Direction of Arrival Estimation of Multiple UWB Signals
V. V. Mani · R. Bose
Abstract :This paper deals with the problem of estimation of direction of arrivals (DOA) of a multiple ultra-wideband (UWB) pulse postion modulation signals incident on a smart antenna in the presence of white Gaussian noise. We transform the received signal into frequency domain in order to split the array output into multiple frequency channels. Corre- sponding frequency channels data of the array is arranged into a model similar to narrowband DOA estimation. Iterative quadratic maximum likelihood algorithm is applied to yield DOA estimates. These separate estimates at different frequencies are combined into a single estimate of DOA for each source in an appropriate manner. The performance of the proposed method is studied via extensive computer simulations. It is seen that the technique can successfully resolve the DOA of the closely-spaced UWB signals.transform中文翻译
Keywords :Ultra-wideband · PPM · Smart antenna · Direction-of arrival · IQML
1 Introduction
Ultra-wideband (UWB) technology has increasingly been used in personal area wireless networks an
d Radio Frequency Identification (RFID). This technology is characterized bythe transmission of extremely short duration pulses, has become a candidate technology for ranging and positioning applications [1,2]. Ultra-wideband signals have fine time and angle resolution capability. By adding multi-antenna techniques to UWB, additional spatial parameters e.g direction of arrival (DOA), time of arrival(TOA) can be extracted, leading to enhanced precision in positioning applications [3]. Unlike narrowband, UWB signalcovers awide range of spectrumthat causesmany narrowband array processingmethods to be ineffective. Moreover, the approach of broadband array processing faces difficulties in such a wide spectrum.
The problem of maximum likelihood (ML) estimation of the DOA’s of multiple UWB sources has been rarely addressed in the literature. Until recently, the papers of Najar et al.,Lee et al., Keshavaraz are a few who dealt this problem [4–6]. However, these authors are mainly concerned with single source DOA & TOA estimation. Kesavaraz [6] proposed the use of weighted signal-subspace method of DOA estimation for UWB pulse postion modulation (PPM) sources impinging on ULA (Uniform linear array). There have been other approaches that consider the estimation of the angle of arrival based on temporal delays as in Pierucci and Roig [7], where authors propose DOA estimation by evaluating the propagation delays impinging from each element array. This approach suffers from constraints as
sociated with high sampling rate requirements. The authors proposed in Navarro and Najar [4] the use of low-complexity frequency domain approach for joint estimation of TOA and DOA for a single UWB source.
This paper, is an attempt to proceed further with the solution of UWB multiple sources DOA estimation problem based on maximum likelihood principles. However, DOA estimators proposed for narrowband signals does not exploit the advantage of large signal bandwidth of UWB. Our objective here is to explore the possibility of computationally efficient algorithm for maximum likelihood UWB DOA estimation. Time shift property of Fourier transforms used to develop a frequency domain model for the data that is similar to models used in DOA estimation in narrowband case. In this respect, we
have been motivated by theapproach proposed by Bresler and Macovski [8] for the parameter estimation of exponential signals in narrow band case with ULA. In this approach, an iterative method, called iterative quadratic maximum likelihood (IQML) has been formulated for obtaining the ML solution.This involves the creation of toeplitz matrix which spans the subspace orthogonal to signal space, and can be par ameterized by the coefficients of a prediction polynomial, whose roots  yield the DOA estimates. UWB data model is transformed into frequency domain and on each bin, IQ
ML is applied to generate DOA estimates.
This solution, however, falls short of providing single ML estimate for the DOA of each UWB source. This is because the IQML search is conducted over an expanded signal spacewhich in turn leads to generation of separate DOA estimates for different frequencies. For applications where we would require a separate DOA estimate for each source. So, separate estimates at different frequencies are combined into a single estimate of DOA for each source in an appropriately average sense. It is seen that procedure leads to very good estimates, as evidenced by simulation results.
