SAR System & Signals

SAR System and Signals Part 2 EE880 Synthetic Aperture Radar M. A. Saville, PhD, PE Summer, 2012 EE880 SAR System & Signals Part 2 Lesson Overview • • • • Imaging radar requirements Array Basics SAR signal modeling Summary EE880 SAR System & Signals Part 2 2 Imaging Radar Requirements • • • • • Resolve scatterers in 1D,2D,3D Construct geospatial image Estimate reflectivity function Estimate RCS of scene scatterers Estimate cross-section coefficient of clutter • Image one uncompressed range cell or voxel (3D case) • Achieve specified resolution in 1, 2 or 3D • Perform above within time and computational constraints EE880 SAR System & Signals Part 2 3 Ideal 2D Radar Imaging Collection • Shown: ground plane imaging • Down-range resolution set by HRR waveform, i.e. bandwidth • Cross-range resolution set by narrow antenna beam • Each echo resolves both dimensions EE880 SAR System & Signals Part 2 4 Realistic Down-range Reconstruction Ideal down-range target profile rect(𝑡) (infinite bandwidth) Time Domain ∆𝑡 Spectral Domain -2000 -1500 -1000 Ideal receiver filtering rect(𝑓) (finite bandwidth) Lost energy -500 0 Time Domain ∆𝑡 500 1000 1500 2000 Profile distortion & spreading Reconstructed down-range target profile is IDFT of windowed rect(𝑡) Note duality and reciprocity in Fourier Transforms. If we start with ideal S, transform to s, window by applying a range-gate and inverse transform, we still observe spread in sw EE880 SAR System & Signals Part 2 5 Down-range Digital Signal Processing • Time/range domain • Frequency domain – finite signal bandwidth B << W – sampling period ΔT – record length T 𝑇 Δ𝑇 = 2𝑊 = 𝑓𝑠 D 1 𝑓𝑠 𝑐Δ𝑇 𝑐 𝑐 Δ𝑅 = = = 2 2𝑓𝑠 4𝑊 – Unambiguous spectrum 𝑊 = fs/2 – spectral resolution Δf D-1 𝑐𝑇 𝑐 𝑅= = 2 2∆𝑓 Δ𝑓 = 𝑐 Δ𝑓 = 2𝑅 1 𝑇 1 2𝑊 = ∆𝑇 range results from scaling time EE880 SAR System & Signals Part 2 6 Realistic Cross-range Reconstruction • Down-range resolved • Cross-range not resolved because of antenna beam • Solution: apply discrete-time Fourier principles to form narrow antenna beam EE880 SAR System & Signals Part 2 7 Cross-range Coordinates End synthetic aperture Θ 𝑅0 1. Collection 4. Scene center reference Start synthetic aperture 2. Coordinate references 3. Synthetic aperture reference Ground plane 𝑅0 Θ EE880 