Sinusoidal frequency estimation by multiple signal ... - IEEE Xplore

IET Signal Processing Research Article

Sinusoidal frequency estimation by multiple signal classification in frequency domain beam-space

ISSN 1751-9675 Received on 6th August 2014 Accepted on 6th October 2014 doi: 10.1049/iet-spr.2014.0246 www.ietdl.org

Wei Guo 1, Guanghao Shen 1, Kangning Xie 1, Xiaoming Wu 1, Chi Tang 1, Juan Liu 1, Min Jia 2, Da Jing 1, Tao Lei 1, Erping Luo 1 ✉ 1

School of Biomedical Engineering, Department of Military Medical Equipment and Metrology, The Fourth Military Medical University, Xi’an, Shaanxi, People’s Republic of China 2 Department of Physiology, The Fourth Military Medical University, Xi’an, Shaanxi, People’s Republic of China ✉ E-mail: [email protected]

Abstract: A novel method is presented to estimate sinusoidal frequency from highly contaminated single channel signals by constructing multi-channel surrogates using multiple signal classification (MUSIC) method in frequency domain beamspace (FB-MUSIC). According to the comparability of sampled data in time domain and observed data in uniform linear array, the FB-MUSIC method is proposed and the explicit expressions for the covariance elements of the estimation errors associated with FB-MUSIC are derived. These expressions are then used to analyse the statistical performance of FB-MUSIC and MUSIC. These expressions for the estimation error covariance are also used to compare the theoretical results and simulation results. Monte-Carlo simulations show that the root-mean-square error of frequency estimation in simulations keep consistent with the theoretical covariance for FB-MUSIC and MUSIC, and the signal-tonoise ratio resolution threshold of FB-MUSIC with reduced dimensionality is lower than that of MUSIC. This method may provide a higher resolution of sinusoidal frequency estimation and lower computation cost as compared with the conventional MUSIC method.

1

Introduction

In many applications, it is of importance to estimate the frequency of multiple sinusoids in additive white Gaussian noise. Various methods are advocated for frequency estimation, such as the fast Fourier transform method, maximum likelihood method [1] and subspace-based method [2, 3], but these methods exhibit their own advantages and limitations. In these methods, multiple signal classification (MUSIC), as a subspace-based method, is able to provide high resolution and precision in frequency estimation. The asymptotic performance studies for MUSIC have been discussed in some literatures [4–14]. However, the limitation of MUSIC is the high cost when computing the eigenvector. This issue could become even more remarkable provided that the dimensionality of eigenvector is large. Another limitation of conventional MUSIC is the notable degradation for its performance at low signal-to-noise ratio (SNR). To overcome these issues, some improved version of MUSIC for various occasions have been previously reported, such as DFT-MUSIC [15], ACM-MUSIC [16], ROC-MUSIC [17], FLOM-MUSIC [18], TR-MUSIC [19], HMUSIC [20], W-MUSIC [21], F-HMUSIC [22], FOC-MUSIC [23, 24], F-MUSIC [25, 26] and the other variants of MUSIC [27–29]. However, no method that was similar to MUSIC in beam-space for frequency estimation to date has been reported. Thus, the aim of the present study is to reduce the dimensionality of eigenvector and to improve the resolution performance at low SNR for frequency estimation in frequency domain beam-space, and another important aim is to provide an idea of estimating frequency using some other well-established methods for direction of arrival (DOA) estimation in beam-space. It is well known that MUSIC in beam-space possesses many advantages as compared with that in element-space [9, 13, 14, 30], such as lower SNR resolution threshold, dimensionality and computation cost. In this paper, we present a signal model that is similar to the array signal model in uniform linear array (ULA) and propose a MUSIC method in frequency domain beam-space for frequency estimation.

IET Signal Process., 2015, Vol. 9, Iss. 4, pp. 357–368 & The Institution of Engineering and Technology 2015

The proposed method significantly reduces the dimensionality of covariance matrix, decreases the computationalcomplexity and improvesthe resolution in frequency estimation. In this paper, Section 2 introduces thesignal model considered for frequency estimation. In the following section, the FB-MUSIC method is proposed by the comparabili ty of the signal model and ULA model. Section 4 analyses the statistical property of FB-MUSIC frequency estimation. Section 5 consists of numerical simulations and the general discussion. The last section contains the main conclusions. In addition, the mathematical notations are denoted as follows: (·)T (·)H (·)* E[·] diag{v} Re{·} (·)i,j δ(·)

2

transpose operation; conjugate transpose operation; complex conjugate operation; expectation operation; diagonal matrix whose diagonal is the vector v; real part of the embraced argument; (i, j)th element of the embraced matrix; Kronecker delta function.

Signal model

The data are assumed to consist of p complex sinusoids in additive complex white noise y(t) =

p


xi (t) + w(t)

(1)

i=1

Here xi (t) = ci ej(2pfi t+ui ) is the ith complex sinusoid. The amplitudes ci and frequencies fi of the complex sinusoids are unknown constants, and the frequency is distinct. The θi is assumed to be uniformly distributed in the interval [0, 2π). It is also assumed that θi and w(t) are independent, and the complex Gaussian white

357

RY = E{Y (n)Y (n)H } = ARX AH + s2 I

noise w(t) has zero mean and variance σ 2, such that ⎧ ⎨ E[w(t)] = 0 E[w(t)w∗ (s)] = s2 d(t − s) ⎩ E[w(t)w(s)] = 0

(2)

In this paper, the number of signals p is assumed to be known or has been estimated by some methods. The technique for estimating p has been well documented in the literatures [31, 32], which will not be discussed herein. The signal is sampled at discrete times t = T, 2T, …, LT. F = 1/T is the sampling rate. The samples can be expressed as y(k) =

p


xi (k) + w(k),

k = 1, 2, . . . , L

Here RX = E{X(n)X(n)H} is the covariance of signal. RX is assumed to be non-singular, and rank (ARXA H) = p. σ 2 is the variance of complex white noise. The covariance matrix RY can be decomposed to RY = ULU H

U H RY U = U H ARX AH U + s2 U H U

(3)

= diag(a21 , a22 , . . . , a2p , 0, . . . , 0) + s2 I



li =

a2i + s2 , i = 1, 2, . . . , p s2 , i = p + 1, . . . , M

Let S = [s1, s2, …, sp] denote the orthonormal eigenvectors of RY associated with l1, l2, …, lp, and G = [g1, g2, …, gM − p] denote the orthonormal eigenvectors of RY which are associated with lp + 1, lp + 2, …, lM. We also define U = [S|G]. Then, we observe that  G H RY G = G H [S|G]L

 0 SH = s2 I G = [0|I] L I GH

(5)

According to (14) and (10), it follows that

where a(fi ) = [1, ej2pfi (d/F) , . . . , ej2pfi ((M −1)d/F) ]T is the frequencybearing vector. We define the spatial interval of frequency as

G H ARX AH G = 0

d (f − fj ), F i

i, j = 1, 2, . . . , p,

i=j

1 fi d 1 ≤ ≤ , 2 F 2

i = 1, 2, . . . , p

(7)

AH G = 0

aH (f i )GG H a(f i ) = 0,

Y (n) = AX(n) + W (n),

n = 1, 2, . . . , N

(9)

Here X (n) = [c1 ej2pf1 (n/F)+u1 , c2 ej2pf2 (n/F)+u2 , . . . , cp ej2pfp (n/F)+up ]T and W(n) = [w(n), w(d + n), …, w((M − 1)d + n)]T are the signal vector and noise vector, respectively. This model is the same as the observed data model of ULA.

