users (SUs) in vinicity trying to access (portions of) the BOI opportunistically. The signal samples yd (t) at SUd can be modeled as follows,

T

R

= E[Z(t)Z H (t)] = AP AH + σ 2 Im

(4)

where Im is an m × m identity matrix. Here, P = S(t)E[HH H ]S H (t) is a diagonal matrix with the diagonal element being |hkd |2 |sk |2 . m is selected such that n < m T . After eigen-decomposition of matrix R, we have R = V ΛV H

(5)

where Λ is a diagonal matrix, whose diagonal elements (λl ’s) are the eigenvalues and the column vectors (Vl ’s) of V are the corresponding eigenvectors. Since A is a full-rank matrix and P is positive definite, if the eigenvalues are arranged in the descending order, i.e., λ1 ≥ λ2 ≥ · · · ≥ λm , we have > σ2 1 ≤ l ≤ n λl (6) = σ2 n < l ≤ m The eigenvectors corresponding to λl for l = 1, . . . , n expand the signal space S = [V1 , V2 , · · · , Vn ] and the noise space composes of the remaining eigenvectors, G = [Vn+1 , Vn+2 , · · · , Vm ]. If the number of PUs (n) is known, we can obtain the estimate of the frequencies {ωk }nk=1 as in [12]. Let S1 = [Im−1 0]S and S2 = [0 Im−1 ]S. Here, Im−1 is a (m − 1) × (m − 1) identity matrix. It can be shown that the estimates of primary carrier frequencies {ˆ ωk }nk=1 are angular positions of the eigenvalues of matrix φ, where, φ = (S1H S1 )−1 S1H S2

(7)

Generally, we do not have the covariance matrix R and we have to estimate it from the data. The sample covariance matrix can be used as an estimate, M ˆ= 1 Z(t)Z H (t) R M t=1 1 We use notation * for conjugate, transpose throughout the paper.

T

for transpose and

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(8) H

for conjugate

ˆ approaches R as M goes to where M = T − m + 1. R infinity. In the context of cognitive radio, the number of PUs (n) is not known beforehand. In fact, this is the information we want to estimate from the sensing data. Since we use a finite sample average to estimate the covariance matrix, the m − n smallest eigenvalues are not exactly equal to σ 2 . It has been shown in [14] that if P = m − n is the correct dimension of the noise space, then the sum square of the P normalized eigenvalues follow a chi-squared distribution with P degrees of freedom. A threshold can be set on this statistic to satisfy the probability of false alarm (the probability of overestimating n). Based on this, an estimate n ˆ d of the number of PUs at SUd may be obtained and can be used in (7) for carrier frequency estimation. If n ˆ d < n, we have missed detection, while if n ˆ d > n, we have false alarms. When n ˆ d = n, we define it as a correct detection. B. Global fusion If some PU signals experience deep fading at certain SU, i.e., |hkd |2 is very small, then there is a high probability that the SU will fail to detect these weak PU signals. If we lower the thresholds (tl ’s) to make the test more sensitive to weak signals, inevitably false alarm increases. Thus, the performance of a single cognitive terminal is very sensitive to channel fading. But fading is assumed to be spatially uncorrelated and it is unlikely that all SUs will miss one particular PU signal or all SUs generate a similar false alarm. Through collaboration of SUs, we expect to achieve more robust estimation of the number of PUs and their carrier frequencies. In this paper, we propose fusion of the information from different SUs to enhance the overall performance. Also, each SU locally performs a subspacebased algorithm to estimate the number and the frequency of the carriers present. Due to fading and finite sample estimate of the covariance matrices, different SUs may provide nonidentical estimates of the number of carriers present. Thus, there is a necessity of data association, i.e., which estimates from different secondary users belong to the same PU signal. Next, we provide a data association algorithm. The problem as how to fuse the estimates from different SUs will be addressed subsequently. ˆ d frequency 1) Data Association: Assume that SUd has n ˆd , d = 1, . . . , D, 0 ≤ n ˆ d < m. The estimates {ˆ ωkd }nk=1 idea behind our data association algorithm is to group those estimates that are close to each other into clusters. Each cluster can have at most one frequency estimate from each SU. The estimate after fusion of the estimates within each cluster will be the final estimate of the PU signal. Fig. 1 shows frequency estimates from 10 SUs, among which SU10 does not detect any signal, SU1−3 detect one, SU4−6 detect two, SU7−8 detect three and SU9 detects four. As can be seen, these estimates form four clusters. The left most cluster has only one estimate, which is very likely to be a false alarm. To be specific, the data can be associated in a sequential manner as follows:

SU

1

SU2 SU3 SU4 SU5 SU6 SU7 SU8 SU9

−1

−0.5

0

0.5

1 1.5 ω(rad/sample)

2

2.5

3

Fig. 1. Illustration of the data association algorithm. SNR=-15dB, D = 10. There are only three PU signals in the source located at 0.31, 1.57 and 2.83 rad/sample, respectively.