SAR System & Signals Part 2 Slant plane 𝑅𝑐𝑟 = 𝑅0 sin Θ Cross range scene extent is set by beamwidth of real aperture 8 SAR Coordinate Reference • SAR coordinates are different from detection and tracking radar applications • Coordinates are referenced to the scene center • Synthetic aperture elements (spacing d and length L) are referenced to scene center in angular coordinates 𝛥𝜃 ← 𝛥𝜃 𝑑, 𝐿, 𝑅 , 𝛩 ← 𝛩 𝑑, 𝐿, 𝑅 • SAR is a receive array antenna Angle scene Angle scene Range radar EE880 SAR System & Signals Part 2 Radar centric Range radar Scene centric 9 Cross-range Digital Signal Processing • Array (angular) sampling: • Cross-range sampling – array defined in linear coordinates 𝑑, 𝐿 – array spacing 𝑑 ← Δ𝜃 – array length 𝐿 = 𝑄𝑑 ← Θ – conceptually: spatial samples 𝐿 = 𝑄𝑑 𝑑 Angles are scaled array length and spacing EE880 SAR System & Signals Part 2 – Unambiguous spectrum Θ = 𝜃3dB – cross-range extent 𝑌 ≈ RΘ – cross-range resolution Δ𝑌 = ℬ −1 Δ𝜃, Θ 𝑌 B B-1 Δ𝑌 𝑌 is based on arc-length, but resolution depends on the operator B and is subject of course 10 Antenna Array Basics • Array - collection of antenna elements • Each element is a single antenna • Typically, elements have identical radiation patterns • Isotropic elements used in analysis for convenience EE880 SAR System & Signals Part 2 AN/SPY-1A 11 Array Antenna (1/4) Isotropic transmit antenna Received P0 Power level (dB) P0 - 6 P0 - 12 P0 - 18 R0 R0 2R0 4R0 8R0 Observation angle ZL Receive antennas Note: Antenna observation is defined in angle coordinates because pattern is range-invariant EE880 SAR System & Signals Part 2 12 Array Antenna (2/4) Array of Q isotropic transmit elements 𝐄2 1 𝑃≈ = 𝑍𝐿 𝑍𝐿 Spherical observation surface 2 𝑄 𝐄𝑞 = 𝐺𝑃0 𝑞=1 ZL 𝐄 = 𝐞𝐸 𝑥, 𝑦, 𝑧, 𝑡 𝐸∈ℂ Electric fields combine in a constructive or deconstructive manner at different points on the observation surface EE880 SAR System & Signals Part 2 13 Array Antenna (3/4) Radiation pattern of array of isotropic elements GP0 Received GP0 - 6 Power level (dB) GP0 - 12 GP0 - 18 G R0 2R0 4R0 8R0 Observation angle ZL 𝐹 𝑅 cos 𝜃 , 𝑅 sin 𝜃 = 𝑒 𝑄−1 −𝑗𝑘𝑅 𝑞=0 𝐼𝑞 𝑒 𝑗𝑞Δ𝜑 Δ𝜑 𝜃 = 𝑘𝑑 sin 𝜃 𝐹 𝜃 = 𝐼0 𝑒 𝑗 𝑄−1 Δ𝜑 𝜃 2 𝑄Δ𝜑 𝜃 2 Δ𝜑 𝜃 sin 2 sin Null-to-null beamwidth 𝜃𝑁𝑁 ≈ 𝜃 𝜃 𝑒 −𝑗𝑘𝑅 2𝜆 𝐿 Half-power beamwidth 𝜃3dB ≈ 0.866𝜆 𝐿 Note: transmit array