3

Proposed FB-MUSIC method

The covariance matrix RY of the sampled data is given in (10) based on the above model

(16)

i = 1, 2, . . . , p

(17)

RY is unknown in practice, but it can be consistently estimated from the sampled data. The equation is as follows N
ˆY = 1 Y (n)Y (n)H R N n=1

(8)

where fmax = max{f1, f2, …, fp}. According to (4) and (5), the data vector Y(n) can be decomposed as

(15)

That is

that is F d≤ 2fmax

(14)

Since the eigenvalues of RX are higher than zero, the following equation is obtained

(6)

It is obvious that the spatial interval of frequency will be enlarged when d increases. To avoid the illegibility of the frequency-bearing vectors, the value of d must follow the expression −

(13)

(4)

where Y(n) is similar to the nth snapshot of ULA, and the array response matrix is shown in (5). M is defined as the number of elements, N is the number of snapshot and d is the spacing of elements. In this paper, we suppose M > p and L ≥ (M–1)d + N

Dv = 2p

(12)

where a21 , a22 , . . . , a2p are the eigenvalues of RX. So, the eigenvalues of RY consist of two parts as follows

We take one from every d samples to form new data vectors

A = [a(f1 ), a(f2 ), . . . , a(fp )]

(11)

Here Λ = diag(l1, l2, …, lM) is a diagonal matrix. l1 ≥ l2 ≥ ⋯ ≥ lM denote the eigenvalues of RY. Since the rank (ARXA H) = p, it follows that

i=1

⎧ ⎪ Y (1) = [y(1), y(d + 1), . . . , y((M − 1)d + 1)]T ⎪ ⎪ ⎪ ⎪ Y (2) = [y(2), y(d + 2), . . . , y((M − 1)d + 2)]T ⎪ ⎪ ⎪ ⎪ .. ⎨ .. . . ⎪ Y (n) = [y(n), y(d + n), . . . , y((M − 1)d + n)]T ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎩ Y (N ) = [y(N ), y(d + N ), . . . , y((M − 1)d + N )]T

(10)

(18)

let Similar to the eigen decomposition of RY, denote the unit-norm {ˆs1 , sˆ2 , . . . , sˆp , gˆ 1 , gˆ 2 , . . . , gˆ M −p } eigenvectors of Rˆ Y , arranged in the descending order of the ˆ are made of {ˆs1 , sˆ2 , . . . , sˆp } associated eigenvalues, and Sˆ and G and {ˆg1 , gˆ 2 , . . . , gˆ M −p }, respectively. According to (16) and (17), it follows that ˆG ˆ H a(f i ) ≃ 0, aH (f i )G

i = 1, 2, . . . , p

(19)

We define PMUSIC (f ) =

aH (f )a(f ) ˆG ˆ H a(f ) aH (f )G

(20)

The MUSIC estimation of frequency is obtained by finding the p peaks of PMUSIC( f ). The range of estimated fes can be obtained IET Signal Process., 2015, Vol. 9, Iss. 4, pp. 357–368

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from (7) and (8) as follows 0 ≤ fes ≤

F 2d

(21)

Since the above-mentioned MUSIC method is similar with MUSIC for estimating DOA in ULA, we suppose that there exists MUSIC method in frequency domain beam-space, which is similar with MUSIC in beam-space. It is assumed that T is an M × B beam-former matrix in frequency domain ( p < B ≤ M ). Each column of T is a beam-former vector. In addition, it is also assumed that the columns of T are orthonormal (T HT = I). B is defined as the number of beams in frequency domain, which follows Y b (n) = T H Y (n) = T H AX (n) + V (n)

(22)

Herein, V(n) = T HW(n) is the additive noise in frequency domain beam-space. The covariance matrix of Yb(n) is shown as follows RYb = E{Y b (n)Y b (n)H } = T H ARX AH T + s2 I

(23)

The detailed descriptions for the technique of selecting beam-former matrix were reported in the previous literatures [33, 34] and will not be discussed herein. In this paper, we use discrete Fourier transform (DFT) beam-former [34]. We define the M × 1 DFT beam-forming weight vector as T(u) = [1, ej2pu(1/M) , . . . , ej2pu((M−1)/M ) ]T

ˆ b = [ˆgb,1 , gˆ b,2 , . . . , gˆ b,B−p ] are the noise eigenvectors of R ˆY and G b which are associated with lˆ b,p+1 , lˆ b,p+2 , . . . , lˆ b,B . Then, the estimated spatial spectrum in frequency domain beam-space is obtained as follows

PFB−MUSIC (f ) =

The FB-MUSIC estimation of frequency can be obtained by picking the p highest values of PFB-MUSIC( f ). The computational complexity of FB-MUSIC algorithm can be determined as follows: 2 × B × M × N multiplications are required for the beam-forming process, 2 × B 2 × N and (4/3)B 3 multiplications are roughly required for the determination of the ˆ Y and the related eigen decomposition. B × B covariance matrix R b The 2 × B × N × (M + B) + (4/3)B 3 multiplications are required for the FB-MUSIC algorithm as a whole. In contrast, the MUSIC algorithm requires roughly 2 × M 2 × N + (4/3)M 3 multiplications ˆ Y . We can obtain B ≤ M/2 by comparing (21) with respect to R with (29), so the computation cost of FB-MUSIC decreases greatly and the real-time performance is improved as compared with that of MUSIC. In general, the computational complexity of FB-MUSIC is in the order of O(B 3), while that of MUSIC is of order O(M 3).

