Introduction to Synthetic Aperture Radar Dr. Armin Doerry Detailed - - PDF document

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Introduction to Synthetic Aperture Radar

• Dr. Armin Doerry

Detailed contact information at www.doerry.us

1

Major Sections

• Introduction
• Electromagnetic Roots
• Signal Processing
• Image Formation
• Radar Equation (Performance)

2

This presentation is an informal communication intended for a limited audience comprised of attendees to the Institute for Computational and Experimental Research in Mathematics (ICERM) Semester Program on "Mathematical and Computational Challenges in Radar and Seismic Reconstruction“ (September 6 ‐ December 8, 2017). This presentation is not intended for further distribution, dissemination, or publication, either whole or in part.

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Synthetic Aperture Radar ‐ Introduction

3

SAR is first and foremost a radar mode that allows creation of images or maps. Each pixel in a SAR image is a measure of radar energy reflected from that location in the target scene. All the usual advantages of radar apply; penetration of weather, dust, smoke, etc. Images are formed taking advantage of coherent processing

• f radar echoes from multiple

pulses, or over extended

• bservation intervals.

SAR allows resolving target scenes to much finer angles/locations than other real‐beam techniques; effectively synthesizing an antenna much larger than what the platform might otherwise carry.

Synthetic Aperture Radar ‐ Introduction

4

Radar frequencies from VHF through THz have been used for SAR. Lower frequencies offer better penetration of weather, foliage, and even the ground. Higher frequencies offer easier processing to finer resolutions. Pulse radars as well as CW radars can be used. We will hereafter assume generally airborne microwave/mm‐wave pulse radars. SAR images can be formed from aircraft, spacecraft, and ground‐based systems.

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Brief History

• Late 19th century

– Heinrich Hertz shows radio waves can be reflected by metal objects

• November 1903

– Christian Hülsmeyer invents “Telemobiloscope” to detect passing ships

• Reichspatent Nr. 165546, initially filed 21 November 1903.
• June 1951

– SAR idea Invented by Carl A. Wiley, Goodyear Aircraft Co.

• April 1960

– Revelation of first operational airborne SAR system

• Airborne Subsystem – Texas Instruments AN/UPD‐1
• Ground processor – Willow Run Research Center
• February 1961

– First publication describing SAR

• L. J. Cutrona, W. E. Vivian, E. N. Leith, and G. O Hall, "A High‐Resolution Radar

Combat‐Intelligence System," IRE Transactions on Military Electronics, pp 127–131, April 1961.

• June 1978

– First orbital SAR system

• SEASAT

5

Select References

• Synthetic Aperture Radar

– Optics & Photonics News (OPN), November, 2004

6

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Electromagnetic Roots for Radar

7

Outline (more‐or‐less)

• Maxwell’s Equations
• Wave Propagation Equation
• Plane‐Wave Propagation
• Plane‐Wave Reflection
• Radar Range/Delay
• Dielectrics
• Point Sources and Reflections
• Complicated Scattering
• Born Approximation
• Antenna Basics

8

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Maxwell’s Equations

9

t t                  D B B E D H J (1) Gauss’ Law (2) Gauss’ Law for magnetism (3) Faraday’s Law (4) Ampere‐Maxwell Law     D E B H E = Electric Field H = Magnetic Field  = charge density J = current density  = permittivity  = permeability Maxwell’s equations relate electric fields and magnetic fields. They underpin all electrical, optical, and radio technologies.  Electric Displacement field  Magnetic Induction field Everything starts here. Everything starts here. Let there be light.

Vector Calculus Identities/Formulae

10

                         

2 2 S l V S

dV                                                       

   

dS dl dS A B C B C A C A B A B C B A C C A B A B B A A B A B A B B A B A A B A A A A A B A B B A A B B A A A A A                      

 

Stokes theorem Divergence theorem

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Free‐Space Propagation

11

In free‐space there are no currents or charges, and no losses. Maxwell’s equations can be manipulated to

2 2

t       E E and in turn, using some identities, to

2 2 2

t      E E Similarly, for the magnetic field

2 2 2

t      H H In Cartesian coordinates, each component of the vectors E and H satisfy a scalar wave equation. Note that these are second‐order differential equations, with solutions that are sinusoids.

12 7

8.854 10 F m 4 10 H m 299,792,458 m s 377

• hms

c c       

 

          In free‐space 1 c         We further identify Propagation velocity Characteristic wave impedance

Free‐Space Propagation

12

Taking the Inverse Fourier Transform of both sides yields the Helmholtz equations

2 2 2 2

k k       E E H H 2 f f k c         Temporal frequency in Hz (cycles/sec.) Wavenumber in radians/meter Wavelength in meters where we also define Poynting’s theorem shows that the direction and magnitude of energy flow is   P E H As seen in the next few slides, Maxwell’s equations reveal that E and H are perpendicular to each other, and both are also perpendicular to the direction of travel. The orientation of E defines the “polarization” of the plane‐wave. Angular frequency in radians/sec. 2 c f k      We further define Solutions have phase that is a function of both time and space. These ‘waves’ travel, with a free‐ space velocity of propagation

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Sinusoidal Plane‐Wave Propagation

13

A propagating wave with a planar wave‐front is a plane‐wave. The electric field of a linearly polarized plane wave is given by

   

, cos t t    r k r E E  where ˆ ˆ ˆ E k r       k k r r E E Direction of propagation Polarization vector Field observation point ˆ   k E = H E and H fields are related as The Poynting vector is in the direction of ˆ k

Sinusoidal Plane‐Wave Propagation

14

If traveling in the direction of the z‐axis, with an electric field oriented parallel to the x‐axis, our field reduces to simply

2 2 2 2 2

1

x x

E E z c t      with a solution

   

1

, cos

x

E t z e t kz 



  Forward/right travelling Backward/left travelling

   

2

, cos

x

E t z e t kx 



  and another solution E H P z

Wave front (right travelling)

ˆ

x

E  x E and the field equation reduces to x y E and H fields are related as ˆ 1 ˆ        z z E H H E with ˆ ˆ ˆ ˆ ˆ    k r z x E

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Propagation in a Dielectric

15

In a dielectric

r r

       

r r

    where relative permittivity relative permeability These relative quantities are typically greater than one. Complex values denote propagation is lossy. Frequency‐dependence implies a “dispersive” media, where the echo may ‘not’ be a faithful reproduction of the incident signal. In a lossless dielectric, it remains true, but with different numerical values, that 1 c         Propagation velocity Characteristic wave impedance with E and H fields still related as ˆ    z E H with comparable electric field solutions

   

1

, cos

x

E t z e t kz 



 

   

2

, cos

x

E t z e t kx 



  1 ˆ     z H E and Note additionally that k and  are affected.

Fields in a Perfect Conductor

16

In a conductor, we observe electric fields causing charge motion, i.e. a current, with density calculated by Ohm’s law  J = E where  = conductivity. Free charges placed within a conductor will disperse towards the conductor surface, instantaneously, leaving none in the interior, until the total electric field inside the conductor is zero. In a perfect conductor    Bottom line: the surface of a perfect conductor cannot support tangential electric fields, or normal magnetic fields. Bottom line: the surface of a perfect conductor cannot support tangential electric fields, or normal magnetic fields.

Applications of Stokes’ theorem to a perfect conductor boundary shows that tangential electric field must be zero at the boundary. Application of Gauss’ law for magnetic fields yields that the normal electric field may exist, but must do so with a corresponding surface charge density. Applications of Stokes’ theorem to a perfect conductor boundary also shows that a tangential magnetic field may exist, but must do so with a corresponding surface current density. Application of Gauss’ law for magnetic fields yields that the normal component is zero. Actually a part of Maxwell’s

• riginal set of equations

At the conductor boundary: (Time‐varying fields)

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Plane‐Wave Reflection from Perfect Conductor

17

E H Let the forward‐travelling field encounter a perfectly conducting planar surface at normal incidence after distance z0.

Recall that at the conductor surface, the tangential electric field must be zero, and the normal magnetic field must also be zero.

For these boundary conditions to be met, we must also have generated at the surface a backward travelling wave such that

   

, ,

x x

E t z E t z

 

  At distance z0 the forward‐travelling wavefront will have travelled a time z t c  Boundary Conditions This backward travelling field will take an additional time t0 to reach the forward wave starting point z 

   

2 ,0 0,0 ,0 2 ,0

x x x x

z E E c z E t E t c

   

                 Consequently, the backward travelling field is related to the forward‐travelling field by Radar Echo

z

z

Plane‐Wave Reflection from Perfect Conductor

18

A fundamental tenet of monostatic radar is that any generated/transmitted field in free‐space that encounters a reflecting boundary will echo a faithful reproduction (in shape) of the incident signal, to arrive at its origin with a round‐trip time delay of 2

delay

z t c 

In free‐space

       

1 1

, cos , cos 2

x x

E t z e t kz E t z e t kz kz z z  

 

       We observe that the incident and reflected fields are Note that the ratio of the magnitudes of these fields is constant (unity), and independent of frequency Furthermore, the field equation is linear, meaning that any signal that can be written as the sum of sinusoids will exhibit the same reflection characteristics, which means pretty much any signal we can realistically create. True for all frequencies. These observations combine to yield the following:

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Plane‐Wave Reflection from Dielectric Boundary

19

Ei Hi Pi Er Hr Pr Et Ht Pt ,  

1 1

,   The boundary conditions are that the tangential components of E and H must be continuous at the dielectric interface.

incident reflected transmitted

We define

1

   Reflection coefficient Transmission coefficient Note that for a perfect “match”

       

1 1 1 1

2

r x i x t x i x

E z E z E z E z                   and for a perfect conductor

1

  With respect to power, we observe

2 2 1

      Relative reflected power Relative transmitted power

Plane‐Wave Reflection from Dielectric Boundary

20

For oblique angles, and lossy dielectrics, reflections and transmission properties are readily calculated. Furthermore, familiar optical properties

• f reflection, refraction, and Snell’s law

apply. Similarly, for interfaces other than a plane, diffraction applies. The “index of refraction” is still defined as

r r

c n c    

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Spherical Wavefronts

21

Consider a radiating field in a lossless dielectric driven by a forcing function; a “ping” in both time and space, namely

       

2 2 2

, , t t r t t          r r E E

   

0 ˆ

, for 4 r t c t r r            r r E E This has a solution This field is travelling in a radial direction, with diminishing field strength. Furthermore, recall that fields are perpendicular to direction of travel. Observations:

• Recall that fields are perpendicular to

the direction of travel

• A small finite‐dimension area becomes

more planar as r increases

• Power/Energy density diminishes as 1/r2
• Total power/energy crossing the

sphere’s surface remains constant; independent of sphere size. E H

Mythical Point Target

22

More complicated targets are often presumed to be merely collections (clouds) of point reflectors. Consider a reflecting object that

• Occupies a point in space
• Intercepts a portion of a radiated

field, and

• Emanates a reflected field from

that point towards a receiver with finite total power Consider the point reflector intercepting a propagating field with power density  W/m2. Let the point reflector reradiate a field with a power density as seen by a receiver of The point then has a “Radar Cross Section” of

 

2

4 r  

2

R C S m      Furthermore, that point reflects all frequencies equally, and instantaneously, without generating a delay more than its range from the wave emitter and receiver. These assumptions allow tractable processing algorithms to be developed. These assumptions allow tractable processing algorithms to be developed. Using real targets that approximate point target reflectors is an indispensable tool for radar performance evaluation.

