1 synthetic-aperture radar (sar) basics. 2 outline spatial resolution range resolution short pulse...

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Synthetic-Aperture Radar (SAR) Basics

2

OutlineSpatial resolution

Range resolution• Short pulse system

• Pulse compression

• Chirp waveform

• Slant range vs. ground range

Azimuth resolution• Unfocused SAR

• Focused SAR

Geometric distortionForeshortening

Layover

Shadow

Radiometric resolutionFading

Radiometric calibration

3

Spatial discriminationSpatial discrimination relates to the ability to resolve signals from targets based on spatial position or velocity.

angle, range, velocity

Resolution is the measure of the ability to determine whether only one or more than one different targets are observed.

Range resolution, r, is related to signal bandwidth, B

Short pulse higher bandwidth

Long pulse lower bandwidth

Two targets at nearly the same range

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Spatial discriminationThe ability to discriminate between targets is better when the resolution distance is said to be finer (not greater)Fine (and coarse) resolution are preferred to high (and low) resolution

Various combinations of resolution can be used to discriminate targets

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Range resolution

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Range resolutionShort pulse radar

The received echo, Pr(t) is

wherePt(t) is the pulse shape

S(t) is the target impulse response

denotes convolution

To resolve two closely spaced targets, r

Exampler = 1 m 6.7 ns

r = 1 ft 2 ns

tStPtP tr

Tx

Rx

T

point target echo

T= 2R/c

2

cor

c

2r

r

7

Range resolutionClearly to obtain fine range resolution, a short pulse duration is needed.

However the amount of energy (not power) illuminating the target is a key radar performance parameter.

Energy, E, is related to the transmitted power, Pt by

Therefore for a fixed transmit power, Pt, (e.g., 100 W), reducing the pulse duration, , reduces the energy E.

Pt = 100 W, = 100 ns r = 50 ft, E = 10 J

Pt = 100 W, = 2 ns r = 1 ft, E = 0.2 J

Consequently, to keep E constant, as is reduced, Pt must increase.

0 t dttPE

Pt

8

More Tx Power??Why not just get a transmitter that outputs more power?

High-power transmitters present problemsRequire high-voltage power supplies (kV)

Reliability problems

Safety issues (both from electrocution and irradiation)

Bigger, heavier, costlier, …

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Simplified view of pulse compression

Energy content of long-duration, low-power pulse will be comparable to that of the short-duration, high-power pulse

1 « 2 and P1 » P2

time

1

Po

we r

P1

P2

2

2211 PP Goal:

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Pulse compressionChirp waveforms represent one approach for pulse compression.Radar range resolution depends on the bandwidth of the received signal.

The bandwidth of a time-gated sinusoid is inversely proportional to the pulse duration.So short pulses are better for range resolution

Received signal strength is proportional to the pulse duration.So long pulses are better for signal reception

B2

c

2

cr

c = speed of light, r = range resolution, = pulse duration, B = signal bandwidth

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Pulse compression, the compromiseTransmit a long pulse that has a bandwidth corresponding to a short pulse

Must modulate or code the transmitted pulseto have sufficient bandwidth, B

can be processed to provide the desired range resolution, r

Example:Desired resolution, r = 15 cm (~ 6”) Required bandwidth, B = 1 GHz (109 Hz)

Required pulse energy, E = 1 mJ E(J) = Pt(W)· (s)

Brute force approach

Raw pulse duration, = 1 ns (10-9 s) Required transmitter power, P = 1 MW !

Pulse compression approach

Pulse duration, = 0.1 ms (10-4 s) Required transmitter power, P = 10 W

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

FM-CW radar block diagram

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FM-CW radarLinear FM sweep

Bandwidth: B Repetition period: TR= 1/fm

Round-trip time to target: T = 2R/c

fR = Tx signal frequency – Rx signal frequency

If 2fm is the frequency resolution, then the range resolution r is

Hz,fc

RB4

Tc

RB4T

2T

Bf m

RRR

m,B2cr

B2

14

FM-CW radarThe FM-CW radar has the advantage of constantly illuminating the target (complicating the radar design).

