This paper addresses the problem of joint time of arrival (TOA) and direction of arrival (DOA) estimation in impulse radio ultrawideband systems with a twoantenna receiver and links the joint estimation of TOA and DOA to the sparse representation framework. Exploiting this link, an orthogonal matching pursuit algorithm is used for TOA estimation in the two antennas, and then the DOA parameters are estimated via the difference in the TOAs between the two antennas. The proposed algorithm can work well with a single measurement vector and can pair TOA and DOA parameters. Furthermore, it has better parameterestimation performance than traditional propagator methods, such as, estimation of signal parameters via rotational invariance techniques algorithms matrix pencil algorithms, and other new jointestimation schemes, with one single snapshot. The simulation results verify the usefulness of the proposed algorithm.
I. Introduction
Impulse radio ultrawideband (IR–UWB) has recently attracted considerable interest for indoor geolocation and sensor networks due to its intrinsic properties, such as immunity to multipath fading, extremely short duration pulse, being carrier free, having a lowduty cycle, wide bandwidth, and lowpower spectral density
[1]
–
[5]
. On account of the high time resolution nature, UWB positioning based on timeofarrival (TOA) estimation methods becomes a superior alternative. Furthermore, if direction of arrival (DOA) can also be estimated in the positioning system it would reduce the number of reference nodes needed to estimate the position of UWB sources.
Recently the joint estimation of TOA and DOA is a hot topic in IRUWB communication systems, and many techniques have been proposed to obtain accurate TOA and DOA estimates. In
[6]
and
[7]
, TOA estimation was accomplished by using a matched filter but had strong practical limitations due to the requirement of extremely high sampling rates and complexity. To reduce the complexity at subNyquist sampling rates, the noncoherent algorithms based on energy detection were proposed in
[8]
and
[9]
. However, the precision of the TOA estimates decreased. Besides, the above algorithms consider the TOA estimation as a timing acquisition problem; TOA estimation was also linked to frequency domain and to superresolution techniques such as the multiple signal classification (MUSIC) algorithm
[10]
,
[11]
, minimumnorm algorithm
[12]
and propagator method (PM)
[13]
. These algorithms are applied after the estimated channelimpulse response is transformed to frequency domain and offers highresolution TOA estimation. However, the estimators obtain the TOA estimates by spectral peak searching, which has high complexity and large computation. In other works, such as
[14]
, a unionofsubspaces approach was proposed to recover the time delays and timevarying gain coefficients of each multipath component from lowrate samples of the received signal.
In addition to TOA, DOA is also one of the significant parameters in UWB communication systems, and the problem of DOA estimation has attracted considerable attention in earlier literatures. In
[15]
, a frequencydomain MUSIC algorithm was presented for the estimation and tracking of UWB signals. In
[16]
, the iterative quadratic maximum likelihood algorithm was applied to yield DOA estimates in UWB communication systems. In
[17]
, a beamspacebased DOA estimation method for directsequence UWB signals was proposed using the frequencydomain frequencyinvariant beamformers algorithm. DOA estimators, however, cannot explicitly exploit the advantage of the large bandwidth of UWB signals. Actually, TOA and DOA are closely related and can be jointly estimated
[18]
–
[21]
. In
[18]
, the matrix pencil algorithm was extended to joint TOA and DOA estimation for UWB positioning. The scheme proposed in
[19]
performs timing acquisition following a twostep approach: a coarse TOA estimator based on a minimum distance criterion, and a fine TOA estimator based on calculation of power delay profile and the selection of a suitable threshold. Finally, the DOA is obtained from the independent TOA measurements at each array antenna by means of a linear estimator. In
[20]
, the joint TOA and angleofarrival (AoA) estimator utilizes an array of antennas, each feeding a demodulator consisting of a squarer and a lowpass filter. Signal samples, taken at Nyquist rate at the filter outputs, are processed to produce TOA and AoA estimates. In
[21]
, a joint spacetime technique for UWB signals based on the extended MUSIC algorithm was presented. All those joint TOA and DOA estimation methods extract the DOA estimation from the TOA, and because of the high time nature of the UWB signals, DOA estimates can be obtained with reasonable accuracy.
