Direct Position Determination with Single Sensor Based on Signal Periodicity

Hindawi
Mathematical Problems in Engineering
Volume 2021, Article ID 6649287, 12 pages
https://doi.org/10.1155/2021/6649287

Research Article
Direct Position Determination with Single Sensor Based on
Signal Periodicity

 Cheng Wang ,1 Ding Wang ,1 Lu Gao ,2 and Bin Yang1
 1
 PLA Strategic Support Force Information Engineering University, Zhengzhou 450001, China
 2
 Beijing Institute of Space Long March Vehicle National Key Laboratory of Science and Technology on
 Test Physics and Numerical Mathematics, Beijing 100076, China

 Correspondence should be addressed to Ding Wang; wang_ding814@aliyun.com

 Received 25 December 2020; Revised 3 March 2021; Accepted 15 April 2021; Published 13 May 2021

 Academic Editor: Ciro Núñez-Gutiérrez

 Copyright © 2021 Cheng Wang et al. This is an open access article distributed under the Creative Commons Attribution License,
 which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 Due to practical limitations on size and cost, aerial vehicles generally cannot equip complicated sensors to form sensor array for
 target localization. In this paper, we investigate the direct position determination (DPD) of stationary source via single moving
 sensor. First, we analyze artificial signal structure and construct the DPD model with the frame periodicity of artificial signal. The
 model incorporates Doppler information extracted from both transformation frames and adjacent samples into target locali-
 zation. Secondly, we consider the effect of oscillator instability and present an iterative solution for joint estimation of target
 location and phase noise caused by oscillator imperfection. The proposed technique fully exploits periodic structure of artificial
 wireless signal, which leads to significant enhancement in localization performance. Both theoretical analysis and simulations are
 presented to confirm its effectiveness.

1. Introduction the positioning accuracy [6, 7]. These single-station methods
 are designed based on the angle information and array
Passive localization of emitters is an important issue in many sensors. For moving station such as unmanned aerial ve-
scientific and engineering applications. Traditional two-step hicle, restrictions on payload dimensions and weight pose
localization methods extract intermediate parameter such as various challenges on the deployment of multichannel re-
angle or time delay of incident signals to determine source ceiver or sensor array.
positions [1]. In past decades, direct position determination To address such issue, a variety of techniques such as
(DPD) technique has become the focus of intensive research. reflector-aided geolocation have been proposed [4]. Among
DPD directly exploits sensor outputs to localize target existing solutions, single-station DPD with Doppler shift is
without estimating intermediate parameter. It outperforms regarded as the most promising one. This approach utilizes
traditional two-step methods especially for low signal-to- Doppler shift caused by sensor motion to determine target
noise ratio (SNR) and/or limited signal samples [2, 3]. location and enables single sensor geolocation [8, 9]. Though
 Among current DPD study, there has been considerable effective and simple, single-station DPD based on Doppler
interest for geolocation based on single moving station. This would encounter two problems in practice. First, complex-
is mainly due to the fact that both synchronization and data modulated symbols would bring phase fluctuation into re-
transmission between stations are not required when all ceived signal, so it demands approaches that could separate
processing is done within a single station [4]. Demissie first unknown symbols and recover Doppler shift [10]. For in-
developed and analyzed a mathematical DPD model with stance, methods proposed in [11, 12] utilize perfect temporal
moving station [5]. Based on array data, Capon iterative coherence of strictly noncircular signals to remove phase
optimization technique and characteristic of noncircular ambiguity caused by complex modulation. Single-channel
source are introduced into DPD, respectively, for improving technique used in [13, 14] adopts phase curve fitting for joint