It maps range into frequency and therefore requires additional signal processing to determine target range.

Targets moving relative to the radar will produce a Doppler frequency shift further complicating the processing.

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Chirp radarBlending the ideas of pulsed radar with linear frequency modulation results in a chirp (or linear FM) radar.

Transmit a long-duration, FM pulse.

Correlate the received signal with a linear FM waveform to produce range dependent target frequencies.

Signal processing (pulse compression) converts frequency into range.

Key parameters:

B, chirp bandwidth

, Tx pulse duration

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Chirp radar

Linear frequency modulation (chirp) waveform

for 0 t

fC is the starting frequency (Hz)

k is the chirp rate (Hz/s)

C is the starting phase (rad)

B is the chirp bandwidth, B = k

C2

C tk5.0tf2cosA)t(s

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Stretch chirp processing

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Challenges with stretch processing

time

TxB Rx

LO

near

farfreq

uen

cy

time

freq

uen

cy near

far

Reference chirp

Received signal (analog)

Digitized signalLow-pass filter

A/D converter

Echoes from targets at various ranges have different start times with constant pulse duration. Makes signal processing more difficult.

To dechirp the signal from extended targets, a local oscillator (LO) chirp with a much greater bandwidth is required. Performing analog dechirp operation relaxes requirement on A/D converter.

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Pulse compression exampleKey system parametersPt = 10 W, = 100 s, B = 1 GHz, E = 1 mJ , r = 15 cm

Derived system parametersk = 1 GHz / 100 s = 10 MHz / s = 1013 s-2

Echo duration = = 100 sFrequency resolution f = (observation time)-1 = 10 kHz

Range to first target, R1 = 150 m

T1 = 2 R1 / c = 1 s

Beat frequency, fb = k T1 = 10 MHz

Range to second target, R2 = 150.15 m

T2 = 2 R2 / c = 1.001 s

Beat frequency, fb = k T2 = 10.01 MHz

fb2 – fb1 = 10 kHz which is the resolution of the frequency measurement

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Pulse compression example (cont.)

With stretch processing a reduced video signal bandwidth is output from the analog portion of the radar receiver.

video bandwidth, Bvid = k Tp (where Tp = 2 Wr /c is the swath’s slant width)

for Wr = 3 km, Tp = 20 s Bvid = 200 MHz

This relaxes the requirements on the data acquisition system (i.e., analog-to-digital (A/D) converter and associated memory systems).

Without stretch processing the data acquisition system must sample a 1-GHz signal bandwidth requiring a sampling frequency of 2 GHz and memory access times less than 500 ps.

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Correlation processing of chirp signals

Avoids problems associated with stretch processing

Takes advantage of fact that convolution in time domain equivalent to multiplication in frequency domain

Convert received signal to freq domain (FFT)

Multiply with freq domain version of reference chirp function

Convert product back to time domain (IFFT)

FFT IFFT

Freq-domain reference chirp

Received signal (after digitization)

Correlated signal

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Signal correlation examples

Input waveform #1High-SNR gated sinusoid, no delay

Input waveform #2High-SNR gated sinusoid, ~800 count delay

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Input waveform #1High-SNR gated sinusoid, no delay

Input waveform #2Low-SNR gated sinusoid, ~800 count delay

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Input waveform #1High-SNR gated chirp, no delay

Input waveform #2High-SNR gated chirp, ~800 count delay

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Input waveform #1High-SNR gated chirp, no delay

Input waveform #2Low-SNR gated chirp, ~800 count delay

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Chirp pulse compression and time sidelobes

Peak sidelobe level can be controlled by introducing a weighting function -- however this has side effects.

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Superposition and multiple targetsSignals from multiple targets do not interfere with one another. (negligible coupling between scatterers)

Free-space propagation, target interaction, radar receiver all have linear transfer functions superposition applies.

Signal from each target adds linearly with signals from other targets.

r = r

range resolution

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Why time sidelobes are a problemSidelobes from large-RCS targets with can obscure signals from nearby smaller-RCS targets.

