1 radar basic -part i 1

RADAR Basics
Part I
SOLO HERMELIN
Updated: 27.01.09Run This
http://www.solohermelin.com

Table of Content
SOLO
Radar Basics
Introduction to Radars
Basic Radar Concepts
The Physics of Radio Waves
Maxwell’s Equations:
Properties of Electro-Magnetic Waves
Polarization
Energy and Momentum
The Electromagnetic Spectrum
Dipole Antenna Radiation
Interaction of Electromagnetic Waves with Material
Absorption and Emission
Reflection and Refraction at a Boundary Interface
Diffraction
Atmospheric Effects

Table of Content (continue – 1)
SOLO
Radar Basics
Basic Radar Measurements
Radar Configurations
Range & Doppler Measurements in RADAR Systems
Waveform Hierarchy
Fourier Transform of a Signal
Continuous Wave Radar (CW Radar)
Basic CW Radar
Frequency Modulated Continuous Wave (FMCW)
Linear Sawtooth Frequency Modulated Continuous Wave
Linear Triangular Frequency Modulated Continuous Wave
Sinusoidal Frequency Modulated Continuous Wave
Multiple Frequency CW Radar (MFCW)
Phase Modulated Continuous Wave (PMCW)

SOLO
Radar Basics
Pulse Radars
Non-Coherent Pulse Radar
Coherent Pulse-Doppler Radar
Range & Doppler Measurements in Pulse-Radar Systems
Range Measurements
Range Measurement Unambiguity
Doppler Frequency Shift
Resolving Doppler Measurement Ambiguity
Resolution
Doppler Resolution
Angle Resolution
Range Resolution

SOLO
Radar Basics
Pulse Compression Waveforms
Linear FM Modulated Pulse (Chirp)
Phase Coding
Poly-Phase Codes
Bi-Phase Codes
Frank Codes
Pseudo-Random Codes
Stepped Frequency Waveform (SFWF)

SOLO
Radar Basics
RF Section of a Generic Radar
Antenna
Antenna Gain, Aperture and Beam Angle
Mechanically/Electrically Scanned Antenna (MSA/ESA)
Mechanically Scanned Antenna (MSA)
Conical Scan Angular Measurement
Monopulse Antenna
Electronically Scanned Array (ESA)

SOLO
Radar Basics
Transmitters
Types of Power Sources
Grid Pulsed Tube
Magnetron
Solid-State Oscillators
Crossed-Field amplifiers (CFA)
Traveling-Wave Tubes (TWT)
Klystrons
Microwave Power Modules (MPM)
Transmitter/Receiver (T/R) Modules
Transmitter Summary
RADAR
BASICS
PART
II

SOLO
Radar Basics
Radar Receiver
Isolators/Circulators
Ferrite circulators
Branch- Duplexer
TR-Tubes
Balanced Duplexer
Wave Guides
Receiver Equivalent Noise
Receiver Intermediate Frequency (IF)
Mixer Technology
Coherent Pulse-RADAR Seeker Block Diagram
RADAR
BASICS
PART
II

SOLO
Radar Basics
Radar Equation
Radar Cross Section
Irradiation
Decibels
Clutter
Ground Clutter
Volume Clutter
Multipath Return
R
A
D
A
R
B
A
S
I
C
S
P
A
R
T
II

SOLO
Radar Basics
Signal Processing
Decision/Detection Theory
Binary Detection
Radar Technologies & Applications
References
R
A
D
A
R
B
A
S
I
C
S
P
A
R
T
II

SOLO
Radar Basics
The SCR-270 operating position shows
the antenna positioning controls,
oscilloscope, and receiver. Photo from
"Searching The Skies"
A mobile SCR-270 radar set. On December 7,
1941, one of these sets detected Japanese
aircraft approaching Pearl Harbor.
Unfortunately, the detection was misinterpreted
and ignored. Photo from "Searching The Skies"

SOLO
Radar Basics
Limber Freya radar
Freya was an early warning
radar deployed by Germany
during World War II, named
after the Norse Goddess
Freyja. During the war over a
thousand stations were built.
A naval version operating on
a slightly different wavelength
was also developed as Seetakt.
Freya was often used in
concert with the primary
German gun laying radar,
Würzburg Riese ("Large
Wurzburg"); the Freya
finding targets at long
distances and then "handing
them off" to the shorter-
ranged Würzburgs for
tracking.

SOLO
Radar Basics
Würzburg mobile radar trailer
The Würzburg radar was the primary ground-based gun laying
radar for both the Luftwaffe and the German Army during
World War II. Initial development took place before the war,
entering service in 1940. Eventually over 4,000 Würzburgs of
various models were produced. The name derives from the
British code name for the device prior to their capture of the
first identified operating unit.
In January 1934 Telefunken met with German radar
researchers, notably Dr. Rüdolf Kuhnhold of the
Communications Research Institute of the German Navy
and Dr. Hans Hollmann, an expert in microwaves, who
informed them of their work on an early warning radar.
Telefunken's director of research, Dr. Wilhelm Runge, was
unimpressed, and dismissed the idea as science fiction. The
developers then went their own way and formed GEMA,
eventually collaborating with Lorenz on the development of
the Freya and Seetakt systems.
Country of origin Germany
Introduced 1941
Number built around 1500
Range up to 70 km (44 mi)
Diameter 7.5 m (24 ft 7 in)
Azimuth 0-360º
Elevation 0-90º
Precision ±15 m (49 ft 2½ in)

SOLO
Basic Radar Concepts
A RADAR transmits radio waves toward an area of interest
and receives (detects) the radio waves reflected from the objects
in that area.
RADAR: RAdio Detection And Ranging
Range to a detected object is determinate by the time, T, it takes the radio
waves to propagate to the object and back
R = c T/2
Object of interest (targets) are detected in
a background of interference.
Interference includes internal and external noise, clutter (objects
not of interest), and electronic countermeasures..
Radar Basics

http://www.radartutorial.eu
Radar BasicsSOLO

SOLO
Return to Table of contents
Radar Basics

SOLO
Electromagnetic Energy propagates (Radiates) by massless elementary “particles”
known as photons. That acts as Electromagnetic Waves. The electromagnetic energy
propagates in space in a wave-like fashion and yet can display particle-like behavior.
The electromagnetic energy can be described by:
• Electromagnetic Theory (macroscopic behavior)
• Quantum Theory (microscopic behavior)
Photon Properties
- There are no restrictions on the number of photons which can exist in a region
with the same linear and angular momentum. Restriction of this sort (The Pauli
Exclusion Principle) do exist for most other particles.
- The photon has zero rest mass (that means that it can not be in rest in any inertial
system)
- Energy of one photon is: ε = h∙f h = 6.6260∙10-34
W∙sec2
– Plank constant
f - frequency
- Momentum of one photon is: p = m∙c = ε/c = h∙f/c
- The Energy transported by a large number of photons is, on the average, equivalent
to the energy transferred by a classical Electromagnetic Wave.
Radar Basics

SOLO
Radio Waves are Electro-Magnetic (EM) Waves, Oscillating Electric and Magnetic
Fields.
The Macroscopic properties of the
Electro-Magnetic Field is defined by
Magnetic Field IntensityH

[ ]1−
⋅mA
Electric DisplacementD

[ ]2−
⋅⋅ msA
Electric Field IntensityE

[ ]1−
⋅mV
Magnetic InductionB

[ ]2−
⋅⋅ msV
The relations between those quantities and the
sources were derived by James Clerk Maxwell in 1861
James Clerk Maxwell
(1831-1879)
1. Ampère’s Circuit Law (A) eJ
t
D
H



+
∂
∂
=×∇
2. Faraday’s Induction Law (F) t
B
E
∂
∂
−=×∇


3. Gauss’ Law – Electric (GE) eD ρ=⋅∇

4. Gauss’ Law – Magnetic (GM) 0=⋅∇ B

André-Marie Ampère
1775-1836
Michael Faraday
1791-1867
Karl Friederich Gauss
1777-1855
Maxwell’s Equations:
Electric Current DensityeJ

[ ]2−
⋅mA
Free Electric Charge Distributioneρ [ ]3−
⋅⋅ msA
z
z
y
y
x
x
111:
∂
∂
+
∂
∂
+
∂
∂
=∇
Radar Basics

SOLO Waves
2 2
2 2 2
1
0
d s d s
d x v d t
− =Wave Equation
Regressive wave Progressive wave
run this
-30 -20 -10
0.6
1.
0.8
0.4
0.2
In the same way for a
3-D wave
( ) ( )
2 2 2 2 2
2
2 2 2 2 2 2 2
1 1
, , , , , , 0
d s d s d s d s d
s x y z t s x y z t
d x d y d z v d t v d t
+ + − = ∇ − =






−=
v
x
tfs





+=
v
x
ts ϕ

 









−=










−=
y
y
v
x
tf
yd
d
td
sd
v
x
tf
yd
d
vxd
sd
2
2
2
2
2
2
22
2
&
1

 









+=










+=
z
z
v
x
t
zd
d
td
sd
v
x
t
zd
d
vxd
sd
ϕ
ϕ
2
2
2
2
2
2
22
2
&
1

EM
Wave
Equations
SOLO
ELECTROMGNETIC WAVE EQUATIONS
For Homogeneous, Linear and Isotropic Medium
ED

ε=
HB

µ=where are constant scalars, we haveµε,
t
E
t
D
H
t
t
H
t
B
E
ED
HB
∂
∂
=
∂
∂
=×∇
∂
∂
∂
∂
−=
∂
∂
−=×∇×∇
=
=






εµ
µ
ε
µ
Since we have also
tt ∂
∂
×∇=∇×
∂
∂
( )
( ) ( )


















=⋅∇=
∇−⋅∇∇=×∇×∇
=
∂
∂
+×∇×∇
0&
0
2
2
2
DED
EEE
t
E
E




ε
µε
t
D
H
∂
∂
=×∇


t
B
E
∂
∂
−=×∇


For Source-less
Medium
02
2
2
=
∂
∂
−∇
t
E
E


µε
Define
meme KK
c
KK
v ===
∆
00
11
εµµε
where ( )
smc /103
10
36
1
104
11 8
9700
×=






××
==
−−
∆
π
π
εµ
is the velocity of light in free space.
2
2
2
0
H
H
t
µε
∂
∇ − =
∂


same way

SOLO
Given a monochromatic (sinusoidal) E-M wave  ( )0 0sin 2 sin
: /
x
E E f t E t k x
c
k c
ω
π ω
ω
  
= − = −  ÷
   
=
  
Period T,
Frequency f = 1/T
Wavelength λ = c T =c/f
c – speed of light
Run This

POLARIZATION
SOLO
Electromagnetic wave in free space is transverse ; i.e. the Electric and Magnetic Intensities
are perpendicular to each other and oscillate perpendicular to the direction of propagation.
A Planar wave (in which the Electric Intensity propagates remaining in a plane –
containing the propagation direction) is said to be Linearly Polarized or Plane-Polarized.
If EM wave composed of two plane waves of equal amplitude but differing in phase
by 90° then the EM wave is said to be Circular Polarized.
If EM wave is composed of two plane waves of different amplitudes and/or the difference
in phase is different than 0,90,180,270° then the light is aid to be Elliptically Polarized.
If the direction of the Electric Intensity vector changes randomly from time to
time we say that the EM wave is Unpolarized.
E

