Synthetic-Aperture Radar (SAR) Basics

Outline Spatial resolution Geometric distortion Radiometric resolution Range resolution Short pulse system Pulse compression Chirp waveform Slant range vs. ground range Azimuth resolution Unfocused SAR Focused SAR Geometric distortion Foreshortening Layover Shadow Radiometric resolution Fading Radiometric calibration

Spatial discrimination Spatial discrimination relates to the ability to resolve signals from targets based on spatial position or velocity. angle, range, velocity Resolution is the measure of the ability to determine whether only one or more than one different targets are observed. Range resolution, r, is related to signal bandwidth, B Two targets at nearly the same range Short pulse  higher bandwidth Long pulse  lower bandwidth

Spatial discrimination The ability to discriminate between targets is better when the resolution distance is said to be finer (not greater) Fine (and coarse) resolution are preferred to high (and low) resolution Various combinations of resolution can be used to discriminate targets

Range resolution

Range resolution Short pulse radar The received echo, Pr(t) is where Pt(t) is the pulse shape S(t) is the target impulse response  denotes convolution To resolve two closely spaced targets, r Example r = 1 m    6.7 ns r = 1 ft    2 ns

Range resolution Clearly to obtain fine range resolution, a short pulse duration is needed. However the amount of energy (not power) illuminating the target is a key radar performance parameter. Energy, E, is related to the transmitted power, Pt by Therefore for a fixed transmit power, Pt, (e.g., 100 W), reducing the pulse duration, , reduces the energy E. Pt = 100 W,  = 100 ns  r = 50 ft, E = 10 J Pt = 100 W,  = 2 ns  r = 1 ft, E = 0.2 J Consequently, to keep E constant, as  is reduced, Pt must increase.

More Tx Power?? Why not just get a transmitter that outputs more power? High-power transmitters present problems Require high-voltage power supplies (kV) Reliability problems Safety issues (both from electrocution and irradiation) Bigger, heavier, costlier, …

Simplified view of pulse compression Energy content of long-duration, low-power pulse will be comparable to that of the short-duration, high-power pulse 1 « 2 and P1 » P2 time t1 Power P1 P2 t2 Goal:

Pulse compression Chirp waveforms represent one approach for pulse compression. Radar range resolution depends on the bandwidth of the received signal. The bandwidth of a time-gated sinusoid is inversely proportional to the pulse duration. So short pulses are better for range resolution Received signal strength is proportional to the pulse duration. So long pulses are better for signal reception c = speed of light, r = range resolution,  = pulse duration, B = signal bandwidth

Pulse compression, the compromise Transmit a long pulse that has a bandwidth corresponding to a short pulse Must modulate or code the transmitted pulse to have sufficient bandwidth, B can be processed to provide the desired range resolution, r Example: Desired resolution, r = 15 cm (~ 6”) Required bandwidth, B = 1 GHz (109 Hz) Required pulse energy, E = 1 mJ E(J) = Pt(W)· (s) Brute force approach Raw pulse duration,  = 1 ns (10-9 s) Required transmitter power, P = 1 MW ! Pulse compression approach Pulse duration,  = 0.1 ms (10-4 s) Required transmitter power, P = 10 W

FM-CW radar Alternative radar schemes do not involve pulses, rather the transmitter runs in “continuous-wave” mode, i.e., CW. FM-CW radar block diagram

FM-CW radar Linear FM sweep Bandwidth: B Repetition period: TR= 1/fm Round-trip time to target: T = 2R/c fR = Tx signal frequency – Rx signal frequency If 2fm is the frequency resolution, then the range resolution r is

FM-CW radar The FM-CW radar has the advantage of constantly illuminating the target (complicating the radar design). It maps range into frequency and therefore requires additional signal processing to determine target range. Targets moving relative to the radar will produce a Doppler frequency shift further complicating the processing.