Courtesy NASA

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Mythical Point Target

23

For real radar‐hardware testing and evaluation, we like to use targets that mimic a point reflector to some extent. These targets are typically large with respect to wavelength, so geometrical

• ptics principles apply.

The RCS of these “canonical” targets can be calculated with relatively high accuracy and precision… with some caveats. Trihedral “corner” reflector Tophat reflector Sphere

Complicated Scattering

24

We now presume that some incident electric field results in a scattered, or reflected, electric field, with

   

, ,

i r

t t   r r E E The total field is the sum of both, namely

     

, , ,

tot i r

t t t   r r r E E E Incident field scattered/reflected field Scattering occurs from dielectric changes, which causes changes in propagation

• velocity. For convenience we acknowledge

this with the model

   

2 2

1 1 c c    r r The wave equations can be manipulated to the Lippmann‐Schwinger integral equation      

2 2

, , 4

r tot

t c t d d                    



r ρ r ρ ρ ρ r ρ E E

Since the total field also contains the scattered field, this becomes an equation that needs to be solved, which is not tractable except for the simplest of geometries. problem

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Born Approximation

25

To make the problem tractable, we ignore the scattered field on the right side of the equation and approximate the problem as

     

2 2

, , 4

r i

t c t d d                    



r ρ r ρ ρ ρ r ρ E E Born Approximation This is the equivalent to assuming that the scattered/reflected field is generally small/weak compared to the incident field. While this makes the problem tractable, it leads to some errors in rendering radar data, often called multipath ‘artifacts.’ While this makes the problem tractable, it leads to some errors in rendering radar data, often called multipath ‘artifacts.’ Incident field only

See development by Cheney & Borden, and Cheney & Borden.

Born Approximation ‐ Artifacts

26

Jet engine inlets often exhibit characteristic multipath effects.

Far range Near range

Side of monument Ground

Ray trace Direct return, Single bounce Double bounce Triple bounce

This image of a tank seems to suggest 3 cannon barrels. However careful analysis shows that along with the direct return, we have multipath effects of double and triple bounces involving the ground.

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Antenna Basics – Hertzian Dipole

27

The task of creating propagating E and M fields from signal voltages/currents is the function of an antenna. Typical antenna design/analysis begins with establishing a current density J as a forcing function to generate the fields. Consider a short linear current element

• f length h and current strength I0

h

I0

r   Spherical field components, using phasor notation, are

2 2 3 2 3

1 sin 4 2 2 cos 4 1 sin 4

jkr jkr r jkr

I h jk H e r r I h E e r j r I h j E e r r j r

 

          

  

                             In the far‐field, where r is large,

0 sin

4

jkr

j I h E e r E H

  

   



 

Field components are perpendicular to each other, and to the direction of travel

Hertzian dipole h  

See development in Ramo, Whinnery, & Van Duzer

Antenna Basics – Hertzian Dipole

28

     

2 2 2 2 2 2 2

sin 32 3 k I h P r I h W          r For our Hertzian dipole Power density Total radiated power The power density related to that of an isotropic antenna is calculated as

 

2

3 sin 2 p   r The power radiated in some directions has been enhanced at the expense of other directions  Antenna Gain More complicated antennas can be analyzed by treating them as collections

• f infinitesimal Hertzian dipoles, and

superposing the results. As a practical matter, at large distances, we may assume the following

• 1. Differences in the radius vectors to the

elemental dipoles are unimportant in their effect on magnitudes.

• 2. All field components decreasing faster

than 1/r are negligible.

• 3. Differences in the radius vectors to the

elemental dipoles ‘are’ important for their phase, but may be approximated.

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Antenna Basics – Linear Aperture

29

Consider a line‐antenna that is long compared to a wavelength, but with constant‐strength current r h   

h

I0

r  

 

sinc cos

jkr peak

h E e E



         r The field pattern can be calculated as an integral of a line of infinitesimal Hertzian dipoles, resulting in the form The main lobe of this response has a nominal angular beamwidth of

bw

h    The power density within this main beam has been enhanced with respect to an isotropic antenna by

 

2

2 2

bw

h p

 

 



  r Longer antenna, shorter wavelength, mean more/higher gain.

   

sin sinc    

Wave‐fronts are still spherical, but strength varies with direction.

where

Antenna Basics – Far‐Field Pattern

30

Note that the current density with shape

 

1 1 2 rect 1 2 1 2 1 2 l h l x l l h h l h                 has far‐field pattern shape cos cos sinc X h h                  These constitute a Fourier Transform pair

 

cos x l X          It is generally true that the far‐field antenna pattern shape is the Fourier Transform of the current distribution on the radiator. It is generally true that the far‐field antenna pattern shape is the Fourier Transform of the current distribution on the radiator. and can be shaped accordingly

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Antenna Basics – Area Aperture

31

r h   

h

J0

r   In two dimensions r w   

w

 

sinc cos sinc cos

jkr peak

h w E e E



                 r The main lobe of this response has nominal angular beamwidths of

bw

h   

bw

w    The power density within this main beam has been enhanced with respect to an isotropic antenna by

 

2 2 2 2

4 4 4

bw bw

hw p A

   

      

 

   r Larger‐area antenna, shorter wavelength, mean more/higher gain, narrower beam Shape is 2D Fourier Transform of current density Actual aperture area

Antenna Basics – Gain and Effective Area

32

Real antennas radiate only a fraction of the power with which they are supplied. The ratio of total radiated‐power to supplied‐ power is the antenna efficiency. The antenna power gain in the center of its main beam is approximated as

2

4 4

A bw bw

G A          efficiency Just as current densities can cause radiated fields, so too can radiated fields cause current densities. This is the duality nature of antennas. The sensitivity of a receiving antenna versus direction is the same as the field shape for field generation. The power generated by an antenna, useable to subsequent processing, is based on the power density incident. Specifically, received power is the incident power density multiplied by

2

4

e A

A G A       Antenna effective area Bigger antennas are more sensitive Bigger antennas are more sensitive

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Antenna Basics – System Parameters

33

From a systems standpoint, the important parameters of an antenna are

• Frequency of operation
• Bandwidth
• Gain versus angles
• Mainlobe beamwidths
• Pattern shape in all dimensions
• Sidelobes
• Efficiency
• Phase center

Where on the physical structure is the center of the spherical wavefront? Answers the question “range from where?”

Antenna – Examples

34

Various internet sources

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Section Summary

• Maxwell’s equations are the root of all radar behavior

and operation

• Radar is about how radiated fields interact with, and

reflect from, dielectric boundaries

• A “point target” is a useful fiction, and can be physically

approximated for radar analysis

• The Born approximation makes radar analysis tractable,

but comes at a price of artifacts in the data rendering

• Antennas are the transducer between signal

voltages/currents and EM fields

• The far‐field pattern is related to aperture current

distribution by Fourier transform

35

Select References

• Margaret Cheney, Brett Borden, Fundamentals of radar

imaging, Society for Industrial and Applied Mathematics, 2009.

• Margaret Cheney, Brett Borden, “Theory of Waveform‐

Diverse Moving‐Target Spotlight Synthetic‐Aperture Radar,” SIAM J. IMAGING SCIENCES, Vol. 4, No. 4, pp. 1180–1199, 2011.

• Constantine A. Balanis, Advanced Engineering

Electromagnetics, ISBN‐13 978‐0471621942, John Wiley & Sons, Inc., 1989.

• Martin A. Plonus, Applied Electromagnetics, ISBN 0‐07‐

050345‐1, McGraw‐Hill, Inc., 1978.

• Simon Ramo, John R. Whinnery, Theodore Van Duzer,

Fields and Waves in Communication Electronics, ISBN 0‐ 471‐87130‐3, John Wiley & Sons, Inc., 1984.

36

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Mathematics of Signal Manipulation and Processing for Radar

37

From EM Fields to Signals

38

The antenna is a transducer that allows converting propagating fields to/from voltages and currents on transmission lines. Our time‐domain “signal” is the voltage function with respect to time. We manipulate these signals to extract useful information from them, subject to the constraints of the components we employ. This manipulation is nothing more than applying desired mathematical functions to the signals of interest. Some of these manipulations will be to analog signals by electronic circuit elements, and others may be to digital data after conversion.

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Time Domain and Frequency Domain

39

Using Maxwell’s equations, the propagating fields are described by a second‐order differential equation, which has at its natural solution sinusoids. Consequently, it becomes useful to decompose our time‐ domain signals into sinusoidal constituents, and assess signal behavior, and perhaps any modifications to the signals, in terms of those sinusoidal basis functions. Hence the prominence of the Fourier Transform in electrical engineering.

   

2 j ft

X f x t e dt

   

 

   

2 j ft

x t X f e df

   

  Forward transform Inverse transform

   

X f x t 

   

, X f x t are Transform pairs, for which we use the shorthand Customary definitions for electrical engineering

Time Domain and Frequency Domain

40

Signal processing is often described in terms

• f manipulating sinusoidal signals, and/or

sinusoidal signal components. Both purposeful Linear and Nonlinear manipulations are used.

10/14/2017 21

Common Radar Signals

41

A typical radar signal as transmitted is time‐limited, and effectively band‐limited. It has a finite time‐duration. It has the bulk of its energy in some finite frequency band.

f f 

B f

 

X f * * B f  Typically

Manipulating Signals

42

Amplification/Attenuation

   

y t x t   if 1   if 1   if 1   then “attenuation” then “unity gain” then “gain” We often use a logarithmic scale when talking gain or attenuation, with units “decibel,” or “dB.”

