Adaptive Wideband Beamforming Based on Digital Delay Filter

Transcription

1 261 Adaptive Wideband Beamforming Baed on Digital Delay Filter Zeehan Ahmad, Song Yaoliang, Qiang Du School of Electronic Engineering & Optoelectronic Technology, Nanjing Univerity of Science & Technology, Nanjing (210094), Jiangu province, P.R.China Abtract A novel adaptive wideband beamforming method i propoed, where beamforming i achieved by LMS baed pacetime adaptive filtering algorithm. Conventional broadband beamforming require deired ignal to be incident from the broadide i.e. direction normal to the array. The new method overcome thi contraint by make ue of digital delay filter which compenate the delay of receiving data, o that the ignal of interet can be treated a if it had arrived from the broadide. Then LMS baed pace-time adaptive filtering algorithm i applied to achieve beamforming. Another advantage of the propoed methodology i it low computational complexity while utaining high reolution. The effectivene and advantage of the propoed methodology i theoretically invetigated, and computational complexity i alo addreed. To verify the theoretical analyi, computer imulation are implemented and comparion with other algorithm are made. Index Term Wideband beamforming, Digital delay filter, LMS algorithm, Space-time adaptive filtering. I. INTRODUCTION The concept of beamforming i to teer the antenna beam in the direction of the deired ignal, whilt uppreing ignal from other direction [1]. Adaptive antenna that operate a purely patial filter are alo bandwidth limited, although not in the ame ene a time-domain, frequency-domain or ubpace proceing approache. For narrowband ignal the time difference of ignal arrival between antenna element can be treated a a phae hift of the received ignal. However, for wideband ignal the difference in the array received ignal complex envelope cannot be neglected. The bandwidth and multi-path limitation of purely patial filter are often overcome with what i known a Space-Time Adaptive Proceing (STAP) [2-5] or Space Frequency Adaptive Proceing (SFAP) [2-4]. Conventional pacecraft control and communication are mainly narrowband communication ytem, i.e. the maximum relative bandwidth of the pace Unified S-band (USB) meaurement and control ytem i 5% [6-8]. The movement toward a globalized pace communication and unprecedented exploion of pace technology worldwide ha opened a new window for broadband communication to develop newer methodologie in the field of pacecraft control and communication. It i thi challenging environment to which broadband communication intend to contribute by

2 262 evolving meaningful and optimal olution to variou problem of antenna array and related application. Due to the low-received pace communication ignal power which make the ytem vulnerable to failure [9], diruption and undeired interference, beamforming baed on patial adaptive filtering i one the main technology to addre and ae the poible failure mode and to develop trategie to detect uch effect and correct them. Reearch on adaptive narrowband beamformer [10] have been gaining ground in the literature, but empirical tudie on wideband beamforming remain relatively marginal and carce. Wideband beamforming i the current direction of ignificant reearch in array ignal proceing. The mot claical approach of broadband beamforming i to ue tapped delay-line (TDL) or FIR filter to accomplih beamforming in time-domain. Thi time domain broadband beamformer i equivalent to a time domain filter, which can form an independent frequency repone to compenate the phae difference of the received ignal with different frequency and the interference ignal i uppreed by pace-time filtering [11]. Frequency domain beamforming i another approach to deal with wideband ignal. Frequency domain beamforming employing leat mean quare (LMS) algorithm i dicued in [12-15]. Another good approach propoed in [1] i to ue enor delay line. Comparion between SDL and TDL beamformer are preented in [16]. A new approach to wideband ignal cloe to end-fire baed on TDL i dicued in [17]. In [18-20], the improved broadband beamforming algorithm propoed i under the aumption that the deired ignal i incident on the array from the normal direction (broadide) i.e. =0, then DMI (Direct Matrix Inverion) algorithm i applied to realize beamforming. To avoid uch contraint and difficultie aociated with conventional broadband beamforming, in thi paper we propoe a novel method of adaptive broadband beamforming baed on LMS algorithm, which i uitable for ignal impinge on array from an angle relative to the broadide. Some other advantage of thi novel methodology include it low computational complexity, high reolution to teer the main lobe to the deired direction and uppre the broadband interference to overcome the hortcoming of conventional algorithm. Thi anti-jamming technology can be ued in the meaurement and control of the pace vehicle a well a the atellite communication application. The ret of thi paper i organized a follow. Section 2 decribe the baic wideband ignal model, wherea Section 3 preent the tructure of wideband beamforming baed on digital delay filter. Section 4 cover the propoed Spatial-Temporal Adaptive Algorithm baed on LMS and ection 5 calculate the computational complexity of the propoed algorithm. To illutrate the validity and performance of the propoed algorithm, computer imulation are conducted, and the reult are given in Section 6. Finally ection 7 offer ome concluion drawn on the bai of imulation reult.