2 Mathematical Problems in Engineering

estimation of both modulated symbols and Doppler infor- Superscripts T, H, and ∗ represent the transpose, conjugate
mation. However, these methods are limited to specific transpose, and complex conjugate, respectively.
 √��� Small letter j
modulations, which makes them pale in practical applica- denotes the imaginary number, i.e., − 1. symbolizes ab-
tion. Another inevitable challenge is the effect of oscillator solute value and ⊗ represents the Kronecker product. Re {·}
instability. Slight phase noise of oscillator will lead to error in and Im {·} signify the real and imaginary parts, respectively
location result and must be considered [15]. ⊙ represents the Hadamard product. represents the Eu-
 In this paper, we consider exploiting signal periodicity to clidean norm. Im denotes m × m identity matrix. 1n denotes
cope with above problem and investigate the DPD algorithm n-dimensional unit vector, and 0m is m-dimensional zero
based on single moving sensor. It is worth noting that most vector.
artificial signals have periodic structure such as periodic
radar pulse and communication signals that are coded in 2. Signal Model and Problem Formulation
frame for transformation. This periodicity allows for time
difference of arrival (TDOA) localization with single moving 2.1. Data Model for Single Receiving Sensor. Consider the
sensor [16, 17]. Similarly, single-channel system presented in scenario as pictured in Figure 1, where a flying sensor re-
[18] virtually rotates receiving antenna with the period of ceives the direct path signal from stationary emitter. The
target signal to provide angle information. Motivated by sensor intercepts signals at K different points along the
previous work, we analyze the signal structure and exploit trajectory and collects each batch of data in a short interval.
signal frame periodicity to abstract Doppler information for After downsampling, the baseband signal collected in the kth
target localization. Moreover, practical environment is interception interval can be written as
considered, where phase noise induced by oscillator im-
 rk (t) � βk s(t)exp j2πfk (p)t + jϕk (t) + nk (t), (1)
perfections is not known exactly but obeys Gaussian dis-
tributions with known correlation coefficients. We establish where p is the target coordinate vector; s(t) represents signal
the DPD model with oscillator phase noise. Then, an iter- waveform; nk (t) is the additive noise; βk stands for complex
ative solution is provided for joint estimation of target lo- fading factor; ϕk (t) denotes random phase noise which is
cation and phase noise. Simulation results show the mainly introduced by oscillator drift [15]; and fk (p) denotes
proposed DPD algorithm can achieve better location ac- the Doppler shift caused by platform motion, that is,
curacy compared with existing methods.
 The main contributions of this paper are listed as follows: fc vTk p − qk 
 fk (p) � �� � (2)
 c ��p − qk ��� ,
 (1) With the analysis of wireless signal structure, we
 divide each signal frame into periodic preamble and where fc represents signal frequency; c is the speed of light;
 data block that contains random information sym- and qk and vk denote coordinate vector and velocity vector
 bols. Then, a novel DPD model is provided for of the receiving sensor at the kth interval, respectively.
 dealing with such partially periodic signal. It in- Equation (2) describes the relationship between fk (p)
 corporates the Doppler information extracted from and source position, which implies that information of
 both transformation frames and adjacent samples source position is involved in the residual frequency of rk (t).
 into target localization. Generally, sensor position qk and velocity vk are considered
 (2) We combine the established signal model with the to be known a priori. However, in contrast with active
 probability distribution of oscillator phase noise. To systems where transmitted signal is under control by the
 avoid multidimensional search, an iterative opti- receiver, s(t) is not available for passive receivers. The
 mizing scheme is then presented for updating target complex-modulated symbols contained in s(t) would bring
 position and phase noise alternately. The algorithm phase ambiguity into rk (t) and restrict the Doppler shift
 fully exploits the location information contained in recovery [10]. Meanwhile, the existence of ϕk (t) also poses
 Doppler shifts and minimizes the effect of oscillator challenges on the abstraction of Doppler shift.
 imperfections.
 (3) The Cramér–Rao bound (CRB) of the source posi-
 2.2. Structure of Artificial Wireless Signal. To get rid of
 tion estimation based on the received signal model is
 complex-modulated sequence s(t), we first analyze the
 derived.
 structure of artificial wireless signal in the sequel. As shown
 The rest of the paper is organized as follows. Section 2 in Figure 2, it is well known that most artificial signals are
introduces the signal model and formulates the problem. characterized by frame transmission and each frame consists
Section 3 presents the DPD method with Doppler infor- of preamble block and data block [19]. Data blocks of signals
mation of received signal. Performance analysis is given via generally contain audio or video data while the preamble
CRB in Section 4. In Section 5, simulation experiments are blocks carry advanced information like data coding pa-
provided for validating the proposed estimator. Finally, rameters, service labels, current date, synchronization se-
conclusion is drawn in Section 6. quence, and so on. Note that both frame structure and
 preamble data remain almost unchanged once communi-
Notation. Throughout the paper, bold lowercase letters are cation service begins [20]. Hence, we can draw the con-
used for vectors, and bold upper case letters are for matrices. clusion that there exists periodicity in artificial signals and

Mathematical Problems in Engineering 3

 Sensor trajectory

 Emitter

 Figure 1: The scenario of source geolocation with single moving sensor.

 Length of frame TF

 · · ·

 Preamble block Data block Preamble block Data block

 Sampling interval Ts

 · · ·
 1 0 1 0

 Figure 2: Typical structure of wireless signals.