Chirp radar Blending the ideas of pulsed radar with linear frequency modulation results in a chirp (or linear FM) radar. Transmit a long-duration, FM pulse. Correlate the received signal with a linear FM waveform to produce range dependent target frequencies. Signal processing (pulse compression) converts frequency into range. Key parameters: B, chirp bandwidth , Tx pulse duration

Chirp radar Linear frequency modulation (chirp) waveform for 0  t   fC is the starting frequency (Hz) k is the chirp rate (Hz/s) C is the starting phase (rad) B is the chirp bandwidth, B = k

Stretch chirp processing

Challenges with stretch processing Reference chirp Received signal (analog) Digitized signal Low-pass filter A/D converter To dechirp the signal from extended targets, a local oscillator (LO) chirp with a much greater bandwidth is required. Performing analog dechirp operation relaxes requirement on A/D converter. Echoes from targets at various ranges have different start times with constant pulse duration. Makes signal processing more difficult. LO near Tx B Rx near time frequency frequency far far time

Pulse compression example Key system parameters Pt = 10 W,  = 100 s, B = 1 GHz, E = 1 mJ , r = 15 cm Derived system parameters k = 1 GHz / 100 s = 10 MHz / s = 1013 s-2 Echo duration =  = 100 s Frequency resolution f = (observation time)-1 = 10 kHz Range to first target, R1 = 150 m T1 = 2 R1 / c = 1 s Beat frequency, fb = k T1 = 10 MHz Range to second target, R2 = 150.15 m T2 = 2 R2 / c = 1.001 s Beat frequency, fb = k T2 = 10.01 MHz fb2 – fb1 = 10 kHz which is the resolution of the frequency measurement

Pulse compression example (cont.) With stretch processing a reduced video signal bandwidth is output from the analog portion of the radar receiver. video bandwidth, Bvid = k Tp (where Tp = 2 Wr /c is the swath’s slant width) for Wr = 3 km, Tp = 20 s  Bvid = 200 MHz This relaxes the requirements on the data acquisition system (i.e., analog-to-digital (A/D) converter and associated memory systems). Without stretch processing the data acquisition system must sample a 1-GHz signal bandwidth requiring a sampling frequency of 2 GHz and memory access times less than 500 ps.

Correlation processing of chirp signals Avoids problems associated with stretch processing Takes advantage of fact that convolution in time domain equivalent to multiplication in frequency domain Convert received signal to freq domain (FFT) Multiply with freq domain version of reference chirp function Convert product back to time domain (IFFT) FFT IFFT Freq-domain reference chirp Received signal (after digitization) Correlated signal

Signal correlation examples Input waveform #1 High-SNR gated sinusoid, no delay Input waveform #2 High-SNR gated sinusoid, ~800 count delay

Signal correlation examples Input waveform #1 High-SNR gated sinusoid, no delay Input waveform #2 Low-SNR gated sinusoid, ~800 count delay

Signal correlation examples Input waveform #1 High-SNR gated chirp, no delay Input waveform #2 High-SNR gated chirp, ~800 count delay

Signal correlation examples Input waveform #1 High-SNR gated chirp, no delay Input waveform #2 Low-SNR gated chirp, ~800 count delay

Chirp pulse compression and time sidelobes Peak sidelobe level can be controlled by introducing a weighting function -- however this has side effects.

Superposition and multiple targets Signals from multiple targets do not interfere with one another. (negligible coupling between scatterers) Free-space propagation, target interaction, radar receiver all have linear transfer functions  superposition applies. Signal from each target adds linearly with signals from other targets. r = r range resolution

Why time sidelobes are a problem Sidelobes from large-RCS targets with can obscure signals from nearby smaller-RCS targets. Time sidelobes are related to pulse duration, . fb = 2 k R/c fb

Window functions and their effects Time sidelobes are a side effect of pulse compression. Windowing the signal prior to frequency analysis helps reduce the effect. Some common weighting functions and key characteristics Less common window functions used in radar applications and their key characteristics

Window functions Basic function: a and b are the –6-dB and - normalized bandwidths

Window functions

Detailed example of chirp pulse compression received signal dechirp analysis which simplifies to sinusoidal term chirp-squared term quadratic frequency dependence linear frequency dependence phase terms sinusoidal term after lowpass filtering to reject harmonics

Pulse compression effects on SNR and blind range SNR improvement due to pulse compression: B Case 1: Pt = 1 MW,  = 1 ns, B = 1 GHz, E = 1 mJ, r = 15 cm For a given R, Gt, Gr, l, s: SNRvideo = 10 dB B = 1 or 0 dB SNRcompress = SNRvideo = 10 dB Blind range = c/2 = 0.15 m Case 2: Pt = 10 W,  = 100 s, B = 1 GHz, E = 1 mJ , r = 15 cm For the same R, Gt, Gr, l, s: SNRvideo = – 40 dB B = 100,000 or 50 dB SNRcompress = 10 dB Blind range = c/2 = 15 km

Pulse compression Pulse compression allows us to use a reduced transmitter power and still achieve the desired range resolution. The costs of applying pulse compression include: added transmitter and receiver complexity must contend with time sidelobes increased blind range The advantages generally outweigh the disadvantages so pulse compression is used widely. Therefore we will be using chirp waveforms to provide the required range resolution for SAR applications.

