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Wireless Sensor Network, 2009, 3, 171-181 doi:10.4236/wsn.2009.13023 ctober 2009 (http://www.SciRP.org/journal/wsn/). Copyright © 2009 SciRes. WSN Published Online O Research on DOA Estimation of Multi-Component LFM Signals Based on the FRFT Haitao QU1, Rihua WANG2, Wu QU3, Peng ZHAO4 1Beijing University of Posts and Telecommunications, Beijing, China 2Communication University of China, Beijing, China 3K-Touch Corporation, Beijing, China 4Beijing Research Institute of China Teleco m Co., Ltd, Beijing, China E-mail: quhaitao2007@gmail.com Received April 10, 2009; revised May 20, 2009; accepted May 25, 2009 Abstract A novel algorithm for the direction of arrival (DOA) estimation based on the fractional Fourier transform (FRFT) is proposed. Firstly, using the properties of FRFT and mask processing, Multi-component LFM sig- nals are filtered and demodulated into a number of stationary single frequency signals. Then the one-dimensional (1-D) direction estimation of LFM signals can be achieved by combining with the tradi- tional spectrum search method in the fractional Fourier (FRF) domain. As for the multi-component LFM signals, there is no cross-term interference, the mean square error (MSE) and Cramer-Rao bound (CRB) are also analyzed which perfects the method theoretically, simulation results are provided to show the validity of our method. The proposed algorithm is also extended to the uniform circular array (UCA), which realizes the two-dimensional (2-D) estimation. Using the characteristics of time-frequency rotation and demodulation of FRFT, the observed LFM signals are demodulated into a series of single frequency ones; secondly, operate the beam-space mapping to the single frequency signals in FRF domain, which UCA in array space is changed into the virtual uniform circular array (ULA) in mode space; finally, the DOA estimation can be realized by the traditional spectral estimation method. Compared with other method, the complex time-frequency cluster and the parameter matching computation are avoided; meanwhile enhances the esti- mation precision by a certain extent. The proposed algorithm can also be used in the multi-path and Doppler frequency shift complex channel, which expands its application scope. In a word, a demodulated DOA esti- mation algorithm is proposed and is applied to 1-D and 2-D angle estimation by dint of ULA and UCA re- spectively. The detailed theoretical analysis and adequate simulations are given to support our proposed al- gorithm, which enriches the theory of the FRFT. Keywords: DOA Estimation, The Fractional Fourier Transform, UCA, ULA, LFM 1. Introduction In various applications of array signal processing such as radar, sonar, communications, and seismology, there is a growing interest in estimating the DOA of LFM signals by dint of time-frequency analysis tools. G. Wang [1] proposed an iterative algorithm based on time-compen- sation, but the initial estimate is necessary. Using inter- polation in the spatial time-frequency distribution matri- ces (STFD’s) [2], Gershman [3] extended the signal subspace technique and estimated effectively DOA of LFM signals, however Gershman’s approach presences model biases in addition to time consuming. The above Wigner-Ville distribution (WVD) based methods conse- quentially suffer from the disturbance of cross-terms in the presence of multi-component signals. Using a new time-frequency analysis tool-FRFT, di- rection estimation of LFM signals has been proposed in Reference [4]. However, only maximal energy concen- tration point is selected as estimate data, easily interfered by surroundings. In this paper, a new FRFT based algo- rithm is proposed. Firstly, Observed signals are separated into a number of single components by adding an adap- H. T. QU ET AL. 172 tive filter in the FRF domain. Secondly, the separated components