IET Radar, Sonar & Navigation

Research Article

Maximum-likelihood angle estimator for multi-channel FM-radio-based passive coherentlocation

ISSN 1751-8784Received on 12th September 2017Revised 14th February 2018Accepted on 26th February 2018E-First on 22nd March 2018doi: 10.1049/iet-rsn.2017.0419www.ietdl.org

Geun-Ho Park1, Dong-Gyu Kim1, Ho Jae Kim1, Hyoung-Nam Kim1 1Department of Electronics Engineering, Pusan National University, Busan, Republic of Korea

E-mail: hnkim@pusan.ac.kr

Abstract: Frequency-modulation-based (FM-based) passive coherent location (PCL) systems can estimate the target location/velocity by exploiting a single FM channel. In order to improve the performance of PCL systems, multi-channel-single-transmittersetup-based PCL systems have been studied recently. However, it has not yet been considered for direction of arrival estimationbased on the multi-channel configuration. Thus, the authors propose a maximum-likelihood angle estimator exploiting themultiple FM-radio channels and derive the Cramer-Rao bound for verifying the convergence of the proposed method in thesense of root-mean-square error (RMSE). In addition, they propose a target association algorithm for solving the ambiguityproblem in selecting the steering vectors of each target. Computer simulations are included to show the estimation performanceof the proposed method and to compare the RMSE of the single-channel case with that of the multi-channel case.

1 IntroductionPassive coherent location (PCL) is a passive radar technique thatexploits the illuminators of opportunity such as FM (frequencymodulated), digital video broadcasting-terrestrial, IEEE 802.11standard-based Wi-Fi, and GSM (global system for mobilecommunications) transmitters [1–7]. The basic structure of the PCLsystem is a bistatic geometry consisting of a receiver, transmitter,and a target. As the receiver is located away from the transmitter, itreceives the target echo signals and the direct-path signals at thesurveillance and the reference channels, respectively.

The location of the target may be estimated by using amultistatic structure with more than three bistatic geometries. Fromthe transmitter–target–receiver distance called the bistatic range,the line of position of the target can be represented by an ellipse,the foci of which are the locations of the transmitter and thereceiver. The target location is determined by the intersection of themultiple ellipses. Thus, it is important to detect the target echosignals emitted from the multiple transmitters [4, 8].

However, the detection of the target echo signals may not beguaranteed even when the targets are near the receiver [8]. Toincrease the detectability of the signals, direction-of-arrival (DOA)estimates can be used for finding the exact location of the target [8,9]. As the DOA estimate of the target echo signal indicates thedirection of the target from the receiver, the target location can bedetermined by finding the crossing point between the DOA and theellipse even when only one target echo signal is detected.

There is much literature regarding DOA estimation for PCLsystems. In [8], a direction-finding algorithm was developed bothto decrease the influence of strong interference signals such as adirect-path signal and clutter echoes in the surveillance channel andto estimate the direction of the target echoes. However, theapplication of this DOA estimation algorithm is limited to theAdcock antenna array. In [10, 11], the multiple signal classification(MUSIC) algorithm [12] was adopted to estimate the DOAs of thetarget echo signals without any modifications for PCL systems.The signal subspace and the noise subspace are decomposed fromthe estimated covariance matrix of the received signal in theMUSIC algorithm. By using a property that the noise subspace isorthogonal to the signal subspace, the multiple DOAs can beestimated [12]. In the PCL systems, the information of thetransmitted signal waveform can be obtained from the referencechannel [9]. Since the estimation accuracy can be enhanced byexploiting the information of the signal waveform [12], the MUSIC

algorithm may not be comparable to the DOA estimation algorithmusing the signal waveform. Falcone et al. [6] proposed anexperimental setup of a Wi-Fi-based multistatic passive radarincluding a DOA estimation block. However, they focused on thePCL system configuration and used a simple method involving theuse of the phase difference between two antennas.

Among the various communication and broadcasting systems,one of the popular illuminators in the area of passive radars is anFM radio transmitter [4]. As the FM transmitter has a suitabletransmitting power level for detecting targets as compared to otherilluminators, the FM-based PCL can realise a wide detectioncoverage. However, the range resolution and range-estimationaccuracy of the FM radio signal still require improvements. Owingto the narrowband property of the FM radio, it is difficult to obtainadequate range resolution and estimation accuracy of theparameters.

Therefore, a multi-channel structure of FM broadcasting hasbeen considered. The conventional PCL system took intoconsideration only a single FM channel owing to the simplicity ofits implementation [4]. However, as multiple carrier frequenciesare allocated to an FM transmitter, as depicted in Fig. 1, there is noreason to limit the configuration to a single channel. Severalstudies on multi-channel FM-based PCL systems have beenrecently conducted in order to improve the performance of theparameter estimation and probability of detection of the target echosignal [13–17]. In addition, the range resolution of the FM signalcan also be improved by exploiting the multi-channel FM structure[18, 19].