4

(25)

Statistical analysis of FB-MUSIC

4.1

⎧ R = E[X (t)X H (t − i)] ⎪ ⎪ ⎪ X ,i ⎪ ⎪  = E[X (t)X T (t − i)] ⎪ R ⎪ ⎪ X ,i ⎨ Qi = E[W (t)W H (t − i)]  i = E[W (t)W T (t − i)] ⎪ Q ⎪ ⎪ ⎪ H ⎪ ⎪ RY ,i = E[Y (t)Y (t − i)] ⎪ ⎪ ⎩ RY ,i = E[Y (t)Y T (t − i)]

(26)

An N × B beam-forming matrix T is composed of B DFT beam-forming vectors in (26) with respective pointing frequency equispaced by the amount Δf = F/Md 1 T = √

[T(m0 ), T(m0 + 1), . . . , T(m0 + B)], . . . , m0 M = 0, 1, . . . , M − 1 − B

(27)

Preliminary results

The following auto-covariance matrices are introduced

Here u = 0, 1, …, M − 1, so we can obtain M × M DFT matrix beam-former as follows T M ×M = [T(0), T(1), . . . , T(M − 1)]

(31)

(24)

The beam pointing frequency of T(u) is uF/Md by comparing (24) with (5). The period of f (x) = ej(2π/M )x is M, so we can obtain T(u) = T(u + M)

[T H a(f )]H [T H a(f )] ˆ bG ˆ bH [T H a(f )] [T H a(f )]H G

(32)

On the basis of assumptions in Section 2, we can easily obtain (see equation (33) at the bottom of the next page) where

Thus, according to (24) and (27), it follows T HT = I

(28)

Herein, the parameter m0 is the beam-starting number, and the range of estimated frequency fb-es is F F m0 ≤ fb−es ≤ (B − 1 + m0 ) Md Md

⎡ ⎢ RX = ⎢ ⎣

In practice, RYb can be estimated from independent snapshots {Yb(1), Yb(2), …, Yb(N)} as follows

0 ..

⎢ D=⎣

.

⎤ ⎥ ⎥ ⎦

c2p

0 ⎡

(29)

c21

ej2pdf1 /F 0

..

0 .

⎤

(34)

⎥ ⎦

ej2pdfp /F

(30)

According to (22), (32) and (33), it follows that (see equation (35) at the bottom of the page)

Let lˆ b,1 ≥ · · · ≥ lˆ b,p ≥ lˆ b,p+1 ≥ · · · ≥ lˆ b,B be the eigenvalues of ˆ Y , and let Sˆ b = [ˆsb,1 , sˆb,2 , . . . , sˆb,p ] denote the signal R b ˆ Y which are associated with lˆ b,1 , lˆ b,2 , . . . , lˆ b,p , eigenvectors of R b

As var[W(n)] = Q0 = σ2I and T HT = I, var[V(n)] = σ2I. Letlb,1 ≥ · · · ≥ lb,p ≥ lb,p+1 = · · · = lb,B be the eigenvalues of RYb . The orthonormal eigenvectors of RYb , Sb = [sb,1, sb,2, …, sb,p] and Gb = [gb,1, gb,2, …, gb,B − p] are associated with {lb,1, lb,2, …, lb,p} and {lb,p + 1, lb,p + 2, …, lb,B}, respectively. The following

N
ˆY = 1 Y (n)Y b (n)H R b N n=1 b

IET Signal Process., 2015, Vol. 9, Iss. 4, pp. 357–368 & The Institution of Engineering and Technology 2015

359

pre-multiplying (43) by S H b , the following equation is obtained

notation is also required ⎡ ⎢ Lb = ⎢ ⎣

lb,1 ..

⎢ ˜ =⎢ L b ⎣

H K −1 = S H bT A

⎥ ⎥ ⎦

.

lb,1 − s2

0 ..

.

⎤

(36)

⎥ ⎥ ⎦

RX AH TX b = I

(45)

Thus, (39) is proved. Then, according to (43) and the definition in (37), it is obvious that Xb must stay in the column space of T HA. Thus

lb,p − s2

0

(44)

According to (37), (42) and (44), we can obtain

lb,p

0 ⎡

⎤

0

So, some results for the eigenvalues and eigenvectors of RYb can be obtained.

X b = T H AH

(46)

According to (45), the square matrix H must follow

Proposition 1: We define

RX AH TT H AH = I

˜ −1 S H T H A X b = Sb L b b

(37)

(47)

which gives H = (AH TT H A)−1 R−1 X and (38) is proved.

then Proposition 2: We define ˜ −1 S H T H A = T H A(AH TT T A)−1 R−1 Sb L b b X A TX b = H

(38)

R−1 X

(39)

bb,k = S Hb T H a(fk )

(48)

then −1

˜ b =x Sb L b b,k b,k

Proof: According to the relations between eigenvalues and eigenvectors, the following equation can be easily obtained

(49)

Here xb,k is the kth column of the matrix Xb defined in (37). RYb S b = S b Lb = T H ARX AH TS b + s2 S b

(40)

˜ −1 S b = T H ARX AH TS b L b

(41)

Proof: Equation (49) can be proved according to (37) and (48). ˆ Y , so the The FB-MUSIC estimation of frequency is derived fromR b following notation is introduced

that is

⎡ ⎢ ˆ =⎢ L b ⎣

Let

⎡

(42)

ˆ =⎢ ⎢ S b ⎣

According to (41) and (42), we can obtain (43)

where T HA is a non-singular matrix. By using S H b S b = I and



..

.

lˆ b,p

lˆ b,p+1

0

⎥ ⎥ ⎦

0 ..

.

0

˜ −1 = T H AK S b = T H ARX AH TS b L b

⎤

0

0

˜ −1 K = RX A TS b L b H

⎧ ⎪ RX ,i = RX ,0 Di = RX Di ⎪ ⎪ ⎪  X ,i = 0 ⎪ R ⎪ ⎪ ⎪ ⎪ ⎪ Qi = s 2 J i ⎪ ⎪ ⎡ ⎧ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ .. ⎪ ⎪ ⎪ ⎨ ⎢ ⎪ . ⎪ Jn = ⎢ ⎪ ⎪ ⎣ ⎨ Ji = ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ 0, ⎪ ⎪   ⎪ ⎪ = 0if |i| ≥ M J ⎪ i ⎪ ⎪ ⎪ T ⎪ ⎪ J ⎪ −i ⎪  = Ji ⎪ ⎪ Qi = 0 ⎪ ⎪ ⎪ R = AR AH + Q ⎪ ⎪ Y ,i X ,i i ⎪ ⎩R  Y ,i = 0

lˆ b,1

⎤

(50)

⎥ ⎥ ⎦

lb,B

ˆ Y  RY (as Owing to the second-order ergodicity of Yb(n), R b b ˆ  Lb and N → ∞) [5, 35]. This implies that Sˆ b  S b , L b

⎤

⎥ ⎥ ⎥ if i = dn(n= 0, 1, 2, . . . ) 1⎦



 here (J n )mn =

1, n = m + n 0, others

 (33)

elsewhere

E[V (n)V H (n − i)] = E[T H W (n)W H (n − i)T] = T H E[W (n)W H (n − i)]T = T H Qi T  iT ∗ E[V (n)V T (n − i)] = E[T H W (n)W T (n − i)T ∗ ] = T H E[W (n)W T (n − i)]T ∗ = T H Q

(35)

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Fig. 1 Normalised spatial spectra for FB-MUSIC and MUSIC with d = 1 and m0 = 0 a SNR = 0 dB b SNR = −10 dB