Compressed sensing (CS), which is a novel theory introduced in
[22]
,
[23]
, unifies signal sensing and compression into a single task and can recover the sparse signal with high probability, from a set of random linear projections using nonlinear reconstruction algorithms. In addition to the signal reconstruction and restoration
[22]
,
[23]
, the CS framework has also been applied to UWB communication systems for signal detection, channel estimation, and TOA estimation
[24]
–
[32]
. In this paper, we propose a joint TOA and DOA estimation algorithm for IRUWB signals, based on the sparse representation framework, with a twoantenna receiver. After transforming the received signals to the frequency domain, the estimation problems are linked to the sparse representation framework. Exploiting this link, an orthogonal matching pursuit (OMP) algorithm is used for TOA estimation in the two antennas, and then DOA parameters are estimated via the difference of the TOAs between the two antennas. The proposed algorithm can work well with one single snapshot and can pair TOA and DOA parameters. Furthermore, it has better parameterestimation performance than the traditional PMs, such as, estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm, matrix pencil algorithm, and other, new, joint schemes in
[19]
and
[20]
, with a single measurement vector (SMV). The simulation results verify the usefulness of the proposed algorithm.
The remainder of this paper is structured as follows. Section II develops the data model, and section III presents the proposed algorithm. The CramérRao bound (CRB) of the joint estimation performance is derived in section IV. In section V, simulation results are presented to verify the improvements for the proposed algorithm, while our conclusions are shown in section VI.
Notation:
Bold lowercase letters represent vectors and bold uppercase letters represent matrices. The symbols (•)
^{*}
, (•)
^{T}
, (•)
^{H}
, (•)
^{1}
and (•)
^{+}
denote the complex conjugation, transpose, conjugatetranspose, inverse, and pseudoinverse, respectively. The symbol ‖ ‖
_{F}
stands for Frobenius norm. A diagonal matrix whose diagonal is the vector
v
is represented by diag(
v
). We denote an estimated expression by (
• ∧
)
II. Data Model
We assume that each information symbol is typically implemented by the repetition of
N_{c}
short pulses with modulation of directsequence binary phaseshift keying (DSBPSK). The transmitted signal can be represented as
(1) s(t)= ∑ i=−∞ +∞ ∑ n=0 N c −1 b i c n p(t−i T s −n T c ) ,
where
b_{i}
∈ {−1, +1} and
c_{n}
∈ {−1, +1} are the information symbols and the userspecific code sequence, respectively. The pulse waveform is referred to as
p
(
t
), which is the second derivative of a Gaussian pulse with
T_{s}
being the symbol duration and
T_{c}
being the chip duration. The transmitted UWB signal passes through a multipath channel, which is modeled, as in
[33]
, by
(2) h(t)= ∑ k=1 K β k δ(t− τ k ) ,
where
β_{κ}
and
τ _{κ}
are the fading coefficient and the propagation delay of the
k
th path, respectively. Without loss of generality, we assume
τ
_{1}
<
τ
_{2}
<…<
τ_{Κ}
. The Dirac Delta function is represented by
δ
(
t
), and
K
is the number of multipath components. Thus, the received signal can be expressed as the summation of multiple delayed and attenuated replicas of the transmitted signal plus the additive Gaussian white noise
w
(
t
), that is
(3) y(t)= ∑ k=1 K β k s(t− τ k ) +w(t) .