This paper has been organized as follows. In Sect. 2 UWB data model is given and in Sect. 3 problem formulation and its mathematical description for direction finding are presented. In Sect. 4 we develop the ML criterion for DOA’s estimation and in Sect. 5 UWB ML estimation algorithm presented. Results and simulation studies are presented in Sect. 6. Finally the paper is concluded in Sect. 7.
2 UWB Data Model
In theUWB communication systemthe transmitted signals of each user consists of sub-nano-
second pulses. Each user transmits one pulse per frame. Each frame of Tf seconds contains Nh hopping chips with duration Tc seconds per chip. In each Tf , one pulse will be placed in one of the Nh slots according to the corresponding user time hopping sequence. The location of the transmitting pulse within the allocated hopping slot is determined by the information bit. A typical transmitted time-hopping PPM signal for qth user is modeled as [9]
()s ()()
q f j tr
q
tr j t t jT d δ∞=-∞=Ω--∑                          (1)
Ωtr (t) represents the transmittedmonocyclewaveformand the receiving antennamodifies the  shape of Ωtr (t) to Ω(t) which is modeled as second derivative Gaussian waveform [10,11]. Tf is the frame interval and is typically hundred or thousand times wider than the monocycle width, resulting in a signal with very low duty cycle. The data sequence (
){d }{0,1}q j ∈ ofthe qth user is a binary
symbol stream that conveys some form of information. When data symbol ()
d q j  is ‘0’, no additional tim
e shift is added whereas when the data symbol is ‘1’, a  time shift o
f δ is added to the monocycle. In the next section we formulate the signal model for findin
g the DOA estimates of UWB sources impinging on smart antenna.
3 UWB Direction Finding
Time Domain: Let Q < M UWB-PPM sources {sq (t), q = 1,..., Q} impinge upon an M element smart antenna array (Fig. 1). The signal received by the mth antenna at time t equals
1()(())()
Q m q m q m q x t s t u t ξθ==-+∑                          (2)
Fig. 1 Smart antenna for DOA estimation of UWB signals  where vm(t) denotes the mth antenna additive Gaussian noise, sq (t) represents the qth UWB
transmitted signal, and ξm(θq ) refers to the qth signal sources propagation delay at the mth  antenna. For a ULA the propagation delay associated to the mth antenna for the qth source is given by
()(1)sin()(1)m q q q d m m c ξθθτ=-=-                      (3)
with d being the distance between antenna elements in the array, c the speed of light and θq direction of arrival of the qth source. The received signal consists of the attenuated and delayed version of the transmitted pulse train caused by multipath during propagation in the medium. The derivation below assumes that multipath profiles for each array element are the  same due to small inter element spacing. The problem of interest is to estimate the direction of arrivals of UWB impinging signals i.e θq ’s. Also, we mak e the assumption that the total number of sources Q impinging on the array are known.
Frequency Domain: Since the frequency range of UWB signal is wide in nature and also to generalize the narrow band DOA estimation techniques to UWB, we are transforming the signal into frequency domain. The received signal xm(t) time- sampled at fs
()
H s w f π≥ ,then apply a K-point discrete Fourier transform (DFT). ()1()()()k m q Q jw m k q k m k q x w s w e
V w ξθ-==+∑                  (4)
where {ωk , k = 0,..., K − 1}∈[ωL ,ωH] and Vm(ωk ) denote the mth antenna noise at  frequency ωk  and Sq (ωk ) refers to the Fourier transform of sq (t). DOA’s are embedded inexponential component i.e ()
k m q jw e ξθ-, q = 1,..., Q which are to be estimated. so, we arrange the corresponding k DFT components of the array in the following vector notation.