radiation pattern is the same as the receive array pattern. EE880 SAR System & Signals Part 2 14 Array Antenna (4/4) • Fields observed far from array • Array pattern looks like I/DFT of rect 𝜃 • Differential phase 𝛼 on elements steers array 𝑅≫𝐿 𝐿 𝜃+𝛼 𝐿 = 𝑄−1 𝑑 𝐹 𝜃 = 𝑄−1 𝑞=0 𝐼0 𝑒 𝑗𝑞Δ𝜑 𝜃 Δ𝜑 𝜃 = 𝑘𝑑 sin 𝜃 + 𝛼 𝐹 𝜃 = planar wave fronts 𝑄−1 𝜑 𝜃 𝑗 2 𝐼0 𝑒 EE880 SAR System & Signals Part 2 𝑄𝜑 𝜃 2 𝜑 𝜃 sin 2 sin Phase shift across dimension of array causes angular shift (translation) to angle 𝛼, i.e. property of DFT. 15 Synthetic Array • Synthetic aperture is a receive aperture • Fields caused by scatterers (targets, clutter) • Differential angle 𝛼 causes differential phase 𝑅≫𝐿 𝐿 𝜃+𝛼 𝐿 = 𝑄−1 𝑑 𝐹 𝜃 = 𝑄−1 𝑞=0 𝐼0 𝑒 𝑗𝑞 Δ𝜑 𝜃 +𝛼 Δ𝜑 𝜃 = 2𝑘𝑑 sin 𝜃 𝐹 𝜃 = planar wave fronts 𝑄−1 𝜑 𝜃 𝑗 2 𝐼0 𝑒 𝜑 𝜃 = Δ𝜑 𝜃 + 𝛼 EE880 SAR System & Signals Part 2 𝑄𝜑 𝜃 2 𝜑 𝜃 sin 2 sin target Synthetic array formed by correcting phases caused by differential ranges. For linear array, DFT along array dimension results in cross-range compression, i.e. resolution. 16 Synthetic Aperture for Cross-range Resolution • SAR spatially samples along array dimension Δ𝜑 𝜃 = 2𝑘𝑑 sin 𝜃 differential phase shift across echoes Incremental path length Point target 𝑆 2𝑘 sin 𝜃 = DFT 𝑠 𝑞Δ𝜑 𝜃 𝑅 =DFT{𝑠[𝑞𝑑]}, 𝑞 = 1, ⋯ , 𝑄 sin −Θ 2 ≤ sin 𝜃 ≤ sin Θ 2 EE880 SAR System & Signals Part 2 Incremental Incremental position angle Cross-range resolution equals arc length 𝑅∆𝜃 17 SAR Signal Modeling Requirements • N-D images require N-D signal representation • Parameterize 2D signals (range,angle) with time • Time has two scales (PRI-𝑇𝑝 , and CPI-𝑀𝑇𝑝 ) • System design must support stable collection method and accurate coherent measurement CPI (inter-pulse sampling) 0 𝑇𝑝 slow time 𝜏 [ms] EE880 SAR System & Signals Part 2 PRI (intra-pulse sampling) 𝑀𝑇𝑝 0 Δ𝑇 𝑇𝑝 fast time 𝑡 [𝜇s] 18 SAR Radar System and Signals • SAR System differs from classic radar system • Collection method (transmit and store), receiver design to support imaging, signal processing TX Differences in CONOP sTX(t) s(t) SAR Simple view TX Ant gc(t) 𝑠TX Env RT , σ RG, σ0 RJ, sjam Differences in receiver RX r(t) yI(t) yQ(t) 𝑅, 𝜎 sRX(t) Differences in RSP 𝑠RX d[n] DB output 𝑅, 𝜎 t, Tp, Fp, τ EE880 SAR System & Signals Part 2 ℎ RX Ant RSP SYNC input DM ℎ−1 SAR is an inverse