Target Fluctuation Models
MAHRUKH LIAQAT
EE-491 RADAR SYSTEMS (SPRING 2023) CHAPTER 57
Introduction
Major task of a radar system is to detect target (when it is present)
Usually accomplished by threshold detection
Non-fluctuating target – discussed in chapter 3
Fluctuating target – Variation in target RCS
Variation in target geometry
Target Vibration
Change in radar frequency
In this chapter
Statistical models of target echoes are discussed
Emphasis on Swerling models
Factor Determining RCS
Aspect Angle and Frequency Dependence of
RCS
The RCS of real targets cannot be modeled as a constant.
RCS is a complex function of aspect angle, frequency, and
polarization
RCS fluctuations → received target power fluctuations.
Common Statistical Models for Radar Cross
Section
Common Statistical Models for Radar Cross
Section
voltage
RCS of Simple targets - Sphere
The simplest radar target is a perfectly conducting sphere
RCS of a sphere is independent of aspect angle
RCS of Simple targets - dihedral or trihedral
Corner reflector such as the dihedral or trihedral
EE-491 Radar Systems (Spring 2020) Chapter 7
Example - Two-scatterer “Dumbbell”
𝑅 (nominal range) ≫ 𝐷 (separation)
Range to the two scatterers
𝐷
𝑅1,2 ≈ 𝑅 ± sin 𝜃
2
Transmitted signal - 𝑎𝑒 𝑗2𝜋𝑓𝑡
Echo from each scatterer will be proportional to
2𝑅1,2
𝑗2𝜋𝑓 𝑡− 𝑐
𝑎𝑒 .
Two-scatterer “Dumbbell”
The voltage, 𝑦(𝑡), of the composite echo is
therefore
2𝑅1 2𝑅2
𝑗2𝜋𝑓 𝑡− 𝑐 𝑗2𝜋𝑓 𝑡− 𝑐
𝑦 𝑡 ∝ 𝑎𝑒 + 𝑎𝑒
2𝑅 𝑗2𝜋𝑓𝐷 sin 𝜃 𝑗2𝜋𝑓𝐷 sin 𝜃
𝑗2𝜋𝑓 𝑡− 𝑐 − +
𝑦 𝑡 ∝ 𝑎𝑒 𝑒 𝑐 +𝑒 𝑐
2𝑅
𝑗2𝜋𝑓 𝑡− 𝜋𝑓𝐷 sin 𝜃
𝑦 𝑡 ∝ 2𝑎𝑒 𝑐 cos
𝑐
Two-scatterer “Dumbbell”
2𝑅
𝑗2𝜋𝑓 𝑡− 𝑐 𝜋𝑓𝐷 sin 𝜃
𝑦 𝑡 ∝ 2𝑎𝑒 cos
𝑐
𝜎 ∝ 𝑃 ∝ 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 2
Taking the squared magnitude
2 𝜋𝑓𝐷 sin 𝜃
𝜎 ∝ 4𝑎 |cos |2
𝑐
𝜋𝐷 sin 𝜃
𝜎 ∝ 4𝑎2 |cos |2
𝜆
𝜋𝐷 sin 𝜃 2
𝜎 ∝ 4𝑎2 |cos ቚ
𝜆
Radar Cross Section Of Complex Targets
Suppose there are 𝑁 scatterers, each with its own RCS 𝜎𝑖 , located at
ranges 𝑅𝑖 from the radar. The complex voltage of the echo will be
𝑁
2𝑅
𝑗2𝜋𝑓(𝑡− 𝑖 )
𝑦 𝑡 = 𝜎𝑖 𝑒 𝑐
𝑖=1 𝑁
𝑅
−𝑗4𝜋𝑓 𝑐𝑖
𝑦 𝑡 = 𝑒 𝑗2𝜋𝑓𝑡 𝜎𝑖 𝑒
𝑖=1
𝑁
𝑅
−𝑗4𝜋 𝑖
𝑦 𝑡 = 𝑒 𝑗2𝜋𝑓𝑡 𝜎𝑖 𝑒 𝜆
𝑖=1
Radar Cross Section Of Complex Targets
𝜍 is given by
𝑁
𝑅
−𝑗4𝜋 𝑖
𝜍≡ 𝑦 = 𝜎𝑖 𝑒 𝜆
𝑖=1
Target RCS 𝜎 is given by
2
𝑁
𝑅
−𝑗4𝜋 𝑖
𝜎 = 𝜍2 = 𝜎𝑖 𝑒 𝜆
𝑖=1
Radar Cross Section Of Complex Targets
2
𝑁
𝑅
−𝑗4𝜋 𝑖
𝜎= 𝜎𝑖 𝑒 𝜆
𝑖=1
Radar Cross Section Of Complex Targets
EE-491 Radar Systems (Spring 2020) Chapter 7
Radar Cross Section Of Complex Targets
A target whose RCS varies strongly with aspect angle or frequency is
called a fluctuating target
Calculations of detection performance for even moderately complex
targets would be sensitive to aspect angle because of large variation
in RCS and SNR
Statistical Description: composite RCS 𝜎 of the scatterers within a
single resolution cell is considered to be a random variable with a
specified probability density function (PDF)
Using a statistical model for RCS does not imply that the actual RCS
of the target is random.