are demodulated into stationary signals. Fi- nally, the 1-D DOA of LFM signals can be estimated by the traditional spectrum search method. This algorithm digs two dimensional time and frequency information without the initial estimate, frequency focusing and pa- rameter partnership. With the increasing of the Sig- nal-to-Noise ratio (SNR), the MSE is quite closed to the CRB [5], for multi-component signals, cross-terms and non-linear optimize operation are also avoided. For the UCA widely used in the third generation mo- bile communication system, the time-frequency charac- teristics of the FRFT are combined with the beamform- ing technology in FRF domain, an algorithm for the 2-D DOA estimation of the multi-component LFM signals is also proposed. Compared with other methods, the preci- sion is enhanced by a certain extent. Simulation verifies the method to be effective in the multipath and Doppler frequency shift existed complex channels. 2. Background Knowledge of FRFT 2.1. Definition and Properties of FRFT Recently the FRFT attracts more and more attention in the signal processing society, in 1980, Namias [6] firstly introduced the mathematical definition of the FRFT. Then Almeida [7] analyzed the relationship between the FRFT and the WVD, and interpreted it as a rotation op- erator in the time-frequency plane. This characteristic makes FRFT especially suitable for the processing of LFM signals [8–9]. As a generalization of the standard Fourier transform, the FRFT can be regarded as a counterclockwise rotation of the signal coordinates around the origin in the time- frequency plane. If the traditional Fourier transform of a signal can be considered as a /2 counterclockwise rotation from the time axis to the frequency axis, the FRFT can be accordingly considered as a counterclock- wise rotation from the time axis to the axis with an angle u , as illustrated by Figure 1. The FRFT of signal() x tis represented as ()[()]() (,) P X uFxtxtKtudt (1) where is called the order of the FRFT, p/2p , denotes the FRFT operator and [ p F](, ) K tu is the kernel function of the FRFT 22 1cot exp( cot 22 (, )csc ), (), 2 (), (21) jtu j Ktu jtu n tu n tu n (2) (, ) s Wtf (,) p s Wuv u v Figure 1. FRFT and WVD. This has the following properties, * (, )(, ) K tuK tu (3) *' ' (, )(,)() K tuKtu dtuu (4) Hence, the inverse FRFT is ()[( )]( )(,) P x tFXu XuKtudu (5) Equation (5) indicates that signal () x t can be inter- preted as decomposition to a basis formed by the or- thonormal LFM functions in the domain, and the domain is usually called the fractional Fourier domain, in which the time and frequency domains are its special cases. The FRFT is a one-dimension linear transform and has the rotation-addition property. Essentially, the repre- sentation of a signal in the fractional domains contains the information in both time and frequency domains of the signal; Thus the FRFT is considered as a time-fre- quency analysis method and has close relationships with other time-frequency analysis tools. u u In Reference [10], some important characteristics are expressed as 22 /2 1tan tan [] exp( 1tan 21tan pjct juc Fe cc ) (6) 2 2 [()](sin )4 exp[(sincoscos )] 2 pjvt p F xteX u vbac v juv (7) 2 [()](cos ) exp[(sincos/ 2sin)] pp Fxt Xu ju (8) 2.2. Discrete FRFT Computation In engineering applications, the discrete FRFT (DFRFT) is usually required. According to the definition of the FRFT, it is obvious that the numerical computation of the DFRFT is much more complicated than that of DFT. So far, there have been several DFRFT algorithms with Copyright © 2009 SciRes. WSN H. T. QU ET AL. 173 x x x Figure 2. Normalized time-frequency support region. different accuracies and different complexities. In this paper, we select the decomposition algorithm proposed in Reference [11]. This algorithm decomposes the com- putation of DFRFT to a convolution which can be com- puted by FFT, and