Slant range vs. ground range Cross-track resolution in the ground plane (x) is the projection of the range resolution from the slant plane onto the ground plane. At grazing angles (  90°), r  x At steep angles (  0°), x   For  = 5°, x = 11.5 r For  = 15°, x = 3.86 r For  = 25°, x = 2.37 r For  = 35°, x = 1.74 r For  = 45°, x = 1.41 r For  = 55°, x = 1.22 r

Azimuth (along-track) resolution Discrimination of targets based on their along-track or azimuth position is possible due to the unique phase history associated with each azimuth position. Note that phase variations and Doppler frequency shifts are analogous since f = d/dt where  is phase. Assuming the phase of the target’s echo is essentially constant over all observation angles, the phase variation is due entirely to range variations during the observation period. Recall that /2 = 2R/ where R is the slant range,  is the wavelength, and the factor of 2 is due to the round-trip path length.

Along-track resolution Consider an airborne radar system flying at a constant speed along a straight and level trajectory as it views the terrain. For a point on the ground the range to the radar and the radial velocity component can be determined as a function of time. Radar position = (0, vt, h), Target position = (xo, yo, 0), Range to target, R(t)

Along-track resolution

Along-track resolution Example Airborne SAR Altitude: 10,000 m Velocity: 75 m/s Five targets on ground All cross-track offsets = 5 km Along-track offsets of -1000, -500, 0, 500, and 1000 m

Along-track resolution

Along-track resolution

Along-track resolution

Along-track resolution Example Airborne SAR Altitude: 10,000 m Velocity: 75 m/s Five targets on ground All along-track offsets = 0 m Cross-track offsets of 5, 7.5, 10, 12.5, and 15 km

Along-track resolution

Along-track resolution

Along-track resolution

Along-track resolution Now solve for R and fD for all target locations and plot lines of constant range (isorange) and lines of constant Doppler shift (isodops) on the surface.

Along-track resolution Isorange and isodoppler lines for aircraft flying north at 10 m/s at a 1500-m altitude. r = 2 m, V = 0.002 m/s, fD = 0.13 Hz @ f = 10 GHz,  = 3 cm

Along-track resolution Without the spatial filtering of the antenna, the azimuth chirp waveform covers a wide bandwidth.

Along-track resolution Samples of phase variations due to a changing range throughout the aperture are provided with each pulse in the slow-time domain. Note that the range chirp has been frequency shifted to baseband.

Along-track resolution Squint-mode operation (or moving targets) will skew the Doppler spectrum. This skew can be detected and accommodated.

Strip-map SAR signal example (no squint) Single point target at center of scene.

Strip-map SAR signal example (no squint) Time domain characteristics of single point target. Magnitude of phase history mapped in azimuth and range. Constant amplitude in range axis indicates uniform pulse amplitude (no windowing). Variation in azimuth represents antenna beam pattern in azimuth plane.

Strip-map SAR signal example (no squint) Time domain characteristics of single point target. Real part of phase history mapped in azimuth and range. Can be shown that contour of constant phase follows: Where K is the pulse chirp rate Ka is the azimuth chirp rate t is the fast time index  is the slow time index  is a constant Positive range chirp (K > 0), negative azimuth chirp (Ka < 0) Contours of constant phase map as hyperbolae

Strip-map SAR signal example (no squint)

Strip-map SAR signal example (no squint) Time domain characteristics of single point target. Real part of phase history mapped in azimuth and range. Can be shown that contour of constant phase follows: Where K is the pulse chirp rate Ka is the azimuth chirp rate t is the fast time index  is the slow time index  is a constant Negative range chirp (K < 0), negative azimuth chirp (Ka < 0) Contours of constant phase map as ellipses

Unfocused SAR Processing SAR phase data to achieve a fine-resolution image requires elaborate signal processing. In some cases trading off resolution for processing complexity is acceptable. In these cases a simplified unfocused SAR processing is used wherein only a portion of the azimuth phase history is used resulting in a coarser azimuth resolution. In unfocused SAR processing consecutive azimuth samples are added together (in the slow-time domain). Since addition a is simple operation for digital signal processors, the image formation processing is much easier (less time consuming) than fully-focused SAR processing.