Performing a Fourier transformation on the received signal in (3), we can obtain
(4) Y(ω)= ∑ k=1 K β k S(ω) e −jω τ k +W(ω) ,
where
Y
(
ω
),
S
(
ω
), and
W
(
ω
) denote the Fourier transformation of
y
(
t
),
s
(
t
), and
w
(
t
), respectively. Then, sampling (4) at
ω_{m}
=
n
Δ
ω
, for
m
= 0, 1,…,
M
−1 and Δ
ω
= 2π/
M
(
M
>
K
), and rearranging the frequency samples
Y
(ω) into vector y = [
Y
(ω
_{0}
)…
Y
(ω
_{M−1}
)]
^{T}
∈ ℂ
^{M×1}
, yields the frequency domain signal model
(5) y=S E τ β+w ,
where
S
∈ℂ
^{M×M}
is a diagonal matrix whose components are the frequency samples
S
(
ω_{m}
), and
E
_{τ}
= [
e
(1) …
e
(
K
)]∈ℂ
^{M×K}
is a delay matrix with the column vectors being
e
(
k
) = [1
e^{jΔωτκ}
…
e^{j(M1)Δωτκ}
]
^{T}
for
k
= 1,…,
K
. The channelfading coefficients are arranged in the vector
β
= [
β_{1}…β_{K}
]
^{T}
∈ℂ
^{K×1}
, and the noise samples are arranged in vector
w
= [
W
(ω
_{0}
)…
W
(ω
_{M1}
)]
^{T}
∈ℂ
^{M×1}
.
III. Joint Estimation of TOA and DOA
 1. The Strategy for Joint Estimation of TOA and DOA
The UWB signal
s
(
t
) propagates through the
K
path fading channel and arrives at an array consisting of two antennas, which is shown in
Fig. 1.
A twoantenna receiver for joint TOA and DOA.
Assume that
τ_{k}
and
ς_{k}
are the TOAs of the
k
th path in antenna 1 and antenna 2 for
k
= 1,…,
K
. According to the above data model, the received signals in the frequency domain of each antenna can be expressed as
(6) y 1 =S E τ β+ w 1 ,
(7) y 2 =S E ς β+ w 2 ,
where
S
= diag([
S
(
ω
_{0}
) …
S
(
ω
_{M1}
)]) is a diagonal matrix whose components are the frequency samples of transmitted UWB signal
s
(
t
). The delay matrices
E
_{τ}
and
E
_{ς}
can be denoted by
(8) E τ =[ 1 1 ⋯ 1 e −jΔω τ 1 e −jΔω τ 2 ⋯ e −jΔω τ K ⋮ ⋮ ⋱ ⋮ e −j(M−1)Δω τ 1 e −j(M−1)Δω τ 2 … e −j(M−1)Δω τ K ]
and
(9) E ς =[ 1 1 ⋯ 1 e −jΔω ς 1 e −jΔω ς 2 ⋯ e −jΔω ς K ⋮ ⋮ ⋱ ⋮ e −j(M−1)Δω ς 1 e −j(M−1)Δω ς 2 … e −j(M−1)Δω ς K ],
where
β
= [
β_{1}…β_{K}
]
^{T}
∈ℂ
^{K×1}
represents the channelfading coefficients. Noise samples of antenna 1 and antenna 2 are arranged in
w
_{1}
= [
W
_{1}
(
ω
_{0}
)…
W
_{1}
(
ω
_{M1}
)]
^{T}
∈ℂ
^{M×1}
, and
w
_{2}
= [
W
_{2}
(
ω
_{0}
)…
W
_{2}
(
ω
_{M1}
)]
^{T}
∈ℂ
^{M×1}
respectively.
Let
Δ τ k ^ = τ k ^ − ς k ^
, which is the difference of the TOAs associated to the
k
th path. From
Fig. 1
, we get
(10) c× Δ τ k ^ =dsin θ k ,
with
θ_{k}
being the DOA of the
k
th path,
d
the distance between the two antennas, and c the speed of light. According to (10), we can obtain the closedform solution of
θ_{k}
, that is
(11) θ k ^ =arcsin( c× Δ τ k ^ d ), k=1,⋯, K.
From (11), we know that to obtain the estimation of DOAs, we must estimate the multipath delays in the two antennas first. So, in the following subsection, we will present the method to estimate the TOAs.
