Sar Tutorial

Presented by:

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Synthetic Aperture Radar Concepts

Prepared for University of Delaware

ELEG 467/667 Introduction to RADAR systems

Joseph C Deroba Radar Applications

I2WD

Distribution Unlimited

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SAR

THE OVERALL CLASSIFICATION OF THIS BRIEFING IS UNCLASSIFIED

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SAR: References

•  Textbooks: –  Carrara, Goodman, Majewski, Spotlight Synthetic Aperture

Radar, Artech House, 1995.

–  Jakowatz, Wahl, Eichel, Ghilia, Thompson, Spotlight-mode Synthetic Aperture Radar: A signal processing approach, Springer, 1996.

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AGENDA

•  Quick Overview

–  Range Resolution

•  Stretch Processing

–  Azimuth Resolution

•  Real Aperture Radar

•  Doppler Beam Sharpening

–  Strip-map SAR

•  Un-focused SAR constraints

–  Phase Error/Resolution

–  Antenna Size/Spot Mode SAR

–  Focused SAR

•  Phase compensation

•  Range Migration

–  Polar Format Image Formation

–  Autofocus

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Range Resolution

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SAR: Range Resolution

• Recall that range resolution is a function of transmitted waveform bandwidth:

–  The larger the bandwidth, the finer the resolution achieved

–  Bandwidth is realized using a chirp waveform

target 1

target 2

during this time the two returns are identical

target 1

target 2

returns are different during this time period now

Chirp rate

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•  How do we process the chirps to locate targets?

–  There are two popular methods

•  Correlation Mixing

–  Typically uses Fast Convolution

–  All processing can be done in “one pass” provided hardware can handle the data sizes and rates


–  Limits processed “Swath” size in order to gain downstream advantages

»  Eases burden on the ADCs

»  Greatly reduces the downstream processing bandwidth

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•  Correlation Mixer with Fast Convolution

–  Convolve the return signals with a matched filter created from a copy of the transmitted signal using the convolution property of the Fourier Transform

•  Target returns are correlation “peaks”

•  A/D rate must be greater than the waveform bandwidth

–  Prohibitive for some applications at higher resolutions

•  Able to process the entire pulse-swath

–  Can perform weighting by multiplying weights to H(n)

•  Number of complex samples can be large due to larger sampling frequency

cos(wt)

sin(wt)

y(t)

A/D

A/D j

FFT

FFT

FFT-1

h(n)

y(n)

X weights

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–  “De-ramp” received signal by mixing with a longer reference signal of the same chirp rate

•  Target returns are “beat frequencies”

•  A/D rate directly related to range window extent and low pass filter

•  Reduced sampling requirements eases downstream processing

–  The frontend hardware must support the additional RF bandwidth required for the “de-ramp reference”

cos(wt)

sin(wt)

y(t)

A/D

A/D j

FFT y(n)

Weights and Compensation Data

r(t)

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Correlation Mixing Stretch Processing

IF Bandwidth

A/D Rate

Num Samples

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SAR: Stretch Processing

•  Most modern imaging radars utilize stretch processing

–  ADCs with higher ENOB (effective number of bits) and better performance are available at lower frequencies

•  Better dynamic range and image quality

•  Let’s take a look at a typical radar time/frequency diagram:

Time

Time

Frequency

Btx

τ

Blpf

τ

Near Edge Returns

Far Edge Returns

Center Range Returns

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•  Let’s take a quick example:

–  Res: 1.0 meter, BW: 150 MHz, Target Range: 990 m, R0: 1000 meters, Tau: 1e-6 sec, Swath: 60 meters

Time

Time

Frequency

Blpf

τ

Near Edge Returns

Far Edge Returns

Target Return = 10 MHz

Swath width

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•  Example Continued:

–  Res: 1.0 meter, BW: 150 MHz, Target Range: 990 m, R0: 1000 meters, Tau: 1e-6 sec, Swath: 60 meters

1usec

150 MHz

0.066 usec delay

1.4 usec FFT

10 cycles in 1.0 usec = 10 MHz

cos

j*sin

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Azimuth Resolution Overview

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SAR: Fundamentals

•  Azimuth Resolution

–  Remember: Range Resolution is higher (better) with increased bandwidth

–  Azimuth resolution becomes higher (better) with increased aperture size

•  If properly constructed, beam-width will decrease with increased aperture size (relative to wave-length)