the result is very close to the output of continuous FRFT. In this algorithm, the signal represen- tation in time domain and frequency domain should be approximately constrained with an interval of [/2T, and a bandwidth of /2]T[/2,/2FF] respectively, viz. the time-bandwidth product of the signal is NTF , and according to the uncertainty principle, con- stantly. If the sampling rate is selected as 1N / sTT N , the discrete representations of the signal in time domain and frequency domain will have the same length, which is called the dimensionless normalized process and the principle can be shown in Figure 2. Therefore, Equation (1) can be expressed as 22 cot2 csccot () () jujut jt X uAeee xtd t (9) where 1cot 2 j A For 0.5 1.5p, signal has a bandwidth which is at most 2cot () jt ex t 2 F and can be represented using Shannon formula 222 cotcot/ ( 2) ()sin 2 22 N jtjn F nN nn ext excFt F F (10) Substituting Equation (10) into Equation (9) and ex- changing the sequence of the integral and the summation, we have 22 cot2csc/ (2)cot/ ( 2) () () 22 P N juj unFjnF nN XuF xt An ee ex By quantizing the variable in the fractional Fourier domain, Equation (11) can be finally discredited as u 222 (2)/(2) () 22 2 N Pjmmnn nN A nn XmF xex F F FF (12) where () X m denotes the DFRFT of signal () x t, cot , csc . This algorithm can be imple- mented by FFT, and has a computation complexity of 2 og )N(lN [11]. 2.3. Two Special FRF Domain WVD is an important non-stationary signal analysis tool, which has a very simple relationship with FRFT; viz. the WVD of FRFT is the coordinate rotation of the original signal’ WVD, while the shape of WVD keeps unchanged in the rotation. Therefore, a lot of the WVD-based signal processing methods can be substituted by FRFT. The relationship of the two time-frequency analysis tools can draw a conclusion that “time width” and “fre- quency width( ()u )v ” will change with the difference of the rotation angle. Considering two extreme cases, 0,uv or0,vu , from the above analysis, the former corresponds to the rotation an- gle 1 cot cot , LFM signal becomes an impact func- tion, which domain is called energy concentrated FRF one. The latter corresponds to the rotation an- gle , LFM becomes a single fre- quency signal, which domain is called demodulated FRF one and is the base of the proposed algorithm in this pa- per. By dint of the time-frequency rotation property of FRFT, the detection, extraction and parameter estimation of LFM signals can be easily achieved. 1/2 2.4. The FRFT of Gaussian White Noise Theorem 1: The FRFT of zero-mean Gaussian white noise is still Gaussian white noise. Proof: let subject to the ()nt2 (0, )N distribution, and is its FRFT, the mean is () p Nu ()[()][()]0 pp p EN uEF ntFEnt (13) Because the FRFT is the linear transform, does not change the distribution characteristics of Gaussian noise. Therefore, the noise is still a zero mean Gaussian noise. 2 F F As for the second-order statistical properties of noise, the correlation of the white noise can be defined as: ()nt (11) *2 ()()()Entnt (14) C opyright © 2009 SciRes. WSN H. T. QU ET AL. 174 The correlation of is defined as: () p Nu * ** ** 2* 2* () () ()(, )( )( ,) ()()(, )( ,) ()(,)(, ) (,) (,) pp pp pp pp pp EN uNuv EntKtudtnKuvd EntnK tuKu vdtd tKtuKuvdtd KuK uvd (15) Submit Equation (2) to Equation (15), and obtain: 2 * 2 20.5(2)cot 2 2 () () 1cot ee 2 1cot 2sin() 2 pp juvv jv EN uNuv jd jv csc (16) Due to Equation (16), we can see that the FRFT does not change the time-domain white characteristics of noise, while noise energy does not be changed. Assume the array noise is the zero-mean airspace one, viz. as for the array element k, the output noise is unrelated: (kl) ** () ()()()0 klk l EntntEnt Ent (17) The cross-correlation of the noise in FRF domain is * ** ()[ ()] ()()(, )(, ) 0 pp kl kl pp EN uN u EntnK tuKudtd (18) The above equation shows, FRFT does not change the airspace white characteristics of noise. Therefore, we can draw a conclusion that FRFT does not change the statis- tical properties of Gaussian white noise, the theorem certification has completed. Inference: as for the M antenna array element, if the array output noise is zero mean and variance2 , the noise covariance matrix in FRF domain is: * () () p Npp RENuNu I 2 M (19) 3. 