Yu et al. EURASIP Journal on Wireless Communications
and Networking (2015) 2015:266
DOI 10.1186/s13638-015-0497-9
R ES EAR CH
Separation and localization of multiple
distributed wideband chirps using the
fractional Fourier transform
Jiexiao Yu, Liang Zhang*, Kaihua Liu and Deliang Liu
Open Access
Abstract
In this paper, we consider the problem of localizing the multiple distributed wideband chirp sources using the fractional
Fourier transform. The model in the time domain and that in the fractional Fourier domain derived by the Taylor series
expansion are presented respectively. The representation of location vector in the Dechirping domain is illustrated which
is only related to the central angle. A novel direction of arrival (DOA) estimation algorithm in the Dechirping domain is
proposed, which is extended from the conventional multiple signal classification (MUSIC) algorithm in the time domain.
Using this algorithm, except for estimating the DOA, the incidence source number can be determined as well which is
allowed to exceed the sensor number in the array. To demonstrate the performance of proposed algorithm, numerical
results are conducted. Compared to the previous FrFT-MUSIC algorithm based on the assumption of point source
model, the proposed algorithm performs a better estimation performance, especially for large angular spread and low
signal-to-noise ratio.
Keywords: Direction of arrival, Fractional Fourier transform, Distributed wideband chirp, Generalized array manifold
1 Introduction
The direction of arrival (DOA) estimation for wideband
chirp signal has been greatly achieved under the point
source assumption [1–3]. However, the signal is always af-
fected by multipath and scattering propagation in the
practical application, which will cause the angular spread,
such as the local scattering source in the mobile multipath
environment, the distributed target reflection wave in the
low-elevation radar target tracing system, the surface and
the bottom reflection signal in the sonar detection of the
shallow sea, tropospheric or ionospheric propagation of
radio waves, the part of detection target in the passive
radar and sonar system [4–6]. In this situation, the re-
ceived signal in an array can be considered as a superpos-
ition of scattered signals originating from the different
direction and the estimation performance of DOA will be
degraded significantly if using the traditional estimators
based on the point source model [7, 8].
For obtaining the exact nominal DOA and angular
spread of a spatially distributed source, the problem of
distributed source model has been widely studied since
* Correspondence: vfleon@163.com
School of Electronic Information Engineering, Tianjin University, Tianjin, China
the early 1990s, and a large number of methods are pro-
posed for the parameter estimation of distributed source
[9–17]. However, most of the models and the estimation
algorithms can be only exploited in the case of narrowband
source, because the location vector in the time domain is
time varying when the incident source is wideband. To
date, the estimation method for wideband source with local
scattering is still in scarcity. M. Ghogho et al. proposed the
distributed wideband source model [18]. Liu et al. [19] pro-
posed a kind of maximum-likelihood (ML) estimates for
finite-bandwidth distributed sources by the perturbation
method. Foroozan and Asif [20] introduced time reversal-
based range and DOA estimators to exploit spatial/multi-
path diversity existing in strong multipath environments,
which has a high complexity in the multi-source scenarios.
Mecklenbrauker et al. [21] extended the Bayesian approach
to a distributed wideband source for the application to
seismic recordings, the kind that should be sparse.
The wideband chirp signal, whose frequency is linearly
increasing with time, is widely used in many applications,
such as communication, radar, sonar, and biomedicine, and
also be used as signal model for a good deal of natural phe-
nomena. However, for the distributed wideband chirps, we
© 2015 Yu et al. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International
License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any
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license, and indicate if changes were made.
Yu et al. EURASIP Journal on Wireless Communications and Networking (2015) 2015:266
Page 2 of 8
have not found much work referring to the problem of
DOA estimation.
In our previous work [22], the properties of the frac-
tional Fourier transform (FrFT), which is a generalized
form of the ordinary Fourier transform (FT), is exploited
to address the problem mentioned above because it pro-
vides a compact representation for chirp signals as it is
based on the decomposition of the signal on the ortho-
normal basis set of the chirp functions [23, 24]. The
models of wideband chirp sources in both time domain
and energy-concentrated domain are studied. And an es-
timation of the spatial parameters of multiple wideband
chirps with local scattering is provided.