–  Narrow beams have better azimuth resolution

–  Azimuth resolution is limited by the beam-width at range

175 m 349 m 2o

5000 m

10000 m

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SAR: Fundamentals

• RAR – Real Aperture Radar

–  The actual antenna (aperture) is used to form images

• Range resolution is typically greater than azimuth (since this depends on the beam width and the range to the target)

–  For a Real Aperture antenna, azimuth resolution is dependent on the size of the physical antenna

ρa = aw λ R / Dant

Dant = real antenna aperture size

–  Example: 1.2 * .018*10000/72 = 3 meters

–  So an antenna would have to be 72 meters long for just 3 meters of azimuth resolution at Ku band

–  High resolution imagery is impractical due to the large apertures required

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SAR: Phase History

• How do we improve angular resolution?

–  Take advantage of the phase information that we collect

• Recall:

–  The argument is the phase in radians

–  Therefore:

–  Knowing how phase changes with time yields frequency

»  Conversely, knowing how phase changes over frequency yields time

• Unprocessed radar data is known as Phase History

–  As we collect each pulse in time, we are recording a time history of the phase of the radar signal

–  From this we can extract frequency information

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SAR: Phase History

•  Frequency (Doppler) Resolution

–  Use the phase information collected on the targets to determine doppler

• Move the radar between pulses

•  The more measurements (pulses) we take the better we are able to resolve frequency

–  The faster we make measurements the larger the frequency span

• Recall: Moving targets produce a doppler shifted return pulse

–  Stationary targets also exhibit “doppler” shifts when the radar is moving (proportional to the radar velocity)

–  Doppler frequency is given by:

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SAR: Doppler

•  Only the motion that is relative to the radar causes a Doppler shift

–  Radars can only measure radial velocities (Range Rates)

•  The component of the target’s Velocity Vector that is in the direction of the radar’s line-of-sight

–  A radar target is either closing or opening to the radar’s position

•  Detection of a target’s doppler highly depends on the geometry to the target

Tree is now tangential and has no relative radial motion to the radar (Pulse comes

back virtually unshifted)

Forward Radar motion creates an apparent closing target out of

the tree (pulse comes back with positive

shift

V

Θ

VR = V*cos(Θ)

Θ

V

Θ VR = V*cos(Θ)=0

V

Θ VR = V*cos(Θ)

V

Continued motion past the tree creates an apparent opening

target out of the tree (pulse comes back

negative shifted

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Doppler Beam Sharpening

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SAR: DBS

• DBS – Doppler Beam Sharpening – Crudest form of SAR processing

–  Allows the radar to resolve angles finer than the azimuth beam width

–  The radar beam illuminates all targets within its beamwidth

• Suppose the beam of width Φ is pointed at look angle Θ

–  Each edge of the beam contributes a doppler return from the “clutter”

»  The doppler bandwidth of the illuminated area follows as:

Φ

V

Θ

Ground clutter from different directions

exhibits different doppler frequencies within the

doppler bandwidth!

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SAR: DBS

• Resolving angle from doppler:

–  Consider the broadside case where theta is 90 degrees

–  Our ability to resolve angle stems from our ability to resolve doppler

Φ

V

Θ Θ’

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SAR: DBS

• Doppler resolution continued:

–  Since we know the platform velocity and the total collection time we calculate the distance the platform traveled as:

–  The angular resolution only depends on the wavelength and the distance we flew, also known as the SYNTHETIC APERTURE!

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SAR: DBS

• Of Course PRF still matters:

–  In order to properly sample the illuminated area the PRF needs to be greater than or equal to the maximum doppler bandwith

• We have to at least sample the beamwidth to avoid ambiguities

• Systems typically have PRFs much higher than necessary

–  Pre-summing of pulses for additional signal gains

–  Reduced effects from ambiguities

»  Higher PRFs “push” ambiguities farther away from the doppler region of interest

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SAR: DBS

• Conclusions on DBS Mapping:

–  Since the synthetic aperture can be much larger than the physical antenna we have effectively narrowed or “sharpened” our beam through the usage of doppler processing

• We can form a series of doppler “sharpened” beams to create a coarse resolution moving map!

• DBS is an early form of SAR Strip-Mapping

–  In our earlier RAR example we needed an antenna that was 73 meters in diameter to achieve a given angular resolution

• With SAR techniques we can just fly 73 meters with a smaller antenna in order to gain the same result!

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SAR: DBS Example

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STRIP-MAP SAR CONCEPTS

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SAR: Strip SAR

Why does strip SAR provide us with a “synthetic” array antenna?