See “Polarization” presentation for more details

POLARIZATION
SOLO
A Planar wave (in which the Electric Intensity propagates remaining in a plane –
containing the propagation direction) is said to be Linearly Polarized or Plane-Polarized.
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://www.enzim.hu/~szia/cddemo/edemo0.htm (Andras Szilagyi(
Linear Polarization or Plane-Polarization
( )
yyzktj
y
eAE 1
∧
+−
=
δω
Run This

POLARIZATION
SOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
If EM wave is composed of two plane waves of equal amplitude but differing in phase
by 90° then the light is said to be Circular Polarized.
http://www.optics.arizona.edu/jcwyant/JoseDiaz/Polarization-Circular.htm
( ) ( )
yx xx zktjzktj
eAeAE 11
2/
∧
++−
∧
+−
+= πδωδω
Run This

POLARIZATION
SOLO Properties of Electro-Magnetic Waves

SOLO
Energy and Momentum
Let start from Ampère and Faraday Laws














∂
∂
−=×∇⋅
+
∂
∂
=×∇⋅−
t
B
EH
J
t
D
HE e





EJ
t
D
E
t
B
HHEEH e





⋅−
∂
∂
⋅−
∂
∂
⋅−=×∇⋅−×∇⋅
( )HEHEEH

×⋅∇=×∇⋅−×∇⋅But
Therefore we obtain
( ) EJ
t
D
E
t
B
HHE e





⋅−
∂
∂
⋅−
∂
∂
⋅−=×⋅∇
This theorem was discovered by Poynting in 1884 and later in the same year by Heaviside.
ELECTROMAGNETICS
John Henry Poynting
1852-1914
Oliver Heaviside
1850-1925

SOLO
Energy and Momentum (continue -1)
We identify the following quantities
-Power density of the current density [watt/m2
[EJe

⋅
( )1 1
2 2
e
m e
J E H B E D E H
t t
w w
S
t t
∂ ∂   
× = − × − × −∇× × ÷  ÷
∂ ∂   
∂ ∂
= − − −∇×
∂ ∂
       







⋅
∂
∂
=⋅= BH
t
pBHw mm

2
1
,
2
1






⋅
∂
∂
=⋅= DE
t
pDEw ee

2
1
,
2
1
( )Rp E H S= ∇× × = ∇×
 
eJ

-Magnetic energy and power densities, respectively [watt/m2
[
-Electric energy and power densities, respectively [watt/m2
[
-Radiation power density [watt/m2
[
For linear, isotropic electro-magnetic materials we can write( )HBED

00 , µε ==
( )DE
tt
D
E
ED 



⋅
∂
∂
=
∂
∂
⋅
=
2
10ε
( )BH
tt
B
H
HB 



⋅
∂
∂
=
∂
∂
⋅
=
2
10µ
Umov-Poynting vector
(direction of E-M
energy propagation)
:S E H= ×
  
John Henry Poynting
1852-1914
Nikolay Umov
1846-1915
S

E

H

( ) EJ
t
D
E
t
B
HHE e





⋅−
∂
∂
⋅−
∂
∂
⋅−=×⋅∇
ELECTROMAGNETICS

SOLO
Run This
-Power density of the current density [watt/m2
[EJe

⋅






⋅
∂
∂
=⋅= BH
t
pBHw mm

2
1
,
2
1






⋅
∂
∂
=⋅= DE
t
pDEw ee

2
1
,
2
1
( )Rp E H S= ∇× × = ∇×
 
eJ

-Magnetic energy and power densities, respectively [watt/m2
[
-Electric energy and power densities, respectively [watt/m2
[
-Radiation power density [watt/m2
[
Energy and Momentum (continue -2)
S
t
w
t
w
EJ em
e

⋅∇−
∂
∂
−
∂
∂
−=⋅





Energy
Radiated
S
Energy
Electric
V
e
Energy
Magnetic
V
m
Energy
Supplied
V
e
VV
e
V
m
V
e
dsSdvw
t
dvw
t
dvEJ
dvSdvw
t
dvw
t
dvEJ
∫∫∫∫∫∫∫∫∫∫∫
∫∫∫∫∫∫∫∫∫∫∫∫
⋅+
∂
∂
+
∂
∂
+⋅=
⋅∇+
∂
∂
+
∂
∂
+⋅=0
∫∫∫ ⋅
V
e dvEJ

∫∫∫∂
∂
V
mdvw
t
∫∫∫∂
∂
V
edvw
t ∫∫ ⋅
S
dsS

Conservation of Energy
Integration over
a finite volume V

SOLO
The Electromagnetic Spectrum

SOLO

SOLO
( ) ( )
φφ ωω
θ
π
θ
π
ω
π
ω
11 0
2
0
2
2
sin1
4
sin
44
∧
−
∧
−






−−=





−= krtjkrtj
ep
rk
j
r
kc
ep
rcr
j
H
( )krtj
ep
r
k
r
kj
r
rccr
j
r
E
r
rr
−
∧∧∧
∧∧∧∧∧






−




 +





+=








−
+
+
+
=
ω
θθ
θθθ
θθθ
πε
πε
θ
ω
πε
θθ
ω
πε
θθ
0
2
23
0
2
0
2
2
0
3
0
111
11111
sinsincos2
1
4
1
4
sin
4
sincos2
4
sincos2
We can divide the zones around the source, as function of the relation between
dipole size d and wavelength λ, in three zones:
Near, Intermediate and Far Fields
The Magnetic Field Intensity is transverse to the propagation direction at all ranges,
but the Electric Field Intensity has components parallel and perpendicular to .r1
∧
r1
∧
E

However and are perpendicular to each other.H

• Near (static) zone: λ<<<< rd
• Intermediate (induction) zone: λ~rd <<
• Far (radiation) zone: rd <<<< λ
Antenna Radiation
Given a Short Wire Antenna. The antenna is oriented along the z axis with its center at the
center of coordinate system. The current density phasor through the antenna is
( ) ( ) zSS
tjm
Se rre
A
I
trj 10,
∧
−=

δω
See “Antenna Radiation”
Presentation, Tildocs # 761172 v1

SOLO Electric Dipole Radiation
Poynting Vector of the Electric Dipole Field
The Total Average Radiant Power is:
( )∫∫ 







==
π
θθπθ
επ
ω
0
22
23
0
2
42
0
sin2sin
42
dr
rc
p
dSSP
A
rad

2
0
2
2
120
123
0
42
0
3/4
0
3
23
0
42
0
40
12
sin
16
0
p
rc
p
d
rc
p
P
c
c
rad 





===
=
=
∫ λ
π
επ
ω
θθ
επ
ω λ
πω
π
ε
π

( ) ( )
3
4
3
2
3
2
cos
3
1
coscoscos1sin
0
3
0
2
0
3
=





−−=





−=−= ∫∫
ππ
π
θθθθθθ dd
HES

×=:
The Poynting Vector of the Electric Dipole Field is given by:
The time average < > of the Poynting vector is: ( )∫→∞
=
T
T
dttS
T
S
0
1
lim

( ) r
rc
p
S 1
2
23
0
2
42
0
sin
42
∧
−= θ
επ
ω
For the Electric Dipole Field:

Radiance Resistence
2
22
0
2
2
8080
2
1
L
I
p
I
P
R
m
m
rad
rad 





=











==
λ
π
λ
π
Average Radiance
22
2
0
2
2
0
2
2
10
4
40
4 r
p
r
p
r
P
S rad
avgr
λ
π
π
λ
π
π
=






==
Gain of Dipole Antenna
θ
λ
π
θ
λ
π
2
22
2
0
2
22
2
0
sin
2
3
10
sin15
===
r
p
r
p
S
S
G
avgr
r
Therefore
G
r
P
GSS rad
avgrr 2
4π
== Radar Equation

http://dept.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_apr_07_2003.shtml#tth_sEc12.1 http://www.falstad.com/mathphysics.html
Electric Field Lines of Force (continue -4)
Run This

SOLO Antenna
Field Regions Relative to Antenna

ElectromagnetismSOLO
In 1888 Heinrich Hertz, created in Kieln Germany a device
that transmitted and received electromagnetic waves.
1888
Heinrich Rudolf Hertz
1857-1894
His apparatus had a resonant
frequency of 5.5 107
c.p.s.
Air
capacitor
Hertz also showed that the
waves could be reflected by a
wall, refracted by a pitch prism,
and polarized by a wire grating.
This proved that the
electromagnetic waves had the
characteristics associated with
visible light.
http://en.wikipedia.org/wiki/Heinrich_Hertz

SOLO Interaction of Electromagnetic Waves with Material
• Reflection
• Refraction
• Diffraction
- the re-radiation (scattering) of EM waves
from the surface of material
- the bending of EM waves at the interface
of two materials
-the bending of EM waves through an
aperture in, or around an edge,
of a material
• Absorption
- the absorption of EM energy is due to
the interaction with the material
Stimulated Emission
& Absorption
Run This

SOLO Absorption and Emission
The absorption of a photon of frequency ν by a medium corresponds to the destruction
of the photon; by conservation of energy the absorbing medium must be excited to a
level with energy h ν1 > h ν0 .
Stimulated Emission
& AbsorptionPhoton emission corresponds to the creation of a photon of
frequency ν; by conservation of energy, the emitting medium
must be de-excited from an excited state to a state of lower
energy than the excited state h ν = h ν2 - h ν1.
Phenomenologically, absorption and emission in gas phase
media composed of atoms, diatomic molecules, and even
larger molecules are restricted to discrete frequencies
corresponding to the difference in the energy levels in the
atoms. Continuous frequencies regimes arise only when the
absorbed electromagnetic frequency is sufficiently high to
ionize the atoms or molecules.
Run This

SOLO Reflection and Refraction at a Boundary Interface
When an electromagnetic wave of frequency ω=2πf is traveling through matter, the
electrons in the medium oscillate with the oscillation frequency of the electromagnetic
wave. The oscillations of the electrons can be described in terms of a polarization of the
matter at the incident electromagnetic wave. Those oscillations modify the electric field in
the material. They become the source of secondary electromagnetic wave which combines
with the incident field to form the total field.
The ability of matter to oscillate with the electromagnetic
wave of frequency ω is embodied in the material property
known as the index of refraction at frequency ω, n (ω).

SOLO Refraction at a Boundary Interface
• If EM wavefronts are incident to a material surface at an angle, then the wavefronts
will bend as they propagate through the material interface. This is called refraction.
• Refraction is due to change in speed of the EM waves when it passes from one material
to another.
Index of refraction: n = c / v
Snell’s Law: n1 sin θ1 = n2 sin θ2
Run This

SOLO Reflection at a Boundary Interface
• Incident EM waves causes charge in material to oscillate, and thus, re-radiates
(scatters) the EM waves.
• If the charge is free (conductor), all the EM – wave energy is essentially re-radiated.
• If the charge is bound (dielectric), some EM – wave energy is re-radiated and some
propagates through the material.