frame length TF is exactly the length of period. Such frame where TF denotes frame length and M is the number of
periodicity phenomenon also occurs in signals of periodic involved frames. σ 2a , σ 2b are the power of sa (t), sb (t),
radar pulse or the radar rotating with constant revolution respectively. In practice, TF can be obtained from signal
speed. standards or directly estimated from received data via
 Moreover, since pulse-shaping filters in digital modulation TDOA measurements [15], so it is assumed to be
are real-valued, the phase ambiguity is mainly bought by in- known to the receiver.
formation symbols of s(t). As such, there exists no phase (A2) Narrow-band signal and a slow-fading environ-
change among samples within each symbol. As demonstrated ment are considered, that is, s(t) remains unchanged in
in Figure 2, adjacent samples of an oversampled signal belong a short period of signal samples and fades indepen-
to the same symbol with a high probability, so phases of these dently among observation intervals. Hence, s(t)satisfies
samples generally remain the same. For narrow-band signal
with low baud rate, this phenomenon is frequently s(t) ≈ s t − nTs , n � 0, 1, . . . , N − 1, (4)
encountered.
 Both frame periodicity and the phase invariance be- where integer N is the number of involved samples and
tween continuous samples has been successfully utilized Ts stands for sampling interval.
in non-cooperative signal processing, such as direction of (A3) The oscillator phase noise ϕk (t), k � 1, . . . , K is
arrival estimation [19] and blind frequency recovery [21]. statistically independent from interval to interval.
Based on above analysis, we list the assumptions in this Within each observation interval, ejϕk (t) is a Gaussian
study: stationary process that satisfies
 (A1) Signal source s(t)can be divided into preamble ⎪
 ⎧
 ⎪
 ⎪ E ejϕk (t) � 1,
 sequence sa (t) and data sequence sb (t), that is, ⎪
 ⎪
 ⎨ τ) ∗
 s(t) � sa (t) + sb (t). sa (t) is independent with sb (t) and ⎪ E ejϕk (t) − 1 ejϕk (t− − 1 � g(τ), (5)
 they satisfy ⎪
 ⎪
 ⎪
 ⎪
 ⎩ E ejϕk (t) − 1 ejϕk (t− τ)
 − 1 � g(τ),
 E sa (t)s∗a t − mTF ≈ σ 2a , m � 0, 1, . . . , M − 1,
 where τ denotes time delay and correlation coefficients
 0, if m ≠ 0,
 E sb (t)s∗b t − mTF ≈ g(τ) and g(τ) are assumed to be known to the receiver.
 σ 2b , if m � 0,
 Above assumptions describe the temporal coherence of
 (3) artificial wireless signals. Though above discussions focus on

4 Mathematical Problems in Engineering

the frame structure of artificial signals, the repetition of 3. Proposed DPD Method
spreading code or cyclic prefix in orthogonal frequency
division multiplexing system also meets Assumption A1. 3.1. DPD Model with Signal Periodicity. Based on As-
Interested readers may refer to [22] for details. Relying on sumption A2, a new observation vector r(t) ∈ CN×1 can be
above assumptions, the problem that we address now is to constructed as follows:
determine target position directly with rk (t) expressed in (1)
and (2).

 T
 rk (t) � rk (t), rk t − Ts , . . . , rk t − (N − 1)Ts 
 T
 ≈ βk s(t)ej2πfk (p)t+jϕk (t) , . . . , s(t)ej2πfk (p)[t− (N− 1)Ts ]+jϕk (t− (N− 1)Ts )
 + nk (t) (6)

 � βk Γ1,k ak (p)s(t)ej2πfk (p)t + nk (t),

 j2πfk (p)(N− 1)Ts T
 ak (p) � 1, e− j2πfk (p)Ts
 , . . . , e− . (7)
where N × 1 noise vector nk (t) � [nk (t), nk (t − Ts ), . . . ,
nk (t − (N − 1)Ts )]T . η1,k � [ejϕk (t) , . . . , ejϕk (t− (N− 1)Ts ) ]T de- According to Assumption A1, rk′(t) ∈ CM×1 can be
scribes the effect of oscillator phase noise and diagonal similarly constructed as
matrix Γ1,k � diag η1,k . The N × 1 steering vector ak (p) has
the following form:

 T
 rk′(t) � rk (t), rk t − TF , . . . , rk t − (M − 1)TF 
 T
 � βk sa (t) + sb (t) ej2πfk (p)t+jϕk (t) , . . . , sa t − (M − 1)TF + sb t − (M − 1)TF ej2πfk (p)[t− (M− 1)TF ]+jϕk (t− (M− 1)TF )
 
 + nk′(t)
 � βk Γ2,k bk (p)sa (t) + bk (p) ⊙ sb (t) ej2πfk (p)t + nk′(t),
 (8)

 j2πfk (p)(M− 1)TF T
 T bk (p) � 1, e− j2πfk (p)TF
 , . . . , e− . (9)
where sb (t) � [sb (t), . . . , sb (t − (M − 1)TF )] . η2,k �
[ejϕk (t) , . . . , ejϕk (t− (M− 1)TF ) ]T and Γ2,k � diag η2,k . The M × With (6)–(9), we can construct array vector
1 steering vector bk (p) has the following form: xk (t) ∈ CMN×1 as

 T
 xk (t) � rTk (t), rTk t − TF , . . . , rTk t − (M − 1)TF 
 T
 � βk Γ1,k ak (p) ⊗ s(t)ej2πfk (p)t+jϕk (t) , . . . , s t − (M − 1)TF ej2πfk (p)[t− (M− 1)TF ]+jϕk (t− (M− 1)TF )
 + wk (t) (10)