 

10

10log dB power gain 

 

10

20log dB voltage gain 

(Caveat: same reference impedance)

This is a Linear processing step.

10/14/2017 22

Manipulating Signals

43

Linear Filtering

Linear filtering is effected to a time‐domain signal by convolution.

     

y t x t h t d  

 

 



In the frequency domain

     

Y f X f H f  If x(t) is a Dirac delta function, then y(t) yields the filter function itself. h(t) is the “impulse response” of the filter H(f) is the “transfer function,” also known as the “frequency response”

• f the filter.

It is important to us that a linear filter exhibits “linearity,” where no mixing products are generated. This is a Linear processing step.

Manipulating Signals

44

Linear Filtering

The linear filter can often be decomposed into sums of derivatives and integrations of the input signal. This may be manipulated to a transfer function of the form

   

 

m m p p

f H f C f     

 

zeros poles In fact, it is often much more common (convenient) to describe linear filters in terms of their transfer function (frequency response) characteristics.

10/14/2017 23

Manipulating Signals

45

Linear Filtering

f f f

 

X f

 

X f

 

X f f

 

X f Low‐pass Band‐pass High‐pass Band‐stop

Manipulating Signals

46

Mixing = signal multiplication

   

1 1 1

cos 2 x t a f t  

   

2 2 2

cos 2 x t a f t  

           

 

 

 

1 2 1 1 2 2 1 2 1 2 1 2 1 2 1

cos 2 cos 2 cos 2 cos 2 2 2 y t x t x t a f t a f t a a a a f f t f f t           Sum frequency Difference frequency The desired component is selected by filtering. Selecting the sum = up‐conversion Selecting the difference = down‐conversion Mixing allows us to translate a band‐limited signal to a different center frequency to allow easier processing. This is a Nonlinear processing step.

10/14/2017 24

Manipulating Signals

47 47

Frequency Multiplication = signal to some power

   

cos 2 x t a f t  

       

 

cos 2 ...

p p

y t x t a p f t         Frequency multiple The desired component is selected by filtering. Frequency multiplication allows us to multiply the bandwidth of a signal; generate wideband signals from narrow‐band signals. This is a Nonlinear processing step.

Signal Processing Diagrams/Schematics

48

We use visual representations of the sequence

• f mathematical operations to communicate

signal processing chains. X  x2 mixer summer frequency doubler amplifier Low‐pass filter Band‐pass filter Signal source Much variability in symbols exists in practice. Symbols often labelled with relevant parameters xxx Special purpose blocks can be labelled as appropriate general linear filters

 

H f

 

h t

10/14/2017 25

Signal Processing Diagrams/Schematics

49

Example: Typical AM Broadcast Radio Receiver

X

Square‐law detector

The arrows mean “tunable” The dotted lines mean ganged, or synchronized. Amplifiers would also typically be labeled with their gain.

535 – 1700 kHz 455 kHz center 10 kHz bandwidth 5 kHz

Antenna input Audio output to speaker

990 – 2155 kHz (this architecture is called a superheterodyne receiver.) Local Oscillator

Modulation/Demodulation

50

We generally do not deal exclusively with single‐frequency sinusoidal tones. Rather we generate/create signals with desired properties and then modify them with signal processing suitable for transmission. This tone modification with interesting signals is called “modulation.” Recovering the original signal (modulation)

• f interest is called “demodulation.”

 

m t = modulation signal,

 

cos 2 f t  = carrier

 

 

 

1 cos 2 m t f t  

 

1 m t 

 

 

cos 2 f t m t   

 

 

cos 2 f t m t dt     Amplitude modulation Phase modulation Frequency modulation

• thers

techniques can be combined

   

cos 2 m t f t  Double‐sideband modulation

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Modulation/Demodulation

51

For high‐power radar transmitters, we prefer phase/frequency modulations, with amplitude modulations typically limited to on/off.

This is because high‐power transmitter components will typically distort the amplitude.

1 2  1 2 1

 

rect  

where  

1 1 2 rect 1 2 1 2 1 2             

A common model for the amplitude envelope is rect

TX

t T      

TX

T  Pulse width of TX pulse

Linear FM Chirp Modulation

52

Envelope

 

2

2 2 2

T T T T T

f d f                   



   

 

, rect cos

n T T T n TX

t t x t n A t t T           For a transmitted (TX) pulse Amplitude modulated with a pulse envelope Frequency modulated with a linear ramp

T

A 

n

t 

TX

T 

 

T t

  Amplitude of TX pulse Reference time of nth TX pulse Pulse width of TX pulse Phase function of TX pulse This is a very popular radar signal modulation, offering some very nice characteristics. This is a very popular radar signal modulation, offering some very nice characteristics. More later. = quadratic phase function t

10/14/2017 27

Linear FM Chirp Modulation

53

 

2

2 2

T T T

f           Recall that the instantaneous frequency is the time‐derivative of phase

   

2 2

T T T

d f d                   Recall also that the finite duration of the chirp stipulates 2 2

TX TX

T T     Customary Properties This implies the frequency ramp between

 

2 2 2

T c c

B B f f                    2

T c TX

B T    where = “chirp bandwidth” Relatively narrow‐band

c

B f  Relatively large time‐bandwidth product 1

TX c

T B  More later.

The chirp bandwidth is merely the span of instantaneous frequencies over which the waveform chirps. It is ‘not’ necessarily the width of the frequency span of significant energy of the Fourier Transform

• f the waveform, although large time‐bandwidth

products approach this.

Quadrature Demodulation

54

In a real bandpass signal, negative frequencies contain the same information as positive frequencies.

f f 

B f

 

X f * *

This is the only part of the spectrum in which we are interested

For easier processing, we will shift the portion of the spectrum in which we are interested to lower, more easier to process, frequencies. We do this with “mixing,” specifically “down conversion,’ and filtering. Easier processing means lower sampling frequency for data sampling, lower data rates, and lesser processing burden. Easier processing means lower sampling frequency for data sampling, lower data rates, and lesser processing burden.

10/14/2017 28

Quadrature Demodulation

55 2 f 

f

 

X f f  * * f

   

LP

X f f H f  * First we mix with a signal Then we use a linear low‐pass filter to trim away undesired energy

2

2

j f t

e

 

The essential information is preserved. The essential information is preserved. This shifts the positive spectra to baseband

Quadrature Demodulation

56

X

2

2

j f t

e

 

   

 

cos 2 x t f t m t    

 

 

2 j m t

y t e



 Note that we began with a real signal, mixed with a complex signal, and output a complex signal, with phase modulation preserved. Although y(t) contains negative frequencies, we say that we are working with the analytic signal of x(t). The problem seemingly is “How do we generate and represent complex signals with real hardware?” Once the signal becomes data, there is no issue. We accomplish this with the following signal processing

10/14/2017 29

Quadrature Demodulation

57

We base our architecture on the Euler formula X

 

2cos 2 f t 

   

 

cos 2 x t f t m t    

 

 

2 j m t

y t e



 X

 

2sin 2 f t  

X  j ADC ADC Analog signal Digital data Analog to digital converter In‐phase channel Quadrature‐phase channel Output data is often called “I/Q” data. cos sin

j

e j



   

Quadrature Demodulation

58

Alternate architecture – Digital Quadrature Demodulation X

 

1

2cos 2 f t 

   

 

cos 2 x t f t m t    

 

 

2 j m t

y t e



 X

 

1

2sin 2 f t  

X  j ADC Analog signal Digital data In‐phase channel Quadrature‐phase channel X

   

1

2cos 2 f f t   f1 is a convenient frequency to make the subsequent digital implementation easier.

This implementation has some advantages with respect to keeping real hardware channels balanced. This implementation has some advantages with respect to keeping real hardware channels balanced. It is particularly convenient when f1 is ¼ the ADC sampling frequency

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Noise

59

The bane of processing real signals is “noise.” These are the perturbations to measured signals resulting from natural random currents and fields, superimposed onto the signals. These obfuscate and yield error to measurements. With some notable exceptions, the noise is modelled as a Gaussian random process, sourced with little or no time correlation. The amount of noise relative to the signal levels we seek to measure is frequently the performance‐limiting factor for radar systems. The amount of noise relative to the signal levels we seek to measure is frequently the performance‐limiting factor for radar systems.

Additive White Gaussian Noise (AWGN)

Signal Similarity

60

The issue is measuring the similarity of two signals. Overwhelmingly, the similarity measures used in radar signal processing are based

• n Euclidean distance, or the L2 norm.

For signals, this manifests as measures of Mean‐Squared Error (MSE). This is tractable because the square of signal magnitude is a power/energy

• measure. Note that energy is a

conserved quantity per the first law of thermodynamics. So MSE is a measure of average, or expected, power/energy in the error. Parseval’s Theorem

   

X f x t  If Then

   

2 2

x t dt X f df

   



 

 

2

X f = Energy Spectral Density (ESD)

10/14/2017 31

Spectral Density and Autocorrelation

61

We identify the Energy Spectral Density as

   

2 X

S f X f  We identify the autocorrelation function as

     

* X

R t x x t dt  

 

 



This measures the distribution of energy with frequency This measures the similarity of a signal with a time‐shifted version of itself The Wiener–Khinchin theorem states that these quantities constitute a Fourier Transform pair

   

X X

S f R t  This is incredibly useful for signal processing, especially

• f random signals.

This is incredibly useful for signal processing, especially

• f random signals.

Matched Filter

62

The question is “Given a signal in AWGN, what filter gives us a minimum MSE in determining the presence of that signal?”

 

h t

   

i

x t n t 

   

• y t

n t  Derivations can be readily found in the

• literature. A typical derivation uses the

Cauchy–Schwarz inequality. In any case, the result is that the optimum filter is

   

*

h t x t   This is just the time‐reversed conjugate

• f the signal we wish to measure, i.e.

“matched” to the desired input. Input AWGN Output Noise

 

h t

 

x t

 

y t With no noise, we have With AWGN input noise, we have

         

*

y t x h t d x x t d      

   

   

 

This is an autocorrelation function, maximum at = 0.