Time sidelobes are related to pulse duration, .

fb = 2 k R/c

fb

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Window functions and their effectsTime sidelobes are a side effect of pulse compression.

Windowing the signal prior to frequency analysis helps reduce the effect.

Some common weighting functions and key characteristics

Less common window functions used in radar applications and their key characteristics

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Window functionsBasic function:

a and b are the –6-dB and - normalized bandwidths

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Window functions

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Detailed example of chirp pulse compression

t0,tk5.0tf2cosa)t(s C2

C

C2

CC2

C )Tt(k5.0)Tt(f2cosatk5.0tf2cosa)Tt(s)t(s

CC2

C2

2C

2

2TfTk5.0tktf2tk2cos

)TkTtk2Tf2(cos

2

a)Tt(s)t(s

2C

2

Tk5.0tTkTf2cos2

a)t(q

after lowpass filtering to reject harmonics

dechirp analysis

which simplifies to

received signal

quadraticfrequency

dependence

linearfrequency

dependencephase terms

chirp-squaredterm

sinusoidal term

sinusoidal term

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Pulse compression effects on SNR and blind range

SNR improvement due to pulse compression: B

Case 1: Pt = 1 MW, = 1 ns, B = 1 GHz, E = 1 mJ, r = 15 cm

For a given R, Gt, Gr, , : SNRvideo = 10 dBB = 1 or 0 dB

SNRcompress = SNRvideo = 10 dBBlind range = c/2 = 0.15 m

Case 2: Pt = 10 W, = 100 s, B = 1 GHz, E = 1 mJ , r = 15 cm

For the same R, Gt, Gr, , : SNRvideo = – 40 dBB = 100,000 or 50 dB

SNRcompress = 10 dBBlind range = c/2 = 15 km

B

FBTkR4

GGPSNR

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2rtt

compress

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Pulse compressionPulse compression allows us to use a reduced transmitter power and still achieve the desired range resolution.

The costs of applying pulse compression include:added transmitter and receiver complexity

must contend with time sidelobes

increased blind range

The advantages generally outweigh the disadvantages so pulse compression is used widely.

Therefore we will be using chirp waveforms to provide the required range resolution for SAR applications.

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Slant range vs. ground rangeCross-track resolution in the ground plane (x) is theprojection of the range resolution from the slant planeonto the ground plane.

At grazing angles ( 90°), r x

At steep angles ( 0°), x For = 5°, x = 11.5 r

For = 15°, x = 3.86 r

For = 25°, x = 2.37 r

For = 35°, x = 1.74 r

For = 45°, x = 1.41 r

For = 55°, x = 1.22 r

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Azimuth (along-track) resolutionDiscrimination of targets based on their along-track or azimuth position is possible due to the unique phase history associated with each azimuth position.

Note that phase variations and Doppler frequency shifts are analogous since f = d/dt where is phase.

Assuming the phase of the target’s echo is essentially constant over all observation angles, the phase variation is due entirely to range variations during the observation period.

Recall that /2 = 2R/ where R is the slant range, is the wavelength, and the factor of 2 is due to the round-trip path length.

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Along-track resolutionConsider an airborne radar system flying at a constant speed along a straight and level trajectory as it views the terrain.

For a point on the ground the range to the radar and the radial velocity component can be determined as a function of time.Radar position = (0, vt, h), Target position = (xo, yo, 0), Range to target, R(t)

22o

2o 0hytvx0tR

22o

2o hyx0R

22

o2

o

o

hytvx

ytvvtR

td

dR

22

o2o

o

hyx

vy0R

22

o2o

oD

hyx

vy20R

2f

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Along-track resolution

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Along-track resolutionExample

Airborne SAR

Altitude: 10,000 m

Velocity: 75 m/s

Five targets on ground

All cross-track offsets = 5 km

Along-track offsets of -1000, -500, 0, 500, and 1000 m

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Along-track resolutionAlt = 10 km

10000

11000

12000

13000

14000

15000

16000

17000

18000

19000

20000

-15000 -10000 -5000 0 5000 10000 15000Along-track position (m)

Ra

ng

e (

m)