 � βk Γ1,k ak (p) ⊗ Γ2,k bk (p)sa (t) + Γ2,k bk (p) ⊙ sb (t) ej2πfk (p)t + wk (t),

 � σ 2b Γ2,k bk (p)bH H
 k (p)Γ2,k ⊙ IM .
where noise vector wk (t) � [nTk (t), . . . , nTk (t− (11)
 T
(M − 1)TF )] . Using properties of Hadamard product and
Assumption A1, we find that Since the diagonal entries of Γ2,k bk (p)bH H
 k (p)Γ2,k are all 1,
 2
 H
 Rb,k becomes the diagonal matrix with σ b as its diagonal
 Rb,k � E Γ2,k bk (p) ⊙ sb (t) Γ2,k bk (p) ⊙ sb (t) entry, i.e., Rb,k � σ 2b IM . Then, with properties of Kronecker
 product and the independence between sa (t) and sb (t),
 � Γ2,k bk (p)bH H H
 k (p)Γ2,k ⊙ E sb (t)sb (t) covariance matrix of xk (t) leads to

Mathematical Problems in Engineering 5

 Rk � E xk (t)xH k (t) 
 2 H
 � βk σ 2a Γ1,k ak (p) ⊗ Γ2,k bk (p) Γ1,k ak (p) ⊗ Γ2,k bk (p) + Γ1,k ak (p)aH H
 k (p)Γ1,k ⊗ Rb,k + Wk
 2 (12)
 H H
 � βk σ 2a Γ1,k ak (p) ⊗ Γ2,k bk (p) Γ1,k ak (p) ⊗ Γ2,k bk (p) + σ 2b Γ1,k ak (p) ⊗ IM Γ1,k ak (p) ⊗ IM + Wk
 2
 � βk σ 2a c1,k p, ηk cH 2 H
 1,k p, ηk + σ b c2,k p, ηk c2,k p, ηk + Wk ,

where c1,k (p, ηk ) � Γ1,k ak (p) ⊗ Γ2,k bk (p). c2,k (p, ηk ) � where vec(·) is a vector obtained by stacking columns of the
Γ1,k ak (p) ⊗ IM . ηk � [ηT1,k , ηT2,k ]T is the column vector with argument on top of each other. Φ1,k (p, ηk ) � vec(c1,k (p, ηk )
dimensions M + N. The noise covariance cH H
 1,k (p, ηk )) and Φ2,k (p, ηk ) � vec(c2,k (p, ηk )c2,k (p, ηk )). Φk
Wk � E[wk (t)wH k (t)] is determined by temporal correlation (p, ηk ) � [Φ1,k (p, ηk ), Φ2,k (p, ηk ), vec(Σ)] and ςk � [σ 2a |βk |2 ,
of noise, so its elements are allowed to differ from each other. In
the following, Wk is supposed to satisfy Wk � Σσ 2k , where σ k σ 2b |βk |2 , σ 2k ]T .
denotes unknown noise power. MN × MN matrix Σ repre- Denote η(b) (b) T (b) T T
 k � [(η1,k ) , (η2,k ) ] as the estimation of ηk .
sents noise covariance matrix. Σ is generally known to the Then, the Markov-like estimate of p with η(b) k is obtained as
receiver, since it can be determined, for example, using sample K
statistics from a number of independent, identical experiments. � arg min γk − γ k H Ck γk − γ k ,
 p (14)
 With above discussions, the Markov-like estimates of p p k�1
with ηk , k � 1, . . . , K can be obtained by fitting covariance k ), and R
 k is
 where Ck denotes weighting matrix, c k � vec(R
matrix of received data to (12) in a weighted least squares
 the covariance matrix constructed by data samples of xk (t).
sense. However, a multidimensional search is essential for
 Generally, Ck is chosen to be the inverse of the asymptotic
estimating target position p and phase noise vector simul-
 covariance of the residuals [23]. Substitution of (13) into (14)
taneously, which has dominant computational complexity
 yields
[23]. To solve the problem, an iteration optimizing scheme is
adopted here. Specifically, the optimization of target position K � ��2
 �
p is performed with fixed ηk , k � 1, . . . , K in each iteration. � arg min ���C0.5
 p (b)
 k ��� ,
 k Φk p, ηk ςk − γ (15)
 p k�1
Then, the refined estimate of ηk , k � 1, . . . , K could be
obtained with respect to the latest estimated p. The updating where C0.5 is a Hermitian square root factor of Ck .
 k
procedure continues iteratively until converged. In the next According to (15), the estimation of ςk that minimizes the
subsection, the above iterative solution will be derived. cost function yields
 − 1
3.2. Updation of Target Position. According to (12), Rk ςk � ΦH (b) (b) H (b)
 k .
 k p, ηk Ck Φk p, ηk Φk p, ηk Ck γ
can be expressed in vector form as (16)
 2 2
 γk � vec Rk � σ 2a Φ1,k p, ηk βk + σ 2b Φ2,k p, ηk βk Substituting (16) into (15) leads to
 + vec(Σ)σ 2k � Φk p, ηk ςk ,
 (13)