In this paper, we focus on the DOA estimation technique with amulti-channel configuration in the PCL system. There is littleliterature on the DOA estimation scheme for the multi-channelFM-based PCL. In [13, 20], the DOA estimation scheme wasproposed in order to integrate the multi-frequency estimates of thetarget echo signals. Moreover, to integrate the multiple DOAestimates and to calculate the multi-frequency-based DOAestimate, a weighted sum of the multiple DOA estimates wasproposed. However, the detailed derivation of the weighted sumwas not revealed in [13, 20].

Multi-frequency-based DOA estimation techniques have beenalso proposed in other areas besides the PCL systems. In [21], theDOA estimation method for multi-band signals was proposed withthe use of the hierarchical structure of the DOA estimates of thelow- and high-frequency signals. Although this algorithm couldimprove the DOA estimation accuracy while preventing the

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occurrence of the spatial aliasing problem, its advantage is limitedto only the case wherein it generates spatial aliasing for a fewcarrier frequencies. In [22], the DOA estimation scheme based onthe compressed sensing technique was proposed for multi-bandsignals. However, this DOA estimation algorithm operates in anenvironment in which the source signals are assumed to beunknown. In addition, its estimation accuracy for multi-bandsignals was not addressed in [22].

In order to improve the estimation accuracy of the direction ofmultiple-target echo signals by exploiting the multi-channelstructure in the FM band, we propose a maximum-likelihood (ML)angle estimator for a multi-channel FM-based PCL system.Although the performance of the ML estimator generallyapproaches the Cramer-Rao bound (CRB), it has not beenconsidered for a multi-channel-based PCL system. We also derivethe CRB for the multi-channel FM structure in this paper. Thederived CRB is used as a performance limit for the estimationaccuracy of the multi-channel-based ML technique.

This paper is organised as follows. Section 2 presents the signalmodel and assumptions for the multi-channel-based FM signal. InSection 3, under the assumption that the direct-path signals areknown, the ML angle estimator for a multi-channel FM signal isproposed and the multi-channel-based CRB is derived. Thesimulation results of the multi-channel ML are presented in Section4, and the conclusions of this study are presented in Section 5.

2 Signal modelPCL geolocation that exploits the multiple FM channels used forthe DOA estimation is presented in Fig. 2. The rth transmitteremits the multiple FM signals using a set of carrier frequencies,f r = f r

(1), . . . , f r(Nch) , where Nch denotes the number of detected

FM signals emitted from the rth transmitter. Since the multi-channel-multi-transmitter setup presented in Fig. 2 can be dividedinto several multi-channel-single-transmitter setups, without loss ofgenerality, we consider only a multi-channel-single-transmittersetup as described in [17, 19]. Thus, the set of frequencies of themultiple FM signals emitted from a single transmitter can berepresented as f = f (1), . . . , f (Nch) by removing the subscript r.

The number of detected target echo signals is assumed to be thesame for all the channels, i.e. the number of detected target echosignals in the frequency band of the multiple channels is equal toNch for all the channels. As the target echo signals from the multi-

channels have an equivalent target direction, all the target echoesmay be used to estimate the target direction. Based on the PCLgeolocation in Fig. 2, we derive a received-signal model of themulti-channel-single-transmitter case.

We also assume that strong interfering signals are removed bythe interference suppression schemes such as ECA (extensivecancellation algorithm), FBLMS (fast block least mean square),NLMS (normalised least mean square), and RLS (recursive leastsquares) [23–26] and only the target echo signals are received fromthe surveillance channel. The receiver is configured with anantenna array of M antennas. When the incident angle of thereceived signal is denoted by θ, the time delay of the narrowbandsignal can be simply modelled using a steering vector, i.e.

a(θ) = [exp(jk ⋅ p1), …, exp(jk ⋅ pM)]T, (1)

where k = 2πe/λ denotes the wavenumber vector of the incidentsignal corresponding to the signal of wavelength λ, pm representsthe position vector of the mth antenna, and e denotes the unitdirection vector of propagation of the target echo signal.

The steering vector of the qth channel can be represented usingthe superscript q as a(q)(θ) (q = 1, …, Nch). When K targets exist,the array manifold matrix having column vectors comprisingsteering vectors is represented by

A(q)(θ) = [a(q)(θ1), …, a(q)(θK)]T, (2)

where θ = [θ1, θ2, …, θK]T denotes the incident angle vector. WhenK target echo signals in the qth channel are denoted bys(q)(tn) = [s1

(q)(tn), …, sK(q)(tn)]T, the received signal vector x(q)(tn) is

represented by

x(q)(tn) = A(q)(θ)s(q)(tn) + w(q)(tn), n = 1, …, N, (3)

where w(q)(tn) represents the white Gaussian random process in theqth channel. As s(q) tn also includes the amplitude components,sk