 2. TOA Estimation Based on Sparse Representation
In this subsection, we formulate the TOA estimation problem as a sparse representation problem. To solve (6) and (7) with a sparse representation, we generalize the delay matrices
E
_{τ}
and
E
_{ς}
to an overcomplete dictionary
E
in terms of all possible TOAs
(12) TOAs { τ 1 ^ , τ 2 ^ , ... , τ N ^ }with τ 1 ^ < τ 2 ^ <⋯< τ N ^ , such that E=[ e( τ 1 ^ ) e( τ 2 ^ ) ... e( τ N ^ ) ],
with
N
being the grid number, and
N
>>
K
,
N
>>
M
. Note that
E
is known and does not depend on the actual multipath arrival times
τ_{k}
and
ς_{k}
in this framework. Thus, the channelfading coefficients vector
β
can be extended to an
N
×1 vector
h
, where the
n
th element
h_{n}
is nonzero and equal to
β_{k}
if the arrival time of the
k
th multipath component is
τ n ^
and zero otherwise. It means that we can estimate the TOA as long as we find the position of nonzero values in
h
. Using the overcomplete dictionary
E
, (6) and (7) become
(13) y 1 =SE h 1 + w 1 ,
(14) y 2 =SE h 2 + w 2 .
Note, that we consider an SMV, that is, a single snapshot in the paper. Let
Ψ
=
SE
and we have
(15) y 1 =Ψ h 1 + w 1 ,
(16) y 2 =Ψ h 2 + w 2 .
For this case,
h
_{1}
and
h
_{2}
are sparse vectors which can be used to improve the TOAs estimation. To use the sparse property as a constraint, we utilize the
l
_{0}
norm to arrive at the following optimization:
(17) h 1 ^ =arg min h 1 ‖ h 1 ‖ l 0 s.t. ‖ y 1 −Ψ h 1 ‖ 2 2 ≤ε,
(18) h 2 ^ =arg min h 2 ‖ h 2 ‖ l 0 s.t. ‖ y 2 −Ψ h 2 ‖ 2 2 ≤ε,
where ‖
h
‖
_{l0}
counts the number of nonzero entries in 
h
 , and
ε
is the maximum acceptable error. These problems can be solved by linear programming techniques, such as in the sparse approximation algorithm OMP. The major advantages of this algorithm are its speed and its ease of implementation
[34]
. The OMP algorithm tries to recover the signal by finding in the measurement signal the strongest component, removing it from the signal, and searching the dictionary again for the strongest atom that is presented in the residual signal. The detailed recovery processing via the OMP algorithm is shown in
Fig. 2.

Input:

AnM×Mdiagonal matrixS.

AnM×Novercomplete dictionaryE.

AnMdimensional data vectory.

Output:

An estimateĥ.

A set containingkelements from {1,…,N}.

AnMdimensional approximationamof the datay.

AnMdimensional residualrk=yak.
Procedure:
1) Initialize the residual
r
_{0}
=
y
, the index set Λ
_{0}
= ø, and the iteration counter
t
= 1.
2) Find the index
λ_{t}
that solves the easy optimization problem
λ_{t}
= arg
max j=1,⋯,N  〈 r t−1 , φ j 〉 
, where φ
_{j}
is the
j
th column vector of
Ψ
. If the maximum occurs for multiple indices, break the tie deterministically.
3) Augment the index set and the matrix of chosen atoms Λ
_{t}
= Λ
_{t1}
∪{
λ
_{t}
} and
F
_{t}
= [
F
_{t1}
φ
_{λt}
]. We use the convention that
F
_{0}
is an empty matrix.
4) Solve a leastsquares problem to obtain a new signal estimate
q
_{t}
=
F
_{t}
^{+}
y
.
5) Calculate the new approximation of the data
a
_{t}
=
F
_{t}
q
_{t}
and the new residual
r
_{t}
=
y

a
_{t}
.
6) Increment
t
, and return to step 2) if
t
<
K
.
7) The estimate
ĥ
for the ideal signal, has nonzero indices at the components listed in Λ
_{k}
. The value of the estimate
ĥ
in component
λ_{j}
equals the
j
th component of
q
_{t}
.
Flowchart of OMP algorithm.
The two
K
sparse vectors
h
_{1}
and
h
_{2}
can be recovered by the above OMP algorithm, and TOA estimates in the two antennas can be equivalent to finding the sufficiently sparse
h
_{1}
and
h
_{2}
provided that the error terms are well suppressed. The TOA estimations
{ τ k ^ } k=1 K
and
{ ς k ^ } k=1 K
are determined from the sparse structure by plotting
h
_{1}
and
h
_{2}
on the grid of time samples.