In order to obtain better estimation precision, we will
further study the problem in another fractional Fourier
domain, the Dechirping domain, in this paper. The rep-
resentation of location vector in the Dechirping domain
is derived based on a number of good properties of chirp
signal in two special fractional Fourier domains, which is
only related to the angular parameter. Then, the stand-
ard MUSIC algorithm combined with the source separ-
ation technique is extended to estimate the incident
angles of multiple sources in the Dechirping domain.
Numerical results illustrate that the proposed algorithm
could separate multiple chirp signals successfully, which
is far more than the number of sensors, and estimate the
DOA of each chirp accurately.
2 Fractional Fourier transform
2.1 Notation and definition
The fractional Fourier transform as a linear integral
transform with kernel Kα(u, t) [25, 26]:
Xα uð Þ ¼ F α x tð Þ
½
¼
ð
K α u; t
Þx tð Þdt
ð1Þ
Z ∞
−∞
2.2 Time-frequency rotating properties
According to the time-frequency rotating properties of
the FrFT, the Wigner-Ville distribution (WVD) of the
FrFT of a signal can be interpreted as the coordinate ro-
tating form of the WVD of this signal [27]. The energy
spectrum of a finite chirp signal shows a fin-shape line
in the fractional Fourier domain, as shown in Fig. 1 and
two special fractional Fourier domains are particularly
the Dechirping domain, whose axis u
noteworthy:
coincides with the line, and its perpendicular domain,
energy-concentrated domain, in which the energy distri-
bution of chirp signal shows an obvious peak.
We assume that a chirp signal is modeled as
y tð Þ ¼ βejπ 2f
ð
ð3Þ
where β ¼ a0ejφ
0 is a constant, a0 symbolizes the ampli-
tude of the chirp signal, φ0 is the initial phase, f0 is the
initial frequency, and μ0 is the chirp rate.
tþμ
t2
Þ
0
0
Then, the rotation angles of the chirp signal in the
Dechirping domain and the energy-concentrated domain
can be written as
αd ¼ tan‐1μ
αe ¼ − cot‐1μ
ð4Þ
ð5Þ
0
0
The relationship between these two rotation angles
can be represented by
αd ¼ αe þ π=2
Y αd uð Þ ¼ F Y αe uð Þ
½
where Fα[⋅] denotes the FT operator.
ð6Þ
ð7Þ
where Fα[⋅] denotes the FrFT operator and kernel
Kα(u, t) is
8<
:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
1−j cotα
δ t−uð
Þ;
Þ;
δ t þ u
ð
ð
K α u; t
Þ ¼
ð
ejπ t2 cotα−2tucscαþu2 cotα
Þ; α≠nπ
α ¼ 2nπ
α ¼ 2n 1
Þπ
ð
ð2Þ
where the domain u is generally known as the fractional
Fourier domain which makes the rotation angle α with
the time domain.
Rather than defining the fraction of the transform as
the rotation angle, α, in the interval [−π, π] radians, a
new variable, p, is defined as the order of the transform
and is valid in the interval [−2, 2]. The zeroth order
transform is simply the function itself, whereas the first
order transform is its Fourier transform.
Fig. 1 The energy spectrum of a chirp signal with a finite length
Yu et al. EURASIP Journal on Wireless Communications and Networking (2015) 2015:266
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0 cosαdu
where
It can be seen that the FrFT of a chirp signal in the
Dechirping domain equals to the FT of the FrFT of
this signal in the energy-concentrated domain. As a re-
sult, when the chirp rate μ0 is unknown which is satis-
fied in most instances, the rotation angle ad can be
acquired using (6) after the determination of the rota-
tion angle ae in the energy-concentrated domain by
searching the coordination of spectral peak in two-
dimensional plane (α,u).
2.3 Time delay representation in Dechirping domain
According to (3), we can obtain the y(t − τ) as follows [25]:
ð
y t−τ
½
Þ ¼ βejπ 2f
¼ βejπ −2f
ð
0
ð
0
Þþμ
t−τ
τþμ
0
Þ2
ð
t−τ
Þej2π f
ð
τ2
0
−μ
τ
0
0
Þtejπμ
0
t2
ð8Þ
where τ is the time delay.