SOLO
Generic Aircraft Model Scattering Center

SOLO
Multiple Bounce Specular Mechanism

SOLO
Wave Propagation Summary (continue)
• Surface Diffraction
- increases at lower frequency, range, and higher surface roughness
• Surface Multipath
• Surface Intervisibility
- increases at lower frequency, range, and lower surface roughness.
Also present at high frequencies for smooth terrain type (asphalt, low sea state,
desert sand, clay,…)
If surface roughness dimension is much less than wavelength, λ, of EM waves,
then scattering is specular, otherwise, scattering is diffuse.

SOLO
• Path difference of the two rays is Δr = 2 h sin γ
• Similarly the phase difference (Δφ) is simply k Δr or 4 π h sin γ / λ
• By arbitrarily setting the phase difference to be less than π /2 we obtain
the Rayleigh criteria for “rough surface”
Other criteria such as phase difference less than π /4 or π /8 are considered
more realistic.
Rayleigh Roughness Criteria
(Multipath/Roughness)

SOLO
• EM waves will propagate isotropically (Huygen’s Principle) unless prevented to do so
by wave interference.
Diffraction

SOLO
• for a circular aperture antenna of diameter D, the half-intensity (3-dB) angular
extent of the diffraction “pattern” is given by:
Radar Diffraction
Antenna Beam-Width (Diffraction Limit)
degrees
D
radians
D
B
λλ
θ 7022.1 ==
We can see that to get
Imaging Resolution of
centimeters, at 10 km,
we need either optical
wavelength λ of micro-
meters for aperture
D of order of foots or
if we use microwaves
λ = 3 cm we need an
Aperture of order of
D ~ 32 km
Resolution cells at a range of 10 kmResolution cells at a range of 10 km

SOLO Radar Diffraction
We saw (previous slide) that to get Imaging Resolution of centimeters, at 10 km, we
need either optical wavelength λ of micro-meters for aperture D of order of foots or
if we use microwaves λ = 3 cm we need an Aperture of order of D ~ 32 km.
For this reason most of Radar Applications deal with blobs of energy returns, not
with imaging.

SOLO Radar Diffraction
To obtain Imaging at Radar Frequencies we must Synthesize a Large Aperture
Antenna, using signal processing. Synthetic Aperture Radar (SAR) is a technique
of “synthesizing” a large antenna (D) by moving a small antenna over some distance,
collecting data during the motion, and processing the data to simulate the results from
a large aperture.

SOLO
Atmospheric Effects
• Atmospheric Absorption
- increases with frequency, range, and concentration of atmospheric particles
(fog, rain drops, snow, smoke,…)
• Atmospheric Refraction
- occurs at land/sea boundaries, in condition of high humidity, and at night
when a thermal profile inversion exists, especially at low frequencies.
• Atmospheric Turbulence
- in general at high frequencies (optical, MMW or sub-MMW), and is strongly
dependent on the refraction index (or temperature) variations, and strong winds.

SOLO
• The index of refraction, n, decreases with altitude.
• Therefore, the path of a horizontally propagating EM wave will gradually bend
towards the earth.
• This allows a radar to detect objects “over the horizon”.
Atmospheric Effects (continue – 1)

SOLO
Sun, Background and Atmosphere (continue – 2)
Atmosphere
Atmosphere affects electromagnetic radiation by
( ) ( )
3.2
1
1 





==
R
kmRR ττ
• Absorption
• Scattering
• Emission
• Turbulence
Atmospheric Windows:
Window # 2: 1.5 μm ≤ λ < 1.8 μm
Window # 4 (MWIR): 3 μm ≤ λ < 5 μm
Window # 5 (LWIR): 8 μm ≤ λ < 14 μm
For fast computations we may use the transmittance equation:
R in kilometers.
Window # 1: 0.2 μm ≤ λ < 1.4 μm
includes VIS: 0.4 μm ≤ λ < 0.7 μm
Window # 3 (SWIR): 2.0 μm ≤ λ < 2.5 μm

SOLO

SOLO
Atmosphere Absorption over Electromagnetic Spectrum

SOLO
Rain Attenuation over Electromagnetic Spectrum
FREQUENCY GHz
ONE-WAYATTENUATION-Db/KILOMETER
WAVELENGTH

SOLO
Basic Radar Measurements
Radar makes measurements in five dimensional-space
• two (orthogonal) angular axes (θ, φ)
• range
• Doppler (frequency)
• polarization
Target information determined by the radar
• size (RCS) - from received power of electromagnetic waves
• range - from time-delay of electromagnetic waves
• angular position - from antenna pointing angles (θ, φ)
• speed (radial) - from received electromagnetic waves frequency
• identification - from amplitude (imagery), frequency, and
polarization of electromagnetic waves
Target
Range
Ground
A.C
RADAR

SOLO Radar Configurations
Monostatic (Collocated) Antennas Bistatic Antennas

SOLO Radar Configuration
antenna
target
Run This

Range & Doppler Measurements in RADAR SystemsSOLO
The transmitted RADAR RF
Signal is:
( ) ( ) ( )[ ]ttftEtEt 0000 2cos ϕπ +=
E0 – amplitude of the signal
f0 – RF frequency of the signal
φ0 –phase of the signal (possible modulated)
The returned signal is delayed by the time that takes to signal to reach the target and to
return back to the receiver. Since the electromagnetic waves travel with the speed of light
c (much greater then RADAR and
Target velocities), the received signal
is delayed by
c
RR
td
21 +
≅
The received signal is: ( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−−= ϕπα 00 2cos
To retrieve the range (and range-rate) information from the received signal the
transmitted signal must be modulated in Amplitude or/and Frequency or/and Phase.
ά < 1 represents the attenuation of the signal

The received signal is:
( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−−= ϕπα 00 2cos
( ) ( ) tRRtRtRRtR ⋅+=⋅+= 222111 & 
We want to compute the delay time td due to the time td1 it takes the EM-wave to reach
the target at a distance R1 (at t=0), from the transmitter, and to the time td2 it takes the
EM-wave to return to the receiver, at a distance R2 (at t=0) from the target. 21 ddd ttt +=
According to the Special Relativity Theory
the EM wave will travel with a constant
velocity c (independent of the relative
velocities ).21 & RR 
The EM wave that reached the target at
time t was send at td1 ,therefore
( ) ( ) 111111 ddd tcttRRttR ⋅=−⋅+=−  ( )
1
11
1
Rc
tRR
ttd 

+
⋅+
=
In the same way the EM wave received from the target at time t was reflected at td2 ,
therefore
( ) ( ) 222222 ddd tcttRRttR ⋅=−⋅+=−  ( )
2
22
2
Rc
tRR
ttd 

+
⋅+
=

SOLO
( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−−= ϕπα 00 2cos
21 ddd ttt += ( )
1
11
1
Rc
tRR
ttd 

+
⋅+
= ( )
2
22
2
Rc
tRR
ttd 

+
⋅+
=
( ) ( )
2
22
1
11
21
Rc
tRR
Rc
tRR
tttttttt ddd 



+
⋅+
−
+
⋅+
−=−−=−
From which:






+
−
+
−
+





+
−
+
−
=−
2
2
2
2
1
1
1
1
2
1
2
1
Rc
R
t
Rc
Rc
Rc
R
t
Rc
Rc
tt d 



or:
Since in most applications we can
approximate where they appear in the arguments of E0 (t-td), φ (t-td),
however, because f0 is of order of 109
Hz=1 GHz, in radar applications, we must use:
cRR <<21, 
1,
2
2
1
1
≈
+
−
+
−
Rc
Rc
Rc
Rc




( )   





−⋅










++





−⋅










+=





−⋅





−⋅+





−⋅





−⋅≈− 2
.
201
.
10
22
0
11
00
2
1
2
1
2
12
1
2
12
1
21
D
Ralong
FreqDoppler
DD
Ralong
FreqDoppler
Dd ttffttff
c
R
t
c
R
f
c
R
t
c
R
fttf

where 21
2
2
1
121
2
02
1
01 ,,,,
2
,
2
dddddDDDDD ttt
c
R
t
c
R
tfff
c
R
ff
c
R
ff +=≈≈+=−≈−≈

( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−+−= ϕπα 00 2cosFinally
Matched Filters in RADAR Systems
Doppler Effect

SOLO
The received signal model:
( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−+−= ϕπα 00 2cos
Matched Filters in RADAR Systems
Delayed by two-
way trip time
Scaled down
Amplitude Possible phase
modulated
Corrupted
By noise
Doppler
effect
We want to estimate:
• delay td range c td/2
• amplitude reduction α
• Doppler frequency fD
• noise power n (relative to signal power)
• phase modulation φ

2-Way Doppler Shift Versus Velocity and Radio FrequencySOLO

Doppler Frequency Shifts (Hz) for Various Radar Frequency Bands and Target Speeds
Band 1m/s 1knot 1mph
L (1 GHz)
S (3 GHz)
C (5 GHz)
X (10 GHz)
Ku (16 GHz)
Ka (35 GHz)
mm (96
GHz)
6.67
20.0
33.3
66.7
107
233
633
3.43
10.3
17.1
34.3
54.9
120
320
2.98
8.94
14.9
29.8
47.7
104
283
Radar
Frequency Radial Target Speed
SOLO

SOLO
Waveform Hierarchy
Radar Waveforms
CW Radars Pulsed Radars
Frequency
Modulated CW
Phase
Modulated CW
bi – phase &
poly-phase
Linear FMCW
Sawtooth, or
Triangle
Nonlinear FMCW
Sinusoidal,
Multiple Frequency,
Noise, Pseudorandom
Intra-pulse
Modulation
Pulse-to-pulse
Modulation,
Frequency Agility
Stepped Frequency
Frequency
Modulate
Linear FM
Nonlinear FM
Phase
Modulated
bi – phase
poly-phase
Unmodulated
CW
Multiple Frequency
Frequency
Shift Keying
Fixed
Frequency

( )tf
2
τ
2
τ
−
A
∞→t
2
τ
+T
2
τ
−T
A
2
τ
+−T
2
τ
−−T
A
t←∞−
T T
A
t
A
t
A
LINEAR FM PULSECODED PULSE
T T
PULSED (INTRAPULSE CODING)
t
( )tf
A
2
τ
2
τ
−T
AA
T T
A
2
2
τ
+T
2
2
τ
−T
A
T T
A
2
τ
− 2
τ
+T
TN
t
( )tf
A
2
τ
2
τ
−T
AA
T T
A
2
2
τ
+T
2
2
τ
−T
A
T T
A
2
τ
− 2
τ
+T
TN
PHASE CODED PULSES HOPPED FREQUENCY PULSES
PULSED (INTERPULSE CODING)
( )tf
2
τ
2
τ
−
A
∞→t
2
τ
+T
2
τ
−T
A
2
τ
+−T
2
τ
−−T
A
t←∞−
T T
NONCOHERENT PULSESCOHERENT PULSES
( )tf
t
A
2
τ
2
τ
−T
AA
T T
A
2
2
τ
+T
2
2
τ
−T
A
T T
A
2
τ
− 2
τ
+T
TN
PULSED (UNCODED)
t
( )tf
A
T
2/τ−
LOW PRF
MEDIUM PRF
PULSED
( )tf
T T T T
2/τ+
τ
HIGH PRF
T
T T T
A Partial List of the Family of RADAR Waveforms