1(1)
(1)(2)(1)()(1)(2)()1122()()()()()()()k k k Q k M k M k M Q k jw jw jw k k k k jw jw jw Q k M k S w V w e e e S w V w x w e e e S w V w ξθξθξθξθξθξθ------⎛⎫⎛⎫⎛⎫ ⎪ ⎪ ⎪ ⎪ ⎪=+ ⎪ ⎪ ⎪ ⎪ ⎪
⎪ ⎪⎝⎭⎝⎭⎝⎭      (5)
The above formulated data model is compactly described by the following vector notation.For notational simplicity we use k in place of ωk :
(6)
where X(k) and V(k) are the M × 1 vectors, S(k) is a Q ×    1 vector
(7)
(8)
(9)
and A(Θ, k) is a M × Q matrix with a block representation
(10)
where Θ=[θ1,...,θQ] T with a(θq , k) is a M × 1 column matrix
(11)
A(Θ, k) is a vendermonde matrix [8] whose columns are the steering vectors of the impinging UWB wave fronts. We use the notation (·)T  to denote transpose operation, (·)H  for Hermitian-transpose operation and (·)* to denote complex conjugate.
The receiver expects signal from the source within a short window of time Tr and both transmitter and receiver are working synchronously. In this formulation the unknown param- eters are (θq )’s and Sq (k)’s. Hence our estimation problem can be formulated as follows.
For the given window of data, estimate (θq )’s and Sq (k)’s where (θq )’s are embedded in  the steering matrix A and S(k) are the components of the vector s(t).
However, this estimation would require the solution of a nonlinear optimization problem in terms of the Q parameters {θq , q = 1,..., Q }.We consider the problem as estimating the spatial vector  , whose components θq contain the required DOA information.
4 The Maximum Likelihood Criterion
As discussed in the previous section, the first step of the problemis concernedwith estimation  of the parameter vector  . The solution of this problem is discussed in this section. It has been shown that under white Gaussian noise, ML estimators and least square estimators are equivalent [12]. Hence ML estimate of the signal parameters can be obtained by solving the non-linear least square problem
2(,)arg min ()(,)()S X k A k S k Θ=-Θ                （12）
where .is the Euclidean norm. This expression may be further simplified by first substituting the least square estimate of S(k) in terms of A as
()(,)?()S k A k X k =Θ                          （13）
where is the pseudoinverse of A(Θ, k).Bysubstituting (13)into(12), the S(k)’s are eliminated and it is reduced to the equivalent formulation [12]
21()arg min ()1arg min {()()}H k P X k A
trace P X k X k A Θ==                  （14）
where A P and 1
P A are the projection matrices onto the column space of A(Θ, k) and onto its
orthogonal complement, respectively, and are given by
-1(,)((,)(,))(,k)H H
A P A k A k A k A =ΘΘΘΘ
-111(,)((,)(,))(,k)H H
P A k A k A k A A =-ΘΘΘΘ        （15）
Once the ML estimate ˆ
ΘML is determined by solving (14), S(k)’s are found by the linear  relationship of (13). Computationally expensive global search is required to minimize (14). Therefore the approach proposed by Bresler and Macovski [8] is adapted here to find the  DOA estimates.
As mentioned in [8] output vector of the ULA obeys a special Auto Regressive Moving Average (ARMA) model, and the ML estimates of its parameters are directly related to its c oefficients. Use of this model converts the problem of estimating the DOA’s to that of estimating special ARMA parameters, fromwhich the DOA’s can be easily derived. Thismethod  consists of the following steps:
(i) Establish a relation to represent the null space of the array steering matrix associated with the model of the space-time staked vector X(k) as defined in (6) in terms of  coefficients bi of the special ARMA model.
(ii) Formulate theminimization problemin terms of a constrained quadraticminimization p rocedure for
obtaining the ARMA coefficients bi .
(iii) Once the coefficients bi of the polynomial are estimated, the unknown DOA’s can be  obtained by finding the roots of b(z) for each spatial frequency (k).
We next present the procedure for the calculation of polynomial coefficients.
As A(Θ, k) is in Vendermondematrix form, there exists a Toeplitzmatrix B of dimension M × M − Q such that [8] (,)0H B A k Θ=                        （16）
The matrix B is given by

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