 K � �� ��2
 (b) − 1 H ��
 � arg min ���C0.5
 p k Φk p, η (b)
 k ΦH
 k p, η(b)
 k C Φ
 k k p, η k Φk p, η (b)
 k C 
 γ
 k k − 
 γ 
 k � �
 p k�1
 K (17)
 − 1
 � arg min γ H k − γ H
 k Ck γ
 (b) H (b) (b) H (b)
 k .
 k Ck Φk p, ηK Φk p, ηk Ck Φk p, ηk Φk p, ηk Ck γ
 p k�1

 As K H
 k�1 c is constant during optimization, equa-
 k Ck c
tion (17) becomes

 K
 − 1
 � arg max γ H
 p (b) H (b) (b) H (b)
 k .
 k Ck Φk p, ηk Φk p, ηk Ck Φk p, ηk Φk p, ηk Ck γ (18)
 p k�1

6 Mathematical Problems in Engineering

 Target position p can be directly updated via grid search where Bk (p) � diag bk (p) . Since Rb,k � σ 2b IM , the ei-
with (18). Next section investigates the procedure for phase genvalue decomposition of covariance matrix of rk′(t)
noise vector updating. leads to

3.3. Updation of Phase Noise Vector. Equation (6) can be R2,k � E rk′(t)r′H H H
 k (t) � λk2 uk2 uk2 + Uk2 Λk2 Uk2 , (26)
rewritten as
 rk (t) � βk Γ1,k ak (p)s(t)ej2πfk (p )t + nk (t)
 (a)
 where vector uk2 denotes the eigenvector related to the
 (19) largest eigenvalue λk2 . Uk2 and Λk2 contain noise eigen-
 � βk Ak (p)η1,k s(t)ej2πfk (p)t + nk (t), vectors and eigenvalues of R2,k , respectively. Similar to (23),
 η2,k , k � 1, . . . , K can be updated by minimizing the fol-
where Ak (p) � diag ak (p) . Then, the eigenvalue decom-
 lowing cost function:
position of its covariance matrix has the following form:
 R1,k � E rk (t)rH H H
 k (t) � λk1 uk1 uk1 + Uk1 Λk1 Uk1 , (20) f η2,k � αηH H
 p)Uk2 UH
 2,k Bk ( k2 Bk ( 
 p)η2,k
 (27)
where vector uk1 denotes the eigenvector related to the largest H
eigenvalue λk1 . Uk1 and Λk1 contain noise eigenvectors and + η2,k − 1M G−2 1 η2,k − 1M .
eigenvalues of R1,k , respectively. Based on subspace theory,
Ak (p)η1,k are orthogonal to the subspace spanned by Uk1 [8]. Taking the complex gradient with respect to η2,k and
Moreover, based on Assumption A3, the covariance matrix of solving gives
η1,k and η2,k is assumed to satisfy
 H − 1
 E η1,k − 1N η1,k − 1N � G1 , 2,k � αBH
 η p)Uk2 UH
 k ( p) + G−2 1 G−2 1 1M .
 k2 Bk ( 
 (28)
 (21)
 H
 E η2,k − 1M η2,k − 1M � G2 , The steps of above target localization scheme are listed
 in Table 1, and related flowchart is shown in Figure 3. As
where the m, nth entry of G1 and G2 is g((m − n)Ts ) and shown in the table, step 3 is engaged to update target
g((m − n)TF ), respectively. location based on received signal, while step 4 is applied
 as the vector containing estimated source
 Denote p for determining phase noise vector η1,k , η2,k , k � 1, . . . , K.
location. With (20) and (21), η1,k , k � 1, . . . , K can be The iteration may be stopped by comparing estimation
updated by minimizing the following cost function: results between adjacent iteration. The main complexity of
 K proposed method is involved in step 3. The computational
 f η1,1 , . . . , η1,K � α ηH H
 p)Uk1 UH
 1,k Ak ( k1 Ak ( 
 p)η1,k complexity for estimating p via grid search is about
 k�1 o(8JKM2 N2 ), where J represents the number of grid
 K �� ��2 points in the search. When there exists no oscillator phase
 + ���G−1 0.5 η1,k − 1N ��� , noise, it merely requires steps 1∼3 to determine target
 k�1
 location.
 (22)
where real constant α is generally selected according to the
noise level. Based on A3, the presumed oscillator phase noise 4. CRB on Position Estimation
is statistically independent from interval to interval. Hence,
 In this section, we use the Cramér–Rao bound (CRB) on
the cost function in (22) can be decoupled as
 position estimation as a performance benchmark. By the
 f η1,k � αηH H
 p)Uk1 UH
 1,k Ak ( k1 Ak ( 
 p)η1,k properties of maximum-likelihood estimators, the as-
 H
 (23) ymptotic distribution of p is Gaussian with covariance
 + η1,k − 1N G−1 1 η1,k − 1N . matrix that equals the CRB.
 Based on Assumption A3, η1,k , η2,k satisfy
 Taking the complex gradient with respect to η1,k and
solving gives T
 − 1 E η1,k − 1N η1,k − 1N � G1 ,
 1,k � αAH
 η p)Uk1 UH
 k ( p) + G−1 1 G−1 1 1N .
 k1 Ak ( 
 (24)
 (29)
 Similarly, equation (8) can be expressed as T
 E η2,k − 1M η2,k − 1M � G2 ,
rk′(t) � βk Γ2,k bk (p)sa (t) + Γ2,k bk (p) ⊙ sb (t) ej2πfk (p)t + nk′(t)
 where the m, nth entry of G1 and G2 is g((m − n)Ts ) and
 � βk Bk (p)η2,k sa (t) + Γ2,k bk (p) ⊙ sb (t) ej2πfk (p)t + nk′(t), g((m − n)TF ), respectively. Substituting (21) into (29) and
 (25) solving gives