(q) tn can be rewritten as

sk(q)(tn) = γk

(q)yk(q)(tn), k = 1, …, K, (4)

where γk(q) is defined as a complex-valued amplitude of the FM

signal reflected from the kth target in the qth channel, and yk(q) tn

denotes the target echo signal that has a normalised power, i.e.N−1∑n = 1

N | yk(q)(tn) |2 = 1. By substituting (4) into (3), (3) can be

rewritten as

x(q)(tn) = D(q)(θ)y(q)(tn) + w(q)(tn), (5)

where

D(q)(θ) = A(q)(θ)Γ(q), (6)

and Γ(q) = diag γ1(q), …, γK

(q) . The received signal model can besimplified as follows:

x(tn) = D(θ)y(tn) + w(tn), (7)

where

x(tn) = x(1)T(tn), …, x(Nch)T(tn)T,

y(tn) = y(1)T(tn), …, y(Nch)T(tn)T,

w(tn) = w(1)T(tn), …, w(Nch)T(tn)T,

Fig. 1  Multi-frequency network for FM broadcasting

Fig. 2  Geolocation of the PCL system when multi-transmitters and multi-channels are exploited

618 IET Radar Sonar Navig., 2018, Vol. 12 Iss. 6, pp. 617-625© The Institution of Engineering and Technology 2018

Γ = blkdiag Γ(1), …, Γ(Nch) ,

and

D(θ) = blkdiag(D(1)(θ), …, D(Nch)(θ)) . (8)

The problem of interest in this paper is the estimation of θ andΓ(q), q = 1, …, Nch under the assumption that the x(q)(tn) is given.Since the signal vector y(q)(tn) can be obtained from the referencechannel and the signal power of the y(q)(tn) is much stronger thanother signal components, y(q)(tn) is assumed to be known in thispaper.

3 ML angle estimator and CRB for multi-channel-based PCLIn this section, the ML angle estimator for the multi-channel-basedPCL is derived. In order to derive the multi-channel-based MLangle estimator, we assume that the number of detected target echosignals are the same for all the channels and the target echo signalsfor each channel are uncorrelated with each other.

3.1 Derivation of the multi-channel-based ML

The log-likelihood function can be expressed as follows [27]:

−ln W − tr [W]−1 1N ∑

n = 1

N[x(tn) − D(θ)y(tn)]

× [x(tn) − D(q)y(tn)]H ,(9)

where tr ⋅ is defined as the sum of the elements of the maindiagonal of a square matrix, and W represents the covariancematrix of w(tn). The array manifold matrix D^ (θ) can be estimatedby minimising the following function:

J = 1N ∑

n = 1

Nx(tn) − D(θ)y(tn) x(tn) − D(θ)y(tn) H . (10)

Let

R^xx = 1

N ∑n = 1

Nx(tn)xH(tn), (11)

R^yx = 1

N ∑n = 1

Ny(tn)xH(tn) (12)

and

R^yy = 1

N ∑n = 1

Ny(tn)yH(tn) . (13)

As the inter-channel covariance matrix of y(q)(tn) and y(r)(tn) forq ≠ r is a zero-valued matrix, the covariance matrices in (11)–(13)can be represented by the block diagonal matrices of R^

yy(q)

, R^yx(q)

, and

R^xx(q)

, where

R^yx(q) = 1

N ∑n = 1

Ny(q)(tn)x(q)H(tn) . (14)

R^yy(q)

and R^xx(q)

can be similarly defined as in (14).Using (11)–(13), the function in (10) can be rewritten as

J = R^xx − D(θ)R^

yx − R^yxH D(θ)H + D(θ)R^

yyD(θ)H

= D(θ) − R^yxH R^

yy−1

R^yy D(θ) − R^

yxH R^

yy−1 H

+R^xx − R^

yxH R^

yy−1

R^yx

(15)

To minimise the cost function J, the array manifold matrix D^ (θ)can be calculated using [27]

D^ (θ) = R^yxH R^

yy−1 . (16)

From the estimate of D^ (θ) in (16), the estimate of the noisecovariance matrix can be obtained by using

W^ = 1N ∑

n = 1

N[x(tn) − D^ (θ)y(tn)][x(tn) − D^ (θ)y(tn)]H

= R^xx − R^

yxH R^

yy−1

R^yx .

(17)

As the cost function J in (15) is defined as a function of θ and γk(q)

for all q and k, the multi-dimensional search over all unknownparameters may be performed on (15). However, it is difficult toperform the multi-dimensional search on (15) owing to theinvolved computational complexity. To reduce the computationalburden, a decoupling of each target echo signal can be applied tothe multi-channel case as in [27, 28]. By decoupling eachcomponent of the targets, the likelihood functions can be separatedfrom each other and the multi-dimensional search need not beperformed.