1) Sort n ˆ d in an ascending order. Without loss of generˆ2 ≤ . . . ≤ n ˆ D . If n ˆ d = 0, it ality, we assume n ˆ1 ≤ n indicates that SUd does not detect any PU signal. For ˆ the first n ˆ d0 = 0, set the final frequency estimates ω ˆ d0 }. to be ω ˆ = {ˆ ωkd0 |1 ≤ k ≤ n 2) For SUd with d = d0 + 1, calculate distances between all possible frequency associations and select ωid − ω ˆ j(d−1) |, 1 ≤ the minimum one: Δ = mini,j |ˆ ˆ (d−1) . If Δ < th where th is a prei≤n ˆd, 1 ≤ j ≤ n ωid − defined threshold,2 then (i0 , j0 ) = arg mini,j |ˆ ˆ i0 d and ω ˆ j0 (d−1) are the ω ˆ j(d−1) |, indicating that ω estimates of the same PU signal. Then, remove this pair from the two frequency estimate sets and find the data association for the next PU: (i1 , j1 ) = ωid − ω ˆ j(d−1) |. Follow these steps arg mini=i0 ,j=j0 |ˆ until all estimates in ω ˆ have been considered and possibly associated. The frequency estimates are updated ˆ (d) } where ω ˆ k(d) is the to ω ˆ = {ˆ ωk(d) |1 ≤ k ≤ n fused estimate3 of the associated estimates so far. Note ˆ d because there may that n ˆ (d) may be different from n be false alarms from previous steps which cannot be associated with any of the current list of estimates. Set d = d + 1 and go back to step 2 until d = D. 3) The final frequency estimates are given by ω ˆ = ˆ (D) }. If ω ˆ k(D) is resulted from {ˆ ωk(D) |1 ≤ k ≤ n only one estimate, it is considered as a false alarm. By removing the frequency estimates that we deem to be false alarms, we obtain the final estimate of the number of ˆ ). PUs (ˆ n) and their carrier frequencies ({ˆ ωk }nk=1 2) Fusion of Estimates: At the fusion center, we have n ˆd (∈ [0, m − 1]) frequency estimates from SUd , d = 1, . . . , D. The probability density function (pdf) of the frequency estimate at individual SUs is very difficult to obtain. Asymptotically, ω ˆ kd converges to ωk in mean square sense [13]. It has also been shown in [13] that the large sample variance of the frequency estimate is inversly propotional to the square 2 This step is to reduce the probability that estimates for two different frequencies be associated. Usually, th can be chosen according to the accuracy of frequency estimation. 3 The fusion algorithm is provided next.


of the SNR, where the proportionality factor depends on m and the relative locations of the frequencies (not the frequency values). Specifically, at SUd , 1 2 2 ∝ SN Rkd = σkd

λkd − σ ˆd2 σ ˆd2

2 (9)

where the noise power σ ˆd2 at SUd can be estimated as follows σ ˆd2

m−ˆ nd 1 = λnˆ d +l m−n ˆd

(10)

l=1

For optimum fusion of the estimates obtained from different SUs, complete knowledge of the statistics of the estimates is required. As analytical results for the statistics of the estimate for finite number of samples is very difficult to obtain, in this paper we assume that the estimate of each SU is unbiased and the variance of the estimate is given by (9). If an SU misses a certain frequency (i.e., n ˆ d < n), we assign an infinite variance to the estimate of that frequency. We show that fusion based on such assumptions performs significantly better than a simple averaging scheme (a scheme in which the final estimate is the average of all the estimates). Under the assumptions provided above, we can rewrite the frequency estimates of the kth PU signal from all SUs as measurements ⎡ ⎤ ⎤ ⎡ ω ˆ k1 − ωk ω ˆ k1 ⎢ω ⎥ ⎥ ⎢ω ⎢ ˆ k2 − ωk ⎥ ⎢ ˆ k2 ⎥ (11) ⎥ ⎢ .. ⎥ = 1D ωk + ⎢ .. ⎦ ⎣ ⎣ . ⎦ . ω ˆ kD − ωk