SOLO
The Fourier transform of a signal f (t) can be written as:
A sufficient (but not necessary) condition for the
existence of the Fourier Transform is:
( ) ( ) ∞<= ∫∫
∞
∞−
∞
∞−
ωω
π
djFdttf
22
2
1
JEAN FOURIER
1768-1830
( ) ( )∫
+∞
∞−
−
= ωω
π
ω
dejF
j
tf tj
2
1
The Inverse Fourier transform of F (j ω) is given by:
( ) ( )∫
+∞
∞−
= dtetfjF tjω
ω

( ) ( )∫
+∞
∞−
−
= ωω
π
ω
dejF
j
tf tj
2
1
Signal
(1) C.W.
( )
2
cos
00
0
tjtj
ee
AtAtf
ωω
ω
−
+
==
0ω - carrier frequency
Frequency
( ) ( )∫
+∞
∞−
= dtetfjF tjω
ω
Fourier Transform
( ) ( ) ( )00
22
ωωδωωδω ++−=
AA
jF
Fourier Transform
SOLO

( ) ( )∫
+∞
∞−
−
= ωω
π
ω
dejF
j
tf tj
2
1
Signal
(2) Single Pulse
( )



>
≤≤−
=
2/0
2/2/
τ
ττ
t
tA
tf
τ - pulse width
Frequency
( ) ( )∫
+∞
∞−
= dtetfjF tjω
ω
Fourier Transform
( ) ( ) ( )
( )2/
2/sin
2/
2/
τω
τω
τω
τ
τ
ω
AdteAjF tj
== ∫−
Fourier Transform
SOLO

( ) ( )∫
+∞
∞−
−
= ωω
π
ω
dejF
j
tf tj
2
1
Signal
( )
( )



>
≤≤−
=
2/0
2/2/cos 0
τ
ττω
t
ttA
tf
τ - pulse width
Frequency
( ) ( )∫
+∞
∞−
= dtetfjF tjω
ω
Fourier Transform
( ) ( )
( )
( )
( )
( )












−



 −
+
+



 +






=
= ∫−
2
2
sin
2
2
sin
2
cos
0
0
0
0
2/
2/
0
τωω
τωω
τωω
τωω
τ
ωω
τ
τ
ω
A
dtetAjF tj
Fourier Transform
(3) Single Pulse Modulated at a
frequency 0ω
ω
( )ωjF
0
τ
π
ω
2
0 +
2
τA
0ω
τ
π
ω
2
0 −
τ
π
ω
2
0 +−
2
τA
0ω−
τ
π
ω
2
0 −−
τ
π
ω
2
20 +
τ
π
ω
2
20 −
SOLO

( ) ( )∫
+∞
∞−
−
= ωω
π
ω
dejF
j
tf tj
2
1
Signal
( )
( )



±±=>−
≤−≤−+
=
,2,1,0,2/0
2/2/cos 0
kkkTt
kTttA
tf
rand
τ
ττϕω
τ - pulse width
Frequency
( ) ( )∫
+∞
∞−
= dtetfjF tjω
ω
Fourier Transform
( ) ( )
( )
( )
( )
( )












−



 −
+
+



 +






=
= ∫−
2
2
sin
2
2
sin
2
cos
0
0
0
0
2/
2/
0
τωω
τωω
τωω
τωω
τ
ωω
τ
τ
ω
A
dtetAjF tj
Fourier Transform
(4) Train of Noncoherent Pulses
(random starting pulses),
modulated at a frequency 0ω
T - Pulse repetition interval (PRI)
SOLO

( ) ( )∫
+∞
∞−
−
= ωω
π
ω
dejF
j
tf tj
2
1
Signal
( )
( )
( ) ( )( ) ( )( )[ ]












−++












+=



±±=>−
≤−≤−
=
∑
∞
=1
000
0
coscos
2
2
sin
cos
,2,1,0,2/0
2/2/cos
n
PRPR
PR
PR
series
Fourier
tntn
n
n
t
T
A
kkkTt
kTttA
tf
ωωωω
τω
τω
ω
τ
τ
ττω

τ - pulse width
Frequency
( ) ( )∫
+∞
∞−
= dtetfjF tjω
ω
Fourier Transform
Fourier Transform
5) Train of Coherent Pulses,
of infinite length,
( ) ( ) ( ){
( ) ( ) ( ) ( )[ ]






+−+−+−−++












+
−+=
∑
∞
=1
0000
00
2
2
sin
2
n
PRPRPRPR
PR
PR
nnnn
n
n
T
A
jF
ωωδωωδωωδωωδ
τω
τω
ωδωδ
τ
ω
T/1 - Pulse repetition frequency (PRF)
TPR /2πω =
SOLO

( ) ( )∫
+∞
∞−
−
= ωω
π
ω
dejF
j
tf tj
2
1
Signal
( )
( )
( ) ( )( ) ( )( )[ ]












−++












+=



±±=>−
≤−≤−
=
∑
∞
=
≤≤−
1
000
22
0
coscos
2
2
sin
cos
2/,,2,1,0,2/0
2/2/cos
n
PRPR
PR
PRNT
t
NT
tntn
n
n
t
T
A
NkkkTt
kTttA
tf
ωωωω
τω
τω
ω
τ
τ
ττω

τ - pulse width
Frequency
( ) ( )∫
+∞
∞−
= dtetfjF tjω
ω
Fourier Transform
Fourier Transform
of finite length N T,
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )



















−−




−−
+
+−




+−












+
+




+
+



















−+




−+
+
++




++












+
+




+
=
∑
∑
∞
=
∞
=
1
0
0
0
0
0
0
1
0
0
0
0
0
0
2
2
sin
2
2
sin
2
2
sin
2
2
sin
2
2
sin
2
2
sin
2
2
sin
2
2
sin
2
n
PR
PR
PR
PR
PR
PR
n
PR
PR
PR
PR
PR
PR
TN
n
TN
n
TN
n
TN
n
n
n
TN
TN
TN
n
TN
n
TN
n
TN
n
n
n
TN
TN
T
A
jF
ωωω
ωωω
ωωω
ωωω
τω
τω
ωω
ωω
ωωω
ωωω
ωωω
ωωω
τω
τω
ωω
ωω
τ
ω
TPR /2πω =
SOLO

Signal
( ) ( )
























+=



±±=>−
≤−≤−
= ∑
∞
=1
1 cos
2
2
sin
21
,2,1,0,2/0
2/2/
n
PR
PR
PR
Series
Fourier
tn
n
n
T
A
kkkTt
kTtA
tf ω
τω
τω
τ
τ
ττ

τ - pulse width
TPR /2πω =
( ) ( )tAtf 03 cos ω=
t
A A
( )tf1
t
2
τ
2
τ
−T
A
T T
2
2
τ+T
2
2
τ−T
T T
2
τ− 2
τ+T
( )tf2
t
TN
2/TN2/TN−
( ) ( ) ( ) ( )tftftftf 321 ⋅⋅=
( ) ( ) ( ) ( )
( )
( ) ( )( ) ( )( )[ ]












−++












+=



±±=>−
≤−≤−
=⋅⋅=
∑
∞
=
≤≤−
1
000
22
0
321
coscos
2
2
sin
cos
2/,,2,1,0,2/0
2/2/cos
n
PRPR
PR
PRNT
t
NT
tntn
n
n
t
T
A
NkkkTt
kTttA
tftftftf
ωωωω
τω
τω
ω
τ
τ
ττω

( )



>
≤≤−
=
2/0
2/2/1
2
TNt
TNtTN
tf ( ) ( )ttf 03 cos ω=
SOLO

SOLO
• Transmitter always on
• Range information can be obtained by modulating EM wave
[e.g., frequency modulation (FM), phase modulation (PM)]
• Simple radars used for speed timing, semi-active missile illuminators,
altimeters, proximity fuzes.
• Continuous Wave Radar (CW Radar)

SOLO • Continuous Wave Radar (CW Radar)
The basic CW Radar will transmit an unmodulated (fixed carrier frequency) signal.
( ) [ ]00cos ϕω += tAts
The received signal (in steady – state) will be.
( ) ( ) ( )[ ]00cos ϕωωα +−+= dDr ttAts
α – attenuation factor
ωD – two way Doppler shift
c
RfR
ff
fc
DDD

0
/
22
&2
0
−=−==
=λ
λ
πω
The Received Power is related to the Transmitted Power by (Radar Equation):
4
1
~
RP
P
tr
rcv
One solution is to have separate antennas
for transmitting and receiving.
For R = 103
m this ratio is 10-12
or 120 db.
This means that we must have a good
isolation between continuously
transmitting energy and receiving energy.
Basic CW Radar

SOLO • Continuous Wave Radar (CW Radar)
The received signal (in steady – state) ( ) ( ) ( )[ ]002cos ϕπα +−⋅+= dDr ttffAts
We can see that the sign of the Doppler is ambiguous (we get the same result for positive
and negative ωD).
To solve the problem of doppler sign ambiguity
we can split the Local Oscillator into two
channels and phase shifting the
Signal in one by 90◦
(quadrature - Q) with
respect to other channel (in-phase – I). Both
channels are downconverted to baseband.
If we look at those channels as the real and
imaginary parts of a complex signal, we get:
has the Fourier Transform: ( ){ } ( ) ( )[ ]DDv ts ωωδωωδπ ++−=F
After being heterodyned to baseband (video band), the signal becomes (after ignoring
amplitude factors and fixed-phase terms): ( ) [ ]tts Dv ωcos=
( ) ( ) ( )[ ] tj
DDv
D
etjtts ω
ωω
2
1
sincos
2
1
=+= ( ){ } ( )Dv ts ωωδ
π
−=
2
F

SOLO • Frequency Modulated Continuous Wave (FMCW)
The transmitted signal is: ( ) ( )[ ]00cos ϕθω ++= ttAts
The frequency of this signal is: ( ) ( )





+= t
dt
d
tf θω
π
0
2
1
For FMCW the θ (t) has a linear slope as seen in the figures bellow

( ) ( ) ( ) ( )[ ]00cos ϕθωωα +−+−+= ddDr ttttAts
α – attenuation factor
ωD – two way Doppler shift
λ
πω
R
ff DDD
2
&2 −==
td – two way time delay
c
R
td
2
=
( ) ( )



−++= dDr tt
dt
d
fftf θ
π2
1
0The frequency of received signal is:
λ – mean value of wavelength

To extract the information we must subtract the received signal frequency from
the transmitted signal frequency. This is done by mixing (multiplying) those signals
and use a Lower Side-Band Filter to retain the difference of frequencies
( ) ( ) ( ) ( ) ( ) Ddrb ftt
dt
d
t
dt
d
tftftf −



−−



=−= θ
π
θ
π 2
1
2
1
The frequency of mixed signal is:
( ) ( ) ( ) ( )[ ]00cos ϕθωωα +−+−+= ddDr ttttAts
( ) ( )[ ]00cos ϕθω ++= ttAts
( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )[ ]ddD
ddDdr
ttttttA
ttttttAts
−++−+++
−−+−−=
θθωωωα
θθωωα
00
2
0
2
cos
2
1
cos
2
1
Lower Side-Band
Filter
Lower SB Filter

The returned signal has a frequency change due to:
• two way time delay c
R
td
2
=
• two way doppler addition λ
R
fD
2
−=
From Figure above, the beat frequencies fb (difference between transmitted to
received frequencies) for a Linear Sawtooth Frequency Modulation are:
D
m
Dd
m
b fR
Tc
f
ft
T
f
f −
∆
=−
∆
=
+ 4
2/
D
m
Dd
m
b fR
Tc
f
ft
T
f
f −
∆
−=−
∆
−=
− 4
2/
( )
28
−+
−
∆
= bbm ff
f
Tc
R ( )
2
−+
+
−= bb
D
ff
f
We have 2 equations with 2 unknowns R and fD
with the solution:

SOLO
• Frequency Modulated Continuous Wave (FMCW)
The Received Power is related to the Transmitted Power by (Radar Equation):
For R = 103
m this ratio is 10-12
or 120 db. This means that we must have a good
isolation between continuously transmitting energy and receiving energy.
4
1
~
RP
P
tr
rcv
One solution is to have separate antennas for transmitting and receiving.