 3. Pair Matching of TOA in Two Antennas
Since the estimated multipath delays
{ τ k ^ } k=1 K
and
{ ς k ^ } k=1 K
are obtained independently, we should associate these estimates so that we can get the DOA estimates via (11). From (6), (7), (13) and (14) we know that the channelfading coefficients vector
β
is extended to two
K
sparse vectors
h
_{1}
and
h
_{2}
, which can be expressed as
(19) h 1 =[ 0 h 1,1 0 h 1,2 0 ... 0 h 1,K 0 ],
(20) h 2 =[ 0 h 2,1 0 h 2,2 0 ... 0 h 2,K 0 ].
Actually, the nonzero elements in
h
_{1}
and
h
_{2}
are the same, that is,
{ h 1,k } k=1 K
and
{ h 2,k } k=1 K
∈ {
β
_{1}
,
β
_{2}
,…,
β_{K}
}. In practice, the estimates
{ h 1,k ^ } k=1 K
and
{ h 2,k ^ } k=1 K
are obtained independently in the case of; so, they are approximately equal. When
h
_{1}
and
h
_{2}
are obtained, we sort
{ h 1,k ^ } k=1 K
and
{ h 2,k ^ } k=1 K
in descending order, respectively, and have
h 1,1 ^ > h 1,2 ^ >⋯> h 1,K ^
and
h 2,1 ^ > h 2,2 ^ >⋯> h 2,K ^
. Then, we can get the TOA estimates
{ τ k ^ } k=1 K
and
{ ς k ^ } k=1 K
according to the positions of nonzero values
{ h 1,k ^ } k=1 K
and
{ h 2,k ^ } k=1 K
in
h
_{1}
and
h
_{2}
, and they are then paired.
 4. Major Steps for Joint Estimation of TOA and DOA
Till now, we have achieved the proposal for joint TOA and DOA estimation in IRUWB systems based on the sparse representation framework. We show the major steps of the proposed algorithm as follows:
1) Transform the transmitted and received signals into frequency domain and obtain
S
,
Y
_{1}
, and
Y
_{2}
.
2) Construct overcomplete dictionaryEand denote (6) and (7) by a sparse representation, which are shown in (13) and (14).
3) Recover the
K
sparse vectors
h
_{1}
and
h
_{2}
via OMP algorithm and then the TOA estimates
{ τ k ^ } k=1 K
and
{ ς k ^ } k=1 K
can be determined from the sparse structure by plotting
h
_{1}
and
h
_{2}
on the grid of time samples.
4) Estimate the DOA estimates
θ k ^
via (11).
Remark 1
: In practice, the information on the number of the multipath rays
K
is always unknown, but it can be estimated through some known techniques
[35]
.
 5. Advantages of Proposed Algorithm
The proposed algorithm has the following advantages:
1) The proposed algorithm can obtain paired TOA estimation, while the matrix pencil algorithm
[18]
cannot.
2) The proposed algorithm can work well with one single snapshot, which will be shown in section V.
3) The proposed algorithm has better parameter estimation performance than that of the matrix pencil algorithm, conventional PM algorithm, ESPRIT algorithm, and the algorithms in
[19]
and
[20]
, which will also be shown in section V.
IV. CRB
In this section, we will derive the CRB of the jointestimation performance based on the data model in the paper. According to
[36]
and
[37]
, we can derive the CRB of TOA estimation as follows:
(21) CR B TOA = σ 0 2 2 [ Re{ D H Π A(τ,ς) ⊥ D⊙ P ^ T } ] −1 ,
where
σ 0 2
represents the noise power,
A(τ,ς)=[ S E τ S E ς ],
,
D = [ ∂a( τ 1 , ς 1 ^ ) ∂ τ 1 ∂a( τ 2 , ς 2 ^ ) ∂ τ 2 … ∂a( τ K , ς K ^ ) ∂ τ K ∂a( τ 1 ^ , ς 1 ) ∂ ς 1 ∂a( τ 2 ^ , ς 2 ) ∂ ς 2 ... ∂a( τ K ^ , ς K ) ∂ ς K ] ,
, and
a( τ k , ς k ^ )
and
a( τ k ^ , ς k )
are the
k
th column of
A(τ, ς ^ ) and A( τ ^ ,ς),
respectively. The orthogonal projection matrix of A is
Π A(τ,ς) ⊥ = I 2M×2M − A(τ,ς) (A (τ,ς) H A(τ,ς)) −1 A (τ,ς) H
and
P ^ = [ P ^ s P ^ s P ^ s P ^ s ]
with
P ^ s =β β H
. Hadamard’s product is represented by ⊙.