According to Theorem 1 in [27], the cost function J isasymptotically equivalent to

J ≃ tr R^yy[D(θ) − D^ (θ)]W^ −1[D(θ) − D^ (θ)]H . (18)

To estimate the DOAs of K targets, the multi-dimensional search of(18) is still required. As the target echo signals are assumed to beuncorrelated, R^

yy can be assumed to be a diagonal matrix.The steering vectors that are associated with the kth target in

D^ (θ) can be exploited to estimate θk. The cost functioncorresponding to the kth target may be written as

Jk = tr F(θk) − F^k

HW^ −1F(θk) − F^

k , (19)

where

F(θ) = blkdiag d(1)(θ), …, d(Nch)(θ) (20)

and

F^k = blkdiag d

^(1)(θk), …, d

^(Nch)(θk) . (21)

F^k includes the estimated steering vectors of the kth target for all

the channels. For example, when K = 3 targets exist and Nch = 2channels are exploited, then F1 is a block diagonal matrix of thesteering vectors for k = 1 target, as shown in Fig. 3. It should benoted that (20) and (21) are defined as matrices with the steeringvectors for all channels as the block diagonal components. From(19)–(21), the unknown parameters such as the amplitudes and theDOAs of the targets can be estimated by using

arg minΓk, θk

tr ΓkZ(θk) − F^k

HW^ −1 ΓkZ(θk) − F^k , (22)

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where Γk = diag γk(1), . …, γk

(Nch) and Z(θk) = Γk−1F(θk). Z(θk)

represents a block diagonal matrix of the steering vectors a(q)(θk)for all the FM channels.

All the amplitude parameters in Γk are inherently uncorrelatedwith each other. From the partial differentiation of (22) withrespect to γk

(q), the amplitude parameters of the target echo signalscan be obtained by using

γ^k(q) =

a(q)H(θ^k) W^ (q) −1b^k(q)

a(q)H(θ^k) W^ (q) −1a(q)(θ^k)

. (23)

The estimate of the DOAs θ^k can be derived by substituting Z(θ)

for (23) and partially differentiating (23) with respect to θk. Wethen obtain

θ^k = arg max

θktr

ZH(θk)W^ −1

F^k

2

ZH(θk)W^ −1

Z(θk). (24)

3.2 Cramer-Rao bound

The main difference between a single-channel and a multi-channelconfiguration is that the inter-channel correlation is consideredusing Fisher information. A log-likelihood function of (5) isrepresented as follows:

ln L = − NMNchln det W

− ∑q = 1

Nch

∑n = 1

Ne(q)H(tn) W(q) −1

e(q)(tn),(25)

where

e(q)(tn) = x(q)(tn) − D(q)(θ)y(q)(tn), (26)

and W = blkdiag(W(1), …, W(Nch)). Let σ(q) = σ1(q), …, σM2

(q) T and

σ = σ(1)T, …, σ(Nch)T T denote the total elements of the noise

covariance matrix, and the unknown parameter vector can bewritten as

ψ = σT, γT, θT T . (27)

Then, the Fisher information matrix I(ψ) can be derived by using

I(ψ) = E ∂ln L∂ψ

∂ln L∂ψ

T. (28)

As the components of σ are uncorrelated with γ and θ [12, 27, 28],we have only to consider γ and θ. The log-likelihood function isequal to the sum of the log-likelihood function of the qth channel,i.e.

ln L = ∑q = 1

Nch

ln L(q) . (29)

The partial derivative of (29) according to γ(q) satisfies thefollowing condition:

∂ln L∂γ(q) = ∂ln L(q)

∂γ(q) (30)

by using the property that γ(q) and γ(r) for q ≠ r are uncorrelatedwith each other.

Assuming that R^ss(q)

approaches Rss(q) asymptotically, then the

CRB of θk is simplified as (see the Appendix for a detailedderivation)

CRB−1(θ^k) = 2N ∑q = 1

Nch

Rss(q)

kk∥ Pa~k(q)

⊥ d~

k(q) ∥

2, (31)

where

Pa~k(q)

⊥ = I − a~k(q)Ha~k

(q) −1a~k

(q)a~k(q)H . (32)

When the uniform linear array with M antennas is considered, then(69) can be rewritten as

CRB(θ^k) = 6σ2

N(M2 − 1)Mcos2(θk)∑q = 1Nch Rss

(q)kk 2πd /λ(q) 2 , (33)

where d denotes the distance between the adjacent antennas, λ(q)

represents the wavelength in the qth channel, and W = σ2IMNch isassumed in (33).

The main difference between the single-channel and the multi-channel cases is the summation term in the denominator of (33).When the target echo signals for the multi-channels are exploited toestimate the DOA, the value of the summation term is increased ascompared to that in the single-channel case. Thus, the value ofCRB(θ^k) is decreased, which causes the variance of the DOAestimate θ

^k to decrease.

When we assume that the wavelengths of each channel λ(q) andthe signal power Rss

(q)kk are approximated using λ and [Rss]kk for

all q, then the following relationship is satisfied:

CRBmulti − channel(θ^k) = CRBsingle − channel(θ

^k)

Nch. (34)

Therefore, under the condition that the estimation accuracy of theML technique approaches the CRB derived in (33), the variance ofθ^k for the single-channel case is approximately equal to Nch times

the variance of θ^k for the multi-channel case.