–  We can take a common antenna building concept and scale it

• Array antennas are a collection of smaller elements

–  Each element has a wide beam pattern

–  Together these beams combine to form a single narrow beam (if properly constructed)

effective array beamwidth an

tenn

a

Element beamwidth

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SAR: Strip SAR

•  It is also possible to collect data from a series of points in space to form a “synthetic” antenna array

•  This is the basis for Synthetic Aperture Radar (SAR)

–  The synthetic beam width is much narrower, offering increased angular resolution

–  At longer ranges we can fly a longer synthetic aperture in order to keep a higher azimuth resolution

Aircraft Position over time

“Synthetic” Beam width

“Synthetic” Array Size

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SAR: Strip SAR

• As range to the target increases, our spatial resolution will decrease for a given angular resolution

–  Azimuth spatial resolution on the ground is given by:

–  At longer ranges we must fly a longer synthetic aperture in order to keep spatial resolution the same

–  Integration angle remains constant

“Synthetic” Array Size

Ran

ge to

targ

et

Integration Angle

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SAR: Strip SAR

•  It would seem that we can form an endlessly long aperture to achieve a given resolution regardless of range, however:

–  The synthetic array size is limited by how long the target is ‘seen’ within the wider beam of the ‘real’ antenna

–  Allows for the generation of azimuth resolutions in strip maps on the order of dreal/2

maximum synthetic aperture

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SAR: Strip SAR Resolution Constraint

–  Remember:

•  For the side-looking Strip mapping process (previous slide) we have to limit the length of the synthetic aperture

–  We can approximate the azimuth radar beam-width as

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SAR: Fundamentals

•  The SAR paradox

–  The wider the beamwidth of the smaller real aperture (antenna), the longer the synthetic aperture (for a given range) can be

–  The (resulting) narrower synthetic antenna beam allows for finer ultimate azimuth resolutions to be realized

• More Bandwidth in our Doppler chirps! (more on this later)

–  Better resolution between target angles!

–  However:

• A smaller antenna is typically lower gain and lower power

–  Shortens range performance

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SAR: Quick Recap

• SAR azimuth resolution was shown to be independent of wavelength and range

• SAR azimuth resolution was shown to depend only on the length of the synthetic aperture

–  The longer the aperture (the greater the integration angle), the finer the resolution

• So we can just create any size aperture we want and expect the commensurate resolution?

–  Not really

»  Longer apertures are more susceptible to phase errors resulting in defocused imagery – compensation is required

–  Unfocused SAR - range and wavelength play a role in determining those resolutions that can be collected and processed with minimal compensation

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UNFOCUSED SYNTHETIC APERTURE RADAR

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SAR: Unfocused SAR

• Recall that we extract our information from the “Phase History”

–  As we move throughout the aperture the target changes positions within the radar beam

• Range (and doppler) changes from pulse to pulse within the CPI

–  What is happening to the Phase?

–  What is happening to the Frequency/Doppler?

Clo

sest

Ran

ge to

targ

et

Range change ΔR

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SAR: Phase

–  Take the pictorial example below:

•  R0 is the range at the center of the aperture (Aperture Reference Point, ARP) to the center of the scene (Scene Reference Point, SRP)

•  R is the instantaneous range from the SRP to the current sensor position in the aperture (Along Track or Slow Time position, x)

–  The phase of the traveling wave changes with R when compared to that which was received at the ARP

•  The phase error associated with the change in range from R0 can be written as

•  It will be shown that this error is quadratic!

ΔR

R0

x ARP

SRP

R

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SAR: Unfocused SAR

•  Ideally we would not have any phase error across the aperture

–  Quadratic errors serve to defocus the array

–  In order to avoid defocus we must limit the magnitude of the errors

•  Generally, phase errors may be tolerated to ¼ of a cycle

–  Recall:

–  The delta range from R0 must be kept within an 8th of a wavelength!!

•  We will use this to solve for the maximum Unfocused SAR aperture

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SAR: Phase Instantaneous Range

–  In order to determine the constraints on aperture we need to solve for R

–  Taking the Taylor Series Expansion:

–  Since x typically << R0 we will ignore the higher order “Others” and evaluate

–  Note that R is quadratic in x

•  So is ΔR!