3 263 θ d 1 2 N Fig. 1. Uniform linear array antenna tructure II. SIGNAL MODEL Conider a uniform linear array with N enor a hown in figure 1. The firt array enor located at the origin of the coordinate i aumed to be the reference enor and d i the pacing between two adjacent enor. Conider the far-field ignal model, aume that the deired ignal and interference ignal are of ame frequency and non-coherent wideband ignal, impinge on the array from an angle relative to the broadide, which refer to the direction normal to the array. The firt array element receive the ignal: M x ( t) ( t) i ( t) n ( t) (1) 1 m 1 m=1 Wherea, im () t i the m -th interfering ignal m 1,2,, M, n () 1 t i noie received on firt array element, t () i the deired ignal received on firt array element, a modulated ignal can be expreed a: t j2 c ( ) m( t)e ft mt () i a baeband ignal, f c i the carrier frequency. (2) The propagation delay of the received ignal from reference array element to the n - th array element can be expreed a: n ( n 1) d in( ) / c (3) Where, n 1, 2,, N, c i the peed of light, then the ignal received on the n th array element can be repreented a: n n m n n m=1 M x ( t) ( t ) i ( t ) n ( t) (4) The received deired ignal t ( n ) i down-converted to baeband ignal ampling. mt before j2 fcn ( n)e The bandwidth B of narrowband ignal atify the condition of B f, and we can approximate m( t ) m( t), o each array element receive the ame ignal envelope. For wideband ignal, n however, the above condition i not atified, o it i not the ame a the narrowband ignal uniform c

4 264 phae hift to compenate for the weight vector [21]. III. WIDEBAND BEAMFORMING BASED ON DIGITAL DELAY FILTER STRUCTURE Compared with the conventional DMI wideband beamforming, the propoed beamformer employ LMS algorithm for calculating tap weight vector, which avoid computing the invere of a matrix and alo ha maller computational complexity. But uch criterion ha a limitation that it can only be ued under the condition that the deired ignal hould impinge on the array from broad ide i.e. =0. So before calculating the weight vector, we need to compenate the delay of the antenna received ignal via digital time delay filter in order to enure that the deired ignal impinge on the array from the direction normal to the array. Once the tap weight vector i obtained, then delay proceing i applied to the optimum weight vector. Finally, we get the main lobe teered to the deired direction and null to the interference. The tructure of digital time delay filter i hown in Figure 2. The core idea i the delay compenation n in the time-domain, which make the ignal normal incident that i the angle of incidence i 0. The Figure 2 how that the tructure of Digital time delay filter i compoed of two part, integer time delay filter and fractional delay filter. x 1 Integer time delay Fractional Delay Filter y 1 x 2 Integer time delay Fractional Delay Filter y 2 LMS Adaptive Filter x N Integer time delay Fractional Delay Filter y N Fig. 2. Structure of Digital time-delay filter Auming the ampling period T and the ignal data received by the antenna after dicretization, the n th array element to compenate time delay can be expreed a: n (5) T L D So the total delay of any FD filter [22] can be plit into an integer time delay of ampling n interval L=round( ) T ; and a fractional delay D -1 2,1 2. They can be calculated a: n n L round( ), D L (6) T T For the ampled data, the integer time delay of the ampling interval i firt carried out, that i, to