problem 19 Detailed SAR Modeling • Signal development from signal processing perspective • Math development from inverse problem perspective • Algorithm processing from linear systems perspective • Outline: – Coordinate systems – Transmit “signal” – Scatterer response – Received signal – Operator representation EE880 SAR System & Signals Part 2 20 Coordinate Systems (1/3) • Lower case letters: global coordinates • Primed lower case letters: local scene coordinates • Upper case letters: local antenna coordinates 𝐙 Antenna position 𝐫𝑎 = 𝐱𝑥𝒂 + 𝐲𝑦𝒂 +𝐳𝑧𝒂 𝐘 𝐫 = 𝐱𝑥 + 𝐲𝑦+𝐳𝑧 𝐫′ = 𝐱′𝑥′ + 𝐲′𝑦’+𝐳′𝑧′ 𝐑 = 𝐗𝑋 + 𝐘𝑌 + 𝐙𝑍 Scene center position 𝐫𝑔 = 𝐱𝑥𝒈 + 𝐲𝑦𝒈 +𝐳𝑧𝒈 𝐗 𝐫𝑎 𝐑 𝐳′ 𝐲′ 𝐳 𝐲 𝐫𝑔 𝐱′ 𝐱 EE880 SAR System & Signals Part 2 Scene center position relative to antenna position 𝐑 = 𝐑 𝟎 = 𝐫𝑔 − 𝐫𝑎 21 Coordinate Systems (2/3) • Local coordinates show variation in position Antenna position Scene center position 𝐫𝑎 + ∆𝐑 𝐫𝑔 + 𝐫′ Scene center position relative to antenna position 𝐑 = 𝐫𝑔 + 𝐫′ − 𝐫𝑎 + ∆𝐑 𝐙 𝐘 ∆𝐑 𝐫𝑎 𝐗 𝐑 • Typically assume ∆𝐑 = 0 • Scene defined by 𝐫′ 𝐳′ 𝐲′ 𝐳 𝐲 𝐱 EE880 SAR System & Signals Part 2 𝐫𝑔 𝐫′ 𝐱′ 𝐑 = 𝐫𝑔 + 𝐫′ − 𝐫𝑎 𝐑 = 𝐑 0 + 𝐫′ • Position parameterized with slow time 𝜏 22 Coordinate System (3/3) • Waveform definition in fast time coordinates 𝑡 • Reference to scene center -- not antenna • Signal has dependency on both 𝑡 and 𝜏 𝑠 𝑡, 𝜏 = ℝ𝕖 𝑝 𝑡 𝑒 𝑗𝜔𝑡 𝑒 −𝑗𝑘𝑅 complex envelope 𝑝 𝑡 = 𝑎 𝑡 cos 𝜓 𝑡 𝑅 𝜏 = 𝐑 𝜏 EE880 SAR System & Signals Part 2 electromagnetic wave behavior + 𝑗𝑎 𝑡 sin 𝜓 𝑡 = 𝐑 0 𝜏 + 𝐫′ 𝜏 𝜏 Can be phase, frequency, or amplitude encoded Assumes 𝐑 𝜏 ≪ 𝑐 Typically, 𝐫 ′ 𝜏 = 0 23 Transmit Signal • Wideband signal (LFM or stepped frequency) • Directional (line-of-sight to scene) 𝑠 𝑡, 𝐤 𝜏 = ℝ𝕖 𝑝 𝑡 𝑒 𝑗𝜔𝑐 𝑡 𝑒 −𝑗𝑘 𝐤∙𝐑 𝜏 𝐑0 𝜏 𝐤 𝜏 = 𝐑0 𝜏 = 𝑅0 𝜏 Cutaway view of a helix Traveling wave tube. (1) Electron gun; (2) RF input; (3) Magnets; (4) Attenuator; (5) Helix coil; (6) RF output; (7) Vacuum tube; (8) Collector. [wikipedia.com] 𝐙 𝐘 𝐗 𝐤 ∙ 𝐑 𝜏 = 𝐑 0 𝜏 ∙ 𝐑 𝜏 ≈ 𝑅0 𝜏 + ∆𝑅 𝜏 𝐑0 ∆𝑅 𝜏 ≈ 𝐤 𝜏 ∙ 𝐫′ Differential path length for arbitrary location in scene EE880 SAR System & Signals Part 2 Flight path 𝐳′ 𝐲′ 𝐑 𝐫′ Scene 𝐱′ 24 Scattered Signal • Clutter & targets, atmospheric and space loss L – In SAR, heterogeneous clutter = “target” – Approximate target signal model is simple sum of