ELECTRONIC WARFARE Q
QUICK REFERENCE GUIDE
Frequency (MHz)
THE ELECTROMAGNETIC SPECTRUM
20
Wavelength (Meters)
RADIO
10
MICROWAVE
3
10
INFRARED
-2
VISIBLE
-5
ULTRAVIOLET
-6
10
10
10
X-RAY
-8
30
100
Frequency (GHz)
200 300
1.5 2
500
3 4 5 6 8 10 15 20 30 40 60 80 100
12 18 27
GAMMA RAY
-10
10
10
VHFF
7 (HF)
8 (VHF)
S
L
UHF
X K*u K K*a
C
9 (UHF)
W
V
Millimeter
11(EHF)
10 (SHF)
Frequency (Hz)
10
8
10
12
10
15
10
16
18
10
10
B
A
20
Band Frequency
Designation Range
12
HF
VHF
UHF
L
S
C
X
Ku
K
Ka
V
W
International Standard Bands
250
4
U.S. Industry Standard Bands
(IEEE Radar Designation)
110
-12
HF
10
200 300 400
E F G H I
D
C
K
J
M
L
Military Standard Bands
* “u” stands for unabsorbed or under K; “a” stands for absorption region or above K
EIRPradar
Target (Target Range - 2HRe)2
=
Height
2Re
radar
}
100 MHz
3.00 m
S
3 GHz
0.10m
X-band
S-band
C
6 GHz
0.05m
Velocity 300 m/s
300 m/s
X
10 GHz
0.03m
Wavelength 0.03 m
0.1 m
Doppler Shift 20 kHz
6 kHz
f(k; n, p)= Pr(X =k) =( n ) pk (1−p)n−k
k
p(r)=
{
r
σ
e
2
−
ln L (θ; x1, ..., xn )= Σ ln f (xi| θ)
i=1
ˆ
ˆ
0.8
4
6
8
10
p(x) =
for (r < 0)
(x−μ)
σ 2π
e − 2σ2
=
=(θ|x) =
(μz=0; σx=1.0)
[ ]
σ = 1.00
v = 0.0
v = 0.5
v = 1.0
v = 2.0
v = 4.0
0.5
0.4
-∞<x<∞
1
0.8
-80
0.6
0.4
-90
0.2
Burn- through
range for SNR =
13 dB
-120
Σ ln f (xi| θ)
i=1
-130
∞
∫z
J
-140
S
-2
-1
0.1
0
1
68.27%
95.45%
99.73%
1-σ
2
z
3
2-σ
e
dt
0.8
0.7
0.6
+∞
x(t) = 1 -∞∫ X(ω)e jωt dω
2π
→
τp
Noise Power in Receiver = kTsBNNf
kTs : = -174 dBm
K: Boltzmann’s constant = 1.38*10-23 J/K
Bn: Noise Bandwidth
Ts: System Noise Temperature
Ts usually set to T0= 290K
Nf : Noise figure of receiver
0.4
Fourier Relationships
PARSEVAL’S RELATION
0.3
0.2
5
10
15
20
25
30
Jammer to Noise Ratio (dB)
35
Radar Processing
SPEED OF LIGHT
40
1 ∫ |X(ω)|2 dω
∫-∞ |x(t)|2 dt = 2π
-∞
+∞
Rmax: Max Radar Range
J/N: Jammer to Noise Ratio
N: Total Noise
k: Boltzmann’s constant
Ts: Receiver Temperature
BN: Receiver Noise Bandwidth
SNR: Radar Signal to Noise Ratio
Nf : Receiver Noise Figure (>1)
+∞
Speed of Light (approx)
Units
3x10^8
300
1.62x10^5
1x10^9
1x10^3
m/sec
m/usec
NM/sec
Ft/sec
Ft/usec
+∞
~
2
2
1 ∫ |x(t)|
|
dt
=
|a
∑
k
To To
k=-∞
Radar Processing
MAX UNAMBIGUOUS RANGE
x(t)
+∞
X(ω) =-∞∫ x(t)e -jωt dt
∫
Modulation
F
s(t) p(t)↔
x(t)
1 [S(ω)P(ω)]
2π
Convolution
F H(ω)X(ω)
h(t)* x(t) ↔
1
F e -jωto X(ω)
x(t-to) ↔
Fourier Relationships
FILTERING
1
e
|X(ω)|
Ideal Lowpass Filter
2
Differentiator
y(t) = dx(t) =>H(ω) = jω
dt
4
-ωc
ωc ω
ω
Doppler
PRF
Unambiguous Range
Ambiguous
Unambiguous
100 kHz
1.5 km
Ambiguous
25 kHz
6 km
Unambiguous Ambiguous
10 kHz
15 km
Low
c: Speed of Light
PRF: Pulse Repetition Frequency
ω
w
x(t)
Radar Processing
SIGNAL TO NOISE RATIO