3.3 Ambiguity problem in the multi-channel-based ML

In this subsection, we consider the association problem of thesteering vectors in D(θ). To estimate θk by using the multi-channel-based ML technique, the kth steering vectors in each D(q)(θ) arerequired to be extracted. However, there is an ambiguity problemin the selection of the steering vectors caused by the kth target.Thus, we are required to find out whether the steering vectors areobtained from the same target.

3.3.1 Problem formulation: If we assume that the knownwaveform y(tn) can be permuted using P, the permuted knownwaveform y⌣(tn) may then be represented by

y⌣(tn) = Py(tn), (35)

where P = blkdiag P(1), …, P(Nch) denotes the block diagonalmatrix of the permutation matrices. The multiplication of thepermutation matrix in (35) indicates that the association to thetargets is unclear. The permutation matrix P may be assumed to beunknown.

We first consider the effect of P in the estimates of D^ (θ), W^ ,and Γ^ . Using (35), the estimate of the covariance matrices in (12)and (13) may be rewritten as

620 IET Radar Sonar Navig., 2018, Vol. 12 Iss. 6, pp. 617-625© The Institution of Engineering and Technology 2018

R^y⌣x = 1

N ∑n = 1

Ny⌣(tn)xH(tn) = PR^

yx, (36)

and

R^y⌣ y⌣ = 1

N ∑n = 1

Ny⌣(tn)y⌣H(tn) = PR^

yyPT ≃ R^yy, (37)

where PT = P−1, and for a large value of N, R^yy may be

approximated using a diagonal matrix. Then, (16) and (17) and(23) may be rewritten as

D^⌣

(θ) = R^y⌣xHR̂ y⌣ y⌣−1 = D^ (θ)PT, (38)

W^⌣

= R^xx − R^

y⌣xH R^

y⌣ y⌣−1

R^y⌣x = W^ , (39)

and

Γ^⌣

= PΓ^ PT . (40)

It should be noted that the amplitude matrix of the target echosignals Γ^ is permuted by the direction of the diagonal components.

The order of the steering vectors for the kth target is mixed in(38) owing to the permutation matrix. For example, when weassume that Nch = 2, K = 2, and

P =

1 0 0 00 1 0 00 0 0 10 0 1 0

, (41)

The permuted array manifold matrix D^⌣

(θ) may then be written as

D^⌣

(θ) =d^(1)

(θ1) d^(1)

(θ2) 0M 0M

0M 0M d^(2)

(θ2) d^(2)

(θ1), (42)

where 0M denotes a zero-valued vector with M components.In order to accurately estimate θ1, we should combine the

steering vectord^(1)

(θ1) with d^(2)

(θ1). As the order of d^(2)

(θ1) andd^(2)

(θ2) is permuted by P(2) in (41), it should be recovered with theestimate of the permutation matrix P^

(2).

3.3.2 Algorithm for target association: As the steering vectorsestimated from the same target are highly correlated, we canestimate the permutation matrix P.

In order to associate the steering vectors of the same target, theinter-channel correlation between each steering vector may besimply calculated using

U(q) = D^⌣ (1)H(θ)D^⌣ (q)

(θ), (43)

where q = 1, …, Nch. As U(q) represents the approximation of the

permutation matrix P(q), the estimate of the permutation matrix P^ (q)

may be obtained by exploiting the indices that represent themaximum values in the column vectors of U(q). When we defineuk

(q) and p^ k(q) as the kth column vector of U(q) and P^ (q)

, respectively,then p^ k

(q) may be calculated using

p^ (q) = T(uk(q)), (44)

where T( ⋅ ) represents a transformation in which the maximumvalue in uk

(q) is set to 1 and the remaining components are changed

to 0. Then P^ (q) can be obtained by using

P^ (q) = p^ 1(q), …, p^ K

(q) . (45)

Thus, P^ can be obtained by using

P^ = blkdiag P^(1), …, P^

(Nch). (46)

Using (35) and (46), the non-permuted source signal y^(tn) mayfinally be estimated by using

y^(tn) = P^ y⌣(tn) . (47)

4 Simulation resultsWe present the root mean square error (RMSE) of the DOAs of thetarget according to the signal-to-noise ratio (SNR). When wedenote the θ

^l (l = 1, …, L) as the estimate of the DOAs at the lth

trial, then the RMSE can be obtained as

RMSE = 1L ∑

l = 1

L(θ − θ

^l)

T(θ − θ^l), (48)

where L represents the number of Monte Carlo simulations.We consider a uniform linear array with M = 8 sensors. In

addition, we assume that the inter-element spacing of the uniformlinear array satisfies d = λmin/2 where λmin denotes the minimumvalue of the wavelength of the multiple channels of interest. Thereceived signals of the multiple channels are sampled withf s = 300 kHz, and the signals are observed for 1 ms. Theobservation time of 1 ms may not be sufficient to achieve thereliable performance of the target localisation. Nevertheless, theobservation time of 1 ms was used to show the convergence of theRMSE to the CRB in the SNR values lying in the range from −35to −5 dB because the short observation time results in distinctconvergence performance. Since the observation time is stronglyrelated to the memory storage size and processing time, its lengthis meaningful for performance comparison. L = 500 Monte Carlosimulations are performed in each simulation.