ΔR

R0

x ARP

SRP

R

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SAR: Instantaneous Range

•  It was shown that the change in range to the scene center is quadratic across a synthetic aperture

–  Since:

•  We can say that the phase error is also quadratic across the aperture

•  What does this tell us about the frequency (doppler)?

–  Remember:

•  So the frequency across the aperture is a linear function

•  As we progress across an aperture we can expect a linear frequency ramp

–  Generates a “chirp” function in doppler (similar to our Tx waveform)

•  Longer aperture = larger chirp = more bandwidth = better resolution!

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SAR: Unfocused SAR Aperture Constraint

–  From our pictorial we know:

–  From the previous slide we know:

–  Therefore:

–  Take x equal to the end point of the aperture so that x-xsrp is equal to half an aperture length, La

–  Note: The constraint for the maximum Unfocused SAR aperture is dependent upon both range and frequency

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SAR: Unfocused SAR Minimum Resolution

•  From our previous discussions we know that limiting the aperture of a SAR data collection will also limit the achievable resolution

•  Recall:

–  Plugging the constrained aperture size into the resolution equation we find

for

•  Clearly, we need a method to focus the array if we are going to generate useful resolutions

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SAR: Unfocused SAR Example

–  Consider a Ku Band radar operating at 11 km slant range

–  Our finest possible resolution is about 7 meters!

•  Recall that a strip SAR’s antenna must be twice the minimum resolution in order to keep the targets in the beam

–  Our antenna must be 14 meters long!

»  How can we alleviate constraints on the physical antenna and synthetic aperture length

»  Spot-mode SAR

»  Phase Compensation

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SAR: Focused SAR Spot Mode

•  Spot-Light SAR

–  If the real antenna beamwidth is too narrow, we might need to slew the antenna in azimuth during an aperture in order to keep the target area illuminated

•  This allows for longer (compensated) maximum synthetic apertures to be formed and alleviates the strip-SAR constraints on physical antenna size

–  Also allows higher gain and narrower beam (real) antennas to be used for high resolution imaging at longer ranges

real beam too narrow real beam moved to scene center

Strip SAR Spot SAR

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SAR: Unfocused SAR Recap

•  Quick Recap:

–  For Strip SAR modes

•  Synthetic aperture length is dependent upon the beamwidth of the physcial antenna

–  The real antenna must be twice the length of the finest resolution that is desired for the beamwidth to cover the necessary aperture

–  For Un-Focused SAR (Strip or Spot)

•  Maximum synthetic aperture length is dependent upon range and wavelength

–  Phase errors are quadratic across an aperture

–  Doppler/Frequency exhibits a linear “chirp” across an aperture

–  Spot mode alleviates the real antenna size constraint for Strip SAR

–  Finest resolution will still be based upon the maximum unfocused SAR aperture (unless we compensate for the increased phase errors)

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FOCUSED SYNTHETIC APERTURE RADAR

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SAR: Focused SAR Overview

•  In order to take advantage of the longer apertures that Spot SAR affords us we need to understand the following:

–  How to focus a synthetic array

–  How doppler is exploited

–  The next level of constraints:

–  Range Migration

»  Interpolation

»  Image Formation

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SAR: Focused SAR Phase Compensation

•  Phase Compensation (also called Motion Compensation)

–  Motion must be measured precisely and accurately

•  Requires high quality INS/GPS system mounted close to the antenna

–  We care about the motion of the antenna phase center

»  Often requires usage of lever arm calculations due to mounting constraints

–  Radars are coherent, the phase of the waveform is preserved from pulse to pulse

•  The system “knows” what the ideal phase return should be

–  We have approximated this return in the previous slides

»  With a perfectly straight and level aperture this error is quadratic

–  Aircraft motion alters this phase from the ideal

»  Reality sets in…Higher order phase errors exist and are generally unavoidable

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–  Recall:

•  Phase error across an aperture is ideally quadratic

–  Causes “smearing” of scatters in the image

•  Must be corrected along with any other variations in the aperture

–  Flying a perfect aperture is impossible

»  Phase error will also have higher order terms

–  Correcting the phase errors will “focus” the array

•  Residual errors often exist

–  Corrected down-stream via “Auto-Focus”

»  We will talk about this later ΔR

R0

x

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•  Motions in the “slant plane” must be controlled/compensated to well within a wavelength

–  Even the slightest deviation from a straight flight path introduces “element” misalignment in the synthetic array

•  Causes the synthetic array to lose “focus”

–  Incorrect measurement of platform velocity also causes errors

•  Remember: Doppler is tied to the platform velocity

–  Doppler provides us with our azimuth resolution

less than precise flight path !

precise flight path !