Compared with (3), it can be seen that the chirp rate
of the signal is invariable after the time delay performs
on this signal. Therefore, the rotation angle of the time-
delayed signal is the same as that of the original signal
in the Dechirping domain.
In Dechirping domain, the FrFT of a chirp signal given
by (3) can be written as:
where
Y αd uð Þ ¼ F αd y tð Þ
½
p
C ¼ β cosαd
¼ Cej2πf
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ j tanαd
ð
e −jπf
2 sinαd cosαd
0
Þ
ð9Þ
ð10Þ
According to the time shifting property of the FrFT,
F αd y t−τ
the FrFT of y(t − τ) in the same Dechirping domain is:
Þ
ÞY αd u−τ cosαd
0 cos2αdτ
ð11Þ
Þ
¼ e j2πτ2 sinαd cosαd
¼ Cej2π f
ð
0 cosαd−τ sinαd
ð
Þe −j2πuτsinαd
ð
Þuejπ τ2 sinαd cosαd−2f
ð
½
ð
½
where τ is the time delay.
In the digital calculation process in FrFT [28], the
chirp rate will be significantly reduced because of the di-
mensional normalization. As a result,
the numerical
value of the rotation angle αd is far from π/2. If τ < < f0,
which is satisfied in most practical applications, we can
obtain that
j
τ sinαd
j << f
j
j
0 cosαd
Thus, (11) can be approximated to
F α y t−τ
Þ
≈Cej2πf
0 cosαuejπ τ2 sinα cosα−2f
½
0 cos2ατ
Comparing (8) and (12), we can obtain:
F α y t−τ
Þ
¼ A τð ÞF α y tð Þ
½
½
½
ð
ð
where
ð12Þ
ð13Þ
ð14Þ
½
0 cos2αdτ
A τð Þ ¼ ejπ τ2 sinαd cosαd−2f
ð15Þ
In (15), A(τ)is only related with the time delay τ , and
the influence of the independent variable u has been
eliminated. This useful conclusion will play an important
role in deducing data model in Dechirping domain in
the next section.
3 Data model
3.1 Data model in time domain
A scenario with Q uncorrelated sources transmitted
wideband chirp signals is considered. Due to multipath
propagation, each source can be seen as a superposition
of Nq scattered point-source components. The uniform
linear array (ULA) with P sensors is taken as an example
for derivation, and the similar conclusions are easy to be
generalized to the arrays with other types. The complex
envelope of the output vector in the array can be mod-
eled as
x tð Þ ¼
βqia θq þ ~θqi; t
sq t−νqi
þ n tð Þ
XQ
XN q
q¼1
i¼1
ð16Þ
βqi: the complex amplitude of the ith scattered source
from the qth source;
a θi þ ~θqi; t
: the time-varying location vector of the
array, θq is the nominal DOA of the qth source and θi
þ~θqi
is the DOA of the ith scattered components from
the qth source;
sq(t): the qth transmitted signal;
vqi: the time delay associated with the ith scattered
source from the qth source;
n(t): the additive zero-mean noise vector, which is as-
sumed to be spatially white and independent of the
transmitted signals.
When the received signals are wideband and with local
scattering, there is no one-to-one relationship between
the DOA parameter and the location vector a θi þ ~θqi; t
which is also dependent on the time and deviation angle
variables. Therefore,
traditional parameter estimation
methods cannot work well in the case of distributed wide-
band chirp. The time-frequency properties of chirp signal
in the two fractional Fourier domain mentioned above
can help us eliminate the influence of time and deviation
angle variable on the location vector. The detailed deriv-
ation will be given in the next section.