1000 m

500 m

0 m

-500 m

-1000 m

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Alt = 10 km, Vel = 75 m/s, = 10 cm

-1500

-1000

-500

0

500

1000

1500


Do

pp

ler

fre

qu

en

cy

(H

z)

1000 m

500 m

0 m

-500 m

-1000 m

-300

-200

-100

0

100

200

300

-1000 -500 0 500 1000

c


42


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Along-track resolutionExample

Airborne SAR

Altitude: 10,000 m

Velocity: 75 m/s

Five targets on ground

All along-track offsets = 0 m

Cross-track offsets of 5, 7.5, 10, 12.5, and 15 km

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Alt = 10 km

10000

11000

12000

13000

14000

15000

16000

17000

18000

19000

20000


Ra

ng

e (

m)

15 km

12.5 km

10 km

7.5 km

5 km

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Along-track resolutionAlt = 10 km, Vel = 75 m/s, = 10 cm

-1500

-1000

-500

0

500

1000

1500


Do

pp

ler

fre

qu

en

cy

(H

z)

15 km

12.5 km

10 km

7.5 km

5 km

-100

-75

-50

-25

0

25

50

75

100

-750 0 750

c

46


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Along-track resolutionNow solve for R and fD for all target locations and plot lines

of constant range (isorange) and lines of constant Doppler shift (isodops) on the surface.

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

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Along-track resolutionWithout the spatial filtering of the antenna, the azimuth chirp waveform covers a wide bandwidth.

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

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

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Strip-map SAR signal example (no squint)

Single point target at center of scene.

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Strip-map SAR signal example (no squint)Time domain characteristics of single point target.Magnitude of phase history mapped in azimuth and range.

• Constant amplitude in range axis indicates uniform pulse amplitude (no windowing).

• Variation in azimuth represents antenna beam pattern in azimuth plane.

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Strip-map SAR signal example (no squint)Time domain characteristics of single point target.Real part of phase history mapped in azimuth and range.Can be shown that contour of constant phase follows:

Where K is the pulse chirp rate Ka is the azimuth chirp ratet is the fast time index is the slow time index is a constant

• Positive range chirp (K > 0), negative azimuth chirp (Ka < 0)

• Contours of constant phase map as hyperbolae

2a

2 KtK

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Strip-map SAR signal example (no squint)

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Strip-map SAR signal example (no squint)Time domain characteristics of single point target.Real part of phase history mapped in azimuth and range.Can be shown that contour of constant phase follows:

Where K is the pulse chirp rateKa is the azimuth chirp ratet is the fast time index is the slow time index is a constant

• Negative range chirp (K < 0), negative azimuth chirp (Ka < 0)

• Contours of constant phase map as ellipses

2a

2 KtK

57

Unfocused SARProcessing SAR phase data to achieve a fine-resolution image requires elaborate signal processing.

In some cases trading off resolution for processing complexity is acceptable.

In these cases a simplified unfocused SAR processing is used wherein only a portion of the azimuth phase history is used resulting in a coarser azimuth resolution.

In unfocused SAR processing consecutive azimuth samples are added together (in the slow-time domain).Since addition a is simple operation for digital signal processors, the image formation processing is much easier (less time consuming) than fully-focused SAR processing.

58

Unfocused SARSumming consecutive samples, also known as a coherent integration or boxcar filtering, is useful so long as the signal’s phase is relatively constant over the integration interval.

ExampleFor a 20-sample interval the central portion of the chirp waveform (zero Doppler) is relatively constant.For the outer portions of the chirp the phase varies significantly and integrating produces a reduced output.

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

60

Unfocused SARThe resolution improves with increased integration length up to a point when oscillations in the signal are included in the integral.

The maximum synthetic aperture length for unfocused SAR is Lu which

corresponds to a maximum phase shift across the aperture of 45º.

The azimuth resolution for L = Lu is

Notice the range- and frequency-dependencies of y.

)m(,2Ry

)m(,2RLu

61

Focused SARTo realize the full potential of SAR and achieve fine along-track (azimuth) resolution requires matched filtering of the azimuth chirp signal.