Additive and Gaussian, but no longer white

10/14/2017 32

Matched Filter

63

If we use the same matched filter, but now delay our input, then we calculate

         

* d d

y t x t h t d x t x t d      

   

     

 

This function is maximum at time

d

t t  This allows us to identify the best estimate of delay time for a signal, perhaps a radar return. The output of a matched filter, when input with a signal to which it its matched, is the signal’s autocorrelation function. The output of a matched filter, when input with a signal to which it its matched, is the signal’s autocorrelation function. This fact is important in radar waveform design. Corollary – The Energy Spectral Density of a radar waveform defines the autocorrelation function, which also defines the

• utput of a matched filter.

Corollary – The Energy Spectral Density of a radar waveform defines the autocorrelation function, which also defines the

• utput of a matched filter.

Shape the ESD  shape the MF output.

Matched Filter ‐ Correlator

64

In practice, the matched filter is often implemented as a correlator. We create a set of correlation kernels with different time offsets, that is

     

*

,

k d k

y t x t g t d   

 

 



   

, k

k

g t x t     Then we correlate against the set of kernels The time offset tk that yields the maximum

• utput is our best match to td.

Or, with explicit input noise

     

 

 

*

,

k d d k

y t x t n t g t d    

 

   



10/14/2017 33

Spectral Density and Autocorrelation

65

Scaling property of the Fourier Transform

   

X X

S f R t 

 

1

X X

f S R t           This means that if we make the energy spectrum wider, the consequence is a narrower autocorrelation function; ‐ sharper autocorrelation peaks ‐ more localized output of the matched filter Wider bandwidths yield finer time‐resolution.

Linear FM Chirp Processing

66

Consider a unit‐amplitude Linear FM chirp waveform

 

2

rect cos 2 2

T T TX

t x t f t t T                   Quadrature‐demodulating to baseband and ignoring the constant phase term

 

2

2

rect

T

j t TX

t y t e T



       This signal gets applied to a filter that is matched to it. The output becomes

     

*

z t y y t d   

 

 



       

sin sinc

c TX TX c c

B t z t T T B t B t    

TX

t T  For typical large time‐bandwidth products 1

TX c

T B  2

T c TX

B T    For small time‐offsets, where the bulk of the output energy will be found

(essentially the stationary‐ phase condition)

Then the matched filter output is adequately approximated as

where    

sin sinc    

where

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Linear FM Chirp Processing

67

Recall that the output of a matched filter, when input with a signal to which it is matched, is the autocorrelation function, Consequently,    

sinc

X TX c

R t T B t 

This means that the ESD is calculated to be approximately  

rect

TX X c c

T f S f B B       

 

X

S f f

c

B

TX c

T B

As the time‐bandwidth product shrinks so that 1

TX c

T B  then the spectrum model degrades  

X

S f f

Spectral Density and Matched Filter Output

68

 

X

S f f

c

B

TX c

T B

In this case, where the chirp bandwidth also describes the width of the spectrum itself, i.e. the signal bandwidth. Note that there are many different definitions for various kinds of signal bandwidth. 1

TX c

T B  Recall the Matched Filter output, which we now write as    

sinc

TX

z t T Bt 

TX

T 1 B

As previously discussed, as bandwidth increases, the match filter output becomes more localized.  Finer time‐resolution

c

B B 

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Spectral Density and Matched Filter Output

69

Furthermore, the Energy Spectral Density does ‘not’ uniquely define the waveform, i.e. a Linear FM Chirp. In fact, ‘ANY’ waveform with the same ESD will generate the same Matched Filter output. Additionally, we may even modify the ESD shape in order to effect desirable properties in the Matched Filter output. Regardless of the waveform, note that we began with a pulse

• f width TTX and ended with a

Matched Filter output with width 1

TX

B T 

We refer to this as “pulse‐compression.” The compression ratio will often reach several orders

• f magnitude, depending on the

radar and/or its application.

TX

T B

Sidelobes

70

sidelobes An artifact of limited bandwidth signals, is that the linear processing of those signals generates “sidelobes” in the Matched Filter output. These sidelobes can mask other legitimate and desired Matched Filter

• utput responses, like a secondary

radar target. We don’t like them.

First sidelobes for a sinc function are only ‐13 dB with respect to the mainlobe peak. 13 dB

We can reduce the sidelobes with linear filtering. This effectively ‘un‐matches’ the Matched Filter somewhat, with some degradations, but does desirably impact the problematic sidelobes.

Matched Filter Sidelobe Filter

Hamming window response

10/14/2017 36

Section Summary

• Radar echo signals may be “processed” to more

convenient frequencies by mixing and demodulation

• The optimum (minimum MSE) filter for measuring

radar echoes is the “Matched Filter”

• The Energy Spectral Density (ESD) of a transmitted

waveform ultimately defines the nature of the Matched Filter output

• Wideband signals allow finer time resolution at the

Matched Filter output

71

Select References

• Noise and Noise Figure for Radar Receivers

– Sandia National Laboratories Report SAND2016‐9649

• Catalog of Window Taper Functions for Sidelobe Control

– Sandia National Laboratories Report SAND2017‐4042

• John R. Klauder, A. C. Price, Sidney Darlington, Walter J.

Albersheim, “The theory and design of chirp radars,” Bell Labs Technical Journal, Vol. 39, no. 4, pp. 745‐808, 1960.

• Generating Nonlinear FM Chirp Waveforms for Radar

– Sandia National Laboratories Report SAND2006‐5856

• Shaping the Spectrum of Random‐Phase Radar

Waveforms

– Sandia National Laboratories Report SAND2012‐6915

72

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SAR Image Formation

73

Imaging Geometry Model

74

x y z s rc,n

n



, g n

 rc,n s = vector defining radar location for nth pulse = vector defining target scatterer location SAR is about “space” more than “time.” Consequently, we need a good definition

• f the geometric

relationships between radar and targets. The radar will emit and collect pulses along its flight path. SRP SRP = Scene Reference Point (nominal center of geometric coordinate frame)

10/14/2017 38

Signal Model – TX Signal

75

Everything begins with a signal model. We shall assume a transmitted (TX) signal

• f a single pulse of the form

   

 

, rect cos

n T T T n TX

t t x t n A t t T          

T

A 

n

t 

TX

T 

 

T t

  Amplitude of TX pulse Reference time of nth TX pulse Pulse width of TX pulse Phase function of TX pulse We have tacitly assumed a rectangular pulse envelope. This doesn’t have to be

• so. Other envelopes can be used.

However, rectangular is a popular pulse shape model for systems where TX power amplifiers are driven into compression. In addition, we have assumed an arbitrary phase function. Many phase functions can be used. It is desirable that the phase function be a modulation that provides a desired Energy Spectral Density for the waveform. One common waveform is the Linear Frequency Modulated (LFM) chirp signal.

 

, 2 , ,

2

T n T T n T n

t t t       

, T n

 

, T n

 

, T n

  Reference phase Reference center frequency Reference chirp rate

Envelope Phase function

Signal Model – RX Echo Signal

76

The received (RX) echo signal from the target scatterer is simply an attenuated and delayed version of the transmitted signal, namely

 

   

, ,

, , rect cos

n s n R R T n s n TX

t t t x t n A t t t T             s

, s n

t  Time delay of RX echo pulse for nth pulse

R

A  Amplitude of RX pulse The echo time delay is related to the geometry by the velocity of propagation

• f the waveform

, ,

2

s n s n

t c  r c 

, , s n c n

   r r s Velocity of propagation Vector from target scatterer to radar The factor of 2 accounts for a round trip

This is one of the most fundamental presumptions in radar, and the real starting point for algorithms.

10/14/2017 39

Signal Model – RX Echo Signal

77

x x x x x x x x x Target scene Flight path Pulse locations (synthetic aperture) Each location within the target scene generates a unique range profile along the synthetic aperture.

, s n

r It is the uniqueness of the profile that allows us to separate these locations from each other.

ap

 The collection of pulse locations along the flight path define the synthetic aperture. Pulse spacing is chosen to meet anti‐aliasing requirements for the illuminated scene – another way of saying that radar PRF is chosen to adequately sample Doppler bandwidth of scene illuminated by antenna.

What is Doppler in SAR?

78

The Doppler effect (or Doppler shift) is the change in frequency of a wave (or

• ther periodic event) for an observer

moving relative to its source. It is named after the Austrian physicist Christian Doppler, who proposed it in 1842 in Prague. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer. – Wikipedia, 21 September 2017

   

2 , , rect ... cos 2

R R n s

x t n A f t t r c                 s Received signal from a static range Received signal from a linearly changing range

     

 

     

2 , , rect ... cos 2 4 4 2 rect ... cos 2 1

R R n s s R s n s s n

x t n A f t t r v t t c f f A f v t t r v t t c c c                                        s

, s n s

r  r

 

, s n s s

r v t t    r Time/Frequency scaling inside of a pulse Pulse‐to‐pulse phase change Consider a CW pulse echo (pulse of fixed‐frequency signal)

Principal exploited effect Often ignored

10/14/2017 40

What is Doppler in SAR?

79

A tacit assumption in SAR is that tn increases linearly with pulse index n, which increases linearly with distance flown along a synthetic aperture. So, a pulse‐to‐pulse phase change becomes a spatially‐dependent phase change, i.e. a wavenumber measure. For SAR, ultimately the velocity, and hence the times at which data are collected, is immaterial. The important factor is “where” the synthetic aperture has been sampled. The concept of “Doppler” is just a means to an end.

Signal Model – Baseband Video Signal

80

The RX signal is typically demodulated to baseband for easier processing. Accordingly, we define a Local Oscillator (LO) signal of the form

   

   

, 2rect exp

n m L L n m LO

t t t x t n j t t t T            

m

t  Reference time delay of LO pulse

LO

T  Pulse width of LO pulse Quadrature demodulation yields a video signal of the form

 

 

 

, ,

, , rect rect exp

T n s n n s n n m V R TX LO L n m

t t t t t t t t t x t n A j T T t t t                                  s

 

L t

  Phase function of LO pulse This is just a product of the envelopes and a difference of the phases. We want to translate the echoes to lower frequencies where they are more easily sampled and processed.

10/14/2017 41

Signal Model – Baseband Video Signal

81

If the radar position changes as a function of pulse index n, mainly by changing the aspect angle , then we may write our model as

   

, ,

2 2 , , rect rect exp

n s n T n s n n m V R TX LO L n m

t t t t t t t c c x t n A j T T t t t                                            r r s Or more explicitly in terms of radar position and target position as

   

, ,

2 2 , , rect rect exp

n c n T n c n n m V R TX LO L n m

t t t t t t t c c x t n A j T T t t t                                              r s r s s Note that everything is known in our video signal except the target amplitude AR and target scatterer location s. Recall that this is for a single point target scatterer.

