Application of vector analysis on study of illuminated area and Doppler characteristics of airborne pulse radar

Transcription

1 Wang and Yang EURASIP Journal on Advanes in Signal Proessing 14, 14:114 RESEARCH Open Aess Appliation of vetor analysis on study of illuminated area and Doppler harateristis of airborne pulse radar Haijiang Wang 1,* and Ling Yang 1 Abstrat In this paper, the appliation of vetor analysis tool in the illuminated area and the Doppler frequeny distribution researh for the airborne pulse radar is studied. An important feature of vetor analysis is that it an losely ombine the geometri ideas with algebrai alulations. Through oordinate transform, the relationship between the frame of radar antenna and the ground, under airraft motion attitude, is derived. Under the time-spae analysis, the overlap area between the footprint of radar beam and the pulse-illuminated zone is obtained. Furthermore, the Doppler frequeny expression is suessfully dedued. In addition, the Doppler frequeny distribution is plotted finally. Using the time-spae analysis results, some important parameters of a speified airborne radar system are obtained. Simultaneously, the results are applied to orret the phase error brought by attitude hange in airborne syntheti aperture radar (SAR) imaging. Keywords: Vetor analysis; Illuminated area; Doppler harateristis 1 Introdution In a radar system, it is neessary to desribe the time-spae relation as learly as possible. The radar geometry study is very important for the dedution of radar eho signal. For example, in airborne radar, if it is required to present the radar eho equation and to obtain the illuminated area with Doppler harateristis, the geometry relations among the airplane, the antenna, and the ground must be analyzed intensively. Researhers have done some studies on the illumination harateristis of radar. In [1], Koraz and Genderen measured the footprint of stepped frequeny ontinuous wave (CW) radar using a probe antenna, but little geometry and time-spae analysis are applied. In [], for airborne radar, Green et al. analyzed the attitude hanges and their influenes on the eho's Doppler spetrum but the timespae analysis was rarely applied and the illuminated area was not obtained. In [], Liu and Arone alulated the radar pulse's wave field on the ground with pseudospetral * Correspondene: whj@uit.edu.n 1 College of Eletroni Engineering, Chengdu University of Information Tehnology, Chengdu, Sihuan 65, China CMA. Key Laboratory of Atmospheri Sounding, Chengdu, Sihuan 65, China time domain method. This artile is written on the hypothesis that the antenna's shape is known and its position is fixed, but there is no further disussion on the radar platform's motion and attitude hanges. The onept and alulation of a vetor were detailed in textbooks sine early times [4-6], and vetor analysis has been used in spae geometry, omputer graphis, and field analysis [7-]. But it has not been used suffiiently in the study of airborne radar geometry. Many researhers just refer to the vetor method for show in the beginning and at the end of disussions, but during the proess, they often rely on figure observation rather than vetor alulation to derive equations [11-1]. In fat, making the full advantage of vetor analysis in geometry attituderelated problems an make geometry onepts learer and improve researh effiieny notieably. In addition, vetorial expressions an be transferred to MATLAB statements diretly, whih are onvenient for the vetor alulation and figure plotting. In this paper, the vetor analysis is adopted to study the geometry onfiguration of airborne pulse radar. Based on the vetor derivation, some important results about the time-spae relations of the airborne radar signal are 14 Wang and Yang; liensee Springer. This is an Open Aess artile distributed under the terms of the Creative Commons Attribution Liense ( whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly redited.