ˆ  s2 I (as N → ∞). It is obvious that S b 

Sˆ bH G b = (Sˆ b − S b )H G b  0 ˆ Hˆ ˆ SH b Gb = (S b − Sb ) Gb  0

and (N  1)

(51)

=

The FB-MUSIC estimation of frequency is computed from the eigenvalues and eigenvectors of Rˆ Yb . Thus, it is evident that their statistical properties will rely on the random fluctuation of these eigenvalues and eigenvectors. It is well known that the estimation errors {ˆsb,k − sb,k } and {lˆ b,k − lb,k } are asymptotically (for sufficiently large N) jointly Gaussian distributed in ULA, where W (n) is a white noise process [5, 8]. In the present case, W(n) is a moving-average process of order (M − 1), and the statistical properties of {ˆsb,k − sb,k }and {lˆ b,k − lb,k } seem to be more complicated. In the following section, the statistical properties of FB-MUSIC estimation of frequencies will be analysed according to the second-order properties of random variables of the following form sb,i − sb,i ), uH b (ˆ

i = 1, 2, . . . , p

(52)

Here u b is a vector in the column space of Gb. The following lemma is derived from these properties. Lemma 1: Let u b, 1 and u b, 2 be arbitrary vectors in the column space of Gb, and we define sb,1 − sb,1 ), uH sb,2 − sb,2 ), . . . , uH sb,p − sb,p )]T , vb,i = [uH b,i (ˆ b,i (ˆ b,i (ˆ i = 1, 2 (53) For sufficiently large N, the covariance matrix of the random vectors vb,1 and vb,2 are given by (see (54))

C(vb,1 , vb,2 ) = E[vb,1 vH b,2 ] =

E(fˆi − fi )2 =

 b,1 , vb,2 ) = E[vb,1 vTb,2 ] C(v d(M −1)
1 ˜ −1 S T Q u∗ × uH Q S L ˜ −1 (N − |l|)L b b l b,2 b,1 l b b N 2 l=−d(M −1)

(55) Proof: See Appendix 1. 4.2

Performance analysis of FB-MUSIC

According to (16) and (17) H aH (fi )TG b G H b T a(fi ) = 0,

i = 1, 2, . . . , p

(56)

and we define ˆ bG ˆ bH T H a(f ) DFB−MUSIC (f ) = aH (f )T G

(57)

The FB-MUSIC estimation of frequency can also be obtained by picking the p values of f for which DFB-MUSIC( f ) is minimised. Here DFB-MUSIC( f ) is used for the analysis of the distribution of the estimation errors {fˆi − fi }. Theorem 1: The FB-MUSIC estimation of the sinusoidal frequencies has the below variances for sufficiently large N (see (58)) H H H H where hb,i = d H (fi )TG b G H b T d(fi ), mb,i = G b G b T d(fi ), G b G b = H H H −1 H I − T A(A TT A) A T, d( f ) = da( f )/df, and bb,j denotes the jth column of Bb as follows

Bb = T H A(AH TT H A)−1

d(M −1)
1 H ˜ −1 T T ∗ ∗ ∗ ˜ −1 (N − |l|)(uH b,1 T Ql Tub,2 ) × Lb S b T RY ,l T S b Lb N 2 l=−d(M −1)

(59)

(54)

 d(M−1)    
s4 c2i −jl2pdfi /F T H ∗ H H H H T H H Re (N − |l|) (bb,i T J l T mb,i ) × (mb,i T J l Tbb,i ) + (mb,i T J l mb,i ) bb,i T J −l Tbb,i + 2 e s 2N 2 h2b,i c4i l=−d(M −1) (58)

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5

Fig. 2 Probability of resolution of FB-MUSIC and MUSIC with d = 1 and m0 = 0

Proof: See Appendix 2.

□

Numerical simulations and discussion

A number of Monte-Carlo simulations are conducted in this section to assess the proposed method. In these trials, the data consist of two complex sinusoids at frequency f1 = 2 and f2 = 2.4 kHz in the white noise. The sampling rate is F = 20 kHz. The amplitudes of the sinusoids are c1 = 1, c2 = 1, respectively, and their phases are chosen randomly. The number of elements is M = 32 and the number of snapshots is N = 100. Fig. 1 shows FB-MUSIC spatial spectrum and MUSIC spatial spectrum with the data when SNRs are 0 and −10 dB, respectively. Fig. 2 shows the probability of resolution of MUSIC and FB-MUSIC with B = 8, 12 and 16 according to 100 Monte-Carlo simulations. From Figs. 1 and 2, it can be seen that the resolution of FB-MUSIC is higher than that of MUSIC, and the SNR resolution threshold of FB-MUSIC is lower than that of MUSIC. Fig. 2 also shows that the SNR resolution threshold of FB-MUSIC has the declining tendency with the decrease of B. Let varB8, varB12 and varB16 be short notation for the variances of FB-MUSIC estimation errors with B = 8, 12 and 16, and varM32 denote the variances of MUSIC estimation errors with M = 32. To estimate the root-mean-square error (RMSE) of frequency f1 = 2 kHz when the SNR varies from −14 to 10 dB, 100 Monte-Carlo

Fig. 3 Theoretical variances and estimated RMSE of frequency 2 kHz against SNR with d = 1 and m0 = 0 a VarFB-MUSIC/VarMUSIC b Theoretical variances and RMSE

Fig. 4 Theoretical variances and estimated RMSE of frequency 2 kHz against SNR with d = 2 and m0 = 2 a VarFB-MUSIC/VarMUSIC b Theoretical variances and RMSE

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Fig. 5 Theoretical variances and estimated RMSE of frequency 2 kHz against SNR with d = 3 and m0 = 8 a VarFB-MUSIC/VarMUSIC b Theoretical variances and RMSE

simulations are conducted. The RMSE is defined as 









n 1
RMSE = 20 × log10 (fˆ − fi )2 n i=1 i

(60)

In addition, let RMSEB8, RMSEB12 and RMSEB16 denote the RMSE of FB-MUSIC estimation errors with B = 8, 12 and 16, respectively, and let RMSEM32 denotes the RMSE of MUSIC estimation errors with M = 32. Fig. 3 shows the theoretical variances and the estimated RMSE with d = 1, m0 = 0 and N = 100.