Unfocused SAR Summing consecutive samples, also known as a coherent integration or boxcar filtering, is useful so long as the signal’s phase is relatively constant over the integration interval. Example For a 20-sample interval the central portion of the chirp waveform (zero Doppler) is relatively constant. For the outer portions of the chirp the phase varies significantly and integrating produces a reduced output.

Unfocused SAR Example (cont.) Over a 38-sample interval phase variations within the central portion of the chirp waveform results in a reduced output (0.8 peak vs. 1). The magnitude of the first sidelobe is also larger (0.4 vs. 0.3). The width of the main lobe is narrower.

Unfocused SAR The resolution improves with increased integration length up to a point when oscillations in the signal are included in the integral. The maximum synthetic aperture length for unfocused SAR is Lu which corresponds to a maximum phase shift across the aperture of 45º. The azimuth resolution for L = Lu is Notice the range- and frequency-dependencies of y.

Focused SAR To realize the full potential of SAR and achieve fine along-track (azimuth) resolution requires matched filtering of the azimuth chirp signal. Stretch chirp processing, correlation processing, tracking Doppler filters, as well as other techniques can be used in a matched filter process. However the range processing is not entirely separable from the azimuth processing as an intricate interaction between range and azimuth domains exists which must also be dealt with to achieve the desired image quality.

Focused SAR Consider the phenomenon known as range walk or range-cell migration. Variations in range to a target over the synthetic aperture not only introduce a quadratic phase change (resulting in the azimuth chirp) but may also displace echo in the range (fast-time) domain.

Focused SAR If the range to the target varies by an amount greater than the range resolution then the range-cell migration must be compensated during the image formation processing. Details on the processing required to achieve fully-focused fine-resolution SAR images will be addressed later.

Focused SAR In SAR systems a very long antenna aperture is synthesized resulting in fine along-track resolution. For a synthesized-aperture length, L, the along-track resolution, y, is L is determined by the system configuration. For a fully focused stripmap system, Lm = azR (m), where az is the azimuthal or along-track beamwidth of the real antenna (az  /ℓ) R is the range to the target For L = Lm, y = ℓ/2 (independent of range and wavelength)

Radiometric resolution -- signal fading For extended targets (and targets composed of multiple scattering centers within a resolution cell) the return signal (the echo) is composed of many independent complex signals. The overall signal is the vector sum of these signals. Consequently the received voltage will fluctuate as the scatterers’ relative magnitudes and phases vary spatially. Consider the simple case of only two scatters with equal RCSs separated by a distance d observed at a range Ro.

Signal fading As the observation point moves along the x direction, the observation angle  will change the interference of the signals from the two targets. The received voltage, V, at the radar receiver is where The measured voltage varies from 0 to 2, power from 0 to 4. Single measurement will not provide a good estimate of the scatterer’s . Note: Same analysis used for antenna arrays.

Fading statistics Consider the case of Ns independent scatterers (Ns is large) where the voltage due to each scatterer is The vector sum of the voltage terms from each scatterer is where Ve and  are the envelope voltage and phase. It is assumed that each voltage term, Vi and i are independent random variables and that i is uniformly distributed. The magnitude component Vi can be decomposed into orthogonal components, Vx and Vy where Vx and Vy are normally distributed.

Fading statistics The fluctuation of the envelope voltage, Ve, is due to fading although it is similar to that of noise. The models for fading and noise are essentially the same. Two common envelope detection schemes are considered, linear detection (where the magnitude of the envelope voltage is output) and square-law detection (where the output is the square of the envelope magnitute). Linear detection, VOUT = |VIN| = Ve It can be shown that Ve follows a Rayleigh distribution where 2 is the variance of the input signal

Fading statistics (linear detection) For a Rayleigh distribution the mean is the variance is The fluctuation about the mean is Vac which has a variance of So the ratio of the square of the envelope mean to the variance of the fluctuating component represents a kind of inherent signal-to-noise ratio for Rayleigh fading.