2 Wang and Yang EURASIP Journal on Advanes in Signal Proessing 14, 14:114 Page of 9 obtained. These results will be applied to solve some problems in airborne radar appliation. The paper is organized as follows. In Setion, under the hypothesis of the airplane's attitude, the relationship between antenna oordinate frame and ground oordinate frame is dedued. Next, in Setion, through the analysis of transmitted and reeived signal paths of the radar, aording to the vetor differential equation, a omplete expression of the radar eho is obtained. Furthermore, the eho area of the airborne pulse radar and the eho's Doppler distribution on the ground are alulated in Setion 4. In addition, the geometry harateristis of airborne radar signal transmission are applied to obtain parameters of airborne radar, and those features are adopted for the phase error orretion in airborne syntheti aperture radar (SAR) imaging in Setion 5. Lastly, the onlusion is presented in Setion 6. The relationship of the oordinate frames In this paper, it is assumed that ground frame is formed by unit vetors, Ŷ, and. In the oordinate, and Ŷ are parallel to the ground level and is perpendiular to it. These three unit vetors form a right-handed system as follows: ¼ ^Y : ð1þ It is designed that the airraft frame is formed by unit vetors ^F (along the fuselage diretion), Ŵ (along the wing diretion), and ^T (along the top diretion), whih form another right-handed helix system. So, they an be expressed as ^T ¼ ^F ^W : ðþ It is assumed the initial attitude of the airraft is ^F 1 4 ^W 5 ¼ ^Y 5; ðþ ^T 1 and the angle of the airraft whih rotates along the top axis ^T is supposed to be ψ, whih is known as the azimuth angle. Through the vetor alulation, the new attitude an be obtained as follows: ^F osψ sinψ ^F 4 ^W 5 ¼ 4 sinψ osψ 54 ^W 5 ¼ PðψÞ4 ^Y 5: ^T 1 ^T ð4þ On the above basis, it is supposed that the airraft rotates along the new wing axis Ŵ by an angle α, whih is alled the elevation angle, and the attitude of the airraft transfers to ^F osα sinα ^F 4 ^W 5 ¼ ^W 5 ¼ AðαÞPðψÞ4 ^Y 5: ^T sinα osα ^T ð5þ On the basis of the above two rotations, suppose that the airraft rotates one more along the fuselage axis by an angle β, whih is alled the rolling angle, and the attitude is obtained as ^F 4 ^W 5 ¼ ^T 1 ^F 4 osβ sinβ 54 ^W sinβ osβ ^T 5 ¼ BðβÞAðαÞPðψÞ 4 ^Y 5: ð6þ For the angles above, the range of elevation angle α is within [π) and the ranges of azimuth angle ψ and rolling angle β are both within [π). These three angles determine all possible attitudes of the airraft. The antenna frame an be derived from the airraft frame as 1 ^x 4 ^y 5 os γ þ π sin γ þ π ^F ¼ 6 4 ^z sin γ þ π os γ þ π 74 5 ^W ^T 1 ¼ ¼ QðγÞ4 sinγ osγ osγ sinγ ^F ^W ^T 5 ^F ^W ^T 5; ð7þ where γ is the depression angle of the antenna's main lobe, ^z represents the diretion of the main lobe's enter line, ^x ¼ ^F is along the fuselage diretion, and ^y ¼ ^z ^x. So, the antenna frame is obtained by rotating the airraft frame at the angle of γ þ π along the fuselage diretion. From the above disussion, the relationship between antenna frame and ground frame an be expressed as ^x 4 ^y ^z 5 ¼ QðγÞBðβÞAðαÞPðψÞ ¼ Πα; ð β; γ; ψ It is dedued as follows: 4 ^Y 5 Þ4 ^Y 5: ð8þ QðÞ ¼ BðÞ ¼ AðÞ ¼ PðÞ ¼ I 1 ¼ 4 1 5: ð9þ 1 Through the algebra alulation, it an be obtained as follows:

3 Wang and Yang EURASIP Journal on Advanes in Signal Proessing 14, 14:114 Page of QðγÞBðβÞ ¼ 4 sinðβ γþ osðβ γþ5 osðβ γþ sinðβ γþ ¼ Cðβ γþ: ðþ In order to keep the generality, the azimuth angle an be assumed as ψ =. Then, the final result an be obtained as Πα; ð β; γ; Þ ¼ QðγÞBðβÞAðαÞ ¼ Cðβ γþaðαþ osα sinα ¼ 4 osðβ γþsinα sinðβ γþ osðβ γþosα 5: sinðβ γþsinα osðβ γþ sinðβ γþosα ð11þ Expression of the airborne pulse radar eho signals Before onsidering the expression of the airborne pulse radar eho signal, the formula for the differential of a vetor's magnitude is neessary to dedue. For any vetor a, Figure 1 Proess of transmission and reeption of a ray. A m ðþ¼ t 1 t τ; elsewhere: ð19þ jaj ¼ a a: ð1þ Conduting the differential operation on both sides of the above equation, it is transferred to So, jajda j j ¼ a da: ð1þ da j j ¼ a da ¼ ^a da: jaj ð14þ This result is very important and will be used later. The transmission and reeption of a ray during the flight of the airraft are shown in Figure 1. Assume that the position vetor of the airraft is as follows: Rt ðþ¼r þ tv; ð15þ where v is the veloity vetor. Assume that a ray in the radar beam is transmitted at the zero time and reahes the ground point and the length of the ray is The time delay between the reeived signal and the transmitted signal is t d ðþ¼ t l 1 þ l ¼ 1 ½j r s RðÞjþ jr s Rt ðþj: ðþ For the radar's motion, l 1, l,andt d are related to t. Aording to Equation 11, the differential oeffiient of t d is dt d dt ¼ 1 ½r s RðÞ jr s RðÞj þ r ½ s RðÞ t v: jr s Rt ðþj ð1þ It is obvious that the time delay at the zero time is obtained by l 1 ¼ jr j ¼ jr s RðÞj: ð16þ If the refleted signal arrived to the radar at time t, then the distane that the refleted signal traverses is l ¼ jrt ðþj ¼ jr s Rt ðþj: ð17þ Suppose the transmitted pulse signal of the airborne radar is where St ðþ¼a m ðþe t jω t ; ð18þ Figure Representation in antenna oordinate frame of a ray vetor to the target.