a
1
- T1
π/2
F 1 X(ω) + πX(0) δ(ω)
∫-∞ x(τ)dτ↔
jω
FaX1(ω)+bX2(ω)
ax1(t)+bx2(t)↔
Range
High
< X(ω)
Integration
t
Linearity
μ: Mean
σ: Standard Difference
A: Distance between the reference point and
the center of the bivariate distribution
PRF
1
X(ω)
|H(ω)|
c
2PRF
X(ω)
1/a
-a
Rmax =
Medium Ambiguous
t
π
w
ω
dx(t) F
jω
dt ↔
sin wt
2πt
t
1/a
-w
Differentiation
2
w
π
1/a √2
H(ω)
μ: Mean
σ: Standard Difference
A: Distance between the reference point and
the center of the bivariate distribution
Radar Processing
NOISE POWER
0.5
SNR=
t
T1
X(ω)
π/4
2T1
2
ω
−a
μ: Mean
σ: Standard Difference
A: Distance between the reference point and
the center of the bivariate distribution
I0: Bessel Function of the first kind with order zero
τ (time)
Fourier Relationships
MODULATION PROPERTY
F
x(t) ↔ X(ω)
3-σ
8
S(t): Complex Baseband Pulse
τ: Time Delay
f: Doppler Shift
Analysis
Synthesis
z e -t2 d t
0
-2
x(τ, t) =∫−∞∞s(t)s*(t-τ)ei2πft dt
Duality Property
-t 2
0.5
-4
2
s(): Transmitted Signal Waveform
fc: Center Frequency
τ: Range Time (fast time)
τp: Pulse Length
b: Chirp Rate
Bp: Pulse Bandwidth
γ: Range Frequency
0.9
Fourier Relationships
CONTINUOUS-TIME FOURIER TRANSFORMATION
1
0.2
Reduction in Radar Detection Range due to JNR
Lrradar: Radar Receiver Losses
Ptradar: Radar Transmit Power
Gtradar: Radar Transmitter Gain
σ: Radar Target Radar Cross Section
BWRadar: Radar Transmit Bandwidth
BWJam: Jammer Transmit Bandwidth
J: Jammer Power
Rmaxjammed: Jammed Radar Range
(Burn through Range)
4
Jself: Self R
Protect
Jammer Power
Skin Return
J/S: Jam to 2Signal Ratio at Radar Receiver
Jammer
R Received Signal Power
S: Radar
Ptjam: Jammer Transmit Power
Gtjam: Jammer Transmit Gain
Rjr: Range between Jammer and Radar
R: Range between Radar Target and Radar
λ: Jammer Transmit Wavelength
Grradar: Radar Receiver Gain
2
τp
determines
signal energy
1
0.1
0
Rmax
Rmax Jammed
103
102
Range (km)
1.5
0.2
-3
Main
Beam
≤τ≤
determines Bp
resolution
Sidelobe
Time Shifting
0.1
6
)
τp
(frequency)
γ
Reduction in Normalized Rmax
erfc(x)
2
±1-σ: P (-1 ≤ z ≤ 1) = 0.6827
±2-σ: P (-2 ≤ z ≤ 2) = 0.9545
±3-σ: P (-3 ≤ z ≤ 3) = 0.9973
0.4
)(
BWradar
BWjam
,-
Bp = bτ p
PtG'tG'r λ2
σ
=
(4π)3(kTsBNNf +J)*SNR*Lr*Lt
-110
n
2
π
)(
4πR2
σ
0.3
0.3
4
2
π
f(z)
2
0.6
1
n
ln L
erfc(z)=1−erf(z)=
erf(z)=
z
fz(z)= 1 e - 2
2π
)
-70
Detection & Estimation Probability
ERROR FUNCTIONS
Standard Normal Curve
1
)(
Rmaxjammed
4
-100
xi : Observations
n: Number of Samples
f: Is one, or joint, probability distribution(s)
θ: Distribution parameters can be vectors
Detection & Estimation Probability
NORMAL
for (A ≥ 0, r ≥ 0)
1
n
0.4
2
Mainlobe
radar
4πR2
σ
)
Assume: J >> N
BWJam = BWRadar