3.2 Approximating the spatial signature
Because the majority of scattered signals distribute in the
vicinity of the transmitted source, the time delay vqi is
Yu et al. EURASIP Journal on Wireless Communications and Networking (2015) 2015:266
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XQ
XN q
q¼1
i¼1
relatively small so that (12) is satisfied and the quadratic
term of vqi can be ignored. Hence, the ith scattered com-
ponent of the qth source received on the conference sen-
sor in the Dechirping domain is given by
≈ejπ vqið Þ2
≈ejπ −2f q cos2αdvqi
ð
sinαd cosαd−2f q cos2αdvqi
ÞF αd sq tð Þ
F αd sq tð Þ
F αd sq t−vqi
ð17Þ
Then, the FrFT with a rotation angle αd performs on
the output vector in (16), and the result can be given by
X uð Þ ¼
q uð Þ þ N uð Þ
Sαd
ð18Þ
βqie−j2πf q cos2αdvqiAq θq þ ~θqi
is the location vector of the qth
q uð Þ is the FrFT of sq (t) and N(u) is the FrFT
where Aq θq þ ~θqi
source, Sαd
of n(t) in the Dechirping domain.
Because FrFT of multiple sources or that of one single
source’s multi-scattering components does not result in
cross-term interference, the conclusion in [31] could be
h
exploited here directly as follows:
¼ 1; A2 θq þ ~θqi
; ⋯; AP θq þ ~θqi
Aq θq þ ~θqi
iT
ð19Þ
Ap θq þ ~θqi
where T is the transpose and
¼ ejπ τpð Þ2
sinαd cosαd−2f
0 cos2αdτp
ð20Þ
ð
Þd sin θq þ ~θqi
where τp ¼ p−1
=c represents the time
delay on the pth sensor, d is the inter-sensor distance of
ULA, and c is the transmission speed.
In order to estimate θq, [29] proposed an approximate
model called the generalized array manifold (GAM) and a
corresponding algorithm using a Vandermonde structure.
Due to the deviation angle ~θqi of each scattered signal is
relatively small, Aq θq þ ~θqi
in (20) can be approxi-
mated through the first-order Taylor series expansion:
Aq θq þ ~θqi
þ ~θqid θq
≈Aq θq
where
d θð Þ ¼ ∂Aq θð Þ
∂θ
Then, we define a variable vq, that is
ð21Þ
ð22Þ
XN q
cqiAq θq þ ~θqi
!
XN q
þ
Aq θq
i¼1
cqi
¼
XN q
!
i¼1
~θqi
cqi
i¼1
vq ¼
XN q
i¼1
¼
cqi Aq θq
d θq
þ ~θqid θq
¼ Aq θq
þ ϕqd θq
ð23Þ
XN q
ci
~θqi and we as-
i¼1
where cqi ¼ βqie−j2πf q cos2αdvqi , ϕq ¼
XN q
sume that
i¼1
Φ ϕð Þ
cqi ¼ 1:
þ D θ
Hence, the compact matrix notation V can be given by
V¼A θ ; ϕ
ð24Þ
¼ A θ
¼ A1 θ1ð
¼ d θ1ð
Þ; …; AQ θQ
Þ; …; d θQ
¼ diag ϕ
; …; ϕQ
¼ ϕ
A θ
D θ
Φ ϕ
θ ¼ θ1; …; θQ
; …; ϕQ
ð26Þ
ð25Þ
T
1
Tϕ
1
where
Finally, the model (18) can be approximated as
X uð Þ ¼ VS uð Þ þ N uð Þ
iT
uð Þ ⋯ Sαd
Q uð Þ
and N uð Þ ¼ N αd
½
1
ð27Þ
uð Þ
h
where S uð Þ ¼ Sαd
⋯ N αd
Q uð ÞT .
1
According to (24), we know that the spatial parame-
are needed to estimate in the approxi-
ters θ and ϕ
mated model.
4 Parameter estimation algorithm
4.1 Single source estimation algorithm
Firstly, we consider one chirp source (Q = 1 in (16)).