Signal Model – Distributed Target Scene

82

More generally, a target scene is composed of many scatterers, and the echo video signal is a superposition (integration) of all of them. We can write this as

     

, ,

2 2 , , rect rect exp

n c n T n c n n m V TX LO L n m

t t t t t t t c c x t n j d T T t t t                                              



s

r s r s s s s Where the single target amplitude is now replaced by

 

  s Target scene reflectivity function

The question now becomes “If we pick some arbitrary scene location, how much echo signal can we measure as coming from that location?”

If the reflectivity function were a single impulse, then the previous model would apply.

10/14/2017 42

SAR Processing – Matched Filter

83

Let us now pick a test location and create the following reference filter function for a particular scatterer location

   

, ,

2 2 ˆ ˆ ˆ , , rect rect exp

n c n T n c n n m TX LO L n m

t t t t t t t c c h t n j T T t t t                                              r s r s s where ˆ  s Test location All quantities in the filter function are known or specified. Note that this is just the expected response from a unit amplitude target at the test

• location. We want to see how well the actual data matches this.

The task at hand is to “filter” or correlate the input video signal against this function.

SAR Processing – Matched Filter

84

The output of this filter (for a particular scatterer location) is calculated as

     

*

ˆ ˆ , , , ,

V n t

y x t n h t n dt  s s s where “*” denotes complex conjugate. Note that we do this for all pulses and over the entire pulse for each pulse. Doing so for an array of interesting/desired locations will yield a SAR image. ˆ s

A direct implementation of this algorithm is very computationally intensive, and generally prohibitively so. As a result, we attempt to select waveforms and algorithms that get us very close to this result, but with a more tolerable computational load. Nevertheless, all SAR image formation algorithms attempt to at least approximate this “matched‐filter” output.

10/14/2017 43

SAR Processing – Resolution

85

The question is now “If the target is indeed an impulse, how well can we localize the target’s response?” The overall function is called the “Impulse Response” (IPR) of the SAR image. The IPR width is its resolution. This is most often characterized in cardinal directions

• f ‘range’ and ‘azimuth’. Neglecting any sidelobe filtering, we can calculate

2

r T

c B    Slant‐range resolution Azimuth resolution

T

B 

,0

2 cos

a ap g

       

ap

  Bandwidth of TX signal Nominal wavelength of TX signal Nominal angle subtended by synthetic aperture in ground plane

,0 g

  Nominal grazing angle Azimuth resolution depends how much data you collect, not the size of the antenna you carry

SAR Processing – A Zoo of Algorithms

86

There are many different SAR image formation algorithms, and usually many variations

• f each. They all have strengths and weaknesses. A partial list might include

Picking one over the other is done by first examining the following

• parameter space of the data, including frequency, bandwidth,

waveform, imaging geometry, etc.

• Processing constraints, including need for real‐time, image size,

processing hardware, etc. Doppler Beam Sharpening Simple 2D‐DFT (2D‐FFT) Processing Polar Format Algorithm Overlapped Subaperture Algorithm Range Migration Algorithm Chirp‐Scaling Algorithm Wavenumber‐Domain Processing Backprojection Processing Forgive me if I left out your favorite one

10/14/2017 44

SAR Processing – A Zoo of Algorithms

87

Less computationally intensive More computationally intensive Lower fidelity image More approximations to data model Higher fidelity image Less approximations to data model Application dependent design point A fairly typical trade space for algorithm selection might be described by Image Formation Algorithm

Spotlight vs. Stripmap Processing

88 Synthetic aperture

Spotlight SAR …

Spotlight SAR creates a single full 2D image from a single synthetic aperture

Classic Stripmap SAR processes a column of pixels from a single synthetic aperture, and then adds/drops data to form the next column of pixels, for an arbitrarily long composite image.

Stripmap SAR

More typically today, Stripmap SAR images are formed by mosaicking individual Spotlight SAR images formed from non‐

• verlapping distinct synthetic apertures.

Images Courtesy of Sandia National Laboratories, Airborne ISR

10/14/2017 45

Pulse Compression – LFM Chirp Correlation

89

f f f t t t range correlator

• utput

TX pulse

near range echo far range echo

Microwave signals Baseband video signals Correlation filters Range‐ compressed signals Pulse compression via correlation filters is indicated when the range swath is a large fraction (or larger) of the TX pulse width, or for coarser range resolutions. Note that the baseband video signal bandwidth is the same as the microwave signal bandwidth. This model applies to

• ther waveforms, too.

LO pulse

Pulse Compression – LFM Chirp Stretch Processing

90

f t TX pulse

near range echo far range echo

Microwave signals f t t range DFT

• utput

Baseband video signals Range‐ compressed signals LO pulse Pulse compression via stretch processing is indicated when the range swath is a small fraction of the TX pulse width, or for finer range resolutions. Note that the baseband video signal bandwidth is less than the microwave signal bandwidth. Stretch processing generally assumes LFM chirp waveforms. After de‐ramping the echo energy with a LO pulse that is also a chirp, range is encoded in baseband video frequency. Consequently a DFT will separate frequencies, and hence ranges.

10/14/2017 46

Doppler Processing

91

f t f t Geometry with targets displaced in azimuth Target Doppler frequency responses Focused Doppler frequencies

Doppler focus function

Focusing Doppler entails removing the Doppler “chirp” characteristic, and causing targets to correspond to unique constant Doppler frequencies. Target locations displaced in azimuth exhibit different range profiles, that manifest as different echo phases. This causes different phase fluctuations as a function

• f pulse position, i.e.

Doppler. Consequently, an analysis

• f Doppler characteristics

will yield a measure of resolution in azimuth.

Stretch Processing the Video Signal

92

We now choose to narrow the scope of our processing to a TX signal that is a LFM

• chirp. Furthermore, we will assume that the LO signal is also a LFM chirp with the

same constant reference phase, constant center frequency, and constant chirp rate. Furthermore, we will digitize (sample) the video signal such that

 

    

,0 ,0 , 2 ,0 ,

, , exp 2

T T s s n m V R T s n m

T i t t x i n A j t t                    s

 

n m s

t t t T i    where

s

T  ADC sample spacing i  ADC sample index within any one pulse 2 2 I i I    The video signal model then becomes a function of the new index i, which we denote Note that all quadratic phase terms have disappeared. The signal has been “de‐chirped.”

In fast‐time, index i

10/14/2017 47

More Geometry

93

     

, , ,

2 2 ˆ

s n m s n m c n m

t t r r c c       r r s We expand the terms that involve time‐delay into ranges as follows where

m

r  Reference range associated with

m

t We may further expand the relative delay term to

 

2 2 , , ,

2 ˆ ˆ 2

s n m c n c n m

t t r c             r s r s For small horizontal target offsets with respect to range, this may be approximated as

   

 

, , , ,

2 cos cos cos sin

s n m c n m y g n n x g n n

t t r s s c          r This simplification is tantamount to presuming planar wavefronts (instead of spherical) at the SRP. It is often called the “far‐field” approximation. Terms ignored in this simplification generate errors, and limit the SAR scene size and resolutions achievable with adequate fidelity. But it is often “good enough.”

 

, ,

x y z

s s s  s where the target location has coordinates

Processing – Simple 2D‐DFT

94

If we further simplify the geometry to allow small‐angle approximations for

   

 

, , ,0 ,0

2 cos cos

s n m c n m y g x g n

t t r s s c         r and assume the pulses are sampled such that

n

d n    with 2 2 N n N   

 

 

 

,0 , ,0 ,0 ,0 ,0

2 2 cos 2 cos , , exp

T c n m T g T s g V R x y

r c d T x i n A j s n s i c c inconsequential error terms                               r s and make further simplifying assumptions about the video signal, including no motion during the pulse, the we can arrive at a model We will assume that is a constant.

m

r

n



10/14/2017 48

Processing – Simple 2D‐DFT

95

We observe that the first term is a phase term that varies with the range to the SRP. If the flight path were straight, this would be predominantly a quadratic term. Compensating for this term will “focus” the SRP, and some neighborhood around it. Ignoring the inconsequential error terms, and compensating the first term, yields the expression

 

,0 ,0 ,0 ,0

2 cos , , exp 2 cos

T g x V R T s g y

d s n c x i n A j T s i c                       s Note that the first phase term here is linear exclusively in index n, and the second phase term here is linear exclusively in index i. This suggests then that a 2D‐DFT can determine the frequencies of the corresponding indices, and therewith identify target spatial coordinates. The independence of the indices allow us to perform the 2D‐DFT as orthogonal independent 1D‐DFTs. This suggests that the data samples are on a rectangular grid

Assuming, of course, lots of simplifications

Processing – Simple 2D‐DFT

96

We perform the 2D‐DFT conventionally with the summations

   

 

2 2 , , , exp

V s i n

y v u x T i n j u n vi N I      



s where u and v are image pixel indices in the azimuth and range dimensions. Performing the 2D‐DFT yields the “image” with pixel values described by

 

,0 ,0 ,0 ,0

2 cos 2 2 , 2 cos 2 2

T g N x R T s g I y

d N W s u c N y v u A T I W s v c I                                                     where the “shape” of the single point‐target response is given by

     

2 2 1 2

sin sin

f f M j m j M M M m M

f W f e M e M f M

 

 

  

       



This is the complete 2D Impulse Response (IPR) Has peak at f = 0, and mainlobe nominal width of 1

10/14/2017 49

Processing – Simple 2D‐DFT – Extensions

97

 

 

 

,0 , ,0 ,0 ,0 ,0

2 2 cos 2 cos , , exp

T c n m T g T s g V R x y

r c d T x i n A j s n s i c c inconsequential error terms                               r s Recall that we made the assumption that that is a constant in the following signal model

m

r If we had allowed the reference range to equal the range to the SRP, namely

• n a pulse‐to‐pulse basis, then the signal model would be

, m c n

r  r

   

,0 ,0 ,0 ,0

2 cos 2 cos , , exp

T g T s g x y V R

d T s n s i x i n A j c c inconsequential error terms                   s and no independent focusing operation would be needed. The data would already be focused to the SRP.

Real‐time motion compensation.