5 265 compenate the L, and then the fractional delay D i compenated. Compenation L can be achieved by delay of an integer multiple of the ampling interval T uing digital delay line, having impler hardware implementation. Compenation of D can be realized by deigning FIR filter. Becaue of it mall value, the order of the filter required i not high, thereby reducing the hardware complexity. The impule repone hn ( ) of the FD FIR filter can be expreed a:[22] h( n) inc( n D) (7) Clearly, for a non-integer D, hn ( ) i infinite a well a non-caual and a filter with uch an impule repone i thu non-realizable [22]. So we approximate the ideal filter by window method. The impule repone of the window function can be decribed a: h( n) W( n) inc( n D), 0 n N (8) Window method are extenively ued in ignal proceing and it related application. The mot ignificant ue of window can be found in deign of digital filter, where a non-caual and infinite ideal impule repone i converted to a finite impule repone (FIR) filter deign [23]. Though, the window method i numerical efficient in the ene that the ideal repone i imply multiplied by a window [24]. However, limitation impoed on thi approach to chooe the optimal window method i quite complicated. Taking into conideration advantage and diadvantage of available window method [22], we have choen Chebyhev window characterized by a minimum main-lobe width for a given ide-lobe attenuation [23]. Chebyhev Window ha the unique property that all it ide-lobe are equal and the ide-lobe height i the ame at all frequencie [25]. Thi effect i hown in Figure 3 below. Fig. 3. Default length N = 64 Dolph-Chebyhev window with 100 db relative idelobe attenuation The Dolph-Chebyhev Window function w ( ) 0 n in dicrete time domain can be repreented by the following equation [26-27]

6 266 1 N 1 i2 nk N 0 ( ) D/ Ch W0 ( k) e (9) N k 0 w n W ( ) 0 k i the Fourier coefficient and can be derived a: k coh[ N coh ( )] (10) -1 co{ N co [ co( )]} k 0 ( ) D/ Ch (-1) N -1 W k Where i a fixed value parameter decribed by the following equation: And 1 N 1 D Ch coh[ coh (10 )] (11) A coh[ coh (10 )] (12) N The width of the main-lobe and the reulting filter tranition-band can be controlled by varying N. The ripple ratio (Side-lobe level) can be controlled by parameter. IV. SPATIAL-TEMPORAL ADAPTIVE ALGORITHM BASED ON LMS Space-Time Adaptive technology can tranform a one-dimenional patial filtering into twodimenional of time and pace, forming a model of a two-dimenional pace-time proceing. After the proce of digital delay line and fractional time delay filtering for the ample data, it will turn into the tructure of pace-time adaptive filtering baed tap delay line, a hown in Figure 4. In it dicrete form, the TDL ytem i replaced by a finite impule repone (FIR) filter and adaptive algorithm can be realized by digital circuit [28]. It will realize the beamforming by adjuting the order of FIR filter/tdl and the weight of the tap. The order of the TDL i decided by the bandwidth of the impinging ignal [29]. Generally, higher the bandwidth, the longer the TDL [30]. A hown in Figure 4, there are N array element in the tructure and each channel i connected to J order FIR filter, and the output i a follow [31]: z t H ( ) ( t) w y (13) Where; w i two dimenion weight vector in pace-time and hold all the NJ enor coefficient, it ize i NJ 1 expreed a follow w = [ w w w ] T (14) 1 2 Wherea, for the j th column vector w j, j 1,2,, J whoe ize i N 1, contain the N complex conjugate coefficient found at the j th tap poition of the N TDL, and i expreed a: J T w j [ w1, j w2, j wn, j ] (15) And y () t i the input data of dimenion NJ 1 obtained after the delay compenation of N 1 dimenional data through TDL tructure. With y n 1,2,, N; j 1,2,. J denote the n th nj

