Frequency Diverse Array Radar Data Processing Yunhan Dong National Security and ISR Division Defence Science and Technology Group Australia yunhan.dong@dst.defence.gov.au Abstract The frequency diverse array (FDA) radar has recently been proposed. However, what left behind is how e FDA data should be correctly, efficiently and systematically processed to maximise its full potential and capacity of search and detection. This paper fills e gap. It shows at overall e processing of e FDA data is even simpler and faster an e processing of e conventional phased-array (CPA) radar to complete wide-area search and detection wiout a need of electronic scan and repeat pulse transmission and reception. The ambiguous range is resolved wiout using multiple pulse repetition frequencies (PRFs). This not only greatly simplifies e radar s design and operation but also makes radar quieter and steal. Keywords frequency diverse array; radar signal processing; radar waveforms. I. INTRODUCTION Frequency diversity technology has recently been proposed and possibly applied to radar systems [1-7]. Having had its roots in wireless telecommunications, it is more popularly discussed in applications for multi-input multi-output (MIMO) radar systems [8, 9]. However, more and more studies on frequency diverse array (FDA) radar have been published in recent years. Wang published two review papers on FDA radar where 125 and 60 papers, respectively (ere were overlaps ough), were cited [10, 11]. Among ose auors, Antonik, Wicks, Griffis and Baker [1] were some of ose who initially introduced is radar technology. Differing from its conventional phased-array (CPA) counterpart, e FDA employs pulsed continuous-wave (CW) waveforms wi some frequency increment across array channels / elements to generate a range, angle and time dependent beam pattern [2, 12, 13] at may have some advantages and different features over e CPA counterpart. While FDA radars have been widely studied and reviewed [10], characteristics, features, advantages as well as limitations of an FDA radar have not been reported oroughly. The most important part being left behind is how e FDA data should be systematically and efficiently processed to explore its full potential and capacity in terms of search and detection. Some claimed advantages, such as achieving e same signal-to-noise ratio (SNR) improvement as e CPA radar and rangeambiguous clutter suppression, were not true. Some data processing techniques, notably, e pulse compression and beamforming, were insufficient or not necessary. The paper focuses more on e systematic processing of e FDA data on receive. Techniques analogous to ose of e CPA data processing are presented. It shows at overall e processing e FDA data is faster and simpler an at of CPA data, but not e verse versa as suggested by most references. The ability of resolve range ambiguity wiout using multiple PRFs is demonstrated. A. Beamforming II. FDA DATA PROCESSING In principle ere could be two ways to carried out e FDA aperture synesis and beamforming. The first way is to separate frequency components in each receiving channel (assuming e separation can be perfectly implemented), synesise each frequency component across e whole array and en combine all frequency components coherently. The second way is to treat e received waveforms in each receiving channel as a whole and synesise all channels appropriately. We can show at e results of ese two ways are identical [14]. In fact, e perfect frequency separation cannot be implemented on receive using bandpass filters for short pulses. We use e second way for e beamforming and data processing. Consider a linear one-dimensional (1D) FDA consisting of N +1 independent transceivers (channels, N is an even number) spaced by d in e azimu direction ( d λ 0 / 2, λ 0 is e waveleng of e carrier frequency f 0 ). Each channel is assigned a unique CW waveform wi a frequency increment f ( N f << f0, so it is a narrowband system). The FDA radar steers at e boresight direction (i.e. e (r,0,0) direction in e r θφ polar coordinate system). The received signal in e k channel echoed by a point target or a clutter patch wi a unit reflection coefficient located at ( r 0, θ, can be written as, 2 γ k, u ( r, θ, = g ( θ, exp j (2r + ( n + k) dϑ) (1) c where ϑ = cosθ sinφ ; f n = f 0 + n f, f = 1/ T and T is e pulse wid, so waveforms in FDA channels are mutually orogonal [5, 6]. Function g ( θ, is e one-way individual transceiver directional gain pattern which is reciprocal for emission and receive, and is assumed to be identical for all channels ( g ( θ, 1 is assumed hereafter for simplicity).
After focusing (focusing at r is achieved by a time delay of t = 2r / c ), e signal received by e k channel is, γ k ( θ, = exp j ( n + k) (2) c Summing up all channels gives, Γ( θ, = γ k ( θ, = exp j ( n + k) (3) k= k= N / 2 c Since e frequency diversity is small compared to e carrier frequency, N f << f0 (a narrowband system), f n can be replaced by f 0 in (3), giving, 2 Γ ( θ, G ( θ, ) (4) 0 φ f0 sin ( N + 1) π 0 = = c G θ φ 0(, ) exp j ndϑ (5) c f0 sin π c Readers can verify at ere are essentially no any noticeable differences between e beamformed patterns given by e exact formula (3) and e approximation (4) for a narrowband case. Actually (4) is identical to e well-known two-way CPA beam pattern. B. FDA Data Processing The received signal in e k channel wi respect to time can be written as 2πf γ k, u ( = A exp j2πf n ( t + j ( n + k) (6) c where A is a complexed value representing amplitude of e target reflection coefficient (e target is located at ( r 0, θ, ); t = 2r 0 / c is e round trip time delay for unambiguous range r 0. Unlike e CPA radar whose emission is coherent and has a constant time-angle pattern, e FDA emission is noncoherent and has a varying time-angle-range pattern [2, 10, 12, 13]. Therefore, e processing of e FDA radar needs focusing to steer radar beam on to e location of interest. The focusing is to multiply a phase term of exp( j2π fn ( 2r + ( n + k) dϑ) / c) to focus at ( r, θ, for e k receiving channel and frequency component of f n. Because data received by a range bin is normally collected after a time delay of t, it means e radar has automatically focused at ( r 0,0,0), where t = 2r 0 / c. If ( r 0,0,0) is e location of interest, ere is no extra focusing processing required. However, if e location of interest is off e boresight, for instance, if ( r 0, θ, is e location of interest, en an extra phase of exp( j2π fn ( n + k) dϑ / c) is required for e k channel and frequency f n, so at e focusing point is changed from e point ( r 0,0,0) to point ( r 0, θ,. The problem is at each receiving channel contains N + 1 frequencies, and hence e above focusing processing is not straightforward as e received individual frequency components in a range bin are unknown. When e time delay t and e carrier frequency are excluded from (6), e received baseband signal in e k channel is, 2πf γ k ( = A exp + j ( n + k) (7) c For a narrowband system, (7) can be simplified to, 2π 2π γ k ( = A exp j kdϑ exp + j ndϑ (8) λ0 λ0 Therefore, for an orientation ( θ,, e pulse compression processing is identical for all channels, wi a matched filter of, 2π h( t, θ, = exp + j ndϑ λ0 / 2 t T / 2 (9) Let, dϑ tϑ = (10) λ 0 f Equation (9) can be write as, h( t, θ, = exp ( j2πn f ( t + t )) = h( t + t,0,0) ϑ ϑ (11) Note at (11) is important and interest as it shows at e matched filter h ( t, θ, for an orientation ( θ, is a circularly rotated copy (by e amount of t ϑ ) of e matched filter h (t,0,0) for e boresight direction. According to e property of convolution, if, T = h( τ, θ, h ( t τ,0,0) dτ en, t * T * tϑ ) = h( τ, θ, h ( t ( τ + tϑ ),0,0) dτ (12) (13) Equations (12) and (13) indicates at e pulse compression processing for an orientation ( θ, can be implemented by e pulse compression processing using a unified matched filter h (t,0,0). However, e resulted range t is being circularly advanced ( t ϑ > 0 ) or delayed ( t ϑ < 0 ), i.e. e range of target is being shifted if ϑ 0 as e result of using e unified filter h (t,0,0). This greatly saves e computational time for e pulse compression processing. Instead of using orientation-dependent matched filters to repeat e pulse compression, a single unified pulse compression is sufficient. After beamforming determines e bearing of e target, e range shift amount tϑ can also be determined, and e correct range is measured accordingly. The output of pulse compression by e use of e unified matched filter for e k channel is, 2π y k ( = Aexp j k t tϑ ) λ k =, L,0, L, 0 (14) Finally, to line up e signals from all N + 1 channels to focus at ( r, θ, we need to use a spatial steering vector of,
vs ( θ, = 1 [ 1 N + 1 exp( j2πdϑ / λ0 ) L exp( j2πndϑ / λ )] H 0 (15) where e superscript H denotes complex conjugate transpose. Finally e pulse-compressed, aperture-synesised and beamformed output is, y out H ( t, θ, = v ( θ, y( (16) s where y ( t ) = [ y ] T N / 2( L y0( L yn / 2( and e superscript T denote transpose. The spatial processing given by (16) is exactly e same as e spatial processing of e CPA data and can be implemented by e fast Fourier transform (FFT). The difference is at for e CPA radar, if transmission steers at e boresight direction, e illumination off e boresight direction is only by e array s sidelobes. Therefore, in general e CPA radar needs to electronically scan its beam for search and detection in different directions. However, ere is no need for e FDA radar to do so, and e steering focusing is achieved by processing e same received data. The intensity of e received signal is independent of e signal arrival ( θ,, under e assumption of e antenna element directional gain g ( θ, = 1. In reality, e region is limited by e antenna element directional gain g ( θ,, e same as e CPA array s electronic scan region. Since e FDA is a narrowband system, for e Doppler processing, e same temporal steering vectors can be used as if e radar is a narrowband CPA radar. Accordingly, e temporal steering vector of FDA is e same as of CPA. For a coherent processing interval (CPI) dataset, e associated temporal steering vector is, 1 v [ ( ) ( )] T t = 1 exp j2πf d / f PRF L exp j2πmf d / f PRF M + 1 (17) where f PRF / 2 fd f PRF / 2 is e target Doppler frequency to be searched, f PRF is e radar s pulse repetition frequency (PRF), M + 1 is e number pulses in a CPI, and fd = 2v a / λ0 where v a is e target s radial velocity. III. DETECTION AND IDENTIFICATION OF RANGE- AMBIGUOUS TARGETS For a radar operated in medium to high PRF modes, range can be ambiguous. This means at echoes from distant targets illuminated by previous pulses can be received by e current pulse repetition interval (PRI), resulting in e so-called rangeambiguous targets. In general, different PRFs have to be used to resolve is range-ambiguous issue. This complexes radar schedules, increases radar load and also increases e exposure of e radar. It is possible for FDA radar to identify e true range of targets wiout using multiple PRFs. Since e transmission of FDA is non-coherent and channel frequencies are orogonal, e combined waveforms can be totally different once an initial random phase is coded to each frequency. The random phase coded pulse m may be defined as, s ( = m exp( j2π ( f + n f )( t mt ) + jϕ( n, m ) / 2 0 r ) T / 2 t T / 2 (18) where 0 ϕ( n, m) 2π is a pseudo-random phase. As an example, Figure 1 shows e cross-correlation of two FDA pulses compared to e auto-correlation of an FDA pulse. Signal level (db) 0-5 -10-15 -20-25 -30-35 -40-5 -4-3 -2-1 0 1 2 3 4 5 Time delay (µs) Figure 1: Autocorrelation of an FDA pulse and e cross-correlation of two FDA pulses. The waveforms are random phase coded as given by (18) ( N +1 = 41, T = 10 µs and f = 100 khz). When e received range samples undergo pulse compression, e target illuminated by e pulse matching wi e pulse used in e pulse compression receives a coherent processing gain of 10log10 ( N + 1) db whereas e target illuminated by oer pulse fails to receive is coherent processing gain. Likewise, e Doppler processing provides a coherent gains of 10log10 ( M + 1) db. Therefore, for e same target, its output SINR can be 10log10 [( N + 1)( M + 1)] db higher if e corresponding pulses are used in e pulse compression and Doppler processing compared to e use of noncorresponding pulses. In oer words, for e same received data snapshot, different pulses should be used in pulse compression, so at all targets in eier unambiguous range or ambiguous range can be detected and eir true range determined. The above statement is slightly over claimed. Realistically, two noise-like pulses wi e same carrier frequency and same bandwid should be treated as interference signals mutually. The cross-correlation, in terms of peak-to-sidelobe level (PSL), has a lower boundary of 10log10 ( N + 1) db which represents e best achievable isolation [15]. In general e peak of e cross-correlation is often higher an is eoretically lower bound (e lower bound is e mean value). The eoretically achievable isolation for e above example is 10log10 41 = 16.1 db (e mean) and e actual isolation achieved is about 12 db (e highest peak). However, is does demonstrate e use of e random-phase coded FDA waveforms to resolve e range-ambiguous targets and determine e true range of targets. The autocorrelation represents e result of pulse compression wi e matched
pulse whilst e cross-correlation represents e result of pulse compression wi an unmatched pulse. In general, e FDA radar needs to refocus at ( r 0 + krmax, θ, raer an ( r 0, θ, for detecting a target located at ( r 0 + krmax, θ,, where r max = ctr / 2, T r PRI, i.e. T r = 1/ f PRF and k =1,2, L is e order of range-ambiguity. However, if mod( n f, f PRF ) = 0, for n = 0, ± 1,..., ±, The focuses at ( r 0 + krmax, θ, and ( r 0, θ, are e same. So ere is no need of extra focus. IV. SIMULATION To demonstrate, a scenario is simulated using e geometric optics. Since e Doppler processing and clutter suppression/cancellation are well known, ey are not included in e simulation, and only single-pulse data were generated. The ermal white Gaussian noise is e only undesired signal considered in e simulation. Parameters used in e simulation are given in Table 1. The meod of geometric optics was used in e simulation, i.e., waveforms transmitted by channels were treated separately according to channels positions and targets locations. The returned waveforms were an combined before adding e ermal noise. Table 1: FDA and target parameters used in e simulation FDA Parameters Carrier frequency, f 0 1.25 GHz (L-band) Number of channels, N + 1 41 Channel interval, d Half waveleng Pulse wid, T 10 µs Frequency diversity, f 100 khz (linear) wi a random initial phase PRF 1500 Hz Target parameters Target 1 49 km range, 30 deg azimu Target 2 51 km range, 0 deg azimu Target 3 150 km range, -30 deg azimu The data collection geometry is depicted in Figure 2. According to e FDA parameters, e maximum unambiguous range is 100 km. Therefore, Targets 1 and 2 are rangeunambiguous targets whereas Target 3 is a range-ambiguous target, and its apparent range is 50 km. The simulated raw data are shown in Figure 3 which is a noise dominant image as e SNR in e raw data for all ree targets were assumed to be 10 db. After pulse compression using e range-unambiguous matched filter (e current pulse) and beamforming, e processed data is shown in Figure 4. It can be seen at bo e range-unambiguous targets are detected: Target 1 ( r = 48. 625 km and cos θ sinφ = 0. 5 ) and Target 2 ( r = 51 km and cos θ sinφ = 0 ). Since Target 1 is off e boresight direction, its range has been shifted. The associated range shift in time, according to (10) is 2.5 µs, i.e. 0.375 km in range. Hence we determine e true range of Target 1 to be 49 km, consistent wi e given simulation conditions. Figure 3: The simulated single-pulse received raw data in FDA channels prior to pulse compression (a noise-dominated image, -10 db SNR in raw data). Figure 4: After pulse compression by e range-unambiguous matched filter (e current pulse) and beamforming, Targets 1 and 2 are focused and detected, and Target 3 becomes an interference signal. 49, = 0, = 30 51, = 0, = 0 FDA Figure 2: Geometry of FDA data collection. 150, = 0, = 30 Figure 5: After pulse compression by e range-ambiguous matched filter (e previous pulse) and beamforming: Target 3 is focused and detected, and Targets 1 and 2 become interference signals. In order to search out any targets in ambiguous range, e raw dataset was reprocessed using e previous pulse as e
matched filter, and e result is shown in Figure 5. It can be seen at in is case Target 3 ( r = 50.375 km and cosθ sinφ = 0.5 ) is focused and detected whereas Targets 1 and 2 become interference signals. Since Target 3 is off e boresight direction, e associated range shift in time, according to (10) is -2.5 µs, i.e. -0.375 km in range. Therefore, e apparent range of Target 3 should be 50 km. Because PRF is 1500 Hz, e true range of Target 3 is 150 km, consistent wi e given simulation parameters. The isolation among pulses is limited by e lower boundary of e cross-correlation of e pulses as discussed. If e isolation level is not satisfied to e system design, oer techniques or algorims need to be considered. One technique is using e frequency-division multiple access (FDMA) pulse by pulse. Anoer possible solution is to use signal-removal algorims. Once a target (especially ose strong ones) is detected, its signal is synesised and removed from e original raw data to eliminate its interference to weak targets. A single CPA pulse data can only produce a 1D range profile and a CPI burst of pulse train data can produce a 2D range-doppler map. However, for e FDA data, a single pulse data can produce multiple 2D range-orientation maps (range-unambiguous, first-order range-ambiguous, etc) and a CPI burst of pulse train data can produce multiple 3D rangeorientation-doppler maps. The concept of space-time adaptive processing (STAP) can be adapted to processing e FDA data, which is out of e scope of is paper. V. CONCLUSIONS The focus of is paper is more at details of how e FDA data should be systematically and efficiently processed, which was largely missing in e previous publications. The detection and identification of range-ambiguous targets by e FDA radar wiout using multiple PRFs is demonstrated. The non-coherent emission enables e FDA radar detection wiout using electronic scan at greatly simplifies e radar design and operation. Hence, e FDA has a great potential to be implemented in simple wide-area search radars and navigation radars wiout compromising its performance. Once e FDA CW waveform are coded wi an initial random phase as defined by (18), e pulse becomes a noiselike one wi varying amplitude and phase. It makes e FDA radar low probability of interception and difficult to be jammed. Because of energy spread, e energy emitted onto a target is 10log10 ( N + 1) db less an at of CPA radar. The FDA radar may be not quite suitable for e long-range detection. Therefore, in comparison wi e conventional phased-array (CPA) radar, e detection area is fan-area like for e former and pencil-beam like for e latter making two radars (or radar operation modes) complementary. REFERENCES [1] P. Antonik, M. C. Wicks, H. D. Griffis and C. J. 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