Processing – Simple 2D‐DFT – Extensions

98

Recall that we assumed that the shape of the IPR in cardinal directions was given by

     

2 2 1 2

sin sin

f f M j m j M M M m M

f W f e M e M f M

 

 

  

       



This function exhibits fairly high processing sidelobes. These sidelobes can be diminished by tapering, or “windowing,” the data prior to DFT processing. For example

   

2 2 1 2 f M j m M M M m M

W f w m e

  

  where

 

M

w m  Window function Popular window functions include Hamming, Hanning, Taylor, and

• Blackman. There are many others.

The cost is a wider mainlobe. This is generally considered a good trade.

• 10
• 8
• 6
• 4
• 2

2 4 6 8 10

• 80
• 70
• 60
• 50
• 40
• 30
• 20
• 10

pixels dBc no window Hamming Hanning

IPR plots

10/14/2017 50

Processing – Simple 2D‐DFT – Extensions

99

If we now include a target with some height offset, that is, then

   

 

, , ,0 ,0 ,0

2 cos cos sin

s n m c n m y g x g n z g

t t r s s s c           r

z

s  This can be rewritten as

     

 

, , ,0 ,0 ,0

2 tan cos cos

s n m c n m y z g g x g n

t t r s s s c           r

 

 

,0 ,0 ,0 ,0 ,0

2 cos 2 2 , 2 cos 2 tan 2

T g N x R T s g I y z g

d N W s u c N y v u A T I W s s v c I                                                       The SAR image is then modified to This indicates that a target that is above the nominal ground‐plane will manifest energy at the equivalent of a ‘nearer’ ground‐range.

“Layover” (foreshortening)

Processing – Simple 2D‐DFT – Limitations

100

The previously termed “inconsequential error terms” in the signal model represent errors can in fact become “consequential” under the right conditions. These errors tend get worse for target scene locations farther away from the SRP, and at finer resolutions. The errors cause excessive blurring in the SAR image. This is due to uncompensated phase errors, as well as residual range migration errors. Degradation tends to be gradual as distance from SRP increases. For simple 2D‐DFT processing, the scene diameter limits are due to residual migration, and are often expressed as

2 ,0

4 cos

y a g

D      4

x a r

D      Azimuth scene diameter limit Ground‐range scene diameter limit For 2 cm wavelength, 1 m resolution, and shallow grazing angles, the scene diameter limit is on the order of 200 m.

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Wavenumbers

101

 

,0 ,0 ,0 ,0

2 cos 2 cos , , exp

T g T s g V R x y

d T x i n A j s n s i c c               s Recall that our focused signal model was expressed as This can be written as

     

 

, , exp

V R x x y y

x i n A j k n s k i s    s

   

,0 ,0 ,0 ,0

2 cos 2 cos

T g x T s g y

d k n n c T k i i c         where These are spatial‐frequency terms, known as “wavenumbers.” The fact that they each depend on an index indicates that they each represent a ‘band’ of wavenumbers. The width of the bands is what allows resolution of the spatial variables. Processing is considerably easier when each is dependent solely on an independent index. We like this.

Processing – Polar Format Algorithm

102

 

    

2 ,0 ,0 ,0 , ,

, , exp 2

T V R T T s s n m s n m

x i n A j T i t t t t                s

   

, , , ,

2 cos cos cos sin sin

s n m y g n n x g n n z g n

t t s s s c         

n

d n   

, m c n

r  r The Polar Format Algorithm (PFA) for SAR image formation “fixes” the principal error sources that limited the 2D‐DFT processing algorithm. We recall our signal model for a sampled constant‐waveform LFM chirp where we are employing stretch processing as We employ motion compensation such that the reference range tracks the range from the radar to the SRP as We also then make the less severe (than for 2D‐DFT processing) approximation and sample with pulse positions at equal angle increments This is still a far‐field planar‐wavefront approximation.

   

 

, , ,0 ,0

2 cos cos

s n m c n m y g x g n

t t r s s c         r

Very simple Less simple

10/14/2017 52

Processing – Polar Format Algorithm

103

The phase term is known as the Residual Video Phase Error (RSVE) and is responsible for the “skew” between near‐range and far‐range in the video signal plot of frequency versus time. It can often be ignored, but in any case can be precisely removed with some preprocessing of the data. The preprocessing involves a DFT followed by a quadratic phase‐error correction, followed by an IDFT. This is sometimes called “deskewing.” In either case, the signal model for the target scene data then becomes

 

2 ,0 ,

2

T s n m

t t  

 

 

, ,0 ,0 , ,

cos sin 2 , , exp cos cos sin

x g n n V R T T s y g n n z g n

s x i n A j T i s c s                                         s

n

d n    where samples are taken at pulse positions This is the starting point for PFA processing.

Processing – Polar Format Algorithm

104

In terms of wavenumbers, we may write the signal model as

 

  

 

  

 

  

,0 ,0 , ,0 ,0 , ,0 ,0 ,

2 , cos sin 2 , cos cos 2 , sin

x T T s g n n y T T s g n n z T T s g n

k i n T i c k i n T i c k i n T i c                    

       

 

, , exp , , ,

V R x x y y z z

x i n A j k i n s k i n s k i n s     s where the orthogonal wavenumber functions are identified as A careful study of these wavenumbers shows that for each data index pair (i,n), the wavenumber triplet (kx,ky,kz) describes a specific location in the 3D Fourier‐space

• f the target scene.

The grazing angle and aperture angle also describe the polar angle in Fourier space, and the radius from DC is described by 





,0 ,0

2

T T s

c T i   

Note that all wavenumbers are functions of both indices. This cross‐coupling

• f indices causes

problematic range migration.

10/14/2017 53

Wavenumbers Again

105

x y z kx ky kz 3D target space 3D wavenumber domain (Fourier‐space of the scene) 3D‐DFT Inverse 3D‐DFT A 3D world nets a 3D wavenumber domain.

Processing – Polar Format Algorithm

106

z

s  Individual data samples are associated with specific locations in the 3D wavenumber

• domain. The collection of all data samples describes a sampled surface in the 3D

wavenumber‐domain of the image. If we know (or presume) that , then we may project these samples onto the kx-ky plane, turning the problem from a 3D problem to a 2D problem. Wavenumber domain

10/14/2017 54

Processing – Polar Format Algorithm

107

Recall that efficient 2D‐DFT processing requires, and assumes that, the samples be on a rectangular grid. In the 2D wavenumber plane, however, the samples are typically not on a rectangular grid. For straight‐line flight with constant waveforms, the samples are on more of a polar grid. However, the data may be resampled from this onto a rectangular grid, usually in two stages, thereby allowing more efficient processing by fast 2D‐DFT techniques. This resampling is called “polar reformatting,” giving name to the algorithm. #1 #2

Processing – Polar Format Algorithm

108

     

 

, , exp

V R x x y y

x i n A j k n s k i s        s This resampling creates new separated indices such that the new resampled signal model can be written for a flat target scene as If we assume that sample spacing is approximately the same as in the center of the

• riginal wavenumber domain data, then performing the 2D‐DFT on this resampled data

again yields the “image” with pixel values described by

 

,0 ,0 ,0 ,0

2 cos 2 2 , 2 cos 2 2

T g N x R T s g I y

d N W s u c N y v u A T I W s v c I                                                    

Wavenumbers are now functions of new independent single indices

10/14/2017 55

Processing – Polar Format Algorithm – Limitations

109

The complete far‐field planar‐wave approximation to the relative delay term yields a higher‐fidelity signal model than for simple 2D‐DFT processing, but is nevertheless still an approximation with some residual phase‐error terms. These phase‐error terms are more pronounced for targets farther away from the SRP, at nearer ranges, and for finer resolutions. These errors also cause excessive blurring in the SAR image, in this case due to mainly uncompensated phase errors, predominantly a quadratic phase‐error. Degradation tends to be gradual as distance from SRP increases. For simple PFA processing, the scene diameter limits are often expressed as

,0

, 4

c x y a

D D     r Azimuth/Range scene diameter limit For 2 cm wavelength, 1 m resolution, and 10 km range, the scene diameter limit is on the order of nearly 900 m. An increase from a simple 2D‐DFT

Processing – Backprojection

110

The Backprojection (BP) image formation algorithm for SAR has its roots in

• tomography. It mitigates errors that limit PFA and 2D‐DFT processing. In fact, it is

essentially a matched‐filter algorithm albeit for range‐compressed data. We recall our stretch‐processing signal model as

 

    

2 ,0 ,0 ,0 , ,

, , exp 2

T V R T T s s n m s n m

x i n A j T i t t t t                s

, m c n

r  r For convenience, we will again employ motion compensation such that the reference range tracks the range from the radar to the SRP as However, now we will abandon any approximations for the relative delay as was done for PFA and 2D‐DFT processing, and specify exactly

 

, s n m

t t 

     

, , , , , ,

2 2 2

s n m s n c n c n c n r n

t t s c c c        r r r s r where

 

, , , r n c n c n

s    r s r = relative slant‐range offset

10/14/2017 56

Processing – Backprojection

111

The RVPE term can be compensated as with PFA processing. Doing so yields the signal model we will use for processing, namely

 

 

 

,0 ,0 ,

2 , , exp

V R T T s r n

x i n A j T i s c       s

 

,0 ,0 , ,

2 2 2 , exp 2

T s T R I r n r n

T I z m n A W s m j s c I c                           Performing range‐compression with a range DFT yields the resulting range‐ compressed signal model Range profile Doppler term We note that m and n are integer indices.

This is really the starting place for BP processing.

Processing – Backprojection

112

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* * *

x The basic idea is to

• 1. select a pixel

location,

• 2. interpolate the

data to the exact range of the pixel,

• 3. align the

phases, and

• 4. accumulate the

result. The data from any

• ne pulse, at any
• ne range, is a

superposition of all targets along that constant‐range arc. It is the different aspect angles that allow separation of targets along that arc.

10/14/2017 57

Processing – Backprojection

113

First we select a pixel location that corresponds to image index pair (v,u). Then we calculate for this pixel a corresponding range‐offset from the SRP for each pulse as ˆ s

 

, , ,

ˆ ˆr n

c n c n

s    r s r Next we calculate a compressed range‐vector fractional index location for each pulse as

,0 ,

2 ˆ 2

T s r n

T I m s c           We then back‐project each pulse by interpolating the compressed‐range vector to the fractional index location and compensating for the Doppler term at that particular range‐

• ffset. Finally this is summed over all pulses for that particular pixel. We calculate this as

   

,0 ,

2 ˆ , , exp

T r n n

y v u z m n j s c         



This is then repeated for every pixel index pair (v,u) in the entire image. (Window functions for sidelobe control can be applied during the initial range‐compression and before the final summation.)

