Adaptive Space/Frequency Proceing for Ditributed Aperture Radar Raviraj Adve a, Richard Schneible b, Robert McMillan c a Univerity of Toronto Department of Electrical and Computer Engineering 10 King College Road Toronto, ON M5S 3G4 Canada b Stiefvater Conultant 10002 illide Terrace Marcy, NY 13403, USA c US Army SMDC untville, AL 35805, USA Abtract- Thi paper detail a preliminary invetigation into pace-time-waveform adaptive proceing for waveform divere ditributed aperture. The large baeline of uch a ditributed radar reult in angular reolution that i order of magnitude better than the reolution of a monolithic ytem (ingle large radar) with the ame power-aperture. Thi capability come at the cot of grating lobe (multitatic with evenly paced aperture) or high idelobe (multitatic with randomly paced aperture). Thi paper develop ome preliminary olution to thee drawbac aociated with ditributed aperture. In particular, the ue of approximately logarithmic pacing with each aperture tranmitting orthogonal waveform provide excellent detection performance. I. INTRODUCTION Thi paper invetigate the application of adaptive proceing to a relatively new concept in radar ytem: waveform divere ditributed aperture. In uch a radar, the tranmit/receive aperture i divided into a number of ubaperture that can be placed in variou location relative to each other. The ditributed radar operate in a multitatic mode with all aperture tranmitting (either the ame ignal, different uncorrelated ignal or orthogonal ignal). Multitatic radar can provide ignificantly improved target tracing and interference rejection becaue of the large baeline between the variou aperture. The large baeline reult in angular reolution that i order of magnitude better than the reolution of a monolithic ytem (ingle large radar) with the ame power-aperture. Thi capability come at the cot of grating lobe (multitatic with evenly paced aperture) or high idelobe (multitatic with randomly paced aperture). In a related paper, we have hown that a new mode of multitatic operation i required to achieve the improved interference rejection while maintaining the ytem urveillance capability. In thi mode, the ubaperture radiate mutually orthogonal waveform, however, each ubaperture receive and procee all orthogonal waveform. Conider a ditributed aperture with N ubaperture. Since each ubaperture receive the return form all tranmitted waveform, there are N N returned ignal for each radar range. A pace-waveform-range data cube therefore replace the uual pace-time-range data cube. In thi paper, the orthogonal waveform are choen to be relatively narrowband ignal offet in center frequency. ere we report on the ue of optimal adaptive pace/waveform proceing for uch a ditributed aperture. In particular, we compare thi ituation to the traditional cae where ubaperture tranmit the ame waveform. In uch a ytem, there are a few unique concept: Adaptive pace/waveform proceing: Traditionally, adaptive proceing ha focued on the pace and time dimenion leading to pace-time adaptive proceing (STAP). The patial teering vector i related to the loo direction while the temporal teering vector i determined by the loo Doppler frequency. In our cae, the time dimenion i replaced with the waveform dimenion. The pace/waveform teering vector i determined by the loo angle uniquely with a different patial teering vector for each tranmit frequency. Spacing of ubaperture/waveform (frequencie): Ditributing the aperture and eparating the tranmit frequencie introduce two new degree of freedom available to the radar deigner: the pacing between the antenna element and the frequencie. Equally paced element with equally paced frequencie can lead to grating lobe that can reduce the effectivene of the adaptive proce. ere we invetigate variou configuration, comparing them in term of grating lobe, mainbeam width, etc. Future wor will invetigate the optimization of thee parameter with repect to ome performance meaure. Target/interference are not necearily in the far field: By common definition, the fair field region i determined by three condition: R > λ, R > D and R > D 2 /λ where R i the radial ditance, D i the total aperture baeline and λ i the frequency of operation [1]. From a phyical point of view, the far field may be defined a the region where the patial teering vector i effectively
independent of the radial ditance. In our example, we chooe D = 200m with a center frequency 10Gz, etting the beginning of the far field at approximately 1500m. The target and interference are therefore not necearily in the far field. Thi ha eriou implication in the type of adaptive proceing cheme choen, including choice of econdary data to etimate the interference covariance matrix. Similar to STAP for bitatic radar, thi range dependent teering vector reduce the econdary data available to etimate the covariance matrix.