From (21), we can rewrite the matrix
CRB
_{TOA}
as
CR B TOA =[ CR B τ κ κ CR B ς ]
with
CRB
_{τ}
being the CRB matrix of
τ
, and
CRB
_{ς}
being the CRB matrix of
ς
. The symbol
κ
denotes a part that is not considered in this paper.
According to (11), the CRB matrix of DOA can be expressed as
(22) CR B DOA = c 2 d 2 ( CR B τ +CR B ς ) ( Ψ −1 ) 2 ,
where
Ψ
= diag([cos(
θ
_{1}
),…, cos(
θ
_{K}
)]) with
θ
_{K}
,
k
= 1,…,
K
being the perfect DOA of the
k
th path.
V. Simulation Results
To assess the parameter estimation performance of the proposed algorithm, we present Monte Carlo simulations. Define the signaltonoise ratio (SNR) and rootmeansquare error (RMSE) as
(23) SNR=10lg ‖ y(t) ‖ F 2 ‖ w(t) ‖ F 2 ,
(24) RMSE= 1 K ∑ k=1 K 1 1000 ∑ m=1 1000  χ k,m ^ − χ k  2 ,
where
y
(
t
) is the received timedomain signal,
w
(
t
) denotes the additive Gaussian white noise, and
χ k,m ^
stands for the estimate of
χ
_{k}
in the
m
th Monte Carlo trial. In the simulations, we assume the UWB pulsewave function is
p
(
t
) = exp(−2π
t
^{2}
/ Γ
^{2}
)(1−4π
t
^{2}
/ Γ
^{2}
) . The main simulation parameters are shown in
Table 1
. The shaping factor for the pulse is represented by Γ. The repetition of every symbol is
N_{c}
= 5, the chip duration is
T_{c}
= 2 ns, and the symbol duration is
T_{s}
=
N_{c}T_{c}
= 10 ns. We plot the transmitted BPSKUWB signal
s
(
t
) and the UWB pulsewave function
p
(
t
) in
Figs. 3
and
4
. Furthermore, we suppose that there are
K
=3 rays of BPSKUWB arriving signals whose arrival times corresponding to the two antennas are (
τ
_{1}
,
ς
_{1}
) = (0.3 ns, 0.2 ns), (
τ
_{2}
,
ς
_{2}
) = (0.46 ns, 0.3 ns) and (
τ
_{3}
,
ς
_{3}
) = (0.62 ns, 0.5 ns), respectively. The UWB multipath channelfading coefficients
β_{k}
are known with
β
= [0.7 0.4
e
^{−jπ/2}
0.2]
^{T}
, and there are
M
=64 frequency samples with the received signal. The grid distance between adjacent grids is 0.002 ns.
Parameter  Value 
Shape factor Γ  0.25 ns 
Chip duration T_{c}  2 ns 
Symbol duration T_{s}  10 ns 
Modulation  BPSK 
Frequency samples M  64 
Number of multipath K  3 
Channelfading coefficients β_{k}  0.7,0.4e^{−jπ/2}, 0.2 
True estimates τ_{k}  0.3 ns, 0.46 ns, 0.62 ns 
True estimates ς_{k}  0.2 ns, 0.3 ns, 0.5 ns 
Space between the antennas d  10 cm 
Distance between adjacent grids  0.002 ns 
UWB pulse waveform p(t).
Transmitted signal s(t).
In
Figs 5
and
6
, we recover the sparse vectors
h
_{1}
and
h
_{2}
with different SNR and plot them on the grid of the time samples. Once
h
_{1}
and
h
_{2}
are obtained, the TOA estimates
τ k ^
and
ς k ^
are determined.