Fig. 6 Theoretical variances and estimated RMSE of frequency 2 kHz for FB-MUSIC with B = 8 and MUSIC with M = 32 against d a Theoretical variance of FB-MUSIC b Estimated RMSE of FB-MUSIC, c Theoretical variance of MUSIC d Estimated RMSE of MUSIC

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Fig. 7 Theoretical variances and estimated RMSE of frequency 2 kHz against snapshot at SNR = 7 dB a VarFB-MUSIC/VarMUSIC b Theoretical variances and RMSE

Fig. 8 Theoretical variances and estimated RMSE of frequency 2 kHz against the number of elements at SNR = 7 dB

Fig. 4 shows the theoretical variances and the estimated RMSE with d = 2, m0 = 2 and N = 100. Fig. 5 shows the theoretical variances and the estimated RMSE with d = 3, m0 = 8 and N = 100. Figs. 3–5 demonstrate that the estimated RMSE keeps consistent with the theoretical variance of frequency 2 kHz for FB-MUSIC and MUSIC, and the estimated RMSE and theoretical variance have the tendency to decrease with the increase of B for two methods. It is obvious that the performance of FB-MUSIC is getting closer to that of MUSIC with the increase of SNR and increase of d from Figs. 3–5. The RMSE of both FB-MUSIC estimation errors and MUSIC estimation errors against d is plotted in Fig. 6. It is notable that the performance of two methods is improved as d increases, as shown in Fig. 6. The results are in consistent with the DOA estimation in ULA. According to (21) and (29), the range of estimated frequency decreases with the increase of d. In practice, it is recommended to select the right d according to the precision requirement and the range of estimated frequency. Fig. 7 shows the results of RMSE at SNR = 7 dB as the number of snapshots N varies from 100 to 800 with d = 1, m0 = 0 and M = 32.

Fig. 9 Estimated RMSE for FB-MUSIC and F-MUSIC against SNR a Estimated RMSE of frequency 2 kHz b Estimated RMSE of frequency 2.4 kHz

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FB-MUSIC method transforms Y(n) of M dimensions in element-space into YB(n) of B dimensions in frequency domain beam-space by beam-former matrix T, which greatly decreases the computational complexity. In this paper, we analysed the statistical property of FB-MUSIC in the frequency estimation and obtained the specific variance expressions. In addition, some Monte-Carlo simulations were conducted to assess the FB-MUSIC method. The results showed that the estimated RMSE kept consistent with the theoretical variance. The SNR resolution threshold of FB-MUSIC was lower than that of MUSIC and F-MUSIC, and the accuracy of FB-MUSIC was close to that of MUSIC, and the computational complexity of FB-MUSIC with the reduced dimensionality is lower than that of MUSIC and the computational complexity of FB-MUSIC is approximately the same as F-MUSIC. It is evident that the proposed FB-MUSIC method can estimate frequency effectively, which provides an idea of estimating frequency in frequency domain beam-space using some other well-established methods for DOA estimation in beam-space. Fig. 10 Probability of resolution of FB-MUSIC and F-MUSIC

From Fig. 7, it is seen that the estimated RMSE and the theoretical variance have the tendency to decrease with the increase of N and the estimated RMSE keeps consistent with the theoretical variance. It is well known that the accuracy for estimating DOA will be improved as the number of elements increases. When M increases, will the performance of the proposed FB-MUSIC and MUSIC for estimating frequency be improved? We conduct 100 Monte-Carlo simulations with d = 1, m0 = 0 and N = 500 to estimate the RMSE of frequency f1 = 2 kHz. Our results are depicted in Fig. 8, which shows that the performance of both FB-MUSIC and MUSIC is improved at SNR = 7 dB when M increases. But the computational complexity will be greatly increased with the increase of M and B. Thus, it is recommended to select the right M and B based on the precision and real-time requirement of the frequency estimation. To compare FB-MUSIC with the state-of-the-art method, another 100 Monte-Carlo simulations with d = 1, m0 = 0 and N = 128 are conducted to estimate the RMSE of frequency 2 and 2.4 kHz for FB-MUSIC and F-MUSIC [25, 26] and the data consist of two complex sinusoids at frequency f1 = 2 and f2 = 2.4 kHz in the white noise. The results are depicted in Fig. 9, showing that the RMSE of FB-MUSIC (B = 16) and MUSIC (M = 32) is superior to the RMSE of F-MUSIC (k1 = 0, kM = 127, MF = 128, m = 16 and m = 32). According to the literatures [25, 26], the computational complexity of F-MUSIC with MF = 128 is of orderO(MF3 ), whereas the computational complexity of the proposed FB-MUSIC with B = 16 is of order O(B3). Therefore the computational complexity of FB-MUSIC is less than that of F-MUSIC. It is also shown in the literature [25] that the selection of MF is not crucial to the performance of F-MUSIC. So, the performance of F-MUSIC with MF = 16 is almost the same as that of F-MUSIC with MF = 128. That is, the performance of FB-MUSIC with B = 16 is still superior to that of F-MUSIC with MF = 16, whereas the computational complexity of FB-MUSIC with B = 16 and F-MUSIC with MF = 16 is almost the same. So we can obtain that the computational complexity of FB-MUSIC with B = 16 is approximately the same as F-MUSIC with MF = 16 and the computational complexity of FB-MUSIC is lower than that of MUSIC. To compare FB-MUSIC with F-MUSIC for frequency resolution, we conduct independent 100 Monte-Carlo simulations with d = 1, m0 = 0 and N = 128 for different SNR and the data consist of two complex sinusoids at frequency f1 = 2 and f2 = 2.4 kHz in the white noise. The result is illustrated in Fig. 10. From the simulations, it can be seen that the frequency resolution of FB-MUSIC with B = 16 is superior to that of F-MUSIC with m = 16 and m = 32.

6

Conclusions

In this paper, the FB-MUSIC method is presented for the frequency estimation in frequency domain beam-space. The proposed