Processing – Backprojection

114

 

,0 ,0 ,0 ,0

2 cos 2 2 , 2 cos 2 2

T g N x R T s g I y

d N W s u c N y v u A T I W s v c I                                                     If we choose a pixel‐spacing similar to what the processing model for PFA and 2D‐DFT processing, then we will end up with a similar image model, namely except that this image model will exhibit higher fidelity over a larger image area and for finer resolutions, etc. In fact, there is no inherent scene‐diameter limitation to the processing itself. This is effectively a matched‐filter result. This fidelity improvement comes at a price of increased computational load over PFA and 2D‐DFT processing.

10/14/2017 58

Some Remaining Issues

• We have tacitly ignored some additional processing issues. These

include

– Compensating for antenna beam pattern roll‐off in the image

• Both azimuth and elevation

– Compensating for range‐loss across the image – Effects of imprecise/inaccurate radar position and motion information

• Employment of “autofocus”

– Effects of time quantization

• Waveforms
• ADC samples

– Effects of Doppler within the pulse itself

• We assumed that the radar was effectively ‘stationary’ during the pulse

– Effects of non‐ideal components/circuits

• Phase errors, amplitude errors
• System delays

– Effects of velocity‐of‐propagation errors

• Atmospheric variations in c

– Selecting more convenient pixel spacing

• Zero‐padding data, etc.

115

Section Summary

• SAR processing attempts to implement a matched‐

filter for each pixel location in the image.

• The different algorithms make various simplifying

assumptions that may or may not be valid, depending on the imaging parameters and fidelity required.

• The LFM chirp has some nice properties for SAR

processing, especially at fine resolutions, but other waveforms can be used, too.

116

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Select References

• Spotlight‐Mode Synthetic Aperture Radar: A Signal Processing Approach,

– Jakowatz, et al., ISBN 0‐7923‐9677‐4

• Spotlight Synthetic Aperture Radar, Signal Processing Algorithms

– Carrara, et al., ISBN 0‐89006‐728‐7

• Basics of Polar‐Format Algorithm for Processing Synthetic Aperture

Radar Images

– Sandia National Laboratories Report SAND2012‐3369

• Range‐Doppler Imaging of Rotating Objects

– J. L. Walker, IEEE Trans. on Aerospace and Electronic Systems, January 1980

• Basics of Backprojection Algorithm for Processing Synthetic Aperture

Radar Images

– Sandia National Laboratories Report SAND2016‐1682

• Computed Tomography – the details

– Sandia National Laboratories Report SAND2007‐4252

• A tomographic formulation of spotlight‐mode synthetic aperture radar

– D. C. Munson, et al., Proceedings of the IEEE, August 1983

• Catalog of Window Taper Functions for Sidelobe Control

– Sandia National Laboratories Report SAND2017‐4042

117

SAR Radar Equation

118

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The Radar Equation

• This is the equation that relates basic radar

performance to basic radar parameters

– i.e. range, resolution, power, etc.

• We will focus on Synthetic Aperture Radar (SAR)

– Monostatic

• Used for performance trade studies

119

Power Radiation Towards Target

The emitted field expands in a spherical manner, diminishing its power density

TX

TX power density (W/m2) at target range

120

2

1 1 1 1 4

T A TX radome atmos

P G L L L R                              

TX Power amplifier

• utput

Loss from TX power amplifier to antenna Antenna gain Range to target Radome loss Atmospheric loss Power dispersed over an ever‐larger spherical surface will diminish over a fixed area.

10/14/2017 61

Interaction With Target

The target captures some of the radiated energy, and then re‐radiates it back towards the receiver, i.e. reflection

121

2

1 1 1 1 4

T A TX radome atmos

P G L L L R                                        Power captured and radiated back towards receiver (W)

TX

Target Radar Cross Section (m2)

Courtesy of Sandia National Laboratories, Airborne ISR

Re‐radiation Back Towards Receiver

The reflected field expands in a spherical manner, diminishing its power density with range = Power density at receiver (W/m2) with range

122

2 2

1 1 1 1 1 1 1 4 4

T A TX radome atmos atmos radome

P G L L L L L R R                                                                     

TX

We are assuming a monostatic radar configuration, i.e. the path from TX to RX is just the reverse of the path from RX to TX.

Power re‐radiated back towards the radar will also expand in an ever‐ larger spherical surface, and therefor diminish

• ver a fixed area

10/14/2017 62

Capture by the Receiving Antenna

The receiving antenna intercepts and collects some of the power in its neighborhood, converting it to power at its terminals. = Power captured by RX antenna (W)

123

2 2

1 1 1 1 1 1 1 4 4

T A e TX radome atmos atmos radome

P G A L L L L L R R                                                                     

TX

Receiver antenna effective area (m2)

Courtesy of Sandia National Laboratories, Airborne ISR

Collecting Terms

Collecting some terms and simplifying yields Signal Power at RX antenna port

124

Combined radar system miscellaneous hardware losses

radar TX radome

L L L   where

TX

 2

4

1 4

T A e r radar atmos

P G A P L L R          

10/14/2017 63

Competing Noise

The received signal power must compete with noise that is also received and/or generated by the radar… This is added to the signal that is received… Noise Power at RX antenna terminals  To receiver

125

r N N

N kTF B   where

23

1.38 10 k



  290 T 

N

F 

N

B  J/K = Boltzmann’s constant K = system reference temperature (nominal scene noise temperature) System noise factor for receiver (referenced to antenna port) Noise bandwidth at antenna port

TX

Noise Sources

• Thermal emissions from the scene to which the antenna is pointed
• Electronic noise in the radar component hardware
• Quantization noise due to the ADC
• Any additional purposeful noise sources used to perhaps ‘dither’ the

ADC data

Not included is “multiplicative” noise due to nonlinear nature

• f radar, integrated sidelobe energy, spurious signals, etc.

126

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Signal to Noise Ratio (SNR) at RX Antenna Port

An important measure of goodness is the ratio of power (energy) of signal to noise. This is a generic form of the Radar Equation that is true for all monostatic radar modes.

127

   

2 4

4

T A e r antenna r radar atmos N N

P G A P SNR N R L L kTF B    

SNR in SAR Image

The SNR can often be improved by signal processing in a manner to “match” the data to our transmitted signal(s). This involves pulse compression and coherently combining multiple pulses, i.e. SAR processing.

DSP

SAR image

128

   

2 4

4

T A e r a image antenna r a radar atmos N N

P G A G G SNR SNR G G R L L kTF B    

r

G  where

a

G  SNR gain due to range processing (pulse compression) SNR gain due to azimuth (Doppler) processing (coherent pulse integration) 

TX

The product GrGa comprises the signal processing gain

10/14/2017 65

The Transmitter

The transmitter generally is constrained by 3 main criteria

• 1. frequency (wavelength) of operation, including bandwidth,
• 2. peak power output, and
• 3. maximum duty factor allowed.

We relate

129

T TX p T avg

P T f P d P    where

TX

T 

p

f  d  TX pulse width Radar Pulse Repetition Frequency (PRF) TX duty factor Average TX power during synthetic aperture

Courtesy of Sandia National Laboratories, Airborne ISR

Survey of TX Power Amplifier Tubes

Higher peak power usually means lower duty factor Higher power also usually means narrower bandwidth.

130

Power Amplifier Tubes

Solid‐state power amplifiers are generally lower‐power than their tube counterparts, typically under 100 W, and more like 10 W to 20 W range (depending on frequency band). However they do offer a possible efficiency advantage, and technology is advancing to the point where these should be considered for relatively short‐range radar applications.

10/14/2017 66

Antenna Details

For a monostatic radar, TX antenna “gain” and RX antenna “effective area” are related to each other by The “effective area” is related to the physical area of the antenna aperture by an efficiency factor For a dish antenna, the physical aperture is the silhouette area of the dish

131

2

4

e A

A G    where   Nominal wavelength of the radar

e ap A

A A  

A

A 

ap

  where Physical area of antenna aperture Aperture efficiency of the antenna (typically on order of 0.5)

Courtesy of Sandia National Laboratories, Airborne ISR

Active Electronic Steered Array (AESA) Antenna

132

While we have thus far explored traditional corporate fed antennas, many newer systems are using AESA antennas. While very promising and attractive on a number of fronts, some care needs to nevertheless be exercised when considering AESA antennas for SAR application

• AESA have gain and beamwidths that are squint‐angle‐dependent
• Wideband waveforms require True‐Time Delay (TTD) steering, or equivalent. Phase

shifters are inadequate.

• AESA beam‐steering does not have the field of regard that a gimballed antenna has.
• AESA antennas still tend to be more expensive than more traditional corporate‐fed

gimballed antennas. AESA antennas that use Digital Beamforming (DBF) techniques on both transmit and receive offer the potential

• f overcoming the wideband operation limitation. They

essentially use DSP techniques to implement TTD steering.

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Signal Bandwidth

The signal bandwidth determines the achievable resolution of the range‐compressed signal. A fundamental rule from Signal Processing is that the Power Spectral Density of a waveform is the Fourier Transform of the Autocorrelation function. The Autocorrelation function is the output of a matched filter when the input is the signal to which it is matched.

133

2

wr T r

a c B   where c 

r

 

wr

a  Velocity of propagation Slant‐range resolution Range IPR broadening factor due to data tapering (windowing) for sidelobe control

Processing Gain – Range Compression

The range processing gain is due to noise bandwidth reduction during the course of pulse compression. This gain is based on matched filter performance.

134

TX N r r

T B G L  where

r

L  Reduction (loss) in SNR gain due to non‐ideal range filtering; a result of using a window taper function for sidelobe control Note that the gain is essentially a time‐bandwidth product. However it is the input noise bandwidth that is important, not the signal bandwidth. The typical presumption is that the input noise bandwidth is wider than the signal bandwidth.