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–  With knowledge, corrections to the signals at the “elements” can be calculated and applied at the time of reception to bring them back into correct phase with one another across the synthetic array (aperture)

–  This can done in hardware with Time Delay Units/Phase Shifters or digitally in the signal processor using properties of a fourier transform

•  Remember the shift property!

–  It can even be done fairly well PRIOR to transmission

•  Requires sophisticated filtering of the antenna’s past and current position in order to estimate the “future” position

•  Can reduce requirements of down stream processing if done correctly

Phase compensation!

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•  Suppose we have a known phase error:

–  Our returned signal containing the error would be

•  Where phi contains the constant and linear phase terms that we are interested in keeping

–  We compensate the phase errors via the creation of a correction vector

•  Multiplying the phasors “shifts” the time domain returns into alignment

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•  Lets take a look at an example:

–  Calculate the radian phase changes across the aperture based on the measured navigation data (n is pulse number)

•  Here we have the theoretical and measured data shown based on the above equation

•  Notice how just using theoretical calculations would miss the large phase perturbation

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•  Lets see what happens to the phase when we multiply in the correction vector

•  The measured data flattens out the phase response nicely!

•  Our purely theoretical calculation did not take into account actual platform motions beyond straight and level

–  The theoretical phase correction does not yield a compensated result

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Phase errors across aperture Un-corrected image result

Phase Corrected Image Result

Compensated Phase

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•  Residual Phase Errors

–  There is a limit to the accuracy and precision of motion measurements

•  Residual errors still exist

–  The images will often remain slightly blurred

•  Auto-focus measures these errors and applies a correction to the imagery to bring it into focus

•  Auto-focus can be done in the processor or post processed on at the ground station

–  Real-time auto-focus typically requires a good motion measurement solution to get it in ballpark, since iterations must be limited due to processing constraints

–  The most popular form of SAR autofocus is Phase Gradient Autofocus

•  Measures phase gradient of bright scatters in an image

•  Iteratively applies corrections and integrates the measurements to converge on the best phase correction vector

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SAR: Focused SAR

•  Quick Recap:

–  Measure the motions and variations in the synthetic aperture

•  Precision Inertial Navigation System (INS)

–  Use the measurements to compensate the phase across the aperture

•  Multiply the correction vector across the array of the pulse data aligning each pulse in phase across the aperture

•  Ok so we have compensated for phase and formed a near perfect synthetic array

–  How do we process that data

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SAR: Focused SAR

Really Quick Revisit to the fundamentals:

•  In SAR modes, the stationary targets exhibit a Doppler “Chirp” as the plane flies past the target.

•  Measurements taken from this chirp allow the radar to place targets on a line of Doppler within the imaged scene

–  From this we can measure the angle to the target

•  The locations where lines of Doppler and Range intercept is how a target is located in the scene

–  Every pixel in a SAR image has a Range and an Angle associated to it

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SAR: Focused SAR

•  Recall that the doppler response due to the quadratic phase was linear

–  By compensating for the quadratic phase you have “dechirped” the azimuth response

•  Allows azimuth image formation to be done with a DFT in the azimuth or cross range direction

•  So we can form a simple SAR image with a 2D-FFT or decouple the range and cross-range FFT for computational reasons

–  We are limited in this approach by two major constraints

•  Range migration

•  Iso-Range Curvature

•  Both affect the size of the largest image that we can form

•  Both have implications in how we sample (and resample) our data

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SAR: Focused SAR Range Migration

•  Range Migration:

–  Motion of targets must be controlled to within a range resolution cell during the course of the aperture

•  More of a problem as scene sizes increase and resolutions decrease

•  We need to compensate for the range migration or else we will still have targets that are “smeared”

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SAR: Focused SAR

•  Ideally, the range and cross-range responses are decoupled

–  Allows a separate range and cross-range FFT to resolve targets to their respective pixel in an image

•  When a target migrates through range cells during the formation of a SAR aperture

–  Target response is now coupled in range and cross range = smearing

n (Slow-time pulses)

i (fa

st-ti

me

sam

ples

)

FFT i FFT n


i (fa

st-ti

me

sam

ples

)

FFT i FFT n

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SAR: Focused SAR

•  Excessive Range Cell Migration or Motion Through Range Cells (MTRC) must be controlled

–  Interpolation of the data to a new data grid can compensate for MTRC

–  One popular SAR image formation technique that uses interpolation is Polar Formatting or the Polar Format Algorithm


FFT i FFT n Resample

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POLAR FORMAT ALGORITHM

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SAR: Spot-Light Polar Format

•  Recall:

–  Most SARs use Stretch Processing to deramp the transmitted pulse

•  Eases hardware and processing constraints

•  Also allows us to analyze SAR data as a projection of a 2 dimensional scene into a one dimensional function

–  Similar to Tomography!!