Figures 5
and
6
illustrate that the elements of the sparse vectors and the estimation of TOA become more accurate in collaboration with SNR increasing.
TOA estimation with SNR=0 dB.
TOA estimation with SNR=10 dB.
Figures 7
and
8
show the TOA estimation results of the proposed algorithm over 50 Monte Carlo simulations with SNR = 0 dB and SNR = 10 dB.
Figures 7
and
8
illustrate that our algorithm is effective for TOA estimation and the estimation precision improves as SNR increases.
TOA estimation with SNR=0 dB.
TOA estimation with SNR=10 dB.
Figures 9
and
10
present the TOA and DOAestimation performance of the proposed algorithm for different values of multipath
K
. It is indicated that the joint TOA and DOA estimation performance becomes better as
K
decreases.
TOAestimation performance with different K.
DOAestimation performance with different K.
Figures 11
and
12
display the TOA and DOAestimation performance comparison of the proposed algorithm with the conventional PM algorithm
[13]
, matrix pencil algorithm
[18]
, ESPRIT algorithm, and other new joint estimation algorithms in
[19]
and
[20]
for the two antennas case. Note that the ESPRIT algorithm is based on the data model in this paper and that all the algorithms are working under the condition of one single snapshot. From
Figs. 11
and
12
, we can draw the conclusion that the proposed algorithm has better joint TOA and DOA estimation performance than the conventional PM algorithm, matrix pencil algorithm, ESPRIT algorithm, and the algorithms in
[19]
and
[20]
.
TOA estimation comparison with different algorithms.
DOAestimation comparison with different algorithms.
The conventional parameterestimation algorithms PM, matrix pencil, and ESPRIT can all achieve reasonable estimation performances when collecting multiple snapshots. In the single snapshot condition, however, all these algorithms have poor estimation performances or even lose their effectiveness. Our algorithm is based on a sparse representation framework, where an
l
_{0}
norm optimization is used based on an OMP algorithm. It works well with one single snapshot and as such, our algorithm has better joint parameterestimation performance than other algorithms.
VI. Conclusion
In this paper, we have presented a new formulation of the joint TOA and DOA estimation problem for IRUWB signals in a sparsesignal representation framework, where a
l
_{0}
norm optimization is used based on an OMP algorithm. The proposed algorithm is able to estimate paired TOA and DOA parameters and has better joint TOA and DOA estimation performance than the conventional PM algorithm, matrix pencil algorithm, ESPRIT algorithm, and the algorithms in
[19]
and
[20]
. Numerical experiments illustrate the accuracy and efficacy of the proposed algorithm in a variety of parameters and scenarios.
This work was supported by China NSF Grants (61371169, 61071164), Jiangsu Planned Projects for Postdoctoral Research Funds (1201039C), China Postdoctoral Science Foundation (2012M521099, 2013M541661), Open Project of Key Laboratory of Underwater Acoustic Communication and Marine Information Technology (Xiamen University), Hubei Key Laboratory of Intelligent Wire1ess Communications (IWC2012002), Open Project of Key Laboratory of Modern Acoustic of Ministry of Education (Nanjing University), the Aeronautical Science Foundation of China (20120152001), Research Innovation Program for College Graduates of Jiangsu Province (CXZZ13_0165), Funding for Outstanding Doctoral Dissertation in NUAA (BCXJ1309), Qing Lan Project, priority academic program development of Jiangsu High Education Institutions and the Fundamental Research Funds for the Central Universities (NZ2012010, NS2013024, kfjj130114).
BIO
wangfangqiu.2008@163.com
Fangqiu Wang was born in Hunan Province, China, on June 25, 1988. He is now a postgraduate student in the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China. His research is focused on array signal processing and UWB communication.
zhangxiaofei@nuaa.edu.cn
Xiaofei Zhang received MS degree in electrical engineering from Wuhan University, Wuhan, China, in 2001. He received his PhD degree in communication and information systems from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2005. Now, he is a professor at the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China. His research is focused on array signal processing and communication signal processing.
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