Performing Fast Fourier Transform (FFT) we obtain fb
+
and fb.
( )
28
−+
−
∆
= bbm ff
f
Tc
R
( )
2
−+
+
−= bb
D
ff
f
From the Doppler Window we get fb
+
and fb
-
, from which:

SOLO
• Frequency Modulated Continuous Wave (FMCW)

The returned signal has a frequency change due to:
• two way time delay c
R
td
2
=
• two way doppler addition λ
R
fD
2
−=
From Figure above, the beat frequencies fb (difference between transmitted to
received frequencies) for a Linear Triangular Frequency Modulation are:
D
m
Dd
m
b fR
Tc
f
ft
T
f
f −
∆
=−
∆
=
+ 8
4/
positive
slope
D
m
Dd
m
b fR
Tc
f
ft
T
f
f −
∆
−=−
∆
−=
− 8
4/
negative
slope
( )
28
−+
−
∆
= bbm ff
f
Tc
R ( )
2
−+
+
−= bb
D
ff
f
We have 2 equations with 2 unknowns R and fD
with the solution:
Linear Triangular Frequency Modulated Continuous Wave

Two Targets Detected
Performing FFT for each of the
positive, negative and zero slopes
we obtain two Beats in each
Doppler window.
To solve two targets we can use the
Segmented Linear Frequency
Modulation.
In the zero slope Doppler
window, we obtain the Doppler
frequency of the two targets fD1
and fD2.
Since , it is
easy to find the pair from
Positive and Negative Slope
Windows that fulfill this condition, and then to compute the respective ranges using:
( )
2
−+
+
−= bb
D
ff
f
( )
28
−+
−
∆
= bbm ff
f
Tc
R
This is a solution for more than two targets.
One other solution that can solve also range and doppler ambiguities is to use many
modulation slopes (Δ f and Tm).

One of the practical frequency
modulations is the Sinusoidal
Frequency Modulation.
Assume that the transmitted
signal is:
( ) ( )




 ∆
+= tf
f
f
tfAts m
m
ππ 2sin2sin 0
The spectrum of this signal is:
( ) ( )
( )[ ] ( )[ ]{ }
( )[ ] ( )[ ]{ }
( )[ ] ( )[ ]{ }
+
−++




 ∆
+
−++




 ∆
+
−++




 ∆
+





 ∆
=
tfftff
f
f
JA
tfftff
f
f
JA
tfftff
f
f
JA
tf
f
f
JAts
mm
m
mm
m
mm
m
m
32sin32sin
22sin22sin
2sin2sin
2sin
003
002
001
00
ππ
ππ
ππ
π
where Jn (u) is the Bessel Function
of the first kind, n order and argument
u.
Bessel Functions of the first kind

Lower Side-Band
Filter
( )ts
( )tr
( )tm
R
c
ff
ftffff m
DdmD
tf
b
dm ∆
+=∆+≈=
<<
+
8
4
1π
A possible modulating is describe bellow, in which we introduce a unmodulated segment
to measure the doppler and two sinusoidal modulation segments in anti-phase.
From which we obtain:
R
c
ff
ftffff m
DdmD
tf
b
dm ∆
−=∆−≈=
<<
−
8
4
1π
The averages of the beat frequency over one-half a modulating cycle are:
28
−+
−
∆
=
bbm
ff
f
Tc
R
2
−+
+
=
bb
D
ff
f
(must be the same as in
unmodulated segment)
Note: We obtaind the same form as for Triangular Frequency Modulated CW

SOLO
Assume that the transmitter transmits n CW frequencies fi (i=0,1,…,n-1)
Transmitted signals are: ( ) [ ] 1,,1,02sin −== nitfAts iii π
The received signals are: ( ) ( ) ( )[ ]dDiiiii ttffAtr −⋅+= πα 2sin
c
R
t
c
R
f
c
R
fff d
i
j
jDi
2
,
22
10
1
0 ≈−≈







∆+−≈ ∑=
where:
1,,2,11 −=∆+= − nifff iii 
Since we want to use no more than one antenna for transmitted signals and one antenna
for received signals we must have
1,,2,10
1
−=<<∆∑=
niff
i
j
j 
We can see that the change in received phase Δφi , of two adjacent signals, is related
to range R by:
( ) c
R
f
c
R
c
R
f
c
R
f
c
R
ff
c
R
f i
cR
iiDDii ii
2
2
22
2
2
2
2
2
2
2
2
1
⋅∆≈⋅⋅∆+⋅∆=⋅−+⋅∆=∆
<<
−
πππππϕ

The maximum unambiguous range is given when Δφi=2π :
i
sunambiguou
f
c
R
∆
=
2
• Multiple Frequency CW Radar (MFCW)

SOLO • Multiple Frequency CW Radar (MFCW)

SOLO • Phase Modulated Continuous Wave (PMCW)
Another way to obtain a time mark in a CW signal is by using Phase Modulation (PM).
PMCW radar measures target range by applying a discrete phase shift every T seconds
to the transmitted CW signal, producing a phase-code waveform. The returning waveform
is correlated with a stored version of the transmitted waveform. The correlation process
gives a maximum when we have a match. The time to achieve this match is the time-delay
between transmitted and receiving signals and provides the required target range.
There are two types of phase coding techniques: binary phase codes and polyphase codes.
In the figure bellow we can see a 7-length Barker binary phase code of the transmitted
signal

SOLO • Phase Modulated Continuous Wave (PMCW)
In the figure bellow we can see a 7-length Barker binary phase code of the received
signal that, at the receiver, passes a 7-cell delay line, and is correlated to a sample
of the 7-length Barker binary signal sample.
-1 = -1
+1 -1 = 0
-1 +1 -1 = -1
-1 -1 +1-( -1) = 0
+1 -1 -1 –(+1)-( -1) = -1
+1 +1 -1-(-1) –(+1)-1= 0
+1+1 +1-( -1)-(-1) +1-(-1)= 8
+1+1 –(+1)-( -1) -1-( +1)= 0
+1-(+1) –(+1) -1-( -1)= -1
-(+1)-(+1) +1 -( -1)= 0
-(+1)+1-(+1) = -1
+1-(+1) = 0
-(+1) = -1
0 = 0
-1-1 -1
Digital Correlation
At the Receiver the coded pulse enters a
7 cells delay lane (from left to right),
a bin at each clock.
The signals in the cells are summed
clock
1
2
3
4
5
6
7
8
9
10
11
12
13
14
+1+1+1+1

SOLO
PRI – Pulse Repetition Interval PRF – Pulse Repetition Frequency
τ – Pulse Width [μsec]
PRF = 1/PRI
Pulse Duty Cycle = DC = τ / PRI = τ * PRF
Paverrage = DC * Ppeak
Pulse Waveform Parameters
Pulse Radars
• Coherent – Phase is predictable from pulse-to-pulse
• Non-coherent – Phase from pulse-to-pulse is not predictable

( )tf
2
τ
2
τ
−
A
∞→t
2
τ
+T
2
τ
−T
A
2
τ
+−T
2
τ
−−T
A
t←∞−
T T
A
t
A
t
A
LINEAR FM PULSECODED PULSE
T T
PULSED (INTRAPULSE CODING)
t
( )tf
A
2
τ
2
τ
−T
AA
T T
A
2
2
τ
+T
2
2
τ
−T
A
T T
A
2
τ
− 2
τ
+T
TN
t
( )tf
A
2
τ
2
τ
−T
AA
T T
A
2
2
τ
+T
2
2
τ
−T
A
T T
A
2
τ
− 2
τ
+T
TN
PHASE CODED PULSES HOPPED FREQUENCY PULSES
PULSED (INTERPULSE CODING)
t
( )tf
A
T
2/τ−
LOW PRF
MEDIUM PRF
PULSED
( )tf
T T T T
2/τ+
τ
HIGH PRF
T
T T T
A Partial List of the Family of RADAR Waveforms (continue – 1)
Pulses

SOLO
Pulse Radars

Coherent Pulse Doppler RadarSOLO
• STALO provides a
continuous frequency
fLO
• COHO provides the
coherent Intermediate
Frequency fIF
• Pulse Modulator
defines the pulse width
the Pulses Rate
Frequency (PRF)
number of pulses in a
batch
• Transmitter/Receiver (T/R) (Circulator)
- in the Transmission Phase directs the Transmitted Energy to the Antenna and
isolates the Receiving Channel
• IF Amplifier is a Band Pass Filter in the Receiving Channel centered around
IF frequency fIF.
• Mixer multiplies two sinusoidal signals providing signals with sum or
differences of the input frequencies
- in the Receiving Phase directs the Received Energy to the Receiving Channel
21 ff >>
2f
1f
21 ff +
21 ff −

Radar Waveforms and their Fourier Transforms

SOLO
The basic way to measure the Range to a Target is to send a pulse of EM energy and
to measure the time delay between received and transmitted pulse
Range = c td/2
Range Measurements in RADAR Systems
Run This

SOLO Range & Doppler Measurements in RADAR Systems

Range Measurement Unambiguity
SOLO
The returned signal from the target
located at a range R from the transmitter
reaches the receiver (collocated with the
transmitter) after
c
R
t
2
=
To detect the target, a train of pulses must
be transmitted.
PRI – Pulse Repetition Interval
PRF – Pulse Repetition Frequency = 1/PRT
To have an unanbigous target range the received pulse must arrive before the transmission
of the next pulse, therefore:
PRF
PRI
c
Runabigous 1
2
=<
PRF
c
Runabigous
2
<

Resolving Range Measurement Ambiguity
SOLO
To solve the ambiguity of targets return
we must use multiple batches, each with
different PRIs (Pulse Repetition Interval).
Example: one target, use two batches
First batch: PRI 1 = T1
Target Return = t1-amb
R1_amb=2 c t1_amb
Second batch: PRI 2 = T2
Target Return = t2-amb
R2_amb=2 c t2_amb
To find the range, R, we must solve for the integers k1 and k2
in the equation:
( ) ( )ambamb tTkctTkcR _222_111 22 +=+=
We have 2 equations with 3 unknowns: R, k1 and k2, that can be solved because
k1 and k2 are integers. One method is to use the Chinese Remainder Theorem .
For more targets, more batches must be used to solve the Range ambiguity.
See Tildocs # 763333 v1