Many DOA estimation algorithms for the narrowband
point source model can also be applied here because
the location matrix in the Dechirping domain has a
one-to-one relationship with the spatial parameters. In
this paper, the classical MUSIC algorithm in the time
domain is extended to estimate spatial parameter of
wideband chirp source with local
scattering. The
spatial estimation spectrum of standard MUSIC algo-
rithm is
V MUSIC θð Þ ¼ a θð Þ^En ^E
a θð Þa θð Þ
na θð Þ
ð28Þ
In the Dechirping domain, the calibrated location
vector is exploited to replace the location vector in
(28), that is
Yu et al. EURASIP Journal on Wireless Communications and Networking (2015) 2015:266
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A θð Þ þ ϕd θð Þ ¼ A θð Þϕ
ð29Þ
T .
where Ā(θ) = [A(θ)
Hence, the estimation spectrum for the proposed model
should be revised to
ϕ θð Þ ¼ 1 ϕ
ϕd(θ)] and
½
V MUSIC θ; ϕð
Þ ¼ ϕ A
A θð Þϕ
θð Þ^En ^E
A θð ÞA θð Þ
n
ð30Þ
The parameters θ and ϕ can be determined by the
spectral peak position.
in their
rotation angles
4.2 Separation of different chirp sources and the source
number determination
As (5) shows, chirp signals with different chirp rates
have different
energy-
concentrated domain in which chirp signal shows an
obvious spectrum peak. The energy-concentrated do-
main of each signal can be determined by the two di-
mension search in the time-frequency plane (α,u). In
each energy-concentrated domain, only one chirp signal
can acquire the best energy-concentrated property
while the energy distributions of the other signals and
noise are dispersing, therefore, this chirp signal can be
separated from others through a band-pass filter. Then,
the parameter estimation algorithm proposed above
can be applied to filtered single source after that is
transformed into the corresponding Dechirping do-
main. The detailed separation process could be found
in [31]. It is noted that the separation process cannot
only resist the interference of noise and other signals,
but also solve the problem of source number restriction
in many traditional DOA estimators. For example, the
MUSIC algorithm cannot be applied when the source
number exceeds the number of the sensors in the array.
However, the proposed algorithm avoids this problem
through the separation processing. The parameters of
each source can be estimated respectively.
5 Simulation study and results
To demonstrate the performance of the proposed algo-
rithm, an ULA with eight sensors is employed to
estimate the spatial parameters of the proposed model.
In the simulation, two uncorrelated wideband chirp
sources are considered as signal emitters. The initial
parameters of the first wideband chirp source are f1 =
100 Hz and μ1 = 10 Hz/s. And that of the second one
are f2 = 200 Hz and μ2 = −10 Hz/s. Besides, the central
angle of the first source is taken as θ1 = 30∘ with the ex-
tension width Δ1 = 2∘ from a uniform angular distribu-
tion, and the central angle of the second source is
taken as θ2 = − 45∘ with the extension width Δ2 = 2∘
from a Gaussian angular distribution of width, respect-
ively. The signal-to-noise ratio (SNR) is set at 10 dB,
and the number of snapshots is 500.
The two-dimensional searching results of two sources
on the reference sensor are shown in Fig. 2. According
to the coordinates of two distinct spectrum peaks, the
rotation angles of their energy-concentrated domain can
be determined, and the source number can be easily ob-
tained by the peak number.
Fig. 2 Two-dimensional spectrum of two wideband chirp sources in the plane (α, u)
Yu et al. EURASIP Journal on Wireless Communications and Networking (2015) 2015:266
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a
c
b
d
Fig. 3 The energy spectrums of the chirp signal and noise on the first sensor in the energy-concentrated domain. a Source 1. b Source 2. c Filtered
source 1. d Filtered source 2
In Fig. 3a, b, the energy spectrums of two sources on the
first sensor in the corresponding energy-concentrated do-
main show an obvious energy concentration respectively.
In this energy-concentrated domain, the chirp signal forms
a sinc function and the majority of the energy spectrum fo-
cuses on its support, and the single peak will appear on the
other sensors as well. Therefore, the desired chirp signal
on each sensor can be separated conveniently using the
band-pass filtering, and the filtered results are illustrated in
Fig. 3c, d.