[2, 3]. In addition to the above, another important conideration i poition error in the array. Due to the large baeline, a relatively mall error in poition may be comparable to the wavelength of operation. Thi i epecially true for radar operating at X-band. Section II decribe the ytem model and the adaptive proceing cheme. Section III preent ome preliminary reult for pace-time-waveform adaptive proceing for waveform divere ditributed aperture. Finally, Section IV preent ome concluion and point to ome future wor. II. SYSTEM MODEL AND ADAPTIVE PROCESSING SCEME The element of the linear array are not equally paced and each element in the array may tranmit at a frequency. Let {x n, n = 0,1,2 N-1} denote the poition of the N element, each with correponding frequency {f n, n = 0,1,2 N-1}. Each element receive and procee the ignal from all N tranmiion. Conider a cenario wherein each element tranmit M pule within a ingle coherent pule interval (CPI) at a pule repetition frequency (PRF) of f r. Due to thee N tranmiion, the return ignal from a unit target at the n th element, th frequency and m th pule, for a target at relative velocity v and relative angle ϕ i given by f d 2π xn f ( n,, m) = exp m exp j, (1) fr c where f d i the Doppler frequency aociated with tranmit frequency f, i.e. 2v f fd =, (2) c 2v f 2π xn f ( n,, m) = exp m exp j. (3) c fr c Thi ignal can be written a a length N 2 M pacewaveform-time teering vector T T T T T ( ) = [... ] T ϕ, (4) v, 0 1 2... N 1 where each length NM vector, i the traditional pace-time teering vector for center frequency f () ( ϕ ) b v a =, (5) b 1 2 v f exp c f M 2 v f exp ( M 1) c fr () r v =, (6) 1 2π x1 f exp j ( ) c a ϕ =. (7) M 2π xn 1 f exp j c Note that both the patial and temporal teering vector are defined in term of the N frequencie of operation f. Alo, unlie the traditional patial teering vector for a linear equipaced array, the patial teering vector here i defined in term of the poition of the element x. The jammer ignal ha a tructure imilar to the target ignal. ere we model Gauian barrage noie jammer. ence, the only difference between the target and jammer model i that the temporal teering vector i replaced by a vector of independent, complex, Gauian random variable. The jammer ignal, for frequency index i modeled a ( ϕ ) = ξ b a, (8) where ξ i the amplitude of the jammer and the temporal vector b i a white, complex Gauian random vector of independent random variable with zero mean and unit variance. The length N 2 M vector of jammer ignal i therefore T T T T T [ ] T = 0 1 2...... ( N 1), (9) Noie i modeled a a white complex Gauian random variable for all frequencie, pule and element. The overall received ignal, i therefore given by, ( ) + n x ξ v,ϕ + = t, (10) where n i complex Gauian noie vector. Uing the ignal in Eqn. (10), we can now implement a pace-time-waveform adaptive proceing algorithm. The algorithm choen here i the traditional optimal approach where the N 2 M element of the received ignal x are combined uing a weight vector w. The weight vector i determined uing the relation w = R -1, (11) where i the pace-time-waveform teering vector of Eqn. (4) and R i the interference plu noie covariance matrix. Note that in practice, thi matrix mut be etimated.
A. Data Generation and Implementation of Adaptive Proce Uing Eqn. (10) above, received data can be generated correponding to the choen cenario. Repeating thi everal time, e.g. (P+1) time, yield a pace-time-waveform-range hypercube, organized a a N 2 M (P + 1) matrix. Each column of thi matrix correpond to a ingle range. To implement an adaptive proce, uing thi data, a pace-timewaveform covariance matrix i etimated. In the imulation preented in Section III, the data for the (P + 1) range i generated without a target, i.e. ξ t = 0. Then a target with choen power i injected into the middle range, p = (P/2+1). For the q th range, an interference covariance matrix i etimated by uing a liding window P + 1 p= 1 R ˆ = x x p p, (12) p q where the upercript repreent the ermitian of a complex matrix and x p repreent one of (P+1) naphot of data. In general, for a reaonably accurate etimate of R, we need P > 2N 2 M [4]. The adaptive weight are obtained uing Eqn. (11). Uing thee weight we define the modified ample matrix inverion (MSMI) tatitic, which a the property of having contant fale alarm rate (CFAR) in Gauian interference, 2 w x p ρ p =. (13) 2 w ( v, ϕ) Thi tatitic i plotted a a function of range. Clearly, if the range correpond to the one with the target, the output tatitic hould be a large a poible, while if the range doe not contain a target, the output MSMI tatitic hould be cloe to zero. Thee weight are alo ued to obtain the output ignal-tojammer-ratio (SR), auming a unit target, a SR w ( v, ϕ) w =. (14) Note that the jammer ignal include the jammer amplitude. In thi analyi, to illutrate jammer uppreion, thi SR i plotted a a function of jammer angle ϕ. Table 1: Parameter for tet cenario N 6 PRF 2z Center Frequency Frequency Offet Radar Baeline Pule in CPI (M) 10Gz ammer-to- Noie Ratio 50dB 100Mz Target SNR 0dB 200m 12 Target velocity (v) Number of range bin (P + 1) 10m/ 1728 The frequency offet given in Table 1 i ued in the cae where different element tranmit on different frequencie. When uing the frequency offet, each tranmiion i eparated by 100Mz. The firt null beam width of uch an array i 0.014 o. A. Statitic Veru Range Figure 1: MSMI tatitic veru range. No frequency offet. III. PRELIMINARY RESULTS Unle tated otherwie, the parameter choen in the tet cenario are given in Table 1.
Figure 4 plot the SR for the cae where each element tranmit at a different frequency. The deep null at the target i viible, however, clearly there i a huge improvement in grating lobe. Off-target null till occur, the null are much hallower and much further away from the target location. To confirm the fact that grating lobe are reduced in thi cae, in Figure 4, the analyi i conducted over a much larger angular extent than in Figure 3. The reolution, however, i the ame. Note that the off-target null in Figure 4 are broader than the off-target null in Figure 3, i.e. while uing multiple frequencie help, the off-target null broaden. Thi i true due to the equal pacing between array element. Figure 2: MSMI tatitic veru range. Uing frequency offet (orthogonal waveform) The firt example illutrate the ue of orthogonal waveform to eparate target from interference. The target i at broadide in range bin 865. A 50dB barrage noie jammer at 0.04297 o, approximately three firt-null beamwidth away, hinder target detection. Figure 1 and Figure 2 plot the MSMI tatitic veru range for two cae conidered here: in Figure 1, all ix element tranmit at the ame frequency wherea, in Figure 2, each element tranmit orthogonal waveform (waveform eparated in center frequency by 100Mz). Both figure plot the MSMI tet tatitic veru range cloe to the range cell where the target wa injected. A i clear from the figure, when all element tranmit at the ame frequency, the target cannot be ditinguihed from the interference. Wherea, when each element tranmit a orthogonal waveform (different frequencie), the target i clearly viible, with approximately a 6dB eparation between target and interference. The ret of the example plot the output ignal-to-jammer ratio (SR) veru jammer angle. Figure 3: Signal-to-jammer ratio. Equal frequencie and element pacing. B. SR veru ammer Angle. Equally Spaced Element The econd example, again, compare the ue of the ame frequency from each element with uing multiple frequencie (orthogonal waveform). The element of the array are equally paced. The jammer are tepped over angle paced by 1.4 10-3 degree. Thi example ue only one pule, i.e. M = 1. Figure 3 plot the output SR veru angle for the cae where all element tranmit at the ame frequency. The output SR i rather high for mot angle. owever, at certain angle for the jammer, the SR how deep null a the null in direction of target. The large null at the target loo direction i expected a the jammer and target cannot be at the ame location. The deep null in the other direction are due to the grating lobe aociated with equal pacing and all element tranmitting at the ame frequency. Figure 4: Signal to jammer ratio veru jammer angle.
approximately logarithmic pacing, coupled with frequency divere waveform, grating lobe are eliminated. The analyi here i undertaen under ideal condition. The mot important extenion would be to include range dependent target and jamming ignal. The extremely long radar baeline et the beginning of the far field beyond any practical range. Similar to bitatic radar, the range dependent data would reduce the econdary data available to accurately etimate an interference covariance matrix. Thi iue lead to another poible extenion, namely the development of new adaptive proceing cheme pecifically for waveform divere ditributed aperture. ACKNOWLEDGMENTS Figure 5: Signal-to-jammer ratio. Log pacing and the ame frequency. Thi effort wa upported in part by US Army SMDC under contract DASGO-02-R-0037. REFERENCES [1] Balani, C., Antenna Theory, Analyi and Deign: ohn Wiley 1997. [2] imed, B.,.. Michel, Y. Zhang. Bitatic STAP performance analyi in radar application. in Proceeding of the 2001 IEEE Radar Conference. pp. 198-203, 2001. Atlanta, GA. [3] Sanyal, P.K., R.D. Brown, M.O. Little, R.A. Schneible, M.C. Wic,. Space-time adaptive proceing bitatic airborne radar. in Proceeding of the 1999 IEEE Radar Conference. pp. 1999. Waltham, MA. [4] Ward,., Space-time adaptive proceing for airborne radar. Technical Report. 1994, MIT Lincoln Laboratory: Cambridge, MA. Figure 6: Signal to jammer ratio. Log pacing and orthogonal waveform. The next example illutrate the ue of unequal pacing, here cloe to log-pacing. The ix element are located at 0m, 20m, 60m, 140m, 190m and 200m. Figure 5 plot the output SR veru jammer angle for the cae where all element tranmit at the ame frequency. In comparing with Figure 3, clearly the grating lobe are ignificantly reduced in number. owever, note that there till exit grating lobe that are paced further away. Figure 6 plot the SR for the cae of uing orthogonal waveform (unequal frequencie). ere the grating lobe are totally eliminated and the output SR i high, except at extremely cloe to the target loo direction. The null i le than 1.4 10-3 degree wide. IV. CONCLUSIONS AND EXTENSIONS Thi paper ha documented a preliminary invetigation into the ue of waveform divere ditributed aperture. The ue of divere waveform, in the form of frequency offet orthogonal ignal, overcome a ignificant drawbac with ditributed aperture, i.e. grating lobe. By chooing an