isotropic point scatterers: amplitude scaled, time, frequency/phase shifted 𝐴𝑞 𝑝 𝑡 − 2𝑇0 𝑒 𝑗𝜔𝑐 𝑠 𝑡, 𝜏 = 𝑡−2𝑇0 𝑒 −𝑗𝑘2 𝑅0 𝜏 +∆𝑅𝑞 𝜏 𝑞 • EM physics (with typical approximations) 𝑠 𝑡, 𝜏 = 𝐿𝑝 𝑡 − 2𝑇0 𝑒 𝑗𝜔𝑐 𝑡−2𝑇0 𝑒 −𝑗2𝑘𝑅0 𝜏 𝜌 𝐫 ′ = 𝛿 𝐫 ′ − 𝐫𝑞 𝜌 𝐫 ′ 𝑒 −𝑗2𝑘𝐤 𝜏 ∙𝐫 ′ 𝑑𝐫 ′ 𝑆𝑐𝑒𝑛𝑒 SAR approximates scene’s reflectivity function EE880 SAR System & Signals Part 2 25 Received Signal (1/2) • Signal comprises all echoes during synthetic aperture 𝑠 ′ 𝑡, 𝜏𝑚 ; 𝑚 = 1, ⋯ , 𝑀 • Inertial navigation system provides motion compensation timing, i.e., compensates for aperture deviation from flight path compensation 𝑠 𝑡, 𝜏𝑚 = 𝑠 ′ 𝑡, 𝜏𝑚 𝑒 𝑗2𝑘𝐤𝑚 ∙∆𝐑𝑚 𝐤 𝑚 = 𝐤 𝜏𝒎 = 𝐑 0 𝜏𝒎 = 𝐑 0,𝑚 Flight path 𝐙 𝐘 ∆𝐑 𝑚 • Slow-time recorded in angle coordinates 𝐗 𝐑 0,𝑚 𝐳′ 𝐲′ 𝐑𝑚 𝜙𝑚 , 𝜃𝑚 = 𝜙 𝜏𝒎 , 𝜃 𝜏𝒎 EE880 SAR System & Signals Part 2 Scene 𝐱′ 26 Received Signal (2/2) • Fast-time signals sampled according to signal bandwidth 𝑠 𝑡𝑛 , 𝜏𝑚 ; 𝑛 = 1, ⋯ , 𝑁 • Signals recorded either with absolute time or relative to initial or middle pulse in collection with respect to scene center • LFM signal recovered using deramp and deskew receiver -- relates sample time to instantaneous frequency EE880 SAR System & Signals Part 2 27 SAR Signal Processing Overview • Signal model after A/D 𝑆𝐼𝐹 𝑡𝑛 , 𝜏𝑚 = 𝑡𝑛 − 𝑚𝑇𝑝𝑟𝑖 − 2 𝑅0,𝑚 𝑐 rect 𝑇𝑝𝑢𝑙𝑠𝑒 𝑚=0 𝑀−1 × 𝑒 𝑗Φ 𝑡𝑛 ,𝜏𝑚 𝜌 𝐫 ′ 𝑒 −𝑗2𝑘𝐤 𝜏 ∙𝐫 ′ 𝑑𝐫 ′ 𝑆𝑐𝑒𝑛𝑒 • LFM transmit phase profile Φ 𝑡, 𝜏𝑚 = 2𝜋𝑓𝑐 𝑡 + 𝜋𝛾 𝑡 − 𝑚𝑇𝑝𝑟𝑖 2 Chirp 𝛾 [Hz/sec] • LFM receive (deramp) phase profile Φ 𝑡𝑛 , 𝜏𝑚 = −𝑗 4𝜋𝛾 𝑓𝑐 2𝑅0,𝑚 + 𝑡𝑛 − 𝑚𝑇𝑝𝑟𝑖 − 𝑐 𝛾 𝑐 EE880 SAR System & Signals Part 2 𝑅𝑞 − 𝑅0,𝑚 + 4𝜋𝛾 𝑅𝑞 − 𝑅0,𝑚 2 𝑐 2 28 Deramp & Deskew Receiver (1/5) • Recall LFM waveform with chirp 𝛾 Hz/sec [Sullivan, 7.2]: transmit 𝑡 − 𝑚𝑇𝑝𝑟𝑖 𝑗 2𝜋𝑓𝑐 𝑡+𝜋𝛾 = 𝐴0 rect 𝑒 𝑇𝑝𝑢𝑙𝑠𝑒 𝑆𝑇𝑥 𝑡, 𝜏𝑚 receive 𝑆𝑅𝑥 𝑡, 𝜏𝑚 𝑡−𝑚𝑇𝑝𝑟𝑖 𝑡 − 𝑚𝑇𝑝𝑟𝑖 − 2 𝑅𝑞 𝑐 = 𝑎𝑞 rect 𝑇𝑝𝑢𝑙𝑠𝑒 x𝑒 𝑗 2𝜋𝑓𝑐 𝑡−2𝑅𝑞 𝑐 +𝜋𝛾 𝑡−𝑚𝑇𝑝𝑟𝑖 −2𝑅𝑞 𝑐 2 2 Reference to Scene Center (motion compensation point) 𝑆𝑅𝑒𝑓 𝑡, 𝜏𝑚 = EE880 SAR System & Signals Part 2 𝑗 2𝜋𝑓𝑐 𝑡−2𝑅0 𝑐 +𝜋𝛾 𝑡−𝑚𝑇𝑝𝑟𝑖 −2𝑅0 𝑐 𝑒 2 29 Deramp & Deskew Receiver (2/5) • Mix reference signal with echo 𝑆𝑅𝑥 𝑡, 𝜏𝑚 X 𝑡 = 𝑡 − 𝑚𝑇𝑝𝑟𝑖 𝑆𝐼𝐹 𝑡, 𝜏𝑚 fast time within PRI conj 𝑆𝑅𝑒𝑓 𝑡, 𝜏𝑚 intermediate frequency 𝑆𝐼𝐹 𝑡, 𝜏𝑚 𝑡 − 2 𝑅𝑞 𝑐 −𝑗4𝜋𝛾 = 𝑎𝑞 rect 𝑒 𝑐 𝑇𝑝𝑢𝑙𝑠𝑒 𝑓𝑐 2𝑅0 + 𝑡− 𝑅𝑞 −𝑅0 𝛾 𝑐 2 4𝜋𝛾 𝑗 2 𝑅𝑞 −𝑅0 𝑒 𝑐 Received pulse train from q-th target 𝑆𝐼𝐹 𝑡, 𝜏𝑚 = 𝑎𝑞 Φ 𝑡, 𝜏𝑚 = −𝑗 EE880 