Stretch chirp processing, correlation processing, tracking Doppler filters, as well as other techniques can be used in a matched filter process.

However the range processing is not entirely separable from the azimuth processing as an intricate interaction between range and azimuth domains exists which must also be dealt with to achieve the desired image quality.

62

Focused SARConsider the phenomenon known as range walk or range-cell migration.

Variations in range to a target over the synthetic aperture not only introduce a quadratic phase change (resulting in the azimuth chirp) but may also displace echo in the range (fast-time) domain.

63

Focused SARIf the range to the target varies by an amount greater than the range resolution then the range-cell migration must be compensated during the image formation processing.

Details on the processing required to achieve fully-focused fine-resolution SAR images will be addressed later.

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Focused SARIn SAR systems a very long antenna aperture is synthesized resulting in fine along-track resolution.For a synthesized-aperture length, L, the along-track resolution, y, is

L is determined by the system configuration.For a fully focused stripmap system, Lm = azR (m), where

az is the azimuthal or along-track beamwidth of the real antenna (az /ℓ)

R is the range to the target

For L = Lm, y = ℓ/2 (independent of range and wavelength)

L2Ry

65

Radiometric resolution -- signal fadingFor extended targets (and targets composed of multiple scattering centers within a resolution cell) the return signal (the echo) is composed of many independent complex signals.

The overall signal is the vector sum of these signals.

Consequently the received voltage will fluctuate as the scatterers’ relative magnitudes and phases vary spatially.

Consider the simple case of only two scatters with equal RCSs separated by a distance d observed at a range Ro.

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Signal fadingAs the observation point moves along the x direction, the observation angle will change the interference of the signals from the two targets.

The received voltage, V, at the radar receiver is

where

The measured voltage varies from 0 to 2, power from 0 to 4.Single measurement will not provide a good estimate of the scatterer’s .

ba Rk2j0

Rk2j0 eVeVV

sin2

dRR,sin

2

dRR 0b0a

sind2

cosV2V 0

Note: Same analysis used for antenna arrays.

67

Fading statisticsConsider the case of Ns independent scatterers (Ns is large) where the voltage due to each scatterer is

The vector sum of the voltage terms from each scatterer is

where Ve and are the envelope voltage and phase.

It is assumed that each voltage term, Vi and i are independent random variables and that i is uniformly distributed.

The magnitude component Vi can be decomposed into orthogonal components, Vx and Vy

where Vx and Vy are normally distributed.

iji eV

je

N

1i

ji eVeVV

s

i

iiyiix sinVVandcosVV

68

Fading statisticsThe fluctuation of the envelope voltage, Ve, is due to fading

although it is similar to that of noise.

The models for fading and noise are essentially the same.

Two common envelope detection schemes are considered, linear detection (where the magnitude of the envelope voltage is

output) and square-law detection (where the output is the square

of the envelope magnitute).

Linear detection, VOUT = |VIN| = Ve

It can be shown that Ve follows a Rayleigh distribution

0V,0

0V,e2

VVp

e

e2V

2e

e

22e where 2 is the variance

of the input signal

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Fading statistics (linear detection)

For a Rayleigh distribution

the mean is

the variance is

The fluctuation about the meanis Vac which has a variance of

So the ratio of the square of the envelope mean to the variance of the fluctuating component represents a kind of inherent signal-to-noise ratio for Rayleigh fading.

22e 2V

2Ve

222

e2e

2ac 429.0

22VVV

dB6.5or66.3VV 2ac

2

e

70


An equivalent SNR of 5.6 dB (due to fading) means that a single Ve measurement will have significant uncertainty.

For a good estimate of the target’s RCS, , multiple independent measurements are required.