1-D DOA Estimation Algorithm 3.1. ULA Array Model Let a ULA of M sensors receive LFM sources from the unknown directions D12 {,,,} D , as illustrated by d () k s t k Figure 3. ULA and array model. Figure 3. The observed signal at the output of the th sensor can be described as i 1 ()[] () D ikik ki x tst n t (20) 1, 2,,1iM 1, 2,,kD where, 2 ()exp[ (/2)] kkk stj tt (21) (1)cos / ik k id c (22) k , k are initial frequency and FM rate, () k s t is the th source in reference sensor k1 x . is the additive white Gauss noise with variance () i nt 2 , which is assumed to be statistically independent with signal sources. ik is the th’s path delay, is light velocity and is sensor spacing. kcd From (20) and (21), we get the direction matrix is time-variant; however the traditional estimation method is merely suitable for time-invariant signal model. Therefore, the traditional method cannot be used to the direction finding of LFM signals directly. 3.2. 1-D Estimation Algorithm Description In this section, the main work is how to make the direc- tion matrix time-invariant. The FRFT is actually a “Ro- tation” of signal in time-frequency plane. An LFM signal can be turned into an impulse in a proper fractional do- main, for the ULA model, signal () k s t ' k will present an impulse while the rotation anglecot k . There will be the energy concentration, consequently a distinct peak will appear in that FRF domain, whereas the noise energy is distributed much more symmetrically in the entire time-frequency plane and will not be concentrated in any FRF domain [12]. Using (20) and (21), we get that path delay can not change the FM rates, so the impulse corresponding rota- tion angles of signal () k s t are same in every sensor. Then Equation (20) is rotated with angle ' k by the FRFT from two sides: '' '' '' ' ()()()() kk kk D iik ili lk Wu YuYuVu ' (23) Copyright © 2009 SciRes. WSN H. T. QU ET AL. 175 where, presents an impulse, and are approximately considered as LFM signal and the white Gauss noise respectively. '' () k ik Yu ) '' () k D il lk Yu '' ( k i Vu Therefore, a mask operation is applied to (23) accord- ing to the peak position , which is a narrowband filter with central frequency , and with a properly selected bandwidth, most energy of the signal will be removed. This procedure can be re- garded as an open loop adaptive time-varying filter whose central frequency varies linearly following the peak position . ik m ik m 2L '' () k ik Yu ik m Signal is performed the FFT (viz. FRFT of ). According to the rotation-addition property [10], the two procedures above are equivalence to one time rotation with angle '' () k ik Yu 1p k viz. 3/2cot kk ; tan kk (24) Using (6), (7) and (8), signal () k s t is rotated with angle k by the FRFT can be expressed as 2 2 () 1tan(sin )tan exp[ ] 1tan2 1tan exp[(sincos/ 2cos)] k k kkkk kk kk kkkkk Su ju ju k 2 1tan exp[(sincos/2)] 1tan exp( cos)exp( cos) kkkk kk kk kk jj juB ju (25) where, 2 1tan exp[(sincos/ 2)] 1tan kkkk kk j Bj (26) From (25) and (26), it can be seen that LFM signal () k s t has been transformed into the single frequency signal in the FRF domain. () k k Su Similarly, the FRFT of path delayed signal () k st with rotation angle k can be expressed as 2 2 [()]exp(sincos/ 2) exp(cos) exp[(cossin)] kkkk kkkk k FstB j jju (27) In practice, is too small viz. sin cos kk 2 exp(sincos/ 2)0 kk j (28) Substituting (28) into (27), we get 2 2 [( )] exp(cos) exp(cos) exp(cos)( ) k k k kk k kkk Fst Bj ju jSu k (29) From the above