Resolving Range Measurement Ambiguity
SOLO
In Figure bellow we can see that using a constant PRF we obtain two targets
Target # 1
Target # 2
By changing the PRF we can see that Target # 2 is unambiguous
Transmitted Pulse
Run This

SOLO
( )ωjF
2
NAτ
ω
TN
π
ω
2
0 +
0ω−
TN
π
ω
2
0 −
PRωω +− 0
PRωω −− 0
T
PR
π
ω
2
=
T
PR
π
ω
2
=
ω0
TN
π
ω
2
0 +
0ω
TN
π
ω
2
0 −
PRωω +0PRωω −0
T
PR
π
ω
2
=
T
PR
π
ω
2
=












2
2
sin
2 τω
τω
τ
n
n
NA
PR
PR
( )
( )
2
2
sin
0
0
NT
NT
ωω
ωω
−




−
( )
( )
2
2
sin
2
2
s in
2
0
0
NT
n
NT
n
n
n
NA
RP
RP
PR
PR
ωωω
ωωω
τω
τω
τ
−−




−−












( )ωjF
( )0
2
ωωδ
τ
−
NA
ω
0ω− PRωω +− 0PRωω −− 0
T
PR
π
ω
2
=
T
PR
π
ω
2
=
ω0
PRωω +0PRωω −0
T
PR
π
ω
2
=
T
PR
π
ω
2
=












2
2
sin
2 τω
τω
τ
n
n
NA
PR
PR












2
2
sin
2 τω
τω
τ
n
n
NA
P R
P R
0
ω P R
ωω 20
+
PRωω 20 −PRωω 20 −−
PR
ωω 30
−−PR
ωω 40
−−
PR
ωω 20
+−
PRωω 30 +− PR
ωω 40
+−
Fourier Transform of an Infinite Train Pulses
Fourier Transform of an Finite Train Pulses of Lenght N
( )P R
P R
P R
NA
ωωωδ
τω
τω
τ
−−












0
2
2
sin
2
( ) ( )tAtf 03 cos ω=
t
A A
( )tf1
t
2
τ
2
τ
−T
A
T T
2
2
τ+T
2
2
τ−T
T T
2
τ− 2
τ+T
( )tf2
t
TN
2/TN2/TN−
( ) ( ) ( ) ( )tftftftf 321 ⋅⋅=
Train of Coherent Pulses,
The pulse coherency is a necessary condition
to preserve the frequency information and
to retrieve the Doppler of the returned signal.
Transmitted Train of Coherent Pulses

SOLO
Fourier Transform of an Finite Train Pulses of Lenght N
2
NAτ
ω
TN
πω 2
0 +
0ω
TN
πω 2
0 −
PRωω+0PRωω−0
T
PR
πω 2
=
T
PR
πω 2
=
2
NAτ
ω
TN
πω 2
0 +
0ω
TN
πω 2
0 −
PRωω+0PRωω−0
T
PR
πω 2
=
T
PR
πω 2
=
















2
2
sin
2 τω
τω
τ
n
n
NA
PR
PR
( )
( )
2
2
sin
0
0
NT
NT
ωω
ωω
−




−
2
NAτ
ω
TN
πω 2
0 +
0ω
TN
πω 2
0 −
P Rωω+0PRωω−0
T
PR
πω 2
=
T
PR
πω 2
=
π
ω
λ 2
&
2
P R
Doppl e rDopple r f
td
Rd
f <






−=
π
ω
λ 2
&
2
P R
Dopple rDopple r f
td
Rd
f >






−=
Fourier Transform of the
Transmitted Signal
Receiveded Signal
with Unambiguous Doppler
Receiveded Signal
with Ambiguous Doppler
Received Train of Coherent Pulses
The bandwidth of a single pulse is usually several order of magnitude greater than the
expected doppler frequency shift 1/τ >> f doppler. To extract the Doppler frequency shift,
the returns from many pulses over an observation time T must be frequency analyzed so
that the single pulse spectrum will separate into individual PRF lines with bandwidths
approximately given by 1/T.
From the Figure we can see
that to obtain an unambiguous
Doppler the following condition
must be satisfied:
PRF
c
td
Rd
f
td
Rd
f PRMaxMax
doppler
=≤==
π
ω
λ 2
22 0
or
0
2 f
PRFc
td
Rd
Max
≤

SOLO
Coherent Pulse Doppler RadarAn idealized target doppler response will
provide at IF Amplifier output the signal:
( ) ( )[ ] ( ) ( )
[ ]tjtj
dIFIF
dIFdIF
ee
A
tAts ωωωω
ωω +−+
+=+=
2
cos
that has the spectrum:
f
fIF+fd
-fIF-fd
-fIF fIF
A2
/4A2
/4 |s|2
0
Because we used N coherent pulses of
width τ and with Pulse Repetition Time T
the spectrum at the IF Amplifier output
f
-fd fd
A2
/4A2
/4
|s|2
0
After the mixer and base-band filter:
( ) ( ) [ ]tjtj
dd
dd
ee
A
tAts ωω
ω −
+==
2
cos
We can not distinguish between
positive to negative doppler!!!
and after the mixer :

SOLO
Coherent Pulse Doppler Radar
We can not distinguish
between positive to negative
doppler!!!
Split IF Signal:
( ) ( )[ ] ( ) ( )
[ ]tjtj
dIFIF
dIFdIF
ee
A
tAts ωωωω
ωω +−+
+=+=
2
cos
( ) ( )[ ]
( ) ( )[ ]t
A
ts
t
A
ts
dIFQ
dIFI
ωω
ωω
+=
+=
sin
2
cos
2
Define a New Complex Signal:
( ) ( ) ( ) ( )[ ]tj
QI
dIF
e
A
tsjtstg ωω +
=+=
2
f
fIF+fd
fIF
A2
/2|g|2
0
f
fd
A2
/2
|s|2
0
Combining the signals after the mixers
( ) tj
d
d
e
A
tg ω
2
=
We now can distinguish
doppler!!!

SOLO
Split IF Signal:
( ) ( )[ ]
( ) ( )[ ]t
A
ts
t
A
ts
dIFQ
dIFI
ωω
ωω
+=
+=
sin
2
cos
2
Define a New Complex Signal:
( ) ( ) ( ) ( )[ ]tj
QI
dIF
e
A
tsjtstg ωω +
=+=
2
f
fd
A2
/2
|s|2
0
Combining the signals after the mixers
( ) tj
d
d
e
A
tg ω
2
=
We now can distinguish
doppler!!!
From the Figure we can see that in this
case the doppler is unambiguous only if:
T
ff PRd
1
=<
Because we used N coherent pulses of
width τ and with Pulse Repetition Time T
the spectrum after the mixer output is

SOLO
Because, for Doppler computation, we used N coherent pulses of width τ and
with Pulse Repetition Interval T, the spectrum after the mixer output is
From the Figure we can see that in this case the doppler is unambiguous only if:
T
ff PRd
1
=<

Resolving Doppler Measurement Ambiguity






+=





+= ambDambD f
T
kf
T
kV _2
2
2_1
1
1
1
2
1
2
λλ
SOLO
To solve the Doppler ambiguity of targets
return we must use multiple batches, each with
different PRIs (Pulse Repetition Interval).
Example: one target, use two batches
First batch: PRI 1 = T1
Target Doppler Return in Range Gate i =
fD1-amb
V1_amb=(λ/2) fD1_amb
To find the range-rate, V, we must solve for the integers k1 and k2
in the equation:
We have 2 equations with 3 unknowns: V, k1 and k2, that can be solved because
k1 and k2 are integers. One method is to use the Chinese Remainder Theorem .
Second batch: PRI 2 = T2
Target Doppler Return in Range Gate i =
fD2-amb
V2_amb=(λ/2) fD2_amb
For more targets, more batches must be used to solve the Doppler ambiguity.

SOLO Range & Doppler Measurements in RADAR Systems

Resolution
Resolution is the spacing (in range, Doppler, angle, etc.) we must have in order to
distinguish between two different targets.
first target
response
second target
response
composite
target
response
greather then 3 db
Distinguishable
Targets
first target
response
second target
response
composite
target
response
Undistinguishable
Targets
less then 3 db
The two targets are distinguishable if
the composite (sum) of the received
signal has a deep (between the two
picks) of at least 3 db.

Doppler Resolution
The Doppler resolution is defined by
the Bandwidth of the Doppler Filters
BWDoppler.
Doppler Dopplerf BW∆ =

Angle Resolution
Angle Resolution
RADAR
Target # 1
Target # 2
R
R
3
θ 





2
cos 3θ
R
3
3
2
sin2 θ
θ
RR ≈





Angle Resolution is Determined by Antenna Beamwidth.
3
3
2
sin2 θ
θ
RRRC ≈





=∆
Angle Resolution is considered equivalent to the 3 db Antenna Beamwidth θ3.
The Cross Range Resolution is given by:

SOLO
Unmodulated Pulse Range Resolution
Range Resolution
RADAR
τ
c
R
RR ∆+
Target # 1
Target # 2
Assume two targets spaced by a range
Δ R and a unmodulated radar pulse of
τ seconds.
The echoes start to be received
at the radar antenna at times:
2 R/c – first target
2 (R+Δ R)/c – second target
The echo of the first target ends
at 2 R/c + τ
τ τ
time from pulse
transmission
c
R2 ( )
c
RR ∆+2
τ+
c
R2
Received
Signals
Target # 1 Target # 2
The two targets echoes can be
resolved if:
c
RR
c
R ∆+
=+ 22 τ
2
τc
R =∆ Pulse Range Resolution
( )
( )


 ≤≤+
=
elsewhere
ttA
ts
0
0cos
: 0 τϕω

SOLO
Range Resolution
Run This

Range Resolution
SOLO Range Measurements in RADAR Systems
Run This

RADAR SignalsSOLO
( )
( )


 ≤≤+
=
elsewhere
ttA
ts
0
0cos
: 0 τϕω
Energy
( ) ( )
2
2cos22cos
1
2
2
000
2
τ
τ
ϕϕτωτ A
E
A
E ss =⇒




 −+
+=
2
τc
R =∆ Pulse Range Resolution
Decreasing Pulse Width Increasing
Decreasing SNR, Radar Performance Increasing
Increasing Range Resolution Capability Decreasing
For the Unmodulated Pulse, there exists a coupling between Range Resolution and
Waveform Energy. Return to Table of contents

Pulse Compression WaveformsSOLO
Pulse Compression Waveforms permit a decoupling between Range Resolution and
Waveform Energy.
- An increased waveform bandwidth (BW) relative to that achievable with an
unmodulated pulse of an equal duration
τ
1
>>BW
22
τc
BW
c
R <<=∆
- Waveform duration in excess of that achievable with unmodulated pulse of
equivalent waveform bandwidth
BW
1
>>τ
PCWF exhibit the following equivalent properties:
This is accomplished by modulating (or coding) the transmit waveform and compressing
the resulting received waveform.

