An oblique projection filtering based DOA estimation algorithm without a priori knowledge

An Oblique Projection Filtering Based DOA Estimation Algorithm Without A Priori KnowledgeHui-jun Hou, Xing-peng Mao, Hong Hong, Ai-jun LiuSchool of Electronics and Information EngineeringHarbin Institute of TechnologyHarbin 150001, P.R. Chinamxp@Abstract—The high resolution multiple signal classification (MUSIC) algorithm provides an efficient way to estimate direc-tion-of-arrival (DOA). However, it performs poorly when weak signals are accompanied with strong ones. To solve this problem, an oblique projection filtering based DOA estimation algorithm is proposed without using a priori knowledge of the sources, such as directions, strength, modulation modes, etc. Numerical results verify the effectiveness of the proposed algorithm. It is shown that a high resolution DOA estimation of the incident sources can be achieved. The detection performances for weak signals are more stable and superior than that of the MUSIC algorithm.Keywords—direction-of-arrival(DOA); oblique projection; weak signal detectionI.I NTRODUCTIONThe subspace based multiple signal classification (MUSIC) [1] algorithm plays an important role in providing the direc-tion-of-arrival (DOA) parameters of multiple incident waves arriving at an antenna array. However, the detection of the weak signal in the MUSIC spectrum is likely to be influenced by adjacent strong ones which lead to the difficulty of DOA estimation. For example, the ability to detect and estimate the weak signals may be lost or severely degraded.One of the standard methods to overcome the influences of strong signals is to divide the field-of-view into sectors. Several well known examples employing such an idea are the beam-space preprocessing method [2], the beam-space MUSIC ap-proach [3], and the CLEAN algorithm [4], etc. They are all computationally efficient for suppressing the out-of-sector sources, whereas these techniques are not suitable to process spatially close signals (i.e., the angular distance between two sources is small).With accurate and prior known source DOA of the strong signal, the interference mitigation techniques, such as the inter-ference jamming DOA estimation algorithm[5], the constrained MUSIC approach [6] and the weighted prior-MUSIC method [7], provide another efficient way to estimate the DOA parame-ters of unknown weak sources. These algorithms pre-subtract the high power components of mixed incident signals, and hence result in an accurate DOA estimation. However, they would lead to an inferior and inaccurate counteraction if the errors of the prior known source directions were introduced.Different from the aforementioned ideas, this work propos-es an oblique projection filtering based DOA estimation algo-rithm. The proposed algorithm does not require any knowledgeof the sources to be known a priori, such as directions, strength,modulation modes, etc. It is effective for estimating directionsof spatially close signals, especially for the signals with greatstrength difference. In this method, an expected direction isfirstly assumed. Then, the oblique projection (OP) [8-9] filter isused to suppress undesired sources which incident from unex-pected directions. After OP filter, only the signal which im-pinges on the array from the expected direction is kept. Finally,ransack all the potential expected directions in the angle do-main, the problem of direction finding is converted into signaldetection. Simulation results show that both the weak and thestrong incident signals can be successfully detected. With greatpower difference, the proposed algorithm is more stable andmore effective for DOA estimation in contrast with the MUSICapproach.II.S IGNAL M ODEL AND O BLIQUE P ROJECTIONA.Signal ModelAssume that D uncorrelated and zero-mean far field sourcesimpinge on an array composed of N output ports. The arrayelements may be disposed discretionarily on a planar or con-formal structure. The complex array response for the k th snap-shot can be written as()()()Fk k k=+X X e (1)where()()()Fk k=X VΘF and()()()1,,Dk f k f k⎡⎤=⎣⎦θθF .The matrix()()()1,,D=⎤⎦VΘvθvθrepresents the ar-ray manifold. ()kvθand()df kθdenote the normalized steering(NS) vector and the complex envelope of the d th incidentsignal, respectively. The DOA parameterkθsymbolizes theazimuth angle for a 1-D array or refers to the elevation andazimuth angles for a 2-D or 3-D structured antenna array,[]T12,,,D=Θθθθ. ()k e is the complex vector of the obser-vation noise. The noise is modeled as a zero-mean, stationary,spatially and temporally white Gaussian process. Furthermore,all the incident signals are assumed to be uncorrelated with theThis work was supported by the National Natural Science Foundation of China (No. 61171180).978-1-4799-4195-7/14/$31.00©2014IEEEarray noise. Then the spatial covariance matrix is given by ()(){}H 2XX XF k k σ==+R X X R I E (2)where {}E i denotes the mathematical expectation and the su-perscript H represents the Hermitian transpose. 