Radar Cross Section Of Complex Targets
If it was possible to describe the target surface shape and materials
in enough detail, and in addition to identify the radar-target aspect
angle accurately enough, then the RCS could in principle be
computed accurately
Statistical models are used because RCS behavior, even for relatively
simple targets like the previous examples, is extremely complex and
very sensitive to aspect angle.
Statistical model is a simple way to capture the complexity of the
target RCS.
Common Statistical Models for Radar Cross
Section
Target with large number of individual scatterers
randomly distributed in space
each with approximately the same individual RCS.
Phase of the echoes from scatterers can then be assumed to be a
random variable distributed uniformly on (0,2π).
Central limit theorem – real and imaginary parts of the composite
echo can be assumed independent, zero mean Gaussian random
variables with the same variance, say 𝛼 2
Exponential (chisquare of degree 2)
In this case, the squared-magnitude 𝜎 has an exponential PDF
The amplitude voltage, 𝜍 = 𝜎 (more appropriate to a radar using a
linear, rather than square law, detector), has a Rayleigh PDF:
Exponential (chisquare of degree 2)
Rayleigh/exponential model is strictly accurate only in the limit of a
very large number of scatterers
In practice it can be a good model for a target having as few as 10 or
20 significant scatterers.
Radar Cross Section Of Complex Targets
Radar Cross Section Of Complex Targets
EE-491 Radar Systems (Spring 2020) Chapter 7
Exponential (chisquare of degree 2)
Exponential (chisquare of degree 2)
Exponential (chisquare of degree 2)
Chi-square of degree 4
RCS versus aspect angle data set for a 20-scatterer target, but with
an additional dominant scatterer added at a random location.
Chi-square of degree 4
Chi-square of degree 4
Chi-square of degree 2m, Weinstock
Chi-square of degree 2m, Weinstock
Rayleigh / Exponential / Chi-square of degree
2m
Rayleigh / Exponential / Chi-square of degree 2m are examples of
one parameter PDFs
Mean completely specifies the pdf
Variance and mean are related
Mean and variance cannot be adjusted independently
Two parameter pdfs – Weibull, log-normal
Can adequately fit a variety of measured rcs distributions due to
two adjustable parameters
Weibull
Weibull
Log-normal
Log-normal
Log-normal
Mean = 1
Log-normal
Mean = 1
Mean = 0.5 for Exponential
Mean = 1 for other pdfs
Comparison RCS variance = 0.5 for all pdfs
EE-491 Radar Systems (Spring 2020) Chapter 7
Comparison
Mean = 0.5 for Exponential
Mean = 1 for other pdfs
RCS variance = 0.5 for all pdfs
Variation in RCS due to Frequency
RCS Correlation Properties
Correlation in time, frequency, and angle.
Change in time, frequency, or angle required to cause the echo
amplitude to decorrelate to a specified degree.
If a fixed target such as a building - same received complex voltage
𝜍.
Relative motion (change of angle or distance) between radar and
target - the composite echo amplitude fluctuates
RCS Correlation Properties
Changing the radar wavelength causes the relative phase of the
contributing scatterers to change, causing fluctuations.
Decorrelation of the RCS is induced by changes in range, aspect
angle, and radar frequency.
RCS Correlation Properties
2𝑀 + 1 scatterers
Total length of target 2𝑀 + 1 Δ𝑥
Nominal distance of target from
Radar 𝑅𝑜
𝑅𝑜 ≫ 2𝑀 + 1 Δ𝑥
𝑅𝑛 ≈ 𝑅𝑜 + 𝑛Δ𝑥 sin 𝜃
RCS Correlation Properties
𝑅𝑛 ≈ 𝑅𝑜 + 𝑛Δ𝑥 sin 𝜃
Target illuminated with the waveform
𝑎𝑒 𝑗𝑤𝑡
Received signal
𝑀
2𝑅
𝑗𝑤(𝑡− 𝑐 𝑛 )
𝑦 𝑡 = 𝑎𝑒
𝑛=−𝑀
𝑀
2𝑅
𝑗𝑤(𝑡− 𝑐 𝑜 )
= 𝑎𝑒 𝑎𝑒 −𝑗4𝜋𝑛Δ𝑥 sin 𝜃 𝑓/𝑐
𝑛=−𝑀