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References

1 Brester, Y., Macovski, A.: ‘Exact maximum likelihood parameter estimation of superimposed exponential signals in noise’, IEEE Trans. Acoust. Speech Signal Process., 1986, 34, (5), pp. 1081–1089 2 Schmidt, R.O.: ‘Multiple emitter location and signal parameter estimation’. Proc. RADC Spectral Estimation Workshop, Rome, NY, 1979, pp. 243–258 3 Roy, R., Paulraj, A., Kailath, T.: ‘ESPRIT – a subspace rotation approach to estimation of parameters of ciscoids in noise’, IEEE Trans. Acoust. Speech Signal Process., 1986, 34, (5), pp. 1340–1342 4 Stoica, P., Nehoral, A.: ‘MUSIC, maximum likelihood, and Cramer–Rao bound’, IEEE Trans. Acoust. Speech Signal Process., 1989, 37, (5), pp. 720–741 5 Stoica, P., So˝ derstro˝ m, T.: ‘Statistical analysis of MUSIC and subspace rotation estimates of sinusoidal frequencies’, IEEE Trans. Signal Process., 1991, 39, (8), pp. 1836–1846 6 Stoica, P., So˝ derstro˝ m, T.: ‘Statistical analysis of MUSIC and ESPRIT estimates of sinusoidal frequencies’. Proc. IEEE ICASSP, Toronto, Canada, 1991, pp. 3273–3276 7 Tichavský, P.: ‘High-SNR asymptotics for signal-subspace methods in sinusoidal frequency estimation’, IEEE Trans. Signal Process., 1993, 41, (7), pp. 2448–2460 8 Kaveh, M., Barabell, A.J.: ‘The statistical performance of the MUSIC and the minimum-norm algorithms in resolving plane waves in noise’, IEEE Trans. Acoust. Speech Signal Process., 1986, 34, (2), pp. 331–341 9 Li, F., Liu, H.: ‘Statistical analysis of beam-space estimation for direction-of-arrivals’, IEEE Trans. Signal Process., 1994, 42, (3), pp. 604–610 10 Kristensson, M., Jansson, M., Ottersten, B.: ‘Further results and insights on subspace based sinusoidal frequency estimation’, IEEE Trans. Signal Process., 2001, 49, (12), pp. 2962–2974 11 Swindlehurst, A.L., Kailath, T.: ‘A performance analysis of subspace-based methods in the presence of model errors, part I: the MUSIC algorithm’, IEEE Trans. Signal Process., 1992, 40, (7), pp. 1758–1774 12 Ferréol, A., Larzabal, P., Viberg, M.: ‘On the asymptotic performance analysis of subspace DOA estimation in the presence of modeling errors: case of MUSIC’, IEEE Trans. Signal Process., 2006, 54, (3), pp. 907–920 13 Xu, X.L., Buckley, K.M.: ‘Statistical performance comparison of MUSIC in element-space and beam-space’. Proc. IEEE ICASSP, Glasgow, UK, 1989, pp. 2124–2127 14 Stoica, P., Nehorai, A.: ‘Comparative performance study of element-space and beam-space MUSIC estimators’, Circuits Syst. Signal Process, 1991, 10, (3), pp. 285–292 15 Karhunen, J.T., Joutsensalo, J.: ‘Sinusoidal frequency estimation by signal subspace approximation’, IEEE Trans. Signal Process., 1992, 40, (12), pp. 2961–2972 16 Karhunen, J.T., Joutsensalo, J.: ‘Robust MUSIC based on direct signal subspace estimation’. Proc. IEEE ICASSP, Toronto, Canada, 1991, pp. 3357–3360 17 Tsakalides, P., Nikias, C.L.: ‘The robust covariation-based MUSIC (ROC-MUSIC) algorithm for bearing estimation in impulsive noise environments’, IEEE Trans. Signal Process., 1996, 44, (7), pp. 1623–1633 18 Liu, T.H., Mendel, J.M.: ‘A subspace-based direction finding algorithm using fractional lower order statistics’, IEEE Trans. Signal Process., 2001, 49, (8), pp. 1605–1613 19 Weng, B., Barner, K.E.: ‘TR-MUSIC – a robust frequency estimation method in impulsive noise’, Signal Process., 2006, 86, pp. 1477–1487 20 Christensen, M.G., Jakobsson, A., Jensen, S.H.: ‘Joint high-resolution fundamental frequency and order estimation’, IEEE Trans. Audio. Speech Lang. Process., 2007, 15, (5), pp. 1635–1644 21 Viberg, M., Lundgren, A.: ‘Array interpolation based on local polynomial approximation with application to DOA estimation using weighted MUSIC’. Proc. IEEE ICASSP, Taiwan, China, 2009, pp. 2145–2148 22 Zhang, J.X., Christensen, M.G., Jensen, S.H., Moonen, M.: ‘A robust and computationally efficient subspace-based fundamental frequency estimator’, IEEE Trans. Audio. Speech Lang. Process., 2010, 18, (3), pp. 487–497 23 Lobos, T., Leonowicz, Z., Rezmer, J., Schegner, P.: ‘High-resolution spectrum-estimation methods for signal analysis in power systems’, IEEE Trans. Instrum. Meas., 2006, 55, (1), pp. 219–225

365

Hua, Z.Z.: ‘The fourth order cumulants based modified MUSIC algorithm for DOA in colored noise’. 2010 Asia-Pacific Conf. on Wearable Computing Systems, 2010, pp. 345–347 25 Zhang, J.X., Christensen, M.G., Dahl, J., Jensen, S.H., Moonen, M.: ‘Robust implementation of the MUSIC algorithm’. Proc. IEEE ICASSP 2009, Taiwan, China, 2009, pp. 3037–3040 26 Zhang, J.X., Christensen, M.G., Dahl, J., Jensen, S.H., Moonen, M.: ‘A robust and computationally efficient subspace-based fundamental frequency estimator’, IEEE Trans. Audio. Speech Lang. Process., 2010, 18, (3), pp. 487–497 27 Christensen, M.G., Stoica, P., Jakobsson, A., Jensen, S.H.: ‘Multi-pitch estimation’, Elsevier Signal Process., 2008, 88, (4), pp. 972–983 28 Zhang, H., Wu, H.-C., Chang, S.Y.: ‘Novel fast MUSIC algorithm for spectral estimation with high subspace dimension’. Proc. IEEE ICNC 2013, San Diego, USA, 2013, pp. 474–478 29 Christensen, M.G.: ‘An exact subspace method for fundamental frequency estimation’. Proc. IEEE ICASSP 2013, Vancouver, Canada, 2013, pp. 6802–6806 30 Bienvenu, G., Tufts, D.W.: ‘Decreasing high-resolution method sensitivity by conventional beamformer preprocessing’. Proc. ICASSP, 1984, pp. 33.2.1–4 31 War, M., Kailath, T.: ‘Detection of signals by information theoretic criteria’, IEEE Trans. Acoust. Speech Signal Process., 1985, 33, (3), pp. 387–392 32 Fuchs, J.-J.: ‘Estimating the number of sinusoids in additive white noise’, IEEE Trans. Acoust. Speech Signal Process., 1988, 36, (12), pp. 1846–1853 33 Trees, H.L.V.: ‘Optimum array processing part IV of detection, estimation and modulation’ (Wiley, New York, 2002) 34 Zoltowski, M.D., Kautz, G.M., Silverstein, S.D.: ‘Beamspace root-MUSIC’, IEEE Trans. Signal Process., 1993, 41, (1), pp. 344–364 35 Stoica, P., So˝ derstro˝ m, T., Ti, F.N.: ‘Over determined Yule-Walker estimation of the frequencies of multiple sinusoids: accuracy aspects’, Signal Process., 1989, 16, pp. 155–174 36 Janssen, P., Stoica, P.: ‘On the expectation of the product of four matrix-valued Gaussian random variables’, IEEE Trans. Autom. Control, 1988, 33, pp. 867–870 24

8

Appendix 1

√

√

ˆ = L + O(1/ N ) For sufficiently large N, Sˆ b = S b + O(1/ N ), L b b √



ˆ = sI + O(1/ N ) (see e.g. [4, 5, 8]). Using these facts, and S b ˆ Y , we can obtain expression (36) and the eigen decomposition of R b   H ˆ ˆ ˆH ˆ ˆ ˆH ˆ SH b RYb G b = S b Sb Lb Sb + Gb Sb Gb G b   2 2ˆ ˆH ˆH ˆ ˆ ˆH ˆ ˆ = SH b Sb Lb Sb + Gb (Sb − s I)Gb + s Gb Gb G b   2 2 ˆH ˆ ˆH ˆ ˆ ˆH ˆ ˆ = SH b Sb Lb Sb + Gb (Sb − s I)Gb + s (I − Sb Sb ) G b   2 ˆH 2 2 ˆ ˆ ˆH ˆ ˆ = SH b Sb (Lb − s I)Sb + Gb (Sb − s I)Gb + s I G b   H 2 ˆ − s2 I)G ˆ −L +L ˜ )Sˆ H + G ˆ ˆ ˆ = SH ( L ( S + s I Gb S b b b b b b b b b

sb,i − sb,i ) = uH b (ˆ

(lb,i − s2 )