7 267 channel to the j th data after the delayed output, the input data vector can be expreed a: T y [ y1( t) y2( t T ) y J ( t JT )] (16) Where the j th column vector y ( t jt ) i the input data of dimenion N 1, expreed a: j T y j ( t jt ) [ y1( t jt ) y2( t jt ) yn ( t jt )] (17) y () t 1 T T T w1,1 w1, 2 w 1,3 w 1, J y () t 2 T T T w2,1 w2, 2 w2, 3 w 2, J zt () y () N t T T T wn,1 wn, 2 wn, 3 w N, J Fig. 4. Space-time adaptive beamforming baed on the TDL The weight vector of the tap can be obtained through a variety of algorithm when the deired ignal i incident from the normal direction. According to the DMI realization principle, the optimal olution olved by the Lagrange multiplier method ha relation to the invere of the correlation matrix R of the input, and can be expreed a follow: 1 1 H 1 1 wopt RC( C RC) f (18) The iterative method of LMS can be derived with pace-time adaptive filter and Frot LMS algorithm in patial filtering [31]. There are J contraint in the pace-time tructure of J-order filter, and the contraint equation i a follow: Where the NJ H C w f (19) J dimenional matrix C i termed a the contraint matrix, and f i the J 1 gain (contant) vector or J being the number of contraint. The iterative equation obtained by Frot LMS algorithm i a follow: w[ k 1] w[ k] ( R w[ k] C [ k]) (20) where i the iterative tep ize, i the Lagrange multiplier. w [ k 1] mut atify the contraint in equation (19), o we ubtitute equation (20) into equation (19) to get [ k]. Then we ubtitute into [ k] the iteration equation in (20) and arrive at: w C C C f + P w R w (21) H 1 [ k 1] ( ) ( [ k] [ k])

8 268 where H 1 H P I C( C C) C (22) The initial weight i H 1 w[0] C( C C) f (23) The better approximate value of the correlation matrix a a time-average can be computed a: 1 K H R y( k) y ( k) (24) K k 1 K i the number of naphot, and then uing the imple approximation of correlation matrix to replace R, the final weight vector can be calculated by following iterative equation a: H 1 * w[ k 1] C( C C) f + P( w [ k] e [ k] y[ k]) (25) R V. COMPUTATIONAL PERFORMANCE ANALYSIS The computational complexity of the DMI algorithm i compared with that of the propoed LMS algorithm. The comparion i baed on a count of the total number of complex multiplication and complex addition involved in each of thee two algorithm a an indication of computational complexity of adaptive algorithm. Thi provide a reaonably accurate bai for comparing the computational complexity of thee two algorithm. Unlike the new algorithm, the conventional DMI algorithm mut eek R, and ubject to a matrix inverion operation to olve the optimal weight vector. The weight vector i calculated according to equation (18) which i computationally intenive. In practice, the covariance matrix i etimated by a finite number K of time domain ample (naphot). In the imulation, we ued the forwardmoothing algorithm to etimate the equation (24). The new algorithm i baed on equation (25) uing the iterative calculation to get the optimal value, without knowing the true econd order tatitic information of the ignal data, and H 1 C( C C) f and P are fixed matrix under known array condition, need to be calculated only once. It hould be noted that the computation of weight by propoed method compute K iteration compared to matrix inverion required by the DMI algorithm and thu the propoed method i computationally efficient. In the tructural model baed on a tapped delay line, R i the covariance matrix of NJ NJ dimenion. According to the conventional method of matrix inverion, any element of the invere matrix R i the diviion of the NJ 1 matrix determinant by 1 R matrix determinant. The matrix inverion require ( NJ ) 3( NJ ) 4( NJ ) NJ complex multiplication and 3 2 ( NJ ) 2( NJ ) NJ 1 complex addition, o only the computation of invere covariance W( N, J ) i