SOLO
• Pulse Compression Techniques
• Wave Coding
• Frequency Modulation (FM)
- Linear
• Phase Modulation (PM)]
- Non-linear
- Pseudo-Random Noise (PRN)
- Bi-phase (0º/180º)
- Quad-phase (0º/90º/180º/270º)
• Implementation
• Hardware
- Surface Acoustic Wave (SAW) expander/compressor
• Digital Control
- Direct Digital Synthesizer (DDS)
- Software compression “filter”

SOLO

SOLO
Return to Table of Contents

SOLO
Linear FM Modulated Pulse (Chirp)
( ) ( )2/cos 2
03 ttAtf ωω ∆+=
t
A
2/τ−
2/τ ( )
222
cos
2
0
ττµ
ω ≤≤−





+= t
t
tAtsi
Linear Frequency Modulation is a technique used to increase the waveform bandwidth
BW while maintaining pulse duration τ, such that
BW
1
>>τ 1>>⋅ BWτ
222
0
2
0
ττ
µω
µ
ωω ≤≤−+=





+= tt
t
t
td
d

Matched Filters for RADAR Signals
( ) ( )
( ) ( )



≤≤−=
= −∗
Ttttsth
eSH
i
tj
i
00
0ω
ωω
SOLO
The Matched Filter (Summary(
si (t) - Signal waveform
Si (ω) - Signal spectral density
h (t) - Filter impulse response
H (ω) - Filter transfer function
t0 - Time filter output is sampled
n (t) - noise
N (ω) - Noise spectral density
Matched Filter is a linear time-invariant filter hopt (t) that maximizes
the output signal-to-noise ratio at a predefined time t0, for a given signal si (t(.
The Matched Filter output is:
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) 0
0
00
tj
iii
iii
eSSHSS
dttssdthsts
ω
ωωωωω
ξξξξξξ
−∗
+∞
∞−
+∞
∞−
⋅=⋅=
+−=−= ∫∫

SOLO
Linear FM Modulated Pulse (continue – 1)
Concept of Group Delay
BW
1
>>τ
τ
BW
1
( )
222
cos
2
0
ττµ
ω ≤≤−





+= t
t
tAtsi
( ) ( )
222
cos
2
0
00 ττµ
ω ≤≤−





−=−=
=
t
t
tAtsth i
t
MF
Matched Filter
( )tsi ( )tso
( ) ( )tsth i
t
MF −=
=00 ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )ωωωωω
ξξξξξξ
∗
=
+∞
∞−
=+∞
∞−
⋅=⋅=
−=−= ∫∫
ii
t
i
ii
t
i
SSHSS
dtssdthsts
0
0
0
0
0
0

SOLO
Linear FM Modulated Pulse (continue – 7)
Linear FM Modulated Pulse (Chirp) Summary
• Chirp is one of the most common type of pulse compression code
• Chirp is simple to generate and compress using IF analog techniques, for example,
surface acoustic waves (SAW) devices.
• Large pulse compression ratios can be achieved (50 – 300).
• Chirp is relative insensitive to uncompressed Doppler shifts and can be easily
weighted for side-lobe reduction.
• The analog nature of chirp sometimes limits its flexibility.
• The very predictibility of chirp mades it asa poor choice for ECCM purpose.

SOLO
Pulse Compression Techniques
Phase Coding
A transmitted radar pulse of duration τ is divided in N sub-pulses of equal duration
τ’ = τ /N, and each sub-pulse is phase coded in terms of the phase of the carrier.
The complex envelope of the phase coded
signal is given by:
( )
( )
( )∑
−
=
−=
1
0
2/1
'
'
1 N
n
n ntu
N
tg τ
τ
where:
( )
( )


 ≤≤
=
elsewhere
tj
tu n
n
0
'0exp τϕ

SOLO
Example: Pulse poly-phase coded of length 4
Given the sequence: { } 1,,,1 −−++= jjck
which corresponds to the sequence of phases 0◦, 90◦, 270◦ and 180◦, the matched filter is
given in Figure bellow.
{ } 1,,,1
*
−+−+= jjck

Pulse poly-phase coded of length 4
At the Receiver the coded pulse enters a 4 cells delay lane (from
left to right), a bin at each clock.
The signals in the cells are multiplied by -1,+j,-j or +1 and summed.
clock
SOLO
Poly-Phase Modulation
-1 = -11 1+
-j +j = 02 1+j+
+j -1-j = -13 1+j+j−
+1 +1+1+1 = 44 1+j+j−1−
-j-1+j = -1
5 j+j−1−
+j - j = 0
6
j−1−
7 1− -1 = -1
8 0
Σ
{ } 1,,,1 −−++= jjck
1− 1+j+ j− {ck*}
0 = 00
0
1
2
3
4
5
6
7
{ } 1,,,1* −+−+= jjck
Run This

SOLO
Bi-Phase Codes
• easy to implement
• significant range sidelobe reduction possible
• Doppler intolerant
A bi-phase code switches the absolute phase of the RF carrier between two states
180º out of phase.
Bandwidth ~ 1/τ
Transmitted Pulse
Received Pulse
• Peak Sidelobe Level
PSL = 10 log (maximum side-lobe power/
peak response power)
• Integrated Side-lobe Level
ISL = 10 log (total power in the side-lobe/
peak response power)
Bi-Phase Codes Properties
The most known are the Barker Codes sequence of length N, with sidelobes levels, at
zero Doppler, not higher than 1/N.

SOLO Pulse Compression Techniques
Bi-Phase Codes
Length
N
Barker Code PSL
(db)
ISL
(db)
2 + - - 6.0 - 3.0
2 + + - 6.0 - 3.0
3 + + - - 9.5 - 6.5
3 + - + - 9.5 - 6.5
4 + + - + - 12.0 - 6.0
4 + + + - - 12.0 - 6.0
5 + + + - + - 14.0 - 8.0
7 + + + - - + - - 16.9 - 9.1
11 + + + - - - + - - + - - 20.8 - 10.8
13 + + + + + - - + + - + - + - 22.3 - 11.5
Barker Codes
-Perfect codes –
Lowest side-lobes for
the values of N listed
in the Table.

Pulse bi-phase Barker coded of length 7
Digital Correlation
At the Receiver the coded pulse enters a
7 cells delay lane (from left to right),
a bin at each clock.
The signals in the cells are multiplied
by ck* and summed.
clock
-1 = -11
+1 -1 = 02
-1 +1 -1 = -13
-1 -1 +1-( -1) = 04
+1 -1 -1 –(+1)-( -1) = -15
+1 +1 -1-(-1) –(+1)-1= 06
+1+1 +1-( -1)-(-1) +1-(-1)= 77
+1+1 –(+1)-( -1) -1-( +1)= 08
+1-(+1) –(+1) -1-( -1)= -19
-(+1)-(+1) +1 -( -1)= 010
-(+1)+1-(+1) = -111
+1-(+1) = 012
-(+1) = -1
13
0 = 014
-1-1 -1+1+1+1+1 { }*
kc
Run This

Bi-Phase Codes
Combined Barker Codes
One scheme of generating codes longer than 13 bits is the method of forming combined
Barker codes using the known Barker codes.
For example to obtain a 20:1 pulse
compression rate, one may use either
a 5x4 or a 4x5 codes.
The 5x4 Barker code (see Figure)
consists of the 5 Barker code, each bit
of which is the 4-bit Barker code. The
5x4 combined code is the 20-bit code.
• Barker Code 4
• Barker Code 5

Bi-Phase Codes

Bi-Phase Codes
Binary Phase Codes Summary
• Binary phase codes (Barker, Combined Barker) are used in most radar applications.
• Binary phase codes can be digitally implemented. It is applied separately to I and Q
channels.
• Binary phase codes are Doppler frequency shift sensitive.
• Barker codes have good side-lobe for low compression ratios.
• At Higher PRFs Doppler frequency shift sensitivity may pose a problem.

Poly-Phase Codes
Frank Codes
In this case the pulse of width τ is divided in N equal groups; each group is
subsequently divided into other N sub-pulses each of width τ’. Therefore the
total number of sub-pulses is N2
, and the compression ratio is also N2
.
A Frank code of N2
sub-pulses is called a N-phase Frank code. The fundamental
phase increment of the N-phase Frank code is: N/360
=∆ ϕ
For N-phase Frank code the phase of each sub-pulse is computed from:
( )
( ) ( ) ( ) ( )
ϕ∆
















−−−−
−
−
2
1131210
126420
13210
00000
NNNN
N
N





Each row represents the phases of the sub-pulses of a group

Poly-Phase Codes
Frank Codes (continue – 1)
Example: For N=4 Frank code. The fundamental phase increment of the
4-phase Frank code is:

904/360 ==∆ ϕ
We have:














−−
−−
−−
⇒














→
jj
jjj
form
complex
11
1111
11
1111
901802700
18001800
270180900
0000
90




Therefore the N = 4 Frank code has the following N2
= 16 elements
{ }jjjjF 11111111111116 −−−−−−=
The phase increments within each row
represent a stepwise approximation of an up-
chirp LFM waveform.

Poly-Phase Codes
Example: For N=4 Frank code (continue – 1).
If we add 2π phase to the third N=4 Frank phase row and 4π phase to the forth
(adding a phase that is a multiply of 2π doesn’t change the signal) we obtain a
analogy to the discrete FM signal.
If we use then the
phases of the discrete linear FM
and the Frank-coded signals are
identical at all multipliers of τ’.
'/1 τ=∆ f

Poly-Phase Codes
Fig. 8.8 Levanon pg.158,159

SOLO
Pseudo-Random Codes
Pseudo-Random Codes are binary-valued sequences similar to Barker codes.
The name pseudo-random (pseudo-noise) stems from the fact that they resemble
a random like sequence.
The pseudo-random codes can be easily generated using feedback shift-registers.
It can be shown that for N shift-registers we can obtain a maximum length sequence
of length 2N
-1.
0 1 0 0 1 1 1
23
-1=7
Register
# 1
Register
# 2
Register
# 3
XOR
clock
A
B
Input A Input B Output XOR
0 0 0
0 1 1
1 0 1
1 1 0
Register
# 1
Register
# 2
Register
# 3
0 1 0
s
e
q
u
e
n
c
e
I.C.
0 0 11
1 0 02
1 1 03
1 1 14
0 1 15
1 0 16
0 1 07
clock
0 0 18
0
Run This

SOLO
Pseudo-Random Codes (continue – 1)
To ensure that the output sequence from a shift register with feedback is maximal length, the biths used in the
feedback path like in Figure bellow, must be determined by the 1 coefficients of primitive, irreducible
polynomials modulo 2. As an example for N = 4, length 2N
-1=15, can be written in binary notation as 1 0 0 1 1.
The primitive, irreductible polynomial that this denotes is
(1)x4
+ (0)x3
+ (0)x2
+ (1)x1
+ (1)x0
1 0 0 1 0 0 0 1 1 1 1 0 1 0 1
24
-1=15
s
e
q
u
e
n
c
e
1 0 0 1 I.C.0
The constant (last) 1 term in every such polynomial
corresponds to the closing of the loop to the first bit in the
register.
Register
# 1
Register
# 2
Register
# 3
XOR
clock
A
B
0 0 0
0 1 1
1 0 1
1 1 0
Register
# 4
Register
# 1
Register
# 2
Register
# 3clock
Register
# 4
1 0 1 0 0
0 0 1 02
0 0 0 13
1 0 0 04
1 1 0 05
1 1 1 06
1 1 1 17
0 1 1 18
1 0 1 19
0 1 0 110
1 0 1 011
1 1 0 112
0 1 1 013
0 0 1 114
1 0 0 115
0 1 0 016
0 0 1 017
Run This

SOLO
0 0 0
0 1 1
1 0 1
1 1 0
Register
# 1
Register
# 2
Register
# n
XOR
clock
A
B
Register
# (n-1)
Register
# m
. . .. . .
2 3 1 2,1
3 7 2 3,2
4 15 2 4,3
5 31 6 5,3
6 63 6 6,5
7 127 18 7,6
8 255 16 8,6,5,4
9 511 48 9,5
10 1,023 60 10,7
11 2,047 176 11,9
12 4,095 144 12,11,8,6
13 8,191 630 13,12,10,9
14 16,383 756 14,13,8,4
15 32,767 1,800 15,14
16 65,535 2,048 16,15,13,4
17 131,071 7,710 17,4
18 262,143 7,776 18,11
19 524,287 27,594 19,18,17,14
20 1,048,575 24,000 20,17
Number of
Stages n
Length of
Maximal Sequence N
Number of
Maximal Sequence M
Feedback stage
connections
Maximum Length Sequence
n – stage generator
N – length of maximum sequence
12 −= n
N
M – the total number of maximal-length
sequences that may be obtained
from a n-stage generator
∏ 





−=
ipN
n
M
1
1
where pi are the prime factors of N.