–  Let’s take a look at the stretch waveform and its output

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•  Recall:

–  Stretch processing

•  Transmitting a signal of the form:

•  And mixing its return (delayed by the range to target and augmented by A, the scatter’s complex reflectivity) of the form:

•  With the local oscillator signal (quadrature):

•  Gives the following signal from the Stretch Processor at the ADC:

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•  If one ignores the squared phase term (called Residual Video Phase) due to its small contribution we are left with a signal of the form:

•  Further, we will consider n = 1 pulse (tn = 0):

–  Consider k scatters in the scene now where: and canceling and ordering terms provides:

–  Extend this to infinite scatters and we get this signal structure:

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•  Consider the fourier transform of A:

–  Compare this to signal we just derived

–  We can write the video signal with the substitution of A(Ω) to find an important property of the stretch processor

–  Remembering that the spatial swath is much smaller than the spatial pulse length for stretch to be useful allows us to cancel some additional terms:

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•  Why is this result important?

•  It shows that TIME domain output of the stretch mixer equates to the Fourier transform of the scattering profile over the frequency interval encompassed by the swept bandwidth of our LFM pulse!

–  So as the radar collects a pulse it is sampling the one dimensional range profile of the two-dimensional scene’s reflectivity (that lies within beam)

–  Due to the projection-slice theorem:

•  Each pulse measurement is actually a band-limited measurement of a particular slice of the scene’s two-dimensional Fourier Transform

–  Ok...maybe we can use the FFT to recover an estimate of the actual scene that was imaged! •  SWEET!

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•  Lets take a look the SAR geometry to visualize the results of the math:

•  Here we have three pulses (beginning, aperture center, end) – Each pulse provides a measurement of a portion of the corresponding “slice” of the fourier domain (sometimes called the “K-space” or spatial frequency domain) of the scene

•  Each measurement carves out a portion of an annular ring segment of the K-Space

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•  Position of the measurements in K-Space depend on the radar collection geometry as well as the radar waveform parameters used to collect the data

•  Provided the K-Space is properly sampled (according to Sampling Theory)…

Tx Bandwidth Center Frequency

Integration Angle

–  Taking the Inverse Transform of the measurements should provide us with a good estimate of the imaged scene

–  The IDFT assumes data on a rectangular grid in the Fourier Domain

–  Clearly, we have collected data on a POLAR GRID

–  We must interpolate the data from the POLAR GRID to a Rectangular one, then Transform

–  This is the basis for the creation of the POLAR FORMAT ALGORITHM

–  Can be used as long as the azimuth scene diameter is does not exceed the limitations of range curvature

–  Typically under 1km for finest resolutions at Ku

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•  PFA Processing Overview

–  Polar to Rectangular Interpolation is the “trick” to PFA

•  Computationally intensive (although not as bad with today’s DSPs)

•  There are several methods

–  Angular Range Interpolation

»  Not a popular method

–  Radial Keystone Interpolation

»  Popular Method

–  Radial Keystone Correction on Transmit

»  Requires greater hardware control

Polar to Rect Interpolation

Aperture window

2D FFT K-Space Data sampled on Polar raster

K-Space Data re-sampled on rectangular grid

Range/Cross Range Image

Windowed Rectangular format data

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SAR: Spot-Light Polar Format: Keystone

•  Since the most popular method in the literature seems to be the Keystone approach we will cover it in more detail

–  The rectangular sample grid must be calculated based on desired scene size and resolution

•  Generally it is desired to have it fit within the annular ring segment that was collected to avoid “odd” sidelobe structures in the final image’s Impulse Response

–  Doing so, does throw away some data negatively affecting the resolution

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•  The existing samples from be re-sampled to the newly formed grid

–  First a sinc interpolation is performed along the range data to align the data along the radials, resulting in a “Key Stone” or trapezoidal grid of data

–  In Radial Keystone Correction, the radar “slips” the center frequency and sampling of each pulse in order to collect the data on this KeyStone grid, thus skipping this step in the processor!