ω ˆ kD

where 1D is a D × 1 all one vector. If a linear unbiased estimator is adopted and we try to minimize the mean square error (MSE), the fused estimate for the kth PU frequency is given by [15] ω ˆ k = (1D Ck−1 1D )−1 1TD Ck−1 XkH

(12)

where Xk = [ˆ ω1k , ω ˆ 2k , . . . , ω ˆ Dk ]T . Ck is the covariance matrix of the estimation errors. Since all SUs collect data independently, Ck is a diagonal matrix and (12) can be simplified as ω ˆk =

1 ˆ k1 2 ω σk1

+

1 2 σk1

1 ˆ k2 + · · · + σ21 ω ˆ kD 2 ω σk2 kD 1 1 + σ2 + · · · + σ2 k2 kD

(13)

which is a weighted sum of the local estimates and the weights are inversely proportional to the estimation variance and hence proportional to the square of the local SNRs. This indicates that the fusion center weighs the local estimates according to the square of their respective SNRs. In the situation that some SUs miss the kth PU signal, mathematically we simply ignore those SUs for the kth PU frequency estimation, by assuming an infinite variance for that estimate.

IV. S IMULATIONS We consider a scenario wherein a group of D secondary users try to estimate the number of primary user signals and their corresponding carrier frequencies. There are n = 3 primary users assumed to be centered at 100Hz, 500Hz and 800Hz.4 Furthermore, each PU has a QPSK modulated signal with unit energy. The signals are subjected to random channel gains and additive complex Gaussian noise with σ 2 varying from 20dB to 0dB. The dimension of the sample covariance matrix, m has to be greater than the maximum number of possible PU signals in the BOI. In the experiment, m is set to be 10. This indicates that the maximum number of signals that can be detected at any SU is 9. Within 0.5s, each SU collects 1000 samples and implements the subspace-based method to estimate the number of PUs and their carrier frequencies. The estimated frequencies and their corresponding SNRs are sent to the fusion center and are combined by the global fusion algorithm presented in Section III.B. The simulation results are based on 105 Monte Carlo runs (106 for σ 2 = 0dB). Fig. 2 shows the histograms of the final estimate of n (the number of PUs) by using one SU, five SUs and ten SUs, respectively. With only one SU, there is a high percentage of missed detections and some false alarms. By increasing the number of SUs, the percentage of correct detection increases dramatically to about 99.74% for ten SUs. For different values of noise power, the percentage of correct detections is shown in Fig. 3 with σ 2 varying from 20dB to 0dB for one, five and ten SUs. The dramatic increase in the probability of correct detection is expected because with the collaboration of multiple SUs, the chance of deep fading at all SUs reduces drastically . Further, due to the randomness of noise, all SUs generating the same false alarm is also very low. Note that missed detections usually occur due to deep fading, while false alarms occur due to random noise. The performance of frequency estimation is evaluated in terms of MSE. Fig.4 demonstrates the MSE performance for all the three PU signals using different number of SUs and different fusion schemes. The proposed weighted fusion scheme performs significantly better than the simple average fusion scheme. Similar to the analysis of estimation of n, collaboration provides more accurate carrier frequency estimation. V. C ONCLUSIONS In this paper, we consider the wideband spectrum sensing problem in cognitive radio and propose a subspace-based cooperative sensing scheme. The task is to estimate the number of PU signals present in the BOI and their carrier locations. At each SU, a preliminary estimate based on its own sample covariance matrix is obtained using the subspace 4 These frequency values are chosen for illustration purpose only. For a more realistic scenario where the BOI is at much higher frequency spectrum, it can be first converted to a lower frequency range using mixers. The sampling rate and duration can be adjusted accordingly to avoid ambiguity.


0

0

10

−1

10

−1

−2

−2

Weighted Sum

−3

10

−4

10

−5

10

−6

10

−6

−15

−10 1/σ2(dB)

−5

10 −20

0

(a) MSE of estimation of ω1 .

−6

−15

−10 1/σ2(dB)

(b) MSE of estimation of ω2 .

1

0.8 0.7

p(ˆ n)

0.6 0.5

10 −20

−15

−10 1/σ2(dB)

−5

0

(c) MSE of estimation of ω3 .

R EFERENCES

0.4 0.3 0.2 0.1 0

1

2

3

4 n ˆ

5

6

7

8

9

Fig. 2. Histogram of estimation of n by using one, five and ten SUs, respectively. σ 2 = 10dB, n = 3. 1 0.9 0.8 0.7 0.6 p(ˆ n = n)

0

number of samples. In our future work, we will explore other communication and fusion schemes to reduce the amount of communication while maintaining/improving the desired detection and estimation performance.