4.1 Root-mean-square error

First, we assume that Nch = 4 channels of the target echo signalsare received, and each channel has carrier frequencies of 95.9,98.7, 106.1, and 107.7 MHz. We also consider that K = 1 targetexists, and the DOA of the target is θ = 0.0873 rad. Fig. 4 showsthe RMSE of the single-channel and the multi-channel MLsaccording to the SNR. As shown in Fig. 4, the RMSE of the multi-channel ML technique has a lower value than that of the single-

Fig. 3  Estimate of the array manifold matrix D^ (θ) for K = 3 and Nch = 2

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621

channel ML. Considering that the RMSE approaches the CRB, thedifference in the RMSE between the qth single-channel and multi-channel can be calculated using (33). When the wavelengths for themulti-channel signals are approximately the same, the difference inthe RMSE can be calculated using 10log10(Nch). For example, inFig. 4, the difference between the RMSE of the single-channel andthat of the multi-channel ML is calculated to be ∼6.02 dB asshown in (34). In addition, the RMSE of the proposed methodconverges to the CRB at lower SNR value than that of the single-channel cases. This means that the proposed method has the betterconvergence performance than the single-channel-based DOAestimation technique.

Second, we increase the number of channels as Nch = 6 and theRMSE is shown in Fig. 5. The carrier frequencies are 89.1, 91.9,95.9, 98.7, 106.1, and 107.7 MHz. As six channels are exploited,the difference between the single-channel and multi-channel MLsis ∼7.78 dB. As in case of Fig. 4, the RMSE of the proposedmethod converges to the CRB at much lower SNR value than thatof the single-channel cases.

Third, the derived RMSE according to the number of snapshotsis shown in Fig. 6. The dotted lines represent the CRBs. Nch = 5channels are exploited and the set of carrier frequencies is the sameas the set of carrier frequencies used in Fig. 5, except for thefrequency of 98.7 MHz, which is not included. For each channel,the SNR values are set as −26, −24, −22, −20, and −18 dB. Asshown in Fig. 6, the RMSE of the multi-channel-based MLapproaches the CRB as the number of snapshots increases. As themulti-channel-based ML exploits all target echo signals of various

SNR values, the RMSE of the proposed method has lower valuesthan that of the single-channel-based ML technique.

4.2 Association problem

We consider the association problem in the multi-channel MLtechnique. Nch = 4 channels are used, and K = 2 targets exist. Theobservation time is set to 2 ms. The carrier frequencies areequivalent to the frequencies in Fig. 4. The permutation matrices ofthe multi-channels are given by

P(1) = P(4) = I2, (49)

and

P(2) = P(3) = 0 11 0 . (50)

The permutation matrices in (49) and (50) represent that d^(1)

(θ1) andd^(4)

(θ1) are associated with d^(2)

(θ2) and d^(3)

(θ2) in the multi-channelML. The RMSE according to the SNR is given in Fig. 7. TheRMSE of the multi-channel ML technique, which is not applied tothe target association algorithm, is diverged. In contrast, the RMSEof the ML technique using the target association algorithmapproaches the CRB. The target association problem can also beobserved in Fig. 8. The SNR values of two target echo signals areset as −20 dB. As shown in Fig. 8a, each likelihood function of thetargets has local maximum values at both θ = − 20∘ and θ = 25∘.On applying the target association technique, each likelihoodfunction indicates the maximum value at the DOA of the target asshown in Fig. 8b.

4.3 Performance comparison

The performance of the multi-channel-based DOA estimationalgorithm presented in [20] is compared to the proposed method. In[20], the phase difference corresponding to each FM channelbetween the adjacent two antennas was derived. This algorithm islimited to antenna configuration only with two antennas. From thephase difference of the multi-channels and the weighted sum of thephase difference, the DOA estimation can be performed.

The phase difference Φ(q) at qth channel between two antennascan be derived by

Φ(q) = arg [χ1(q)(τ, ν)]∗[χ2

(q)(τ, ν)] , q = 1, …, Nch, (51)

where χm(q)(τ, ν) denotes the bistatic range-velocity cross-correlation

functions obtained by qth channel and mth antenna at the range τ

Fig. 4  RMSE of the single-channel ML, multi-channel ML (solid line), andthe CRB (dotted line) according to the SNR, Nch = 4

Fig. 5  RMSE of the single-channel ML, multi-channel ML (solid line), andthe CRB (dotted line) according to the SNR; Nch = 6

Fig. 6  RMSE of the single-channel ML, multi-channel ML (solid line), andthe CRB (dotted line) according to the number of snapshots

622 IET Radar Sonar Navig., 2018, Vol. 12 Iss. 6, pp. 617-625© The Institution of Engineering and Technology 2018

and the velocity ν. To integrate the multi-channel-based phasedifference Φ(q), the weighted sum of the phase differencecorresponding to the wavelength can be written as