Reduction in Radar Detection Range due to JNR
-60
σ = 0.5
σ =1
σ=2
σ =3
σ=4
0.6
Detection & Estimation Probability
RICIAN
(
Log-Likelihood
n
Average Log-Likelihood
1
0
EIRPjam
J
= EIRP
S
radar
-150
101
0.2
μ: Mean
σ: Standard Difference
0
A: Distance between the reference point and 0
the center of the bivariate distribution
p: Success probability of each trial
k: Number of successes
n: Number of trials
p(r)= σ
0
] })
-1
1.2
(r < 0) (0≤r≤∞)
2
2
r e − (r +A2 )I0 ( Ar2 )
σ
2σ
2
∂ ln p(x, θ)
∂θ
T
Detection & Estimation Probability
RAYLEIGH
r2
2σ 2
maxjammed
4
If BWjam ≥ BWradar
x: Observations
p: Probability distribution function (or joint)
θ: Distribution parameters can be vectors
Detection & Estimation Probability
BINOMIAL
J/N ~ ( R
Rmax
s(τ) = e j2π(fcτ+2
→
VHF
][
(
σ
λ 2 Grradar
4πRjr ) Lr
EIRPjam
J
= EIRP
S
radar
i=1
1 bτ2)
↓
f
c: Speed
f: Frequency
( {[
CRB = E
∂ ln p(x, θ)
∂θ
n
L(θ; x1, ..., xn )= f (x1, x2, ..., xn | θ)= Π f (xi| θ)
Detection & Estimation Probability
CRAMER RAO LOWER BOUND
fd = –2vr / λ
Wavelength
Likelihood
Gr λ2
(4π)3 R4
Jself = Ptjam Gtjam(
H: Horizon
Re: Earth Radius ~ 6,371 km
RF Propagation
DOPPLER SHIFT
Band
2
radar
f (x1, x2, ..., xn | θ)= f (x1 | θ) x f (x2 | θ) x ... x f (xn | θ)
Normalized Maximum Radar Range
H: Horizon
Re: Earth Radius ~ 6,371 km
λ= c
f
0
S=
Joint Density Function
Gt
EIRPjam
RF Propagation
WAVELENGTH
0.0
Pt
↓
Dh= 2HRe
Radar Processing
RADAR AMBIGUITY FUNCTION
Radar Processing
LINEAR FM WAVEFORM
Normalized Maximum Radar Range
2
Pr: Received Power
Pt: Transmit Power
Gt: Transmit Gain
Gr: Receive Gain
R: Range
{
Electronic Warfare
NOISE JAMMING
Detection & Estimation Probability
MAX LIKELIHOOD ESTIMATION
→
RF Propagation
TARGET VISIBILITY
}
λ
Pr =Pt Gt Gr 4πR
RF Propagation
RADAR HORIZON
→
RF Propagation
FRIIS TRANSMISSION EQUATION
3–30 MHz
30–300 MHz
300–1,000 MHz
1–2 GHz
2–4 GHz
4–8 GHz
8–12 GHz
12–18 GHz
18–27 GHz
27–40 GHz
40–75 GHz
75–110 GHz
sin ωT1
ω
Pr: Received Power
Pt: Transmit Power
Gt: Transmit Gain
Gr: Receive Gain
R: Range
No: Noise Power
L: Losses
− π/4
− π/2
π
T1
PR PtGtGrσλ2GpL
=
No (4π)3R4kBTsBnNf
t
-23
K: Boltzmann’s constant = 1.38*10 J/K
Bn: Noise Bandwidth
Ts: System Noise Temperature
Ts usually set to T0= 290K
Nf : Noise figure of receiver
Convolution Property
Antennas
ANTENNA BEAMWIDTH
Phased Array, Radians
θBW3dB ∼ 0.886
Antennas
ANTENNA DIRECTIVITY
λ
Nd cos θ0
x(t)
X(ω)
b
D ≈ 4π
Parabolic, Radians
(
180
π
)
2
θ1d θ2d
40000
Gant =
≈ θ1d θ2d
θBWnull ∼ 1.22 λ θBW3dB ∼ 0.88 λ
d
d
θ1d: Half-power beamwidth in one principal plane (degrees)
θ2d: Half-power beamwidth in the other principal plane (degrees)
λ: Wavelength
d: Antenna Diameter
RF Propagation