2σis the va-riance of the noise and I stands for an identity matrix. Thenoise-free spatial covariance XF R is given by ()(){}()()H HXF F F F k k ==R X X V ΘR V ΘE (3) where ()(){}{}12H 222diag ,,,D F k k σσσ==θθθR F F E (4) is the covariance of the incident signals, and 2dσθrefers to the power of the d th signal.B. Oblique Projection The oblique projection whose range is <>A and whose null space contains <>B can be written as [8]()1H H |−⊥⊥=A B B B E A A ΠA A Π (5)where H 1H()⊥−=−B ΠI B B B B . <>A and <>B are disjoint. Both A and B are full column rank complex matrix of size N t × and N s ×, respectively. Then, we have|||=A B A B A B E E E (6) ||,==A B A B E A A E B 0 (7)III. A N O BLIQUE P ROJECTION F ILTERING B ASED DOAE STIOMATION A LGORITHMA. The Principle of the Proposed AlgorithmRecall that the model introduced in Section II is constructed by D sources, and the DOA parameters of the incident signals are collected in set Θ. In this work, none of the sources’ DOAs are known a priori. An expected direction of the impinging signal is firstly assumed. Then, the oblique projection filter is used to suppress incident signals which impinge upon the array from unexpected directions. Meanwhile, the signal which im-pinges on the array from the expected direction is kept.Without loss of generality, the DOA of the expected signalis assumed to be θ. Note that the expected direction θ maybeexactly equals to one of the sources DOA or maybe not. For convenience, a new set ϑ is defined as follows:{}{}{},for \,for ⊄⎧⎪=⎨⊂⎪⎩ΘθΘΘθθΘϑ (8) where {}i and \ refer to mathematical set and difference set operation, respectively. With non-overlapping sets {}θand ϑ, oblique projection operator ()()|v θJ E ϑcan be given by()()()()()()()()()1H H |−⊥⊥=v θJ J J E v θv θΠv θv θΠϑϑϑ (9) where ()v θrefers to the NS vector of the expected signal; and ()J ϑdenotes a set of NS vectors constructed by ϑ. Further,()()()()()()()||,==v θJ v θJ E v θv θE J 0ϑϑϑ (10)According to (10), the output of the oblique projection filteron the k th noise-free snapshot can be given by()()()()()()()()(){}{}||,for ,for F k k f k =⊂=⊄⎪⎩v θJ v θJ θE X E V ΘF θθΘ0θΘϑϑ (11) where ()f k θrefers to the complex envelope of the expectedsignal. Herein, if no practical signal impinging from the ex-pected direction, ()f k θis considered to be zero. Further, the strength of the expected signal can be evaluated by ()()()(){}{}221|11,for 1,for KK k Fk f k K k NK ==⎧⊂⎪=⎨⎪⊄⎩∑∑θv θJ θΘE X 0θΘϑ (12) As the expected direction θis ransacked in the angle do-main, an angle-power spectrum (APS) is developed. i.e.,()()()()()()()()()2|1H ||11trace KF k XFp k NK N===∑v θJ v θJ v θJ θE X E E R ϑϑϑ (13) where ()trace i represents the matrix trace, and2XF XX σ=−R R I (14)Thus, the problem of direction finding is converted into finding the peaks of the APS, i.e.,()ˆarg max p =θθθ (15) Note that the impinging signals may have different strength. If weak signals are accompanied with strong ones, direction finding of the weak signals may be influenced. To solve this problem, in Section III.C, we further define an adap-tive weight ()μθ to balance the peaks’ highlight in the APS. i.e., a weak signal’s spectrum will be weighted with a relative large coefficient and vice versa. By this way, both the weak and the strong incident signals can be successfully detected.B. The Calculation of the OP Operator According to the proposed algorithm, the key is to calculatethe OP operator ()()|v θJ E ϑas well as the noise variance 2σ. In this work, the variance of the noise is estimated by [1]211ˆN l l D N D σλ=+=−∑ (16) where 1λ, 2λ, ,N λare the eigenvalues of the spatial covari-ance matrix sorting in descending order.The calculation of operator ()()|v θJ E ϑmeans to estimate ()⊥J Πϑwhich can be given by [7] ()H HT T T T ⊥=−=J ΠI G G G G ϑ (17)where T G and G are the right-singular vectors of matrix T Rwith size T N D ×and ()T N N D ×−, respectively. i.e.,()()()2HˆT XF XX T TT σ⊥⊥==−⎡⎤=⎣⎦v θv θR ΠR ΠR I U ΛG GStrictly speaking, T D equals to 1D −for {}⊂θΘand T Dequals to D for {}⊄θΘ. Nevertheless, (17) and (18) indicate that a larger value of T D (with constraint T D N <) only results in an augmented null space, i.e., ()T <>⊂<>J G ϑfor any 1T D D >−with {}⊂θΘ. Thus, the expression (10) still