4 Wang and Yang EURASIP Journal on Advanes in Signal Proessing 14, 14:114 Page 4 of 9 Figure Illuminated area and Doppler frequeny distribution when α =,β =,γ = 45, σ x = 5, σ y = 15, and H = 5. t d ðþ ¼ r ; dt d dt ¼ ^r v ¼ v r : t ¼ So, the one-order expansion of t d an be obtained as t d ðþ t r v r t: ðþ ðþ The seond item in the above equation is muh smaller than the first item, so in the amplitude expression, it an be negleted as A m ðt t d Þ ¼ A m t r : ð4þ But in the phase expression, the seond item in Equation annot be negleted, beause after being multiplied by ω, it an also reah the magnitude order π. So, the following expression of the airborne pulse radar eho signal an be obtained as st ðþ¼ Σ A m t r G ðϕ; θþλðr s Þexp jω 1 þ vr t r d Σ: r ð5þ Figure 4 Illuminated area and Doppler frequeny distribution when α =,β = 5, γ = 45, σ x = 5, σ y = 15, and H = 5.

5 Wang and Yang EURASIP Journal on Advanes in Signal Proessing 14, 14:114 Page 5 of 9 Figure 5 Illuminated area and Doppler frequeny distribution when α = 5, β =,γ = 45, σ x = 5, σ y = 15, and H = 5. 4 The airborne pulse radar-illuminated area and the Doppler harateristi distribution Expressing with the radar's antenna frame, the vetor ^r is ^r ¼ sinθ osϕ ^x þ sinθ sinϕ ^y þ osθ ^z; ð6þ whih is shown in Figure as Beause of r s = R() + r and r s ¼, so it an be dedued as ½RðÞþr ¼. Suppose RðÞ ¼ H, then r ¼ r^r ¼ RðÞ ¼ H. Based on Equation 6, r an be obtained as r ¼ H ^r H ¼ sinθ osϕ sinα sinθ sinϕ osα osðβ γþþ osθ osα sinðβ γþ : ð7þ Simultaneously, it is easy to determine the oordinate position of the ground refletion point as Figure 6 Illuminated area and Doppler frequeny distribution when α = 5, β = 5, γ = 45, σ x = 5, σ y = 15, and H = 5.

6 Wang and Yang EURASIP Journal on Advanes in Signal Proessing 14, 14:114 Page 6 of μs 77.1μs 67.μs Figure 7 Illuminated area and Doppler distribution of an airborne pulse radar with known parameters. X ¼ r s ¼ r ¼ r ^r H sinθ osϕ osα þ sinθ sinϕ sinα os β γ ¼ ½ ð Þ osθ sinα sin ð β γ Þ ; sinθ osϕ sinα sinθ sinϕ osα osðβ γþþ osθ osα sinðβ γþ Y ¼ r s ^Y ¼ r ^Y ¼ r ^r ^Y H½sinθ sinϕ sinðβ γþþ osθ osðβ γþ ¼ sinθ osϕ sinα sinθ sinϕ osα osðβ γþþ osθ osα sinðβ γþ : ð8þ For the antenna's beam, the Gaussian model an be used as "!# Gðϕ; θþ ¼ G exp θ os ϕ σ þ sin ϕ x σ : ð9þ y Define the half-power area as Gðϕ; θþ ¼ p G ffiffi : From Equation 9, it an be obtained as "!# exp θ os ϕ σ þ sin ϕ x σ ¼ p 1 ffiffi y So, θ os ϕ σ x Express θ by ϕ as ðþ ð1þ! þ sin ϕ σ ¼ log y : ðþ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi θ ¼ σ x σ y log : σ y os ϕ þ σ x sin ϕ ðþ The above equation indiates the relationship between θ and ϕ, and the beam edge an be obtained from the equation. Assume that ϕ varies within [, π) and substitute Equation into Equation 8, and the beam outline on the ground an be obtained. The area inside the outline an be expressed as Gðϕ; θþ p G ffiffiffi : ð4þ Aording to Equation 19, A m t r is not zero only under the following ondition: t r So, τ: ð5þ Table 1 Calulated parameters of the airborne pulse radar speified by Table Calulated parameters Nearest range Farthest range Earliest reeption time Latest reeption time Doppler range v Values μs 88.6 μs λ :986