•  Leaves only a single interpolation to perform at the processor!! •  Requires Pulse to Pulse control of the waveform which has significant impact on

system timing and hardware

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•  Once the Keystone grid is sampled another sinc interpolation is performed to align the data in the Kx dimension

–  Now we have was is called an “Inscribed” region of support, since the grid was created to fit within the polar collected data

–  Typically the data will be multiplied with a 2D window function

•  -35 dB Taylor is typical

–  Slight Loss of Resolution

–  Good sidelobe containment –  Finally, Take the 2D FFT!!

•  Oh and don’t forget to “AutoFocus”

Magnitude of 2D window

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SAR 201: Autofocus

Auto Focus

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SAR: Spot-Light Auto Focus

•  Quick Recap:

–  We learned how to resample our polar collected data onto a pre-determined rectangular grid

•  Allows us to make efficient use of the FFT

–  We now have an image

•  It is an estimate of the complex reflectivity of the 2D scene

–  But its still “Blurry”

•  Sometimes we just cannot get the motion compensation completely right in real time

–  Need to take the image data and analyze it to determine what went wrong and make the necessary corrections

»  AUTOFOCUS

»  So called since there is no user input required

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•  There are several popular types of Auto Focus routines in the literature:

–  Prominent Point

•  Highly Accurate and resolved all orders of phase error

•  Not “Auto” and High Computation requirements

–  Map Drift •  Simple to implement (Low Computation)

•  Only measures quadratic phase errors

–  Multiple Aperture Map Drift

•  Measures some higher order phase errors

•  High Computation

–  Phase Difference

•  Simple, Low Computation, non-iterative

•  Only measures quadratic phase errors –  Phase Gradient

•  Can measure high order phase errors, fairly robust

•  Moderate Computation (depends on iterations)

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•  Phase Gradient Autofocus is probably the most popular

–  Robust performance in diverse clutter environments

•  May have trouble in areas without many prominent scatters (fields, etc)

–  Highly Depends on particular implementation

–  Estimates arbitrarily high orders of phase error

•  Depends on the minimum extent of window used in the processing

–  Often used towards the end of the Image formation chain after most other processing is complete

•  Some image formation processors use one of the other methods to help estimate residual quadratic errors early in the processing chain

–  Gets you “close”

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•  Phase Gradient Autofocus Processing Steps:

–  Input Range Compressed Image data

–  Select Range bins with the brightest scatters (or use them all)

–  Azimuth FFT (we have an image of our particular range bins)

–  Register the brightest points in each range bin (Circular Shift)

–  Window around the peak (Center)

•  Window can be calculated with a threshold from the peak or iteratively reduced by some chosen percentage

–  Here is where implementation matters for scenes that do not have prominent “peaks”

–  Estimate the Phase Gradient

–  Integrate the Phase Gradient

–  Take the result and “exp(-j)” it to create a Phase correction vector

–  Apply the phase correction to input range compressed data (and repeat)

–  Azimuth FFT the corrected data to obtain the focused image

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•  How do you estimate the gradient? –  Several methods

•  Perhaps Two are “popular”

–  LUMVE – Linear Unbiased Minimum Variance Estimate

–  MLE – Maximum Likelihood Estimate

•  LUMVE

–  Where G(u,v) is the shifted and windowed range compressed image data

•  MLE – Based on Shear Averaging

–  Where “a” is a “shift” parameter (typically one pixel or so)

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•  Phase Gradient Autofocus Processing Visuals:

Input Image Extracted Range Bins Shifted Bins Windowed Bins

“a” Pixel shift

G(u,v)

G(u,v-a)

conj

Sum

S(v) – Shear Average

Angle

Smooth data (if you want)

Remove Linear Trends and Biases

Integrate

Store correction, Apply Correction,

Iterate and sum new corrections until converges,

Apply final correction to original input image

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•  Example image is corrupted with high magnitude Quadratic and Sinusoidal errors

0 iterations 4 iterations 8 iterations

12 iterations 16 iterations

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SAR: Spot-Light Image Formation

•  Ok so now we can form an estimate of the 2D radar reflectivity of the scene and focus it

–  Sounds like taking a picture!

•  Is it a picture?