SAR System & Signals Part 2 𝑀−1 𝑚=0 rect 𝑡 − 2 𝑅𝑞 𝑐 𝑗Φ 𝑒 𝑇𝑝𝑢𝑙𝑠𝑒 4𝜋𝛾 𝑓𝑐 2𝑅0 +𝑡− 𝑐 𝛾 𝑐 𝑡,𝜏𝑚 𝑅𝑞 − 𝑅0 + 4𝜋𝛾 𝑅𝑞 − 𝑅0 2 𝑐 2 30 Deramp & Deskew Receiver (3/5) • Signal phase Φ 𝑡, 𝜏𝑚 = −𝑗 4𝜋𝛾 𝑓𝑐 2𝑅0 +𝑡− 𝑐 𝛾 𝑐 𝑅𝑞 − 𝑅0 + 4𝜋𝛾 𝑅𝑞 − 𝑅0 2 𝑐 linear phase, easily compensated 2 quadratic phase, not easily corrected, often dismissed as phase error term • For a fixed target range, the instantaneous received frequency is 𝑓 𝑡, 𝜏𝑚 1 𝑑Φ 2𝛾 = =− 𝑅𝑞 − 𝑅0 2𝜋 𝑑𝑡 𝑐 constant range-dependent frequency is dechirped or deramped EE880 SAR System & Signals Part 2 31 Deramp & Deskew Receiver (4/5) Adapted from [SUL,7.2] frequency 𝑐𝑇𝑝𝑢𝑙𝑠𝑒 𝑇𝑝𝑢𝑙𝑠𝑒 𝑓𝑐 𝐵 time Near Scene 𝑇𝑝𝑢𝑙𝑠𝑒 2𝑆 𝑐 Scene Center 𝑆 Far Scene before deramp frequency after Targets at different ranges have different frequencies 𝐵𝐼𝐹 < 𝐵 time Deramping also reduces A/D sampling speeds EE880 SAR System & Signals Part 2 32 Deramp & Deskew Receiver (5/5) • Each echo contains multiple tones from scatterers at different ranges in the scene that occur at different times 2 𝑅𝑞 − 𝑅0 𝑓𝑞 𝑡𝑞 = 𝑐 =− 𝛾 • SAR processing requires one-to-one mapping of frequency to sample time, i.e. no time-delay • Correct as Φ ← Φ 𝑡𝑞 , 𝜏 𝜋𝑓𝑞 2 −𝑗 𝛾 𝑒 • IFT each echo to recover frequencies EE880 SAR System & Signals Part 2 frequency Deramped and deskewed 𝐵𝐼𝐹 < 𝐵 time 33 Operator Modeling ℒ 𝜎𝒦 𝑝𝑇𝑥 𝒦 𝑝𝑇𝑥 TX ENV 𝑝𝑇𝑥 𝜎 ℳ1 ℒ 𝜎𝒦 𝑝𝑇𝑥 RX RSP MF ℱ −1 𝑆𝑅𝑥 ℳ2 ℒ 𝜎𝒦 𝑝𝑇𝑥 PFA, CBP 𝜎 𝒦 represents antenna radiation of signal from transmitter ℒ represents scattering from scatterer ℳ represents receiver front end (mixing, matched filtering, etc…) 𝑆𝑅𝑥 = ℳ1 ℒ 𝜎𝒦 𝑝𝑇𝑥 These operations can be approximated as a forward Fourier transform EE880 SAR System & Signals Part 2 ≈ ℱℴ The approximation depends on simple linear superposition of scatterers and far field reception 34 Summary of SAR Systems & Signals Part 2 • • • • Imaging requirements Antenna array SAR signal modeling Operator modeling EE880 SAR System & Signals Part 2 35 Lesson References • [Levanon] N. Levanon, Radar Signals, Wiley-IEEE Press, 2004. • [Stimson] G. Stimson, Introduction to Airborne Radar, SciTech Publishing Inc., 1998. • [Sullivan] R. Sullivan, Foundations for Imaging and Advanced Concepts, SciTech Publishing Inc., 2004. EE880 SAR System & Signals Part 2 36

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