9 W( N, J) ( NJ ) 2( NJ ) 2( NJ ) 1 (26) From equation (26), it can be een that the magnitude of computational cot in computing the invere covariance R ha reached to 1 4 ( NJ ). The value of R i related to the value of K. The higher the value of K, the more cloer the etimated R i to the true covariance matrix, o the value of K i not too mall. The ize of W( N, J ), number of array element N and filter order J, are related to the number of naphot K and it value i large. The optimal weight vector of DMI algorithm and the propoed algorithm i calculated by complex multiplication and addition operation, in accordance with equation (18) and (25) repectively. The total amount of computation W 1 required by the DMI algorithm i a follow: W1 ( N 1) J ( 2N 6N 2N 2) J [(2K 2) N 2N 2] J NJ 1 (27) The total amount of computation W 2 of the new algorithm i a follow: W 2 2 J (2N 8N 4) J [2N K 2] J 5NKJ K 2 (28) Comparing equation (27) and (28), we find that the computational complexity of W 1 i much higher than W 2, o the propoed algorithm not only reduce the computational complexity, but alo eay to implement in real-time engineering application. VI. SIMULATION RESULTS In thi ection, the propoed beamformer i evaluated by computer imulation. A uniform linear array compoed of 8 element with 7-order tap delay of digital filter i imulated in MATLAB. Element are aumed to be omni-directional, and pacing between the adjacent array element i d min 2. Auming the pace ha two ignal, deired ignal come from the look direction 1 20 and the wideband interference i incident on the array at The ampling frequency i f 400MHz. In high SNR imulation environment, we ue a center frequency of f 100MHz, the modulating ignal bandwidth (B) of 50MHz, interference ignal i non-coherent wideband ignal of the ame frequency and ignal bandwidth and the noie i random white noie, obeying the normal ditribution law. After receiving data through digital delay filter, DMI algorithm and propoed LMS algorithm are ued to realize the broadband ignal beamforming. Figure 5 i the 3-D beam pattern of DMI algorithm. x axi i the angle of arrival, the earch range [ 90,90 ], y - axi i the normalized frequency f [0,1], correpond to the frequency range [0, f / 2]. The 50MHz deired ignal correpond to the frequency range [75,125]MHz, i.e. y axi range interval of [0.375,0.625]. From Figure 5, it can be oberved that the main lobe i pointing in the c

10 270 =20 direction. The mot deep null are in the range of y [0.3,0.7], and the null in the interference direction = 20ignificantly uppree the wideband interference. Fig. 5. A 3-D wideband Beam Pattern of DMI algorithm Fig. 6. A 3-D wideband Beam Pattern of Propoed algorithm Figure 6 i the 3-D beam pattern under the propoed methodology. From the figure it can be oberved that the main lobe i teered to the deired ignal direction that i =20. In the frequency

11 271 range y [0.35,0.68], the deepet null in the direction = 20, effectively uppreed the broadband interference. Thi indicate the ucceful wideband beamforming operation by LMS algorithm baed on digital delay filter. Like DMI algorithm, the new methodology i alo high reolution to teer the main beam in the deired ignal direction and place deeper null in the interferer direction. So, the performance the propoed beamformer i comparable with that of the conventional DMI beamformer. It hould be noted that if the deired ignal i incident from the direction other than broadide, the conventional wideband beamformer without the digital delay filter or pre-teered delay i unable to do beamforming. Thi effect i hown in the Figure 7 below. The deired ignal come from the look direction 1 20 and the wideband interference i incident on the array at The conventional wideband beamformer without delay filter form the main beam at 0 degree, while the propoed methodology employing digital delay filter i able to teer the main beam in the direction of deired ignal (20 degree). Fig. 7. The comparion of between the conventional beamformer and propoed beamformer for deired ignal incident from the non-broadide direction. To oberve the computation performance of the algorithm, equation (23), and (24) how that the amount of computation of the DMI algorithm and propoed algorithm are related with the value of N, J and K. By varying thee three variable, the computational complexity of both the algorithm are compared. The computational complexity of DMI algorithm and propoed algorithm are plotted in Figure 8 with fixed number of tapped delay-line J =7. y axi repreent the amount of computation i.e. computational complexity and x axi repreent the number of enor.