–  Sort of

–  There are specific differences in the physics behind optical and radar imaging

•  Affects the interpretability of a radar image

–  They look very similar (especially at higher radar resolutions)

»  However

»  Radar imagery is not as intuitive as an optical image to an untrained eye

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SAR 201: SAR Imagery

SAR IMAGERY

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•  SAR imagery is formed in Range/Angle (Range - Az) Space where as Optical Imagery is Angle/Angle (Az – El) space

–  SAR imagery is typically viewed in the Slant Plane

•  Makes SAR imagery less intuitive to an untrained viewer

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Radar 101: Fundamentals SAR Imagery

• Since SAR imagery has a “range” dimension it cannot be thought of as a photograph

–  There are a number of differences

• Slant Plane vs Ground Plane

• Range Layover and Foreshortening

• Shadowing

• Multipath Effects

• Movers

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Radar 101: Fundamentals SAR Imagery

•  Slant Plane vs Ground Plane Viewing

–  Grazing angle changes with range in an image

•  Farther range = Lesser grazing angle

–  Cosine of the grazing angle determines the difference between ground and slant ranges

Δr = Δy cos Ψ

•  Resolution and range spacing (in the slant plane) change slightly within an image

Ψ Δy

Δr

Slant Plane

Ground Truth

SAR Image

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SAR 201: Fundamentals SAR Imagery

•  Slant plane vs Ground Plane illusion leads to images appearing upside-down when viewed near-range down as they are typically processed and displayed in the radar processor

–  This is evidenced by the mountain imaged here

Ran

ge

Ran

ge

Ran

ge

Ran

ge

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Target Site: Shadowing due to structure

Foreshortening of the mountain face:

In the image the distance between the top and bottom of the mountain will appear to be small

Target Site: Lay-over from tower makes top appear to be nearer than bottom in image Image Plane

Ground Plane

•  Here is a pictorial view of a fairly complex SAR imaging geometry that we will use to discuss some other radar-specific imaging concerns

–  Notice how the “Range” dimension affects the position of objects in the scene when you compare the Image Plane with the Ground Plane

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•  Range Layover and Shadowing

–  The SAR measures lines of constant range in the slant plane

•  When an elevated target is detected its height makes it appear closer to the radar (layover) and it obscures the beam behind it (shadow)

–  However; layover and shadows allow for assessment of target height!

Shadowing becomes worse with smaller grazing angle while layover effect is lessened

Shadow Layover

Grazing angle Depression angle

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•  Large structures such as buildings or mountains will often cause severe shadowing in imagery that is taken at shallow depression angles

•  They will also appear “foreshortened”

Example of severe

shadowing due to lack of RF line of sight

(mountains)

Front edge of the mountain appears “short” compared to what is expected in an optical image

“Comic-Book Effect”

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• Multipath Effects

–  Multipath is caused from a pulse taking multiple bounces to arrive back at the radar receiver

•  Increased time-till-arrival creates an illusion of increased range to target and often changes the azimuth offset (due to doppler)

–  Leads to artifacts in imagery

Multipath from jet-fighter engine components

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•  Moving Targets

–  Movers in a static SAR image cause streaking

–  Each point in a SAR image has a particular range and doppler associated with it

•  Movers inject additional doppler which “spreads” across the image

–  Cars will often be seen as streaks offset from roads and rotating targets will spread in azimuth

rotating antenna Moving

Cars

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SAR 201

Questions?

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Backup

• BACKUP

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SAR: Alternate Delta R derivation

–  Use the Pythagorean theorem (a^2 + b^2 = c^2)

R

L/2 L/2 ΔR

R

SRP

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•  Image Resolution – What is it really?

–  The 3 dB width of the Impulse Response (IPR) of the radar system

•  Typically measured with a point scatterer (a corner reflector)

•  Higher resolutions allow better signal to clutter ratios and better detection of small targets

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• Multiplicative Noise

–  Why do we care?

• Some target returns may be too bright for the radar to handle correctly

–  Hardware saturation and nonlinearities

• We measure the multiplicative noise ratio (MNR) as a hardware performance metric

Image without Multiplicative Noise

Same Image with Multiplicative Noise

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• DTED – Digital Terrain Elevation Data

–  Having accurate knowledge of the imaged terrain is vital to correcting imagery for layover-like effects when geo-location accuracy is important

–  Steep grazing angles require greater DTED accuracy for the corrections

Range error Range error

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•  Levels of DTED data

Typical Capability Desired Levels for Tactical Use

90 m 30 m 10 m 3 m 1 m

Level I Level II Level III Level IV Level V

Documents

Sar Tutorial