The FrFT-MUSIC algorithm proposed in [30], which is
applied to estimate the DOA of the wideband chirp source
based on the point source model, and the method in [22] is
chosen as the comparing algorithm to demonstrate the per-
formance of the proposed algorithm for the wideband chirp
source with local scattering. For each figure, 100 Monte
Fig. 4 The estimation RMSE for the central angle versus the SNR. a Source 1. b Source 2
a
b
Yu et al. EURASIP Journal on Wireless Communications and Networking (2015) 2015:266
Page 7 of 8
Fig. 5 RMSE for the central angle versus the extension width. a Source 1. b Source 2
a
b
Carlo simulations were run to estimate the root mean-
square error (RMSE) of the estimates. We focus on the
estimation of the DOA, while the other parameters are
considered mainly as nuisance parameters.
Figure 4 shows the RMSE for the DOA estimates ver-
sus the different SNRs using three algorithms. For both
sources, the RMSE of the proposed algorithm is less
than that for the other two algorithms. Figure 5 clearly
demonstrates that the proposed estimator has excellent
performance versus the different extension widths. With
the increasing extension width, the proposed algorithm
has better location accuracy than the others.
Finally, the resolution performance of the proposed al-
gorithm versus the different source number is demon-
strated. In this experiment, the chirp rate interval Δμ of
each source is chosen as 1 Hz/s. The numerical results
can be seen in Fig. 6. The proposed algorithm can remain
the good resolution performance even though the source
number is far more than the sensor number.
6 Conclusions
In this paper, parametric localization of multiple wideband
chirp sources with local scattering have been considered.
The models in both the time and fractional Fourier domain
were proposed. In the Dechirping domain, a novel DOA
estimator combined the properties of the chirp signal and
the Taylor series expansion was addressed. The DOA of
each source and the source number can be determined
using the proposed method. Besides, the source number is
allowed to exceed the sensor number in the array. The
simulation study demonstrated that the proposed algo-
rithm provided a superior spatial resolution performance
for wideband chirp sources with local scattering than the
previous FrFT-MUSIC algorithm.
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
This work is funded by the National Natural Science Fund of China (No.
61501322).
Received: 7 September 2015 Accepted: 15 December 2015
Fig. 6 RMSE for the central angle versus the source number
2.
3.
4.
5.
References
1.
AB Gershman, MG Amin, Wideband direction-of-arrival estimation of
multiple chirp signals using spatial time-frequency distributions. Signal
Process IEEE Trans. 7(6), 152–155 (2000)
F Sellone, Robust auto-focusing wideband DOA estimation. Signal Process.
86(1), 17–37 (2006)
PH Leong, T Abhayapala, TA Lamahewa, Multiple target localization using
wideband echo chirp signals. Signal Process IEEE Trans 61(16), 4077–4089
(2013)
KI Pedersen, PE Mogensen, BH Fleury, Power azimuth spectrum in outdoor
environments. Electron. Lett. 33(18), 1583–1584 (1997)
CF Patrick, Angular propagation descriptions relevant for base station
adaptive antenna operations. Wireless Pers. Commun. 11(1), 3–29 (1999)
Yu et al. EURASIP Journal on Wireless Communications and Networking (2015) 2015:266
Page 8 of 8
7.
8.
6. M Tapio, Direction and Spread Estimation of Spatially Distributed Signals via
the Power Azimuth Spectrum, (IEEE Acoustics, Speech, and Signal Processing
(ICASSP) (Orlando, FL, USA, 2002), pp. 3005–3008
D Astely, B Ottersten, The effect of local scattering on direction of arrival
estimation with MUSIC. Signal Process IEEE Trans. 47(12), 3220–3234 (1999)
J Tabrikian, H Messer, Robust Localization of Scattered Sources (IEEE Statistical
Signal and Array Processing, Pcono Manor, PA, 2000), pp. 453–457
G Fuks, J Goldberg, H Mesor, Bearing estimation in a Ricean channel-Part 1:
inherent accuracy limitations. IEEE Trans. Signal Processing 49(5), 925–937
(2001)
S Valaee, B Champagne, P Kabal, Parametric localization of distributed
source. Signal Process IEEE Trans. 43(9), 2144–2153 (1995)
10.
9.