Mathematical Problems in Engineering 7

 Table 1: Implementation of the proposed method.
(1) Choose a small constant e and set η(b)
 1,k � 1N , η(b)
 2,k � 1M .
 ′
(2) Stack the received data vector rk (t), rk(t), xk (t) and then compute related covariance matrices and subspace.
(3) Substitute η(b) (b)
 1,k , η2,k into (18) to estimate p via grid search.
(4) Insert p into (24) and (28) for estimating η(a) (a)
 1,k , η2,k , k � 1, . . . , K, respectively.
 K (a) (b) 2 (a) (b) 2
(5) If k�1 [‖η1,k − η1,k ‖ + ‖η2,k − η2,k ‖ ] is smaller than e, regard p as the position estimation. Otherwise, set η(b) (a) (b) (a)
 1,k � η1,k , η2,k � η2,k and
return to step 3.

 Import received data vectors
 rk (t), r′(t),
 k xk(t), set η1,k
 (b)
 = 1N,
 (b)
 η2,k = 1M and choose a small constant e

 Compute pˆ with η(b) (b)
 1,k, η2,k

 (b) (a)
 Set η1,k , η1,k,
 Compute η(a) (a)
 1,k, η2,k with estimated p
 ˆ
 η(b) (a)
 2,k, η2,k

 K
 (a)
 If || η1,k – η(b) (a) (b) 2
 1,k || + || η2,k – η2,k || < e No
 k=1

 Yes Export p̂

 Figure 3: Flowchart of the proposed method.

 (Re(G1 ) + Re(G1 ))/2 (Im(G1 ) − Im(G1 ))/2
 T Re G1 + Re G1 where G1′ � (Im(G1 ) + Im(G1 ))/2 (Re(G1 ) − Re(G1 ))/2
 ,
 E Re η1,k − 1N Re η1,k − 1N � ,
 2 (Re(G2 ) + Re(G2 ))/2 (Im(G2 ) − Im(G2 ))/2
 G2′ � (Im(G ) + Im(G ))/2 (Re(G2 ) − Re(G2 ))/2
 , and 2K(M + N) ×
 2 2
 Re G2 + Re G2 
 E Re η2,k − 1M ReT η2,k − 1M � , 1 vector η � [1TN , 0TN , 1TM , 0TM , . . .]T .
 2
 According to investigated DPD model and the proba-
 Re G1 − Re G1 bility distribution of the presumed phase noise vector, the
 E Im η1,k − 1N ImT η1,k − 1N � , log-likelihood function yields
 2
 K
 1 �� ��2
 Re G2 − Re G2 f γk |ζ � − 2 ��C0.5 k ��
 k Φk p, ηk ςk − γ
 E Im η2,k − 1M ImT η2,k − 1M � , k�1 σ k
 2 (32)
 �� − 0.5 ��2
 Im G1 − Im G1 − ���G′ (η − η)��� ,
 E Re η1,k − 1N ImT η1,k − 1N � ,
 2
 where ζ � [p, ς, η]T , ς � [ς1 , ς2 , . . . , ςK ]T and G′ �
 Im G2 − Im G2 blkdiag G1′, G2′, . . . , G1′, G2′ .
 E Re η2,k − 1M ImT η2,k − 1M � , With conclusions in [24], the Fisher information matrix
 2
 for all the real unknowns ζ is computed by
 Im G1 + Im G1 
 E Im η1,k − 1N ReT η1,k − 1N � , K
 zΦk p, ηk ςk
 H
 zΦ p, ηk ςk
 2 Jζ � 2 − 0.5 ⊥ − 0.5
 Ck Πk Ck k 
 k�1
 zζ zζ
 Im G2 + Im G2 
 E Im η2,k − 1M ReT η2,k − 1M � .
 2 O O
 (30) + ⎡⎢⎢⎣ ⎤⎥⎥⎦,
 − 1
 O G′
 Denote 2K(M + N) × 1 column vector η � [Re ηT1,1 ,
 (33)
Im ηT1,1 , Re ηT2,1 , Im ηT2,1 , . . . , Re ηT1,K , Im ηT1,K , Re
 where Π⊥k � IMN − C− 0.5 Φk (p, ηk )[ΦH (p, ηk )C− 1 Φk
 ηT2,K , Im ηT2,K ]T . With Assumption A3 and (30), the co- k