By averaging several independent samples of Ve, we

improve our estimate, VL

where

N is the number of independent samples

Ve is the envelope voltage sample

N

1ieiL V

N

1V

71


The mean value, VL, is unaffected by the averaging process

However the magnitude of the fluctuations are reduced

And the effective SNR due to fading improves as 1/N

As more Rayleigh distributed samples are averaged the distribution begins to resemble a normal or Gaussian distribution.

eL V2V

N429.0V 22ac

72

Fading statistics (square-law detection)

Square-law detection, Vs = Ve2

The output voltage is related to the power in the envelope. It can be shown that Vs follows an exponential distribution

Again the mean value is found

and the variance is found(note that )

Again for a single sample measurement yields a poor estimate of the mean.

0V,0

0V,e2

1Vp

s

s2V

2s

2s where 2 is the variance

of the input signal

2

s

2

s2s

2sac VVVV

22es 2VV

2

s2s V2V

dB0or1VV 2sac

2

s

73


An equivalent SNR of 0 dB (due to fading) means that a single Vs measurement will have significant uncertainty.

For a good estimate of the target’s RCS, , multiple independent measurements are required.

By averaging several independent samples of Vs, we

improve our estimate, VL

where

N is the number of independent samples

Vs is the envelope-squared voltage sample

N

1isiL V

N

1V

74


The mean value, VL, is unaffected by the averaging process

However the magnitude of the fluctuations are reduced

And the effective SNR due to fading improves as 1/N.

As more exponential distributed samples are averaged the distribution begins to resemble a 2(2N) distribution.For large N, (N > 10), the distribution becomes Gaussian.

2L 2V

NVV2

s2sac

75

Independent samplesFading is not a noise phenomenon, therefore multiple observations from a fixed radar position observing the same target geometry will not reduce the fading effects.

Two approaches exist for obtaining independent sampleschange the observation geometry

change the observation frequency (more bandwidth)

Both methods produce a change in which yields an independent sample.

Estimating the number of independent samples depends on the system parameters, the illuminated scene size, and on how the data are processed.

76

Independent samplesIn the range dimension, the number of independent samples (NS) is the ratio of the range of the illuminated

scene (Wr) to the range resolution (r)rS RN

77

Independent samplesWhen relative motion exists between the target and the radar, the frequency shift due to Doppler can be used to obtain independent samples.

The number of independent samples due to the Doppler shift, ND, is the product of the Doppler bandwidth, fD, and

the observation time, T

The total number of independent samples is

In both cases (range or Doppler) the result is that to reduce the effects of fading, the resolution is degraded.

TfN DD

DS NNN

78

Independent samples

N = 1 N = 10

N = 50 N = 250

79

Radiometric calibrationTranslating the received signal power into a target’s radar characteristics (cross section or attenuation) requires radiometric accuracy.

From the radar range equation for an extended target

we know that the factor affecting the received signal power include the transmitted signal power, the antenna gain, the range to the target, and the resolution cell area.

Uncertainty in these parameters will contribute to the overall uncertainty in the target’s radar characteristics.

43

2t

2

rR4

AGPP

80

Radiometric calibrationTransmit power, Pt

Addition of an RF coupler or power splitter at the transmitter output permits continuous monitoring of the transmitted signal power.

Antenna gain, GThe antenna’s radiation pattern must be well characterized. In many cases the antenna must be characterized on the platform (aircraft or spacecraft) as it’s immediate environment may affect the radiation characteristics. Furthermore the characterization may need to be performed in flight.

Target range, RRadar’s inherent ability to measure range accurately minimizes any contribution to radiometric uncertainty.

Resolution cell area, ADifficult to measure directly, requires measurement data from extended target with known .

81

Radiometric calibrationCalibration targets

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

82

Radiometric calibrationFlat plate

83

Radiometric calibrationDihedral and trihedral corner reflectors

84

Radiometric calibrationDihedral and trihedral corner reflectors

85

Radiometric calibrationLuneberg lens

86

Radiometric calibrationLuneberg lens

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Radiometric calibrationActive radar calibrator [Brunfeldt and Ulaby, IEEE Trans. Geosci. Rem. Sens., 22(2),

pp. 165-169, 1984.]

89

Radiometric calibrationActive radar calibrator

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91

Radiometric calibrationRCS of some common shapes

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1 synthetic-aperture radar (sar) basics. 2 outline spatial resolution range resolution short pulse...

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