1 SU decen 5SUs decen 10SUs

0.9

0.5 0.4 0.3 1 SU decen. 5 SUs decen. 10 SUs

0.2 0.1 0 −20

−5

MSE performance as a function of 1/σ 2 of all three PU signals using one, five and ten SUs, respectively.

Fig. 4.

0

Weighted Sum

−5

10

10 −20

Average

−3

10

−4

Weighted Sum

10

−5

10

Average MSE(ω3)

MSE(ω2)

MSE(ω1)

−3

10

1 SU Weighted sum, 5 SUs Weighted sum, 10 SUs Average, 5 SUs Average, 10 SUs

−1

10

−2

10 Average

−4

10


10

10

10

0

10


−15

−10 2 1/σ (dB)

−5

0

Fig. 3. The percentage of correct detections as a function of 1/σ 2 using one, five and ten SUs, respectively. n = 3.

method. At the fusion center, after data association, local estimates are combined using an unbiased linear minimum mean square estimator. Simulation results show that the proposed algorithm can effectively identify the number of PUs present and their frequency locations. The detection and estimation performance are improved significantly through the collaboration among secondary users. There are still several issues worth further pursuing in future. The amount of communication of our proposed scheme include transmission to the fusion center the local frequency estimates and their SNRs. Also, the proposed fusion rule is based on several assumptions which may not be completely valid for estimates obtained from small

[1] M. A. McHenry, “NSF Spectrum Occupancy Measurements Projects Summary,” Shared Spectrum Company Report, Aug. 2005. [2] S. Haykin,“Cognitive radio: Brain Empowered Wireless Communications,” IEEE Journal Selet. Area on Communications, vol. 23, no. 2, pp. 201-20, Feb. 2005. [3] D. Cabric, A. Tkachenko and R. W. Brodersen, “Experimental Study of Spectrum Sensing based on Energy Detection and Network Cooperation,” ACM 1st International Workshop on Technology and Policy for Accessing Spectrum, Aug. 2006. [4] M. Ghozzi, F. Marx, M. Dohler and J. Palicot, “CyclostationarityBased Test for Detection of Vacant Frequency Bands,” 1st International Conference on Cognitive Radio Oriented Wireless Networks and Communications, June 2006. [5] A. Sahai and D. Cabric, “Spectrum sensing: fundamental limits and practical challenges,” IEEE International Symposium on New Frontiers in Dynamic Spectrum Access Networks, Nov. 2005. [6] Z. Quan, S. Cui, A. Sayed and H. V. Poor, “Wideband Spectrum Sensing in Cognitive Radio Networks,” ICC, May, 2008. [7] Z. Tian and G. B. Giannakis, “A wavelet approach to wideband spectrum sensing for cognitive radios,” 1st Int. Conference on Cognitive Radio Oriented Wireless Networks and Communications, Jun. 2006. [8] Y. Hur, J. Park, W. Woo, K. Lim, C.-H. Lee, H. S. Kim and J. Laskar, “A wideband analog multi-resolution spectrum sensing technique for cognitive radio systems,” IEEE International Symposium on Circuits and Systems, May 2006. [9] A. Wagstaff and N. Merricks, “A Subspace-Based Method for Spectrum Sensing,” 2007. [10] J. Unnikrishnan and V.V. Veeravalli, “Cooperative Sensing for Primary Detection in Cognitive Radio,” IEEE Journal of Selected Topics in Signal Processing, Vol. 2, No. 1, pp. 18-27, Feb. 2008. [11] Z. Quan, S. Cui and A.H. Sayed, “Optimal Linear Cooperation for Spectrum Sensing in Cognitive Radio Networks,” IEEE Journal of Select. Topics in Signal Processing, Vol. 2, No. 1, pp. 28-40, Feb. 2008. [12] R. Roy and T. Kailath, “ESPRIT-estimation of signal parameters via rotational invariance techniques,” IEEE Trans. on Acoustics, Speech, and Signal Processing, Vol. 37, No. 7, pp. 984-995, Jul 1989. [13] P. Stoica and T. Soderstrom, “Statistical Analysis of MUSIC and Subspace Rotation Estimates of Sinusoidal Frequencies,” IEEE Trans. Signal Processing, vol. 39, no. 8, pp 1836-1847, 1991. [14] J.-J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Trans. on Signal Processing, vol. 36, no. 12, pp. 18461853, Dec. 1988. [15] S. M. Kay, Fundamentals of Statistical Signal Processing, Vol. I Estimation Theory, Prentice Hall, 1993.