variance matrix of η can be rewritten as (p, ηk )]− 1 ΦH
 k ((p, ηk )C
 − 0.5
 }/σ 2k and Odenotes zero matrix.
 Without loss of generality, we suppose weighting matrix C �
 H
 E (η − η)(η − η) � blkdiag G1′, G2′, . . . , G1′, G2′ , (31) IMN and denote

8 Mathematical Problems in Engineering

 K
 zΦk p, ηk ςk
 H generator with zero mean and the variance of 1. The cor-
 ⊥ zΦ p, ηk ςk
 Ypp � Πk k , relation coefficient √� is supposed to decrease linearly with τ,
 zp zp
 k�1 i.e., g(τ) � σ 2g / τ , where σ 2g denotes the oscillator noise
 H
 power. Unless otherwise stated, it is fixed at 0 dB. Root mean
 K
 zΦk p, ηk ςk ⊥ zΦ p, ηk ςk square error (RMSE) is adopted to evaluate the localization
 Ypς � Πk k ,
 k�1
 zp zς estimation, which is computed by
 �����������������
 K H 
 zΦ p, ηk ςk ⊥ zΦ p, ηk ςk 1 1000
 Ypη � k Πk k , RMSE � (i)‖2 ,
 ‖p − p (37)
 k�1
 zp zη 1000 i�1
 (34)
 K H
 zΦk p, ηk ςk ⊥ zΦk p, ηk ςk 
 Yηη � Πk ,
 zη zη where p (i) is the estimation result in the ith Monte Carlo
 k�1
 trial.
 K H Firstly, we contrast proposed method with the two-step
 zΦk p, ηk ςk ⊥ zΦ p, ηk ςk
 Yςς � Πk k , localization algorithm (designated as two-step) presented in
 k�1
 zς zς [10] and the array-based DPD algorithm (designated as
 DPD-array) introduced in [9]. DPD-array determines target
 K H
 zΦ p, ηk ςk ⊥ zΦ p, ηk ςk localization with both Doppler information abstracted from
 Yςη � k Πk k .
 zς zη adjacent samples and the angle estimated with sensor array.
 k�1
 Here DPD-array is applied with a uniform linear array with
Then, Jζ can be rewritten as 3 antennas, each spacing half-wavelength apart. The two-
 step method utilizes nonlinear transformation to recover
 Ypp Ypς Ypη
 ⎢
 ⎡
 ⎢ ⎤⎥⎥⎥ Doppler information from QPSK modulated signal samples.
 ⎢
 ⎢
 ⎢ T ⎥⎥⎥
 Jζ � 2⎢
 ⎢
 ⎢ Ypς Yςς Yςη ⎥⎥⎥. (35) Figure 5 demonstrates position estimation results, and it
 ⎢
 ⎢
 ⎢ ⎥⎥⎦ shows that RMSE of the proposed method coincides with
 ⎣ − 1
 YTpη YTςη Yηη + 0.5G′ CRB when input SNR is larger than 3 dB. The result in
 Figure 5 indicates that the proposed method has the ability
 With matrix inversion lemma, CRB in the presence of to achieve better estimation accuracy than both DPD-array
oscillator noise is given as and two-step methods. The improvement mainly results
 1 − 1 from the employment of signal frame periodicity. Since
 CRB(p) � Ypp − Ypς Y−ςς1 YTpς − Ζ , (36) nonlinear transformations may amplify additive noise or
 2
 − 1 produce interfering components, the achievable perfor-
 1
where Ζ � (Ypη − Ypς Y−ςς1 Yςη )(Yηη + 0.5G′ − YTςη Y−ςς1 Yςη )− mance of two-step method cannot be satisfied at low SNR
(Ypη − Ypς Y−ςς1 Yςη )T . region.
 Secondly, we evaluate the location RMSEs of tested
5. Numerical Results methods with different source positions when SNR is 5 dB.
 As Figure 6 shows, the performance of each algorithm would
In this section, simulation results are provided to demon- degrade with the increase of X coordinate, since the
strate the effectiveness of proposed algorithm. Without loss transmitter and observer are spaced further apart with larger
of generality, we assume a planar geometry and depict both X. The result implies that the proposed method is able to
sensor trajectory and target position in Figure 4. The sensor attain better estimation accuracy with worse location
moves from [− 20, 20]T (km) to [0, 20]T (km) with constant geometry.
velocity v � [250, 0]T (m/s). It intercepts the transmitted Thirdly, we discuss some important factors with respect
signal once per 4 kilometer. The target is fixed at [20,0]T to the proposed algorithm. Specifically, we change the
(km). number of preamble symbols to alter preamble percentage in
 To enable DPD based on frame periodicity, target signal Figure 7 and alter the involved signal frames M in Figure 8.
is assumed to have the transmission frame of 3.5 ms and each Other parameters including frame length remain un-
frame contains 12% periodic preamble symbols. Symbols are changed. Figure 7 shows the influence of percentage of
transmitted using QPSK modulation with 6 kBd. Signal is preamble symbols on proposed estimator. Since nonperiodic
received with the sampling rate of 2.5 MHz and its carrier data symbols cannot produce bk (p) defined in (9) for lo-
frequency fc � 1.5 GHz. Parameters related to target signal calization, the proposed method achieves lower RMSE with
are summarized in Table 2. When adopting DPD processing, less preamble symbols involved in Figure 7. When SNR is
data related to 3 signal frames (M � 3) are collected for 5 dB, the achievable RMSE with 5% preamble symbols ap-
localization during each observation interval and signal proaches 0.5 km. RMSEs versus the number of involved
waveform is assumed to remain unchanged within 3 adja- signal frames M are shown in Figure 8. The results dem-
cent samples (N � 3). onstrate that the performance of proposed method degrades
 Fading factor βk , k � 1, . . . , K is independently and with less signal frames. The phenomenon is due to the fact
randomly drawn from a complex Gaussian random that we exploit Doppler information for target localization.