(64)

for any vector ub in the column space of Gb. Owing to the fact that x(t) and w(t) are independent random variables, with x(t) having zero mean, and following the property that the expectation of product of four complex Gaussian random variables {x1, x2, x3, x4} of which at least one is zero mean [36] E(x1 x2 x3 x4 ) = E(x1 x2 )E(x3 x4 ) + E(x1 x3 )E(x2 x4 ) + E(x1 x4 )E(x2 x3 ) (65) H H Since uH b T A = 0 and ub si = 0(i = 1, 2, . . . , p), using the above observation and properties, it follows that

(lb,i − s2 )(lb,j − s2 )C i,j (vb,1 , vb,2 ) H ˆ H ˆ = E[(uH b,1 RYb sb,i )(ub,2 RYb sb,j ) ]   H  N N

H 1 H H 1 H = E ub,1 (Y (t)Y b (t))sb,i ) ub,2 (Y (p)Y b (p))sb,j N t=1 b N p=1 b   N
N 1
H H H H H =E 2 u T W (t)Yb (t)sb,i × W (p)Tub,2 sb,j Y b (p) N t=1 p=1 b,1

=

N
N 1
H H H H H E uH b,1 T W (t)W (p)Tub,2 [sb,j T AX (p)X (t) N 2 t=1 p=1

H H H H H × AH Tsb,i + sH b,j T AX (t)W (p)Tsb,i + sb,j T W (p)X (t) ! H H ×AH Tsb,i + sH b,j T W (p)W (t)Tsb,i ]

=

N
N " 1
uH T H E[W (t)W H (p)]Tub,2 2 N t=1 p=1 b,1 H H H × sH b,j T AE[X (p)X (t)] × A Tsb,i

! H H H H H + E[uH b,1 T W (t)W (p)Tub,2 ×sb,j T W (p)W (t)Tsb,i ]

Hˆ ˆ ˆH ˆ ˜ ˆH = SH b Sb Lb Sb G b + S b Sb (Lb − Lb )Sb G b

=

2 H ˆH ˆ ˆ + SH b Gb (Sb − s I)Gb G b + S b G b

ˆ uH b RYb sb,i

Hˆ ˆ ˜ ˆH ˆ ˆH = SH b (Sb − S b + S b )Lb Sb G b + S b Sb (Lb − Lb )Sb G b

N
N 1
H H H (uH T H Qt−p Tub,2 × sH b,j T ARX ,t−p A Tsb,i N 2 t=1 p=1 b,1 H H H H + uH b,1 T Qt−p Tub,2 × sb,j Q p−t sb,i + ub,1 Qt−p ub,2 × sb,j Q p−t sb,i

2 ˆH ˆ ˆ + SH b Gb (Sb − s I)Gb G b

˜ Sˆ H G + S H (Sˆ − S )L ˜ Sˆ H G + S H Sˆ (L ˆ − L )Sˆ H G =L b b b b b b b b b b b b b b b 2 ˆH ˆ ˆ + SH b Gb (Sb − s I)Gb G b

H + uH b,1 Qo sb,i × sb,j Q0 ub,2 )

=

(61) √

Neglecting the terms O(1/ N ) in (61), we can obtain

H H H H + uH b,1 T Ql Tub,2 × sb,j T Qk Tsb,i )

=

˜ ˆH ˆ SH b RYb G b = Lb Sb G b

N 1
H H H H H (N − |l|)(uH b,1 T Ql Tub,2 × sb,j T ARX ,l A Tsb,i N 2 l=−N

d(M−1)
1 H H H H (N − |l|)uH b,1 T Ql Tub,2 × sb,j T RY ,l Tsb,i N 2 l=−d(M −1)

(66)

that is Hˆ ˜ −1 ˆ GH b RYb S b Lb = G b Sb

(62)

or, equivalently GH sb,i − sb,i ) = b (ˆ

ˆ GH b RYb sb,i (lb,i − s2 )

implying that, asymptotically

,

i = 1, 2, . . . , p

(63)

where Ci, j (vb,1, vb,2) denotes the (i, j)th element of the matrix C (vb,1, vb,2) defined in (54). Thus, for sufficiently large N, it follows that

C i,j (vb,1 , vb,2 ) =

H H H H d(M −1)
uH 1 b,1 T Ql Tub,2 ×sb,j T RY ,l Tsb,i (N −|l|) 2 2 N l=−d(M−1) (li − s )(lj − s2 )

(67)

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or, in matrix form C(vb,1 , vb,2 ) =

d(M −1)
1 H ˜ −1 T T ∗ ∗ ∗ ˜ −1 (N − |l|)uH b,1 T Ql Tub,2 Lb S b T RY ,l T S b Lb N 2 l=−d(M −1)

Appendix 2

In this appendix, we will use the symbol ≃ to denote an equality in which we have neglected higher order terms. As fˆi is a minimum point of DFB-MUSIC( f ), we must have D′FB−MUSIC (fˆi ) = 0

(68) Here

which proves (54). Similar calculation to (66) gives

D′FB−MUSIC (f ) =

 i,j (vb,1 , vb,2 ) (li − s2 )(lj − s2 )C =



N
N 1
H H ∗ T ∗ H E[uH b,1 T W (t)W (p)Tub,2 × sb,i Y b (t)Y b (p)sb,j ] N 2 t=1 p=1 N
N 1
H H ∗ E uH b,1 T W (t)W (p)Tub,2 N 2 t=1 p=1

A Taylor series expansion of (72) around fi gives 0 = D′FB−MUSIC (fˆi ) = D′FB−MUSIC (fi ) + D′′FB−MUSIC (fi )(fˆi − fi ) + · · ·

+

! +sTb,i T T W ∗ (t)W H (p)Tsb,j ] =

+

D′′FB−MUSIC (fi ) =

(75) By neglecting higher order terms in (74) and using the fact that ˆ b  0, fˆi  fi in mean squares sense ˆ bG ˆ bH  G b G H AH T G G b, as N tends to infinity, the following expression can be obtained for sufficiently large N

N
N 1
H H ∗ E[uH b,1 T W (t)W (p)Tub,2 N 2 t=1 p=1

0 ≃ D′FB−MUSIC (fi ) + 2hb,i (fˆi − fi )

(76)