Φ^ = 1Nch

∑q = 1

Nch

α(q)λ(q)Φ(q), (52)

where ∑q = 1Nch α(q) = 1 and the weight vector α = α(1), …, α(Nch) T

canbe obtained by various criteria. In [20], the estimated SNR at eachchannel is proposed to be used as the weight vector α. The multi-channel-based DOA estimation can be estimated by

θ^ = sin−1 Φ^

2πd . (53)

To calculate the estimation error, we consider an array with M = 2antennas and the number of channels with Nch = 5. The SNRvalues of each FM channel are set as −30, −27, −24, −21, and −18 dB and the carrier frequencies are the same as the frequencies inFig. 6. To derive the estimation results proposed in [20], thefollowing weight vectors α1, α2, and α3 are used:

α1 = 1, 1, 1, 1, 1 T, (54)

α2 = Γα1, (55)

α3 = Γ2α1, (56)

where α1 denotes the uniform weighting, α2 is weighted by theamplitude of the target echo signal and α3 is weighted by the signalpower of the target echo signal.

Fig. 9 shows the estimation performance of the proposedmethod and the estimation method in [20]. The estimationperformance of the proposed method outperforms the weighted-sum methods of (54)–(56). Although the number of snapshotsincreases, the RMSEs of the weighted-sum methods cannotapproach the multi-channel-based CRB.

5 ConclusionsWe derived the multi-channel-based ML angle estimator in theapplication of an FM-radio-based PCL system. By exploiting theproperty of the multi-channel-single-transmitter setup, we showedthat the estimation accuracy can be improved through the use ofmulti-channel configuration.

Furthermore, the target association problem that may be causedby the ambiguity in the target echo signals was also taken intoconsideration. To solve this association problem, we proposed analgorithm for estimating the permutation matrix, which resolvedthe issue of the ambiguity of the target echo signal. We alsodetermined the relationship between the single-channel- and multi-channel-based CRB.

From the simulations, we found that the multi-channel-basedML angle estimator has a lower RMSE than that of the single-channel-based ML technique. We also found that the RMSE of themulti-channel signal asymptotically approaches the CRB.

6 AcknowledgmentsThis work was supported by the Basic Science Research Programof the National Research Foundation of Korea (NRF) funded bythe Ministry of Education under grant no.2017R1D1A1B04035230.

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[3] Inggs, M., Tong, C., Nadjiasngar, R., et al.: ‘Planning and design phases of acommensal radar system in the FM broadcast band’, IEEE Aerosp. Electron.Syst. Mag., 2014, 29, (7), pp. 50–63

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Fig. 7  Comparison of the RMSE according to whether the targetassociation is applied to the multi-channel ML technique(a) θ = 25∘, and, (b) θ = − 20∘

Fig. 8  Likelihood function for K = 2 targets(a) Without target association and, (b) With target association

Fig. 9  RMSE of the multi-channel-based DOA estimation algorithmsaccording to the number of snapshots

IET Radar Sonar Navig., 2018, Vol. 12 Iss. 6, pp. 617-625© The Institution of Engineering and Technology 2018

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[28] Li, J., Compton, R.T.: ‘Maximum likelihood angle estimation for signals withknown waveforms’, IEEE Trans. Signal Process., 1993, 41, (9), pp. 2850–2862

8 Appendix In this appendix, the CRB of the DOA estimation for the multi-channel-based PCL is derived. The first-order derivative of the log-likelihood function in (25) according to the unknown parameter ψis given by

∂ln L∂θ = 2Re ∑

q = 1

Nch

∑n = 1

NS(q)H(tn)C

~(q)H(θ)e~(q)(tn) ,

∂ln L∂Re γ(q) = 2Re ∑

n = 1

NY(q)H(tn)D

~ (q)H(θ)e~(q)(tn) ,

∂ln L∂Im γ(q) = 2Im ∑

n = 1

NY(q)H(tn)D

~ (q)H(θ)e~(q)(tn) ,

(57)

where Y(q)(tn) = diag y1(q)(tn), . . . , yK

(q)(tn) , S(q)(tn) = Γ(q)Y(q)(tn),

C~(q)(θ) = W(q) −1/2

C(q)(θ),C(q)(θ) = c(q)(θ1), …, c(q)(θK) ,

(58)

and

c(q)(θk) = ∂d(q)(θk)∂θk

. (59)

It should be noted that x~(q) for an arbitrary vector x(q) satisfies

x~(q) = W(q) −1/2x(q) . (60)

Remarks 1–3 in [28] can be exploited to obtain the second-orderderivative of the log-likelihood function according to the unknownparameter. The difference of the multi-channel and the single-channel case is that the error term e~(q)(tn) is uncorrelated with theerror term in the other channel such as e~(r)(tn), where r ≠ q, i.e.