11. R Raic, J Goldberg, H Messer, Bearing estimation for a distributed source:
modeling inherent accuracy imitation and algorithms. IEEE Trans. Signal
Processing 48(2), 429–441 (2000)
12. M Bengtsson, N Ottersten, Low complexity estimation for distributed
13.
sources. IEEE Trans. Signal Process. 48(8), 2185–2194 (2000)
S Shahbazpanahi, S Valaee, AB Gershman, A covariance fitting approach to
parametric localization of multiple incoherently distributed sources. IEEE
Trans. Signal Process 52(3), 592–600 (2003)
14. BT Sieskul, S Jitapunkul, An asymptotic maximum likelihood for joint
estimation nominal angle and angular spreads of multiple spatially
distributed sources. IEEE Trans. Veh. Technol. 59(3), 1534–1538 (2006)
J Lee, J Joung, JD Kim, A method for the direction-of-arrival estimation of
incoherently distributed sources. IEEE Trans. Veh. Technol. 57(5), 2885–2893
(2008)
15.
16. A Zoubir, Y Wang, Performance analysis of the generalized beamforming
estimators in the case of coherently distributed sources. Signal Process 88,
428–435 (2008)
K Han, A Nehorai, Nested array processing for distributed sources. IEEE
Signal Process. Lett. 21(9), 1111–1114 (2014)
17.
18. M Ghogho, TS Durrani, Broadband direction of arrival estimation in
19.
20.
presence of angular spread. Electronics Letters 37(15), 986–987 (2001)
SJ Liu, Q Wan, J Yang, YN Peng, Asymptotic performance analysis of bearing
estimate for spatially distributed source with finite bandwidth. IEEE Electron.
Lett. 38(24), 1600–1601 (2002)
F Foroozan, A Asif, Time reversal based active array source localization. IEEE
Trans. Signal Process. 59(6), 2655–2668 (2011)
21. CF Mecklenbrauker, P Gerstoft, H Yao, Bayesian Sparse Wideband Source
22.
Reconstruction of Japanese 2011 Earthquake (IEEE Computational Advances in
Multi-Sensor Adaptive Processing (CAMSAP), San Juan, 2011), pp. 273–276
L Zhang, J Yu, K Liu, D Liu, A Novel DOA Estimation Algorithm for Wideband LFM
Source with Local Scattering (International Conference on Communications,
Signal Processing, and Systems, Hohhot, China, 2014), pp. 491–499
23. DMJ Cowell, S Freear, Separation of overlapping linear frequency
modulated (LFM) signals using the fractional Fourier transform. IEEE Trans.
Ultrason., Ferroelect., Freq. Contr 57(10), 2324–2333 (2010)
24. R Chen, Y Wang, Universal FRFT-based algorithm for parameter estimation of
chirp signals. Journal of Systems Engineering and Electronics 18(4), 495–501
(2012)
25. HM Ozaktas, MA Kuta, D Mendlovic, Introduction to the fractional Fourier
transform and its applications. Adv. Imag. Electron. Phys. 10(6), 239–291
(1999)
26. V Ashok Narayanan, KMM Prabhu, The fractional Fourier transform: theory,
implementation and error analysis. Microprocessors and Microsystems 27,
511–521 (2003)
LB Almeida, The fractional Fourier transform and time-frequency
representations. IEEE Trans. Signal Process 42(11), 3084–3091 (1994)
28. HM Ozaktas, O Arikan, MA Kutay, G Bozdagi, Digital computation of the
27.
fractional Fourier transform. IEEE Trans. Signal Process. 44(9), 2141–2150 (1996)
29. D Asztely, B Ottersten, AL Swindlehurst, Generalised array manifold model
for wireless communication channels with local scattering. Proc. Inst. Electr.
Eng.—Radar, Sonar Navigat 145(1), 51–57 (1998)
30. R Tao, Novel method for the direction of arrival estimation of wideband
linear frequency modulated sources based on fractional Fourier transform.
Transaction of Beijing Institute of Technology. 25(10), 895–899 (2005)
JX Yu, L Zhang, KH Liu, Coherently distributed wideband LFM source
localization. IEEE Signal Processing Letters 22(4), 504–508 (2015)
31.
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