RCS Correlation Properties
𝑧 = 𝑓 sin 𝜃, 𝛼 = 4𝜋Δ𝑥/𝑐
𝑦 𝑡, 𝑧
𝑀
2𝑅
𝑗𝑤 𝑡− 𝑐 𝑜
= 𝑎𝑒 𝑎𝑒 −𝑗𝑛𝛼𝑧
𝑛=−𝑀
Z contains both f and 𝜃
Calculate auto-correlation of 𝑦 w.r.t
z
𝛼 2𝑀+1 Δz
2𝜋𝑎2 sin[ ]
𝑠 Δ𝑧 = 2
𝛼Δ𝑧
𝛼 sin[ 2 ]
RCS Correlation Properties
𝛼 2𝑀+1 Δz
2𝜋𝑎2 sin[ ]
𝑠 Δ𝑧 = 2
𝛼Δ𝑧
𝛼 sin[ 2 ]
Decorrelation criteria - Δ𝑧
corresponding to first null of
correlation function
𝛼 2𝑀+1 Δz
When =𝜋
2
𝑐
Δ𝑧 =
2𝐿
where 𝐿 = 2𝑀 + 1 Δ𝑥, 𝛼 = 4𝜋Δ𝑥/𝑐
𝑅𝑜 ≫ 2𝑀 + 1 Δ𝑥
RCS Correlation Properties
Amount of aspect angle rotation
required to decorrelate the target
echoes
𝜆 𝑐
Δ𝜃 = =
2𝐿 2𝐿𝑓
Aspect angle changes occur because
of relative motion between radar
and the target
Airliner flying past an airport
𝑅𝑜 ≫ 2𝑀 + 1 Δ𝑥
RCS Correlation Properties
Amount of frequency step required
to decorrelate the target echoes
𝑐
Δ𝑓 =
2𝐿 sin 𝜃
𝐿 sin 𝜃 is projection along the LOS
Generalized to 2-D target
𝑐
𝐿1 × 𝐿2 ∶ Δ𝑓 =
2(𝐿1 | sin 𝜃 +𝐿2 cos 𝜃|)
𝜆 𝑐
Δ𝜃 = =
RCS Correlation Properties 2𝐿 2𝐿𝑓
𝑐
Δ𝑓 =
2𝐿 sin 𝜃
• Consider a target the size of an automobile (5 m)
• At L-band (1 GHz), the target signature can be expected to decorrelate
3×108
in
2×5×10 9 = 30 𝑚𝑟𝑎𝑑 (1.7◦) of aspect angle rotation
• At W-band (95 GHz), this is reduced to only 0.018◦.
• The frequency step required for decorrelation with an aspect angle of
45◦ is 42.4 MHz. This result does not depend on the transmitted
frequency.
𝑓 = 10 𝐺𝐻𝑧
EE-491 Radar Systems (Spring 2020) Chapter 7
𝜆
Δ𝜃 =
2𝐿
RCS Correlation Properties Δ𝑓 =
𝑐
2𝐿 sin 𝜃
𝑐
𝐿1 × 𝐿2 ∶ Δ𝑓 =
2(𝐿1 | sin 𝜃 +𝐿2 cos 𝜃|)
Fixed aspect angle 20 deg. Starting from 10GHz, Δ𝑓 = 18.48 MHz
𝑐
𝐿1 × 𝐿2 ∶ Δ𝑓 =
2(𝐿1 | sin 𝜃 +𝐿2 cos 𝜃|)
Target Fluctuation Models (Swerling Models)
Nonfluctuating -> Swerling 0, Swerling 5, Marcum
Scan-to-scan decorrelation
Pulse-to-pulse decorrelation
Target Fluctuation Models (Swerling Models)
Choice be made between two PDFs and two correlation models
Choice of PDF – from RCS characteristics of target of interest
Choice of correlation model – RCS correlation properties
Frequency Agility – forces decorrelation b/w pulses
Use Swerling 2 or 4
Else – determine whether aspect angle changes by more than Δ𝜃
If Yes – use Swerling 2 or 4 (Pulse to Pulse)
If not – then use Swerling 1 or 3 (Scan to Scan)
Example
Consider a 10m long complex aircraft viewed with a stationary X-
band (10 GHz) radar from a range of 30 km.
The radar is not frequency agile.
Suppose the aircraft is flying at 200 m/s in a crossing direction (i.e.,
orthogonal to the radar LOS).
An exponential PDF is assumed due to the scattering complexity of
the aircraft.
The decorrelation angle is 0.086◦ (1.5 mrad).
Example
The angle between the radar and aircraft will change by 1.5 mrad
when the aircraft has traveled (0.0015)(30 × 103 ) = 45 m, which
occurs in 45/200 = 225 ms.
If the radar collects a dwell in less than 225 ms, Swerling 1 model
should be assumed.
If the PRI is longer than 225 ms, then a Swerling 2 model should be
selected.
Extended Models of Target RCS Statistics
The strategy of the Swerling models can easily be extended to other
target models.
For example, model that uses a log-normal PDF for the target fluctuations
with either pulse-to-pulse or scan-to-scan decorrelation
“long-tailed” distributions are often a better representation of observed
radar data statistics than exponential and chisquare for high range
resolution systems
Small resolution cells isolate one or a few scatterers, undermining the
many-scatterer assumption of the traditional models
A number of results are available in literature
Shnidman’s extension of Swerling models
Doppler Spectrum Of Fluctuating Targets
Doppler Spectrum Of Fluctuating Targets