E e~(q)(tn)e~(r)H(tn) =IM, q ≠ r,0M, q = r, (61)

where IM represents the identity matrix and 0M represents a zero-valued square matrix of size M. (see (62))

By using (57), (61), and Remarks 1–3 in [28], the expectationof the second-order derivatives of the log-likelihood functionaccording to the unknown parameter can be obtained by using (see(63)) where δqr denotes the Kronecker delta. The expectation of thesecond derivative in (63) can be simplified by using the followingcompact form:

G1(q) = ∑

n = 1

NS(q)H(tn)C

~(q)H(θ)C~(q)(θ)S(q)(tn)

= N C~(q)H(θ)C~(q)(θ) ⊙ R^

ss(q)T,

(64)

G2(q) = ∑

n = 1

N[Y(q)H(tn)D

~ (q)H(θ)C~(q)(θ)S(q)(tn)],

= N D~ (q)H(θ)C~(q)(θ) ⊙ R^

ys(q)T,

(65)

and

G3(q) = ∑

n = 1

NY(q)H(tn)D

~ (q)H(θ)D~ (q)(θ)Y(q)(tn)

= N D~ (q)H(θ)D

~ (q)(θ) ⊙ R^yy(q)T .

(66)

In (64)–(66), the operator ⊙ represents the element-wisemultiplication of two matrices. From (63)–(66), the Fisherinformation matrix in (28) can be rewritten as shown in (62).

For the derivation of the CRB of θ, Remark 4 in [28], whichrepresents one method for deriving the inverse matrix, can be used.Remark 4 [28] may be used for the non-singular complex matrix Xas

624 IET Radar Sonar Navig., 2018, Vol. 12 Iss. 6, pp. 617-625© The Institution of Engineering and Technology 2018

Re(X) −Im(X)Im(X) Re(X)

−1

=Re(X−1) −Im X−1

Im(X−1) Re X−1 . (67)

Finally, using (63)–(67), the CRB of θ can be obtained as

CRB−1(θ) = 2Re ∑q = 1

Nch

G1(q) − G2

(q)H[G3(q)]−1

G2(q) (68)

Assuming that R^ss(q)

approaches Rss(q) asymptotically, then the CRB of

θk is simplified as

CRB−1(θ^k) = 2N ∑q = 1

Nch

Rss(q)

kk∥ Pa~k(q)

⊥ d~

k(q) ∥

2, (69)

where

Pa~k(q)

⊥ = I − a~k(q)Ha~k

(q) −1a~k

(q)a~k(q)H . (70)

When the uniform linear array with M antennas is considered, then(69) can be rewritten as

CRB(θ^k) = 6σ2

N(M2 − 1)Mcos2(θk)∑q = 1Nch Rss

(q)kk 2πd /λ(q) 2 , (71)

where d denotes the distance between the adjacent antennas, λ(q)

represents the wavelength in the qth channel, and W = σ2IMNch isassumed in (71).

CRB−1({Re[γ], Im[γ], θ}) = 2

Re∑q = 1

Nch G1(q) ReT G2

(1) ImT G2(1) ⋯ ReT G2

(Nch) ImT G2(Nch)

Re G2(1) ReT G3

(1) −ImT G3(1) ⋯ 0 0

Im G2(1) ImT G3

(1) ReT G3(1) ⋯ 0 0

⋮ ⋮ ⋮ ⋮ ⋮Re G2

(Nch) 0 0 ⋯ ReT G3(Nch) −ImT G3

(Nch)

Im G2(Nch) 0 0 ⋯ ImT G3

(Nch) ReT G3(Nch)

. (62)

E ∂ln L∂θ

∂ln L∂θ

T= 2Re ∑

q = 1

Nch

∑n = 1

NS(q)H(tn)C

~(q)H(θ)C~(q)(θ)S(q)(tn) ,

E ∂ln L∂Re[γ(q)]

∂ln L∂θ

T= 2Re ∑

n = 1

NY(q)H(tn)D

~ (q)H(θ)C~(q)(θ)S(q)(tn) ,

E ∂ln L∂Im[γ(q)]

∂ln L∂θ

T= 2Im ∑

n = 1

NY(q)H(tn)D

~ (q)H(θ)C~(q)(θ)S(q)(tn) ,

E ∂ln L∂Re[γ(q)]

∂ln L∂Re[γ(r)]

T= 2Re ∑

n = 1

NY(q)H(tn)D

~ (q)H(θ)D~ (r)(θ)Y(r)(tn) δqr,

E ∂ln L∂Im[γ(q)]

∂ln L∂Re[γ(r)]

T= 2Im ∑

n = 1

NY(q)H(tn)D

~ (q)H(θ)D~ (r)(θ)Y(r)(tn) δqr,

E ∂ln L∂Im[γ(q)]

∂ln L∂Im[γ(r)]

T= 2Re ∑

n = 1

NY(q)H(tn)D

~ (q)H(θ)D~ (r)(θ)Y(r)(tn) δqr,

q, r = 1, …, Nch,

(63)

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