Mathematical Problems in Engineering 9

 Table 2: Parameters of target signal.
 Percentage of preamble
Carrier frequency (GHz) Modulation type Baud rate (kBd) Frame length (ms) Sampling rate (MHz)
 symbols (%)
1.5 QPSK 600 3.5 12 2.5

 Observing
 position

 ······

 ctory
 r traje
 Senso Target position
 40
 20 20
 0 10
 Y (k 0
 m) –20
 –40 –10 m)
 –20 X (k
 Figure 4: Scenario of the target and receiving sensor.

 5
 4.5
 4
 3.5
 3
 RMSE (km)

 2.5
 2
 1.5
 1
 0.5
 0
 –15 –10 –5 0 5 10 15
 SNR (dB)
 CRB Two-step
 Proposed method DPD-array
 Figure 5: Estimated RMSEs versus SNR.

The accuracy of Doppler shift estimation is mainly deter- the level, the frame periodicity assumption given in (A1)
mined by signal collection time, which corresponds to MTF would become invalid and the Doppler extracted from
in the proposed method. transformation frames cannot be utilized for DPD. Note that
 Moreover, we evaluate the proposed method when signal the proposed method still works well when Te < 1 μs, which
period is not exactly known to the receiver, i.e., indicates that it provides certain robustness to such im-
 F � TF + Te , where Te is the error caused by oscillator or
T perfections. Figure 10 shows RMSEs versus the power of
other imperfections. Figure 9 shows location estimates with oscillator phase noise when SNR � 5 dB. As phase noise
distinct period errors. From the figure, we find that the brings ambiguity for localization, it can be seen from Fig-
performance of the proposed method degrades significantly ure 10 that RMSEs would become larger with the increase of
with error larger than 1.5 μs. This phenomenon can be noise power. For phase noise smaller than 5 dB, its influence
predicated. When period error and symbol duration are of can be ignored.

10 Mathematical Problems in Engineering

 1.2
 101

 1

 100
 0.8
 RMSE (km)

 RMSE (km)
 0.6 10–1

 0.4

 10–2
 0.2

 0 10–3
 14 16 18 20 22 24 26 28 30 –15 –10 –5 0 5 10 15
 X (km) SNR (dB)
 CRB Two-step M=2
 Proposed method DPD-array M=3
 M=4
 Figure 6: Estimated RMSEs versus different source positions.
 Figure 8: Estimated RMSEs with different number of involved
 frames.

 4.5 9

 4 8

 3.5 7
 3 6
 RMSE (km)

 RMSE (km)

 2.5 5
 2
 4
 1.5
 3
 1
 2
 0.5
 1
 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
 0
 Percentage of preamble symbols –15 –10 –5 0 5 10 15 20
 SNR (dB)
 SNR = –5 dB
 SNR = 0 dB Te = 0.8 µs
 SNR = 5 dB Te = 1.6 µs
 Te = 2.4 µs
Figure 7: Estimated RMSEs versus percentage of preamble
symbols. Figure 9: Estimated RMSEs with period error.

Mathematical Problems in Engineering 11

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