12 272 Fig. 8. The comparion of computational complexity between the propoed algorithm and the DMI algorithm under a fixed J A can be een from the figure above, the computational complexity of both the algorithm increae with the increaing value of N and K. So an increaing trend i oberved in the complexity of both the DMI and propoed algorithm under fixed number of tap and with the increaing number of enor and naphot. Under the ame value of K, the new algorithm i relatively le computational complex than DMI algorithm. Even, when N 33, the propoed algorithm with K i le computational complex than the DMI algorithm with K Clearly, the propoed algorithm ignificantly reduce the computational complexity a compared to DMI algorithm, for higher value of N. Fig. 9. The comparion of computational complexity between the propoed algorithm and the DMI algorithm under a fixed N

13 273 The computational complexity of DMI algorithm and propoed algorithm are plotted in Figure 9 with fixed number of array element N 20. The computational complexity of both the algorithm how an increaing trend with the increaing value of J and K. The comparion of the propoed algorithm and DMI algorithm under the ame value of K how that the new algorithm i relatively le computational complex. For J 29, the propoed algorithm with K i le computational complex than DMI algorithm with K Clearly, for higher value of J, the propoed algorithm ignificantly reduce the computational complexity a compared to DMI algorithm. VII. CONCLUSION In thi paper, we have propoed a new adaptive wideband beamforming methodology. The propoed methodology employ LMS algorithm baed on the formation of a digital delay filter. The concept behind thi new methodology i to overcome the contraint that ignal of interet hould be impinged on the array from the broadide. Simulation have demontrated the uperiority and validity of the propoed methodology. The appealing advantage of the propoed methodology lie in that it not only achieve the broadband interference uppreion, high reolution to teer the main beam in the direction of deired ignal and null the interference, but i alo le computational complex by reducing the amount of computation to be eaily realized in practical ytem. ACKNOWLEDGMENT The author gracefully acknowledge the upport of the National Natural Science Foundation (NSFC) Project ( ) and project ( ) of China. REFERENCES [1] Liu W. Adaptive wideband beamforming with enor delay-line. Signal Proceing, 2009, 89(5): [2] R. L. Fante, J. J. Vaccaro. Wideband Cancellation of interference in a GPS Receive Array. IEEE Tranaction on Aeropace and Electronic Sytem, 2000, 36(2): [3] W. L. Myrick, M. D. Zoltowki, J. S. Goldtein. Anti-jam pace-time preproceor for GPS baed on multitage neted Weiner filter. Military Communication Conference Proceeding, 1999, 1: [4] R. L. Fante, J. J. Vaccaro. Cancellation of Jammer and jammer multipath in a GPS receiver. IEEE Aeropace and Electronic Sytem Magazine, 1998, 13(11): [5] G. F. Hatke. Adaptive array proceing for wideband nulling in GPS ytem. Conference Record of the Thirty- Second Ailomar Conference on Signal, Sytem & Computer, 1998, 2: [6] Kreng, J.; Sue, M.; Sieu Do; Krikorian, Y.; Raghavan, S., "Telemetry, Tracking, and Command Link Performance Uing the USB/STDN Waveform," Aeropace Conference, 2007 IEEE, vol., no., pp.1,15, 3-10 March 2007 [7] Liu Suxiao; Xiong Huagang; Feng Wengquan; Zhao Hongbo, "A New Subcarrier Demodulator of Satellite Telemetry Approaching to the Ideality Baed on the Digital Signal," Future Information Technology and Management Engineering, FITME '09. Second International Conference on, vol., no., pp.191,194, Dec [8] Oleki, P.J.; Patton, R.W.; Bharj, S.S.; Thaduri, M., "Tranmit receive module for pace ground link ubytem (SGLS) and unified S-band (USB) atellite telemetry, tracking and commanding (TT and C), and communication," Military Communication Conference, MILCOM IEEE, vol.2, no., pp.880,885 Vol. 2, 31 Oct.-3 Nov [9] Jianhong Xiang, Lili Guo, Qingling Liu. Study of GPS Nulling Antenna Baed on Space-Time Proceing Algorithm. 5th International Conference on Wirele Communication, Networking and Mobile Computing, WiCom '09, 2009, 1-4. [10] Yong Wang and Yihua Hu Performance of UWB Satellite Communication Sytem under Narrowband Interference. In Proceeding of the 2012 International Conference on Electronic, Communication and Control (ICECC '12). IEEE Computer Society, Wahington, DC, USA,

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