SOLO
Pseudo-Random Codes Summary
• Longer codes can be generated and side-lobes eventually reduced.
• Low sensitivity to side-lobe degradation in the presence of Doppler frequency shift.
• Pseudo-random codes resemble a noise like sequence.
• They can be easily generated using shift registers.
• The main drawback of pseudo-random codes is that their compression ratio
is not large enough.

SOLO
Waveform Hierarchy

SOLO Coherent Pulse Doppler Radar

SOLO
• Stepped Frequency Waveform (SFWF)
The Stepped Frequency Waveform is a Pulse Radar System technique for
obtaining high resolution range profiles with relative narrow bandwidth pulses.
• SFWF is an ensemble of narrow band (monochromatic) pulses, each of which
is stepped in frequency relative to the preceding pulse, until the required
bandwidth is covered.
• We process the ensemble of received signals using FFT processing.
• The resulting FFT output represents a high resolution range profile of the
Radar illuminated area.
• Sometimes SFWF is used in conjunction with pulse compression.

SOLO
• Stepped Frequency Waveform (SFWF)

SOLO
• Steped Frequency Waveform (SFWF)

SOLO RF Section of a Generic Radar
Antenna – Transmits and receives Electromagnetic
Energy
T/R – Isolates between transmitting and receiving
channels
REF – Generates and Controls all Radar frequencies
XMTR – Transmits High Power EM Radar frequencies
RECEIVER – Receives Returned Radar Power, filter it
and down-converted to Base Band for
digitization trough A/D.
Power Supply – Supplies Power to all Radar components.
Return to Table of Content

Antenna
Antenna performs the following essential functions:
• It transfers the transmitter energy to signals in space with the required distribution
and efficiency. This process is applied in an identical way on reception.
• It ensures that the signal has the required pattern in space. Generally this has to be
sufficiently narrow to provide the required angular resolution and accuracy.
• It has to provide the required time-rate of target position updates. In the case of a
mechanically scanned antenna this equates to the revolution rate. A high revolution
rate can be a significant mechanical problem given that a radar antenna in certain
frequency bands can have a reflector with immense dimensions and can weigh
several tons.
The antenna structure must maintain the operating characteristics under all
environmental conditions. Radomes (Radar Domes) are generally used where relatively
severe environmental conditions are experienced.
• It must measure the pointing direction with a high degree of accuracy.

Antenna pattern
Figure 1: Antenna pattern in a polar-coordinate graph
Figure 2: The same antenna pattern in
a rectangular-coordinate graph
Most radiators emit (radiate) stronger radiation in one
direction than in another. A radiator such as this is referred to
as anisotropic. However, a standard method allows the
positions around a source to be marked so that one radiation
pattern can easily be compared with another.
The energy radiated from an antenna forms a field
having a definite radiation pattern. A radiation pattern
is a way of plotting the radiated energy from an
antenna. This energy is measured at various angles at
a constant distance from the antenna. The shape of
this pattern depends on the type of antenna used.
Antenna Gain
Independent of the use of a given antenna for transmitting or
receiving, an important characteristic of this antenna is the gain.
Some antennas are highly directional; that is, more energy is
propagated in certain directions than in others. The ratio between
the amount of energy propagated in these directions compared to
the energy that would be propagated if the antenna were not
directional (Isotropic Radiation) is known as its gain. When a
transmitting antenna with a certain gain is used as a receiving
antenna, it will also have the same gain for receiving.

SOLO Antenna
Beam Width
Figure 1: Antenna pattern in a polar-coordinate graph
Figure 2: The same antenna pattern in
a rectangular-coordinate graph
The angular range of the antenna pattern in which at least
half of the maximum power is still emitted is described as a
„Beam With”. Bordering points of this major lobe are
therefore the points at which the field strength has fallen in
the room around 3 dB regarding the maximum field strength.
This angle is then described as beam width or aperture angle
or half power (- 3 dB) angle - with notation Θ (also φ). The
beam width Θ is exactly the angle between the 2 red marked
directions in the upper pictures. The angle Θ can be
determined in the horizontal plane (with notation ΘAZ) as
well as in the vertical plane (with notation ΘEL).
Major and Side Lobes (Minor Lobes)
The pattern shown in figures has radiation concentrated
in several lobes. The radiation intensity in one lobe is
considerably stronger than in the other. The strongest
lobe is called major lobe; the others are (minor) side
lobes. Since the complex radiation patterns associated
with arrays frequently contain several lobes of varying
intensity, you should learn to use appropriate
terminology. In general, major lobes are those in which
the greatest amount of radiation occurs. Side or minor
lobes are those in which the radiation intensity is least.

Radar Antenae for Different Frequency Spectrum

SOLO Antenna
Summary Radar Antennae
1. A radar antenna is a microwave system, that radiates or receives energy in the form
of electromagnetic waves.
2. Reciprocity of radar antennas means that the various properties of the antenna apply
equally to transmitting and receiving.
3. Parabolic reflectors („dishes”) and phased arrays are the two basic constructions of
radar antennas.
4. Antennas fall into two general classes, omni-directional and directional.
• Omni-directional antennas radiate RF energy in all directions simultaneously.
• Directional antennas radiate RF energy in patterns of lobes or beams that extend
outward from the antenna in one direction for a given antenna position.
5. Radiation patterns can be plotted on a rectangular- or polar-coordinate graph.
These patterns are a measurement of the energy leaving an antenna.
• An isotropic radiator radiates energy equally in all directions.
• An anisotropic radiator radiates energy directionally.
• The main lobe is the boresight direction of the radiation pattern.
• Side lobes and the back lobe are unwanted areas of the radiation pattern.

r
MAXr
S
S
G =:
Antenna
Bϕ
Bϑ
ϕD
ϑD
Antenna
Radiation
Beam
Assume for simplicity that the Antenna radiates all the power into the solid angle
defined by the product , where and are the angle from the
boresight at which the power is half the maximum (-3 db).
BB ϕϑ , 2/Bϕ± 2/Bϑ±
ϑϑ
λ
η
ϑ
D
B
1
=
ϕϕ
λ
η
ϕ
D
B
1
=
λ - wavelength
ϕϑ DD , - Antenna dimensions in directionsϕϑ,
ϕϑ ηη , - Antenna efficiency in directionsϕϑ,
then
( ) eff
BB
ADDG 22
444
λ
π
ηη
λ
π
ϕϑ
π
ϕϑϕϑ ==
⋅
=
where
ϕϑϕϑ ηη DDAeff =:
is the Effective Area of the Antenna.
2
4
λ
π
=
effA
G
SOLO
Antenna Gain

Antenna
Transmitter
IV
Receiver
R
1 2
Let see what is the received power on an
Antenna, with an effective area A2 and range R
from the transmitter, with an Antenna Gain G1
Transmitter
VI
Receiver
R
1 2
2122
4
AG
R
P
ASP dtransmitte
rreceived
π
==
Let change the previous transmitter into a receiver and the receiver into a
transmitter that transmits the same power as previous. The receiver has now an
Antenna with an effective area A1 . The Gain of the transmitter Antenna is now G2.
According to Lorentz Reciprocity Theorem the
same power will be received by the receiver; i.e.:
122
4
AG
R
P
P dtransmitte
received
π
=
therefore
1221 AGAG =
or
const
A
G
A
G
==
2
2
1
1
We already found the constant; i.e.: 2
4
λ
π
=
A
G
SOLO

AntennaSOLO
There are two types of antennas in modern fighters
1. Mechanically Scanned Antenna (MSA)
In this case the antenna is gimbaled and antenna servo is used to move the
antenna (and antenna beam) in azimuth and elevation.
For target angular position, relative to antenna axis two methods are used:
• Conical scan of the antenna beam relative to antenna axis (older technique)
• Monopulse antenna beam where the antenna is divided in four quadrants
and the received signal of those quadrants is processed to obtain the
sum (Σ) and differences in azimuth and elevation (ΔEl, ΔAz) are
processed separately (modern technique)
2. Electronic Scanned Antenna (ESA)
The antenna is fixed relative to aircraft and the beam is electronically steered in
azimuth and elevation relative to antenna (aircraft) axis. Two types are known:
• Passive Electronic Scanned Array (PESA)
• Active Electronic Scanned Array (AESA) with Transmitter and Receiver (T/R)
elements on the antenna.

SOLO Airborne Radars

Stimson, G.W.,
“Introduction to Airborne
Radar”, 2nd
ed., Scitech
Publishing, 1998
Conically Scanned Antenna

Conical scan radar
SOLO Conical Scan Angular Measurement

Target Angle φ Detector

ERROR DETECTION CONTROL-SCAN RADARERROR DETECTION CONTROL-SCAN RADAR
CONTROL-SCAN TRACKINGCONTROL-SCAN TRACKING CONTROL-SCAN BEAM RELATIONSHIPSCONTROL-SCAN BEAM RELATIONSHIPS
ENVELOPE OF PULSESENVELOPE OF PULSES

Stimson, G.W.,
Radar”, 2nd
ed., Scitech
Publishing, 1998
Monopulse antenna

Monopulse Angle Measurement

SOLO
Monopulse Angular Track

SOLO
Electronically Scanned Array (ESA)
Stimson, G.W., “Introduction to Airborne Radar”, 2nd
ed., Scitech Publishing, 1998

Electronically Scanned Array

Electronically Scanned Array
Stimson, G.W., “Introduction to Airborne Radar”, 2nd
ed., Scitech Publishing, 1998
http://www.ausairpower.net/APA-Zhuk-AE-Analysis.html

Electronic Scanned Antenna
Stimson, G.W.,
Radar”, 2nd
ed., Scitech
Publishing, 1998

SOLO
Antenna

SOLO
Radar Basic
Continue to
Radar Basic- Part II

January 14, 2015 195
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA

1 radar basic -part i 1

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