analysis, Using (25) and (29), the ob- served signals described by (20) are performed the FRFT with rotation angle k from two sides () ()() kk k ik ikik X uSuNu (30) 1, 2,,1iM Equation (30) can be compactly represented by matrix form as follows ()() () kkkk kkkk X uASuNu (31) 1 [,,,, ] kT kkikMk Aaaa (32) where, denotes the transpose of matrix. T 2 2 exp(cos ) 2 exp(cos(1) cos) ikik kk kk aj jid (33) 12 () [(),(),,()] kkk k kkkMk X uXuXuXu 12 () [(),(),,()] kkk k kkkMk NuNuNuNu (34) From (32) and (33), the direction matrix k k A is only relative to the direction information k , so the observed signal model has been time-variant in the FRF domain. In the FRF domain, the covariance matrix of the ob- served signal can be defined as 2 [() ()] kk kkkk HH XXkkkSS k REXuXuARA I (35) where, H denotes the conjugate transpose of matrix. k SS R is the auto-correlation matrix of signal sources. The composite covariance matrix (35) has the same structure as the covariance matrix arising in the case of stationary signals. Therefore, the DOA can be estimated by per- forming eigendecomposition to k X X R . Using the signal subspace k N S and the noise subspace k N E , the space spectrum function of the th source in the FRF domain can be given by [13] k ()1/( ) kkkk HH kkNN PAEE k A (36) () k P is performed an 1-D search and k can be obtain by the maximal peak rotation angle. Similarly, all the Direction of LFM signals can be estimated in turn. This k C opyright © 2009 SciRes. WSN H. T. QU ET AL. 176 algorithm is considered as FRFT based demodulation method. To summarize, the proposed algorithm can be formu- lated as follows: 1) The observed signals at all sensors are rotated with a continuously variable angle by the FRFT; per- form a 2-D peak search in the(, )m plan to obtain the maximal peak position and corresponding rotation angle ik m ' k respectively. 2) Mask operations are applied according to at every sensor, then the filtered ik m 2 L points are per- formed the FFT to obtain stationary signals conse- quently. 3) Get the covariance matrix of the stationary signals and perform eigendecomposition in the FRF do- main, construct the spectrum function() k P ac- cording to (36). 4) Perform 1-D peak search to() k P and obtain the DOA of the th LFM signal. k 5) For multi-component LFM signals, all the direction can be estimated by repeating the above proce- dures. 4. 2-D DOA Estimation Algorithm Using UCA 4.1. Introduction UCA has many advantages which the linear array cannot match. E.g. UCA can be implemented with all-direction- funding; its precision measurement does not change with the azimuth significantly and is fit for the system cor- recting. UCA is the main receiving antenna of base sta- tion system in the third generation mobile communica- tion system. Thus, the UCA based DOA estimation has been a research hotspot in array signal processing. Mathwes [14] proposed an UCA-RB-MUSIC method, which can be only suitable for the stationary signals; however, the actually existed signals are non-stationary ones which are represented by LFM. Tao ran [4] pro- posed an algorithm of LFM signal DOA estimation. However, the method does not apply to the UCA. Due to the above analysis, we propose a novel DOA estimation algorithm based on FRFT using UCA, as for the multi-component LFM signals, using the characteris- tics of time-frequency rotation and demodulation of FRFT. Firstly, the observed signals are demodulated into a series of single frequency ones; secondly, operate the beam-space mapping to the single frequency signals in FRF domain, which UCA in array space is changed into the virtual ULA in mode space; finally, the DOA estima- tion can be realized by the traditional spectral estimation method. The proposed algorithm mines the time, fre- quency and spatial information maximally; compared with other method, the complex time-frequency cluster and the parameter matching computation are avoided; meanwhile enhance the precision [15]. As for the multi-component LFM signals, there is no cross-term interference, the proposed algorithm is also applicable for the multi-path and Doppler frequency shift channels. 