Next, note that

N
N 1
= 2 {uH T H E[W (t)W H (t)]Tsb,i N t=1 p=1 b,1

ˆ bG ˆ bG ˆ bH = AH T G ˆ bH G ˆ bG ˆ bH AH T G ˆ bG ˆ bH Gb GH ≃ AH T G b

H H × uH b,2 T E[W (p)W (p)]Tsb,j

= AH T(I − Sˆ b Sˆ bH )G b G H b

H H H H H + uH b,1 T E[W (t)W (p)]Tsb,j × ub,2 T E[W (p)W (t)]Tsb,i }

= −AH T Sˆ b Sˆ bH G b G H b

N
N 1
H uH T H Qt−p Tsb,j × uH b,2 T Q p−t Tsb,i 2 N t=1 p=1 b,1

(77)

≃ −AH TS b Sˆ bH G b G H b = −AH TS b (Sˆ b − S b )H G b G H b

d(M−1)
1 H H H = 2 (N − |l|)uH b,1 T Ql Tsb,j × ub,2 T Q−l Tsb,i N l=−d(M −1)

(69)

Inserting (77) in (73) for D′FB-MUSIC( fk), the following expression is obtained D′FB−MUSIC (fi ) ≃ −2Re[aH (fi )TS b (Sˆ b − S b )H mi ]

Hence, for sufficiently large N, it follows that H (sTb,i T T Ql T ∗ u∗b,2 )(uH b,1 T Ql Tsb,j )  i,j (vb,1 , vb,2 ) = 1 C (N −|l|) 2 2 N l=−d(M −1) (lb,i − s )(lb,j − s2 )

∗ T = −2Re[bH i vi ] = −2Re[bi vi ]

d(M−1)


(70) or, in matrix form

(78)

where

bb,i = S Hb T H a(fi ),

mb,i = G b G Hb T H d(fi )

sb,1 − sb,1 ), . . . , mH sb,p − sb,p )]T vb,i = [mH b,i (ˆ b,i (ˆ

 b,1 , vb,2 ) C(v d(M −1)
1 ˜ −1 S T T T Q T ∗ u∗ uH T H Q TS L ˜ −1 (N − |l|)L = 2 b l l b b b b,2 b,1 N l=−d(M −1)

(71)

IET Signal Process., 2015, Vol. 9, Iss. 4, pp. 357–368 & The Institution of Engineering and Technology 2015

f =fi

ˆ bG ˆ bG ˆ bH T H d(fi ) + aH (fi )T G ˆ bH T H d ′ (fi )] = 2Re[d (fi )T G

sTb,i T T W ∗ (t)X H (p)AH Tsb,j

And the proof of (55) is also finished.

# d2 DFB−MUSIC (f )## # # df 2 H

× sTb,i T T W ∗ (t)W H (p)Tsb,j ]

=

(74)

where

× [sTb,i T T A∗ X ∗ (t)X H (p)AH Tsb,j sTb,i T T A∗ X ∗ (t)W H (p)Tsb,j

(73)

ˆ bG ˆ bH T H d(f )] = 2Re[a (f )T G

 N N
1
H H 1 H =E (Y (t)Y b (t))sb,i )(ub,2 (Y (p)Y b (p))sb,j ) N t=1 b N p=1 b   N
N 1
H H H H H H =E 2 u T W (t)Y b (t)sb,i × ub,2 T W (p)Y b (p)sb,j N t=1 p=1 b,1

=

dDFB−MUSIC (f ) df H

H ˆ ˆ E[(uH b,1 RYb sb,i )(ub,2 RYb sb,j )]

(uH b,1

=

(72)

According to (76) and (78), it follows that, asymptotically (fˆi − fi ) =

Re[bTb,i vb,i ] hb,i

(79)

For the statistical property of {fˆi − fi }, note that for two

367

complex-valued variables, z1 and z2, we have 1 Re(z1 )Re(z2 ) = [Re(z1 z2 ) + Re(z1 z∗2 )] 2

Similarly (80)

According to (79), (80) and Lemma 1, it follows that

 b,i , vb,k )bb,k bTb,i C(v =

d(M −1)
1 ˜ −1 S T T T Q T ∗ m∗ mH T H Q TS L ˜ −1 (N −|l|)bTb,i L b l l b b bb,k b b,k b,i N 2 l=−d(M −1)

covFB−MUSIC (fˆi , fˆk ) =

1 E{[Re(bTb,i vb,i )][Re(bTb,k vb,k )]} hb,i hb,k

=

d(M −1) a4
H (N −|l|)xTb,i T T J l T ∗ m∗b,k mH b,i T J l Txb,k 2 N l=−d(M −1)

=

1 ∗ E[Re(bTb,i vb,i vTb,k bb,k ) + Re(bTb,i vb,i vH b,k bb,k )] 2hb,i hb,k

=

=

1  b,i , vb,k )bb,k ) + Re(bTb,i C(vb,i , vb,k )b∗b,k )] [Re(bTb,i C(v 2hb,i hb,k

d(M −1)
a4 H (N −|l|)(bTb,i T T J l T ∗ m∗b,k )(mH b,i T J l Tbb,k ) N 2 c2i c2k l=−d(M −1)

(83)

(81) Next, we use Propositions 1 and 2 and (54) to write (see (82))

bTb,i C(vb,i , vb,k )b∗b,k =

Inserting (82) and (83) into (81), we obtain (58), and the proof of theorem 1 is finished.

d(M −1)
1 2 H T ˜ −1 T ∗ ∗ ˜ −1 ∗ (N − |l|)(mH b,i s T J l T mb,k )bb,i Lb S b RY ,l S b Lb bb,k N 2 l=−d(M −1)

=

d(M −1) s2
H T H l H 2 H ∗ ∗ (N − |l|)(mH b,i T J l T mb,k )xb,i [T ARX D TA + s T J l T] xb,k 2 N l=−d(M −1)

=

d(M −1) s2
H H H l H 2 H H (N − |l|)(mH b,i T J l T mb,k )xb,k [T ARX D TA + s T J l T] xb,i N 2 l=−d(M −1)

d(M −1) s2
H −1 −l −1 2 H H T = 2 (N − |l|)(mH b,i T J l T mb,k )[(RX D RX RX )k,i + s xb,k T J l Txb,i ] N l=−d(M −1)

=

 d(M −1) s2
1 −jl2pdfk /F s2 H H H H (N − |l|)( m T J T m ) d e + (b T J Tb ) l b,k ik 2 −l b,i b,i N 2 l=−d(M −1) ck c2k c2i b,k

=

 d(M −1)
s4 c2i −jl2pdfk /F H H H H (N − |l|)( m T J T m ) d e + (b T J Tb ) l b,k ik −l b,i b,i b,k 2 2 s2 N 2 ck ci l=−d(M−1)

(82)

IET Signal Process., 2015, Vol. 9, Iss. 4, pp. 357–368

368

& The Institution of Engineering and Technology 2015