holdseven if a larger T D shows with {}⊂θΘ. Thus, in this paper, we let T D D =for any {}θ.C. The Adaptive Weighting Factor of the APSAccording to the signal model (1) and the APS expression (13), we have()()()()()()()22||110KKk k k k p NK NK ==≤≤+∑∑v θJ v θJ E X E e θϑϑ (19)()()()()()()()22||11KKk k k k p NKNK==≥−∑∑v θJ v θJ E X E e θϑϑ (20)Note That()()()()()()()()22H |||11trace Kk k K σ==∑v θJ v θJ v θJ E e E E ϑϑϑ (21)()()()()()()()()()()()()()22||112H||11trace K Kk k k Np k K K Np σ==≤+=+∑∑v θJ v θJ v θJ v θJ EX θE e θE E ϑϑϑϑ(22)()()()()()()()()()()()()()22||112H||11trace KKk k k Np k KKNp σ==≥−=−∑∑v θJ v θJ v θJ v θJ E X θE e θEE ϑϑϑϑ(23)So, (19) and (20), respectively, indicate that()()()()()()()2H||20trace p p Nσ≤≤+v θJ v θJ θθE E ϑϑ (24) ()()()()()()()2H ||2trace p p Nσ≥−v θJ v θJ θθE E ϑϑ (25) From (24) and (25), it can be seen that the peaks of the APS are influenced by ()μθwhich is defined by()()()()()()()()()()()(]H H ||H H =1trace 101T T μ⊥==−∈v θJ v θJ JθE E v θΠv θv θG G v θϑϑϑ (26) Fig. 1 depicts the relationships between ()μθand signal-to-interference ratio (SIR). 200 snapshots and a half-wavelength spaced uniform linear array with 12 sensors are used in this simulation. The DOA of the first signal is 20 and the SNR is 10 dB. The incident angle of the second signal is 25, and eachsimulation is based on 500 Monte Carlo trails. It is shown that a stronger strength of the second signal makes a relatively larg-er μof the first signal; and the product results of ()μθand 2σθ are stable under different SIR. i.e., with fixed direction, a signal with a large strength will result in a relative small μand vice versa.Thus, we proposed to use ()μθas an adaptive weightingfactor to balance the highlight of the APS. In this way, we have ()()()()220p p N σμμ≤≤+θθθθ (27)()()()()22p p Nσμμ≥−θθθθ (28) After weighting, the peaks formed by all the incident signalscan be balanced adaptively. And both of the peaks formed by the weak and the strong signals can be almost evenly described in the weighted APS. To sum up, the proposed approach to estimate DOA para-meters of the incident signals can be summarized as follows:STEP 1: calculate ()p θand ()μθthrough (13) and (23), respec-tively, for all the discrete angles in the angle domain; STEP 2: find the DOA estimates {}1ˆˆ,,D θθ through searchingimpinge on the array, and the incident angles are 20 , 25 and 50 , respectively. The SNR used for the incident signal com-ing form 25 is 10 dB. The simulation results of Fig. 2(a) and Fig. 2(b) show that the strength and the DOA values of each signal are accurately estimated by the proposed approach, andall the signals have distinct large peaks. The simulation results of Fig. 2(b) indicate that when using the MUSIC algorithm to estimate DOA, the peaks of the weak signals are sheltered by the adjacent strong ones. However, both the peaks of the weak signals and the peaks of the strong ones are balanced in the weighted spatial spectrum. Thus, the proposed approach is Array much more efficient than the MUSIC algorithm in detecting weak signals.Fig. 3 illustrated the comparison of the detection and esti-mation performances between the proposed approach and the MUSIC method under different SIR and different incident an-gle intervals. Two signals are considered in the simulation. The DOA of the first signal is20 , and the angle interval between the second and the first signal isΔ. The SNR of the first signal is 10dB and each simulation is based on 500 Monte Carlo tri-als. A successful detection is made if both of the two signals’ DOA estimation errors are smaller than 1 and the normalizedpeaks are no smaller than 0.01. The simulation results indicatecan be achieved with different angle intervals. Compared with the MUSIC algorithm, the proposed approach is more stable and more efficient in detecting the weak signal, and it has a good robustness against DOA estimation with different SIR.Fig. 4 illustrates the RMSE of the estimated DOA. The si-mulation results are based on 500 Monte Carlo trials, and the practical DOAs of the impinging signals are20 and25 , re-spectively. Simulation result of the first signal is discussed here. In Fig. 4, comparison between the MUSIC algorithm and the proposed approach is made under different number of snap-shots, SNR and SIR. The comparison results show that the es-timation accuracy of the proposed approach is very close to that of the MUSIC method and the DOA estimation error of the incident signal is stable under different SIR.V.C ONCLUSIONIn this paper, a high resolution algorithm for DOA estima-tion is proposed. 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