4.2. UCA Array Model Assuming independent LFM signals and the pitch and azimuth angle is D 1122 {( ,),(,),,(,)} DD N re- spectively, the array element number of UCA is and radius is , the center is the reference point of receiving antenna, as shown in Figure 4. Then the output of the th sensor is: r i 1 ()[] () D ikik ki x tst n t (37) 1, 2,,iN 1, 2,,kD where, 2 ()exp[(/ 2)] kkk stj tt (38) sincos()/ ikkk i rc (39) 2( 1)/ iiN (40) () k s t is the kth LFM source, and k and k are the initial frequency and FM rate respectively, ik is the path delay and is the light velocity. is the ad- ditive white Gaussian noise with zero mean and variance c() i nt 2 , which is independent with signals. From the Equations (37) and (38), the direction matrix of observed signals is time-varying in UCA, while the traditional DOA estimation algorithm is only suitable for the time-invariant model, which cannot be used to deal with LFM signal directly. 4.3. 2-D Estimation Algorithm Description From Equations (26) and (28), operate the FRFT to Equa- tion (37) with the rotation angle k from two sides: () ()() kk k ik ikik X uSuNu (41) DOA X Y Z r k θ k i Figure 4. Uniform circular array. Copyright © 2009 SciRes. WSN H. T. QU ET AL. 177 The matrix form of Equation (41) is: ()() () kkkk k X uASuNu (42) 1 [,,,, ] kT kikNk Aaaa (43) where, donates the transpose of matrix. T 2 2 exp(cos ) exp(2 sin cos()cos/) ikik kk kki k aj jr (44) From Equations (43) and (44), in appropriate FRF domain, direction matrix k A is only related to angle information , , viz. the observed signals have been transformed into unvaried smooth signal model. There- fore, the mode excitation method can be used to estimate the DOA of LFM signals. The spatial beam former H r F in FRF domain is de- fined as H H rH F QCeR (45) where, H denotes the conjugated transpose of matrix. 101 ,,, ,,, M Cediag jjjjj M (46) (,,,,) H H MoM RNQQQ (47) Select the central Hilbert matrix, 0 ' 1[(),,(),, ()] MM Qv vv M (48) 0 () [, ,,,,,] jMjj jjM ve eeee (49) ' 2/ t M (50) [,tMM ] where, the largest model number , M kr'21 M M . Wave number 2/,k is the initial frequency corresponding center wavelength of LFM signals. H r F can change the UCA in the array space into the virtual ULA in the mode space, and finally, the DOA estimation can be achieved by the eigendecomposition based search method. Summarize the above and the main steps are as fol- lows: 1) The observed signals are continuously operated by FRFT; perform a 2-D peak search in the (, )m plan to obtain the maximal peak position ik m and corresponding rotation angle ' k of the kth LFM signal respectively. 2) Select 2L points whose center is ik m and cal- culate the FFT (FRFT with1p), obtain the kth single frequency signal() k ik X u . 3) Let () k ik X u pass the beam switch H r F , viz. () () () () kk kk k H ikr ik HH rk r Yu FXu F AS uFN u , And calculate its covariance matrix [()() kk ] H Yikik REYuYu . 4) Define Re( ) Y RR , perform eigendecomposition to R and obtain the signal subspace S and noise subspace G. Construct: 1 (,)(,) (,) kk TT ik k kik k k PaGGa , where, (,)(,) H ikk krk k aFa k , perform 2-D spectrum search and obtain and k . 5) As for the multi-component LFM signal, repeat the above process and obtain all the DOA of signals respectively. 