Fading statistics (linear detection) An equivalent SNR of 5.6 dB (due to fading) means that a single Ve measurement will have significant uncertainty. For a good estimate of the target’s RCS, , multiple independent measurements are required. By averaging several independent samples of Ve, we improve our estimate, VL where N is the number of independent samples Ve is the envelope voltage sample

Fading statistics (linear detection) The mean value, VL, is unaffected by the averaging process However the magnitude of the fluctuations are reduced And the effective SNR due to fading improves as 1/N As more Rayleigh distributed samples are averaged the distribution begins to resemble a normal or Gaussian distribution.

Fading statistics (square-law detection) Square-law detection, Vs = Ve2 The output voltage is related to the power in the envelope. It can be shown that Vs follows an exponential distribution Again the mean value is found and the variance is found (note that ) Again for a single sample measurement yields a poor estimate of the mean. where 2 is the variance of the input signal

Fading statistics (square-law detection) An equivalent SNR of 0 dB (due to fading) means that a single Vs measurement will have significant uncertainty. For a good estimate of the target’s RCS, , multiple independent measurements are required. By averaging several independent samples of Vs, we improve our estimate, VL where N is the number of independent samples Vs is the envelope-squared voltage sample

Fading statistics (square-law detection) The mean value, VL, is unaffected by the averaging process However the magnitude of the fluctuations are reduced And the effective SNR due to fading improves as 1/N. As more exponential distributed samples are averaged the distribution begins to resemble a 2(2N) distribution. For large N, (N > 10), the distribution becomes Gaussian.

Independent samples Fading is not a noise phenomenon, therefore multiple observations from a fixed radar position observing the same target geometry will not reduce the fading effects. Two approaches exist for obtaining independent samples change the observation geometry change the observation frequency (more bandwidth) Both methods produce a change in  which yields an independent sample. Estimating the number of independent samples depends on the system parameters, the illuminated scene size, and on how the data are processed.

Independent samples In the range dimension, the number of independent samples (NS) is the ratio of the range of the illuminated scene (Wr) to the range resolution (r)

Independent samples When relative motion exists between the target and the radar, the frequency shift due to Doppler can be used to obtain independent samples. The number of independent samples due to the Doppler shift, ND, is the product of the Doppler bandwidth, fD, and the observation time, T The total number of independent samples is In both cases (range or Doppler) the result is that to reduce the effects of fading, the resolution is degraded.

Independent samples N = 1 N = 10 N = 50 N = 250

Radiometric calibration Translating the received signal power into a target’s radar characteristics (cross section or attenuation) requires radiometric accuracy. From the radar range equation for an extended target we know that the factor affecting the received signal power include the transmitted signal power, the antenna gain, the range to the target, and the resolution cell area. Uncertainty in these parameters will contribute to the overall uncertainty in the target’s radar characteristics.

Radiometric calibration Transmit power, Pt Addition of an RF coupler or power splitter at the transmitter output permits continuous monitoring of the transmitted signal power. Antenna gain, G The antenna’s radiation pattern must be well characterized. In many cases the antenna must be characterized on the platform (aircraft or spacecraft) as it’s immediate environment may affect the radiation characteristics. Furthermore the characterization may need to be performed in flight. Target range, R Radar’s inherent ability to measure range accurately minimizes any contribution to radiometric uncertainty. Resolution cell area, A Difficult to measure directly, requires measurement data from extended target with known .

Radiometric calibration Calibration targets Radiometric calibration of the entire radar system may require external reference targets such as spheres, dihedrals, trihedrals, Luneberg lens, or active calibrators.

Radiometric calibration Flat plate

Radiometric calibration Dihedral and trihedral corner reflectors

Radiometric calibration Dihedral and trihedral corner reflectors

Radiometric calibration Luneberg lens

Radiometric calibration Luneberg lens

Radiometric calibration

Radiometric calibration Active radar calibrator [Brunfeldt and Ulaby, IEEE Trans. Geosci. Rem. Sens., 22(2), pp. 165-169, 1984.]

Radiometric calibration Active radar calibrator

Radiometric calibration

Radiometric calibration RCS of some common shapes

Radiometric calibration

Radiometric calibration