10/14/2017 68

Processing Gain – Pulse Integration

The pulse‐integration processing gain is about coherent summation of multiple pulses. This gain is based on matched filter performance

135

2

p wa a a a x a

f Ra N G L v L     where N 

a

L 

x

v 

wa

a  The total number of pulses integrated Radar velocity component, horizontal and normal to the direction to SRP Azimuth IPR broadening factor due to data tapering (windowing) for sidelobe control Reduction (loss) in SNR gain due to non‐ideal azimuth filtering; a result of using a window taper function for sidelobe control

Taylor Window

Window functions are about ‘detuning’ the filters to get some better sidelobe responses. The downsides are 1. is slightly worse SNR performance, because after all the net filter isn’t precisely matched, and 2. Slightly broadened IPR width, i.e. slightly worse resolution. One popular window for SAR processing is the Taylor window, for which some parameters must also be specified. awa, awr Lr, La

136

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Radar Cross Section (RCS)

Even for simple targets, a variety of frequency dependencies exist.

137

RCS frequency dependence

Courtesy of Sandia National Laboratories, Airborne ISR

2

f

Radar Cross Section (RCS)

Size matters…

138

RCS of a sphere as a function of the circumference normalized by the signal wavelength. In fact, for very small targets, shape doesn’t matter. 10 20 30 10 0.1 1 10 Circumference / wavelength RCS / silhouette area (dB) Rayleigh region Resonance region Optical region

10/14/2017 70

Radar Cross Section (RCS)

SAR usually is interested in the RCS of “distributed” clutter, which is resolution dependent.

139

cos

r a g

              where   Distributed clutter reflectivity, measured in terms of RCS per m2 Frequency dependence is typically characterized by

0, n ref ref

f f            where f is the nominal frequency of interest, and 0,ref is the reflectivity at reference frequency fref. Distributed clutter is usually modelled as a Gaussian‐distributed random process. The characteristic reflectivity is the variance of the associated random variable.

Typical Clutter Reflectivity Values

Reflectivity 0 values typically depend on frequency, grazing angle, polarization, etc. Typical values at Ku‐band (16.7 GHz) might be 5 to 10 dBsm/m2 for urban areas or rocky areas 10 to 15 dBsm/m2 for cropland or forest areas 15 to 20 dBsm/m2 for grasslands 20 to 30 dBsm/m2 for desert areas or road surfaces

140

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Geometry Effects

Note that 0 also generally has a dependency on grazing angle g. This is sometimes embodied in a “constant‐gamma” model for clutter. In this case the reflectivity is modelled as

141

Typically, the radar is specified to operate at a particular height above the ground. Consequently, grazing angle depends on this height, and the slant‐range of operation. For a flat earth this is calculated as sin

g

h R  

2

cos 1

g

h R         where h = height of the radar above the target sin

g

    where   Clutter “gamma”

Typical effects of grazing angle on clutter reflectivity

Miscellaneous Losses

• Signal Processing Losses

– Range & azimuth processing losses due to window functions [typically 2‐3 dB or so] – Straddling losses (target smeared across several pixels) [typically ignored in SAR]

• Radar Losses

– Radar plumbing (between TX amplifier and TX antenna) [typically 1‐2 dB or so] – RX signal path losses often included in system noise factor (figure) [typically 1‐2 dB or so] – Radome [typically 0.5 dB or so]

• Atmospheric Losses

– Due to the less‐than‐clear atmosphere

• Worse losses in adverse weather
• Very frequency dependent

142

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Atmospheric Losses

Atmospheric loss‐rates are very nonlinear with frequency; generally getting worse at higher frequency, with notable peaks at 23 GHz (water absorption line) and about 60 GHz (oxygen absorption lines).

143

We identify the overall atmospheric loss as

10

10

R atmos

L



 where  = two‐way atmospheric loss rate in dB per unit distance

X Ku Ka W

There are some “windows” in the transmission spectrum where we find favored radar bands.

Atmospheric Losses

There are some “windows” in the transmission spectrum where we find favored radar bands. X Ku Ka W

144

10 20 30 40 50 70 100 200 300 400 Frequency (GHz) 0.01 0.1 1 10 100 Attenuation (dB/km) H20 H20 H20 02 02

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Atmospheric Absorption

145

Atmospheric Absorption

146

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Atmospheric Absorption

147

Frequency Dependence

The optimal radar band depends on operating conditions, including geometry and weather conditions.

148

Optimum radar band as a function of range and altitude, assuming 4 mm/Hr rain, n=1, and constant antenna aperture area.

10/14/2017 75

Putting it All Together

There are a number of ways to write the Radar Equation. Some of these include

149

 

  

2 0, 2 3 10

8 1 10

n avg ap A r ref wa ref image R x N radar r a

f P A f a f SNR h cv kTF L L L R R



                          

• r perhaps

   

2 3 0, 3 3 10

cos 1 2 4 10

r n avg A ref g wr wa image R r a ref x N wr radar

P G a a f SNR L L f R v kTF a L



                                                 Each has its own utility.

Comments

• SNR does not depend on azimuth resolution
• PRF can be traded for pulse‐width to keep Pavg constant
• For constant ground‐range resolution, there is no SNR overt

dependence on grazing angle, although 0,ref may itself exhibit some dependence on grazing angle as previously discussed, and atmospheric loss depends on height and range.

• Input noise bandwidth BN has no direct effect on ultimate image
• SNR. Signal bandwidth does not explicitly impact SNR directly, but

rather through a somewhat looser dependence on range resolution and perhaps window loss.

• The expressions in the square brackets are typically nearly unity,
• r at least often presumed to be so, and so are often ignored. If

so, then processing losses should not be double‐counted elsewhere.

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Range Performance

151

At constant grazing angle and neglecting atmospheric losses, SAR image SNR depends essentially on R3. This is because we allow the synthetic aperture to grow with range to keep azimuth resolution constant. However, within any one image formed from a constant‐length synthetic aperture, the SNR remains dependent on R4. R1 R2 image Synthetic aperture R1 R2 image Synthetic aperture R3 dependence R4 dependence

Noise Equivalent Reflectivity (NER)

We define an entity that answers the question “What equivalent clutter level does the noise look like?” This goes by several names

• Sigma‐N
• Sigma‐Noise
• Noise Equivalent Reflectivity (NER)
• Noise Equivalent Sigma 0 (NES0)

In fact, the NER is the clutter reflectivity for which SNR goes to 0 dB.

152

   

3 3 2 3

2 4 cos

x g N radar atmos r a N image avg A r wa

R v kTF L L L L SNR P G a        

0dB

image

N SNR

 



 A typical minimum acceptable NER for X‐band and Ku‐band is 25 dBsm/m2, ‐‐ lower for lower frequencies.

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‐30 ‐25 ‐15 ‐20

Courtesy of Sandia National Laboratories, Airborne ISR

NER Dependencies

We can write the NER equation as

Radar operating geometry Radar hardware limitations Radar signal processing

We can probably argue some of these.

154

 

3 3 2 3

256 cos

T N radar atmos r a N x g wr wa avg A

B F L L L L kT R v c a a P G                          

Constants

Some of these we can control, and some of them we can’t control.

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Radar PRF

The Doppler bandwidth of the data depends on frequency (wavelength) and antenna beamwidth. We need a PRF that sufficiently oversamples the Doppler bandwidth.

155

2

Doppler x az

B v   

where

az

  Antenna nominal azimuth beamwidth (presumed to be small)

p a Doppler

f k B 

where

a

k  Doppler oversample factor, typically about 1.5

(Accounts for antenna beam roll‐off)

Extending SAR Range

• Increase average TX power
• Increase Antenna area

– and/or efficiency

• Select better operating geometry

– Fly higher

• Operate at coarser resolution
• Select lower‐loss operating frequency
• Decrease velocity

– Squint mode

• Decreasing Radar Losses,

Signal Processing Losses, and System Noise Factor

• Easing Weather Requirements
• Changing Reference Reflectivity

156

cos sin

x aircraft pitch squint

v v   

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Geometry Limits vs. Resolution

157

Blake Chart

A tabular form of the Radar Equation is often referred to as a “Blake Chart.” Some sources reserve this term for the Radar Equation rearranged to solve for maximum range.

num den 256*pi^3 7937.607 38.9969 k J/K 1.38E‐23 ‐228.601 T K 290 24.62398 c m/s 3.00E+08 8.48E+01 Slant Range km 2.65E+01 1.33E+02 velocity m/s 36 15.56303 altitude kft 2.50E+01 grazing angle deg 16.68589 ‐0.18683 resolution m 0.1016 aw 1.1822 signal bandwidth Hz 1.74E+09 92.41584 noise figure dB 4 4 Radar losses dB 2 2 Atmospheric loss rate dB/km 0.2 5.31E+00 Ppeak W 320 25.0515 Antenna gain dBi 30.4 60.8 duty factor % 25% ‐6.0206 frequency GHz 16.7 ‐5.24E+01 Lr 0.92 0.92 La 0.92 0.92 awr*awa 1.45E+00 sigma_noise (NER)

‐25

dBsm/m2

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A Note About Dynamic Range

159

Consider a SAR operating at shallow grazing angles at 0.1 m resolution, with a NER of 35 dBsm/m2, very common for high‐fidelity images. This equates to a noise floor with pixel values corresponding to about 55 dBsm. Note that even in rural and suburban clutter, data indicates that we may expect specular targets exhibiting RCS greater than +45 dBsm at least once per square mile. This suggests that the SAR image may easily render with more than 100 dB of dynamic range, and indeed will be required to do so for some exploitation techniques. Consequently, SAR images are often displayed with Dynamic Range compression techniques, like Look‐Up Tables (LUTs). A popular function is square‐root of magnitude. Note that the Human Visual System can only perceive about 42 dB or so of dynamic range in any single image.

Section Summary

• A common measure for image ‘goodness’ with respect

to SNR is the Noise‐Equivalent Reflectivity (NER)

– The clutter level that yields SNR of 0 dB

• The ‘range’ limit for SAR is often calculated when the

NER raises to 25 dB (for X and Ku bands)

– However images can still be useful with more noise than this

• Optimum radar frequency is range dependent

– Longer ranges favor lower frequencies for better penetration – Shorter ranges favor higher frequencies for higher gains

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Select References

• Performance Limits for Synthetic Aperture Radar –

second edition

– Sandia National Laboratories Report SAND2006‐0821

• Performance Limits for Exo‐Clutter Ground Moving

Target Indicator (GMTI) Radar

– Sandia National Laboratories Report SAND2010‐5844

• Performance Limits for Maritime Inverse Synthetic

Aperture Radar (ISAR)

– Sandia National Laboratories Report SAND2013‐9915

• Noise and Noise Figure for Radar Receivers

– Sandia National Laboratories Report SAND2016‐9649

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