5. Performance Analysis and Simulation 5.1. FRFT Property Simulation 5.1.1. Simulation of FRFT and WVD As we all know, WVD is also one of the most important and most widely used time-frequency analysis tool, which is bound to FRFT with the existence of close ties. The derivation process is relatively complex; however, there is a very simple relationship between FRFT and WVD, that is, FRFT of WVD is the coordinate’s rotation form of WVD of original signal [10]. In order to validate the relationship between the FRFT and WVD, experiments of compute simulations are given. We assume a wideband LFM signal () s t with a length of 1024, which is modeled as: initial frequency and FM rates are9 M Hz ,0.7/ M Hz s , sample frequency is 50 s f MHz . The WVD of () s t is shown in Figure 5 (a), () s t 5 is performed the FRFT by the rota- tion angle 0.1 and get the transformed signal . The WVD of the transformed signal is shown in Figure 5(b). Compared the two figures, it can be found that the WVD of is just the rotation of the WVD of 0.15 ()Su 0.15 ()Su 0.15 ()Su () s t by angle 0.15 , meanwhile the figure shape is invariable. So the FRFT is testified a kind of rotation arithmetic operators in the time-frequency plane. 5.1.2. Two Special FRF Domain Simulation 2 ( )exp[(2)]stjtt , signal model is: 19 M Hz , 11400000 / M Hz s . Sampling rate 50 s f MHz, the number of snapshots is 1024. Perform continuous FRFT to signal and operate spectrum peak search, in the appro- priate FRF domain, () s t shows the ergy property of en C opyright © 2009 SciRes. WSN H. T. QU ET AL. 178 (a) (b) Figure 5. (a) The WVD of The WVD of ()t, (b) s0.15π() S u. (b) Figure 6. (a) Energy concentration property of FRFT, (b) Demodulated property of FRFT. concentration, as shown in Figure 6(a). The signal con- tinues to be rotated 2 ain, in FRF domain, viz. in the demodulated FRF dom() s t shows the demodulated property, as shown in Figure 6(b). 5.1.3. Gaussian White Noise Simulation Assume the complex Gaussian white noise is: ( )(1,1024)(1,1024)w nrandnjrandn continuous FRFT to it, the energy distrib and perform ution of in different FRF domain is shown in Figure 7. We can see that the Gaussian white noise does not show energy concentration property in any FRF domain and can still be regarded as white noise.s theorem 1 is verified. ()wn Thu Figure 7. Energy distribution in different FRF domain. (a) Copyright © 2009 SciRes. WSN H. T. QU ET AL. 179 5.2. 1-D DOA Estimation Simulation 5.2.1. MSE and CRB Analysis The FRFT is a 1-D linear transform [10]. In the FRF domain is approximately considered as the additive Gauss white noise. Therefore, the probability density of signal () k k Nu function() k k X u represents normal school and the corresponding likelihood function can be expressed as 2 2 1 [] (2)(/ 2) 1 exp{[][]} k kkk kkk kMM H kkk kkk LX XASXAS (51) Using Reference [5], the CRB of the proposed method in the FRF domain can be represented as 1()RB 1 2kkkk k HH H 2Re ()( )()( ) k H kkkkk k kk C SdwIAAAAdwS k (52) where, 2 is the noise variance and I is unit matrix, () / k kk d wdAdw . Similarly, the MSE of the ed algorithm in the FRF domain can be represented as propos 1 ()VAR 1 2 121 11 11 2[ ()[()] ()]/{[ ][ ()]} kk kk kkkkk HHH MU kkkkkk H kXXXXkkX dw IAAAA dwRRA AR where, 11X (53) k X X R is covariance matrix of the observed signals, e MSE of the proposed method will be more and more closed to the CRB with the increasing of the sensor number and the SNR. 5.2.2. MSE and CRB Simulation tod u impinging from 11 denotes the first row and first line element of matrix. From (52) and (53), it can be obtain that th [] In order validate the proposed metho, experiments of compute simulations are given. We assme the ULA of 6M 2D far field wideband ar LFM signals with a length of 1024, which is modeled as: initial frequency and FM rates e1200 H z ， 1900 / H zs ;2200 H z ，2300 / H zs 0 270 respec- , and angles tively. of arri Sample fre val are 1 quency is 0 30 s900 f Hz e FRF dom , the mask snap- in. The input Sshots are 23L00 in thaNR varies fr15dB to 29dB with an interval 2dB, at each level of the SNR, we run 100 Monte-Carlo experiments, MSE of the proposed method and original method are Figure 8. MSE of proposed and original method. Figure 9. MSE and CRB of proposed method. shown in Figure 8. Obviously, the accuracy of our method has certain improvement comparing with the method proposed in the Reference [4]. In same assumption, the input SNR various from –15dB to 6dB with an interval 3dB, 100 times Monte-Carlo simulations are performed at each level of the SNR, MSE of the first signal and CRB are shown in Figure 9. It can be seen, the MSE of proposed method is closed to the CRB even at the lower SNR. 5.3. 2-D DOA Estimation Simulation 5.3.1. 2-D Estimation RMSE Simulation 2D two far-field LFM sources shoot the 20N 00 50 ), 1 UCA (3 om the with the angle information }. The signa 11 {(60 , l model is: 00 11 0, 70) C opyright © 2009 SciRes. WSN H. T. QU ET AL. 180 200 H z , 1300 / H zs ; 2200 H z , 2 900 / H zs . Sampling rate is s f 900 H z, number of snapshots is 1024, and the cover filter length is 2300L. The Figure 10(a) gives the 2-D DOA tio RMSE (root ean square error, RMSE) comparison curves of the be seen in igure 10(b). The accuracy of our method has certain thm. estima- n of signal one in the 0dB SNR. Change the input SNR range from 0dB to 20dB with the interval 5dB, firstly perform big step search to obtain the rough DOA estimation. Then run the high differen- tiation search with the 0.001rad step. Run 300 time Monter-Carlo experiment respectively, the m proposed algorithm and literature one can F improvement compared to the original algori (a) (b) Figure 10. (a) 2-D DOA estimation using UCA, (b) RMSE comparison curves using UCA. 5.3.2. 2-D Estimation Performances in Complex Channel and Simulation In mobile communication system, the proposed algo- rithm is applied to the complex channel which the multi- path and Doppler shift is existed simultaneously. In the same simulation conditions, viz. the random signal source model is: (54) where, 2 1 ()exp() exp[(()()/2)] E keekeke e stMjftj tt 12 1, 0.9MM , 12 2ff Doppler frequency shift is 0, , multi-path delay is 12 0,1/ 900 . When the SNR is 0dB, the simulation result of signal one in most powerful path can be shown in Figure 11(a). (a) (b) Figure 11. (a) 2-D DOA estimation in complex channel, (b RMSE curves in complex chl. ) anne Copyright © 2009 SciRes. WSN H. T. QU ET AL. Copyright © 2009 SciRes. WSN 181 Change the input SNR range from –21dB to 0dB with the interval 3dB. Run 300 time Monter-Carlo experiment respectively, the RMSE comparison curves of signal one can be seen in Figure 11(b), which can show that the proposed algorithm is also effective in complex channel. 6. Conclusions Analyzing the definition and characteristics of the FRFT, a novel DOA estimation algorithm has been presented the implementation of the method, mask operation is introduced to simply the filtering procedure with no ac- curacy degradation. Demodulation operation is us extend the application range of the traditional estimate method without performance loss. Compared with other methods, the veracity has certain improvement while th cross-terms and interpoided. The prop is also expanded to thetimation using UCA, addition, the pro- po aking this method more reliable in theory and in prac- rich the principle and applicatio he optimization, the 2-D Cramer-R is- 2000. . S. Zhou, “A novel method for the DOA ” IEEE Transactions on ASSP, Vol. p. 2395– . In 37, No. 5, May 1989. [6] V. Namias, “The fractional Fourier transform and its ed to application in quantum mechanics [J],” IMA Journal of Applied Mathematics, No. 25, pp. 241–265, 1980. [7] L. B. Almeida, “Fractional Fourier transform and time-frequency representations [J],” IEEE Transactions e olation are av 2-D DOA es osed on Signal Processing, Vol. 42, No. 11, pp. 3084–3091, 1994. [8] Y. Q. Dong, R. Tao, S. Y. Zhou, et al., “SAR moving target detection and imaging based on fractional Fourier which is suitable for the multi-path and Doppler fre- quency shift complex environment. 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