7 Wang and Yang EURASIP Journal on Advanes in Signal Proessing 14, 14:114 Page 7 of 9 Table System parameters of an airborne pulse radar System parameters Flight height Radar depression angle Beam width in azimuth diretion Beam width in range diretion Values μs Pulse duration t τ ð Þ r t : ð6þ p Let ρ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p X þ Y ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi r H, so the above equation an be rewritten as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t τ ð Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H t H ρ : ð7þ Equation 4 desribes an area like a long ellipse, and Equation (7) desribes a ring area. These two areas overlap to an area, whih are shown in Figures, 4, 5, 6 and denoted by blak olor. There is a Doppler frequeny in the eho signal in Equation 5, whih is expressed as f d ¼ ωv r π ¼ 4π fv r π ¼ v r λ ¼ v ð λ ^r ^x Þ: ð8þ In Figures, 4, 5 and 6, ^r ^x, named as Doppler distribution, is plotted. 5 Appliation of the above method and results In airborne pulse radar, when the system parameters, suh as the platform height, radar beam width, and pulse duration, are determined, some other parameters an be alulated aording to the above analysis method. Assume that the parameters are as follows: By the above vetor analysis method, the illuminated area and Doppler distribution are plotted, as shown in Figure 7. Combining the vetor alulation tool and figures, some important dedutions, suh as the reeption parameters, an be obtained. Aording to Equations 8 and, the outline of the beam footprint an be obtained, so its nearest and farthest points an be aquired. On this basis, the reeption time interval an be obtained aurately. Furthermore, aording to Equation 8, the Doppler frequeny of eah point on the outline an be alulated, so the Doppler range of the beam footprint area an be obtained. In Table 1, some alulated parameters are listed for the radar platform, whih is speified by thedataintable. The above disussion indiates that by vetor analysis, the geometry relationship of the airborne pulse radar is lear and some important parameters an be alulated. In airborne SAR imaging, an invariable veloity and an unhanging attitude for the airraft are expeted for ensuring the image quality. But in the real experiment environments, inevitably, there are some attitude disturbanes for various fators, suh as air turbulene and instable steering. These attitude disturbanes will hange the illuminated area and its Doppler frequeny distribution. And orrespondingly, the eho signal will lose phase ohereny with the referene signal and the (a) SAR signal after range ompression (b) SAR signal after azimuth ompression Figure 8 Imaging results of point targets without attitude hange.

8 Wang and Yang EURASIP Journal on Advanes in Signal Proessing 14, 14:114 Page 8 of 9 (a) SAR signal after range ompression (b) SAR signal after azimuth ompression - Figure 9 Imaging results of point targets with attitude hange fousing performane of the SAR will be degraded. To solve this problem, several motion ompensation methods and algorithms were proposed [14-17]. Most of these methods fous on the eho signal itself and rely on the estimation auray of Doppler entroid. In fat, the above researh on illuminated area and Doppler frequeny distribution an be applied to arry on phase error orretion between the eho signals and the referene signals aused by radar platform motion. In order to illustrate the influene of attitude hange to SAR imaging, the imaging simulations are arried out for three point targets. Firstly, the ase of no attitude hange is onduted, and the imaging results are shown in Figure 8. Then, assume that the airraft has an attitude hange relative to the intended flight path and the pithing angle and the rolling angle are both set at.1. Besides, a zero-mean white Gaussian noise with a variane of.5 is added to the angles. The simulated imaging resultsareshowninfigure9. (a) SAR signal after range ompression (b) SAR signal after azimuth ompression Figure Imaging results of point targets after motion orretion.

9 Wang and Yang EURASIP Journal on Advanes in Signal Proessing 14, 14:114 Page 9 of 9 From Figure 9, it is shown that there is a defousing reord in the imaging proess when the radar platform's attitude keeps hanging. The real-time attitude is assumed to be reorded, but there is a zero-mean white Gaussian noise with a variane of.5 ompared to the real value of the parameters, suh as the pithing and rolling angle. In order to orret the imaging deviation, firstly, with the reorded attitude parameters, Doppler distribution is alulated by the vetor analysis method proposed in Setion 4. And then, aording to the Doppler frequeny and the distane to the radar antenna of eah ground point, the phase of the referene signal is orreted. The imaging results with the orreted referene signal are shown in Figure. Comparing Figure to Figure 9, it is notieable that the fousing performane is improved signifiantly after phase orretion based on vetor analysis. 6 Conlusions In airborne pulse radar, some problems, espeially the tasks involving geometry relationships, are diffiult to solve by the salar tool. However, as disussed in the proposed approah, the parameter results an be derived expediently and naturally by the vetor method. From the disussion in the above setions, it an be seen that with the vetor method, the time-spae relationships, suh as the expression of the eho signals, the illuminated area, and the Doppler harateristi distribution are obtained intuitively. Furthermore, the analysis results with the vetor method an be applied in radar parameter alulation and SAR imaging. So, vetor analysis is effetive for airborne pulse radar parameter setting and solving and airborne pulse radar signal proessing tasks. 11. A Aprile, A Mauri, D Pastina, Real Time Rotational Motion Compensation Algorithm for Fousing Spot-SAR/ISAR Images in Case of Variable Rotation- Rate, European Radar Conferene(EURAD) (4) 1. J Balke, SAR Image Formation for Forward-Looking Radar Reeivers in Bistati Geometry by Airborne Illumination, IEEE Radar Conferene, 8, pp AD Lazarov, TP Kostadinov, SAR Signal Modeling and Imaging of a Moving Target, 11 IEEE CIE International Conferene on Radar, Vol. 1, 1, pp P Guione, C Cafforio, Motion Compensation Proessing of Airborne SAR Data, IGARSS'8 Vol., 8, pp Z Ding, L Liu, T Zeng, W Yang, T Long, Improved motion ompensation approah for squint airborne SAR. IEEE Trans. Geosi. Remote. Sens. 51(8), (1) 16. YJ Zhang, KY Han, YP Wang, WX Tan, W Hong, Study on Motion Compensation for Airborne Forward Looking Array SAR by Time Division Multiplexing Reeiving, Syntheti Asia-Paifi Conferene on Aperture Radar (APSAR)1, pp M Arii, Effiient motion ompensation of a moving objet on SAR imagery based on veloity orrelation funtion. IEEE Trans. Geosi. Remote Sens. 5(), (14) doi:.1186/ Cite this artile as: Wang and Yang: Appliation of vetor analysis on study of illuminated area and Doppler harateristis of airborne pulse radar. EURASIP Journal on Advanes in Signal Proessing 14 14:114. Reeived: 8 February 14 Aepted: 11 June 14 Published: 19 July 14 Referenes 1. E Koraz, P Van Genderen, Antenna Footprint Measurements of Stepped Frequeny CW Radar on the Air/Ground Interfae, IEE Antenna Measurements and SAR (AMS'4), 4, pp JW Green, TB Hale, MA Temple, JT Bukreis, Inorporating Pulse-to-Pulse Motion Effets into Side-Looking Array Radar Models, Fourth IEEE Workshop on Sensor Array and Multihannel Proessing, 6, pp L Liu, SA Arone, Near-Surfae Radar Pulse Propagation in Complex Terrain Environments: Preliminary Results, th International Conferene on Ground Penetrating Radar (4), pp JG Coffin, Vetor Analysis: an Introdution to Vetor-Methods and Their Various Appliations to Physis and Mathematis (J. Wiley & sons, New York, 1911) 5. B Hague, An Introdution to Vetor Analysis (Methuen & Co. Ltd, London, 1951) 6. N Kemmer, Vetor Analysis: A physiist's Guide to the Mathematis of Fields in Three Dimensions (Cambridge University Press, London, 1977) 7. AB Sproul, Derivation of the solar geometri relationships using vetor analysis. Renew. Energy (7), (7) 8. JR Miller, Vetor geometry for omputer graphis. IEEE Comput. Graph. Appl. 19(), 66 7 (1999) 9. JR Miller, Appliations of vetor geometry for robustness and speed. IEEE Comput. Graph. Appl. 19(4), 68 7 (1999). BM Notaros, Geometrial approah to vetor analysis in eletromagnetis eduation. IEEE Trans. Edu. 56(), 6 45 (1) Submit your manusript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publiation on aeptane 7 Open aess: artiles freely available online 7 High visibility within the field 7 Retaining the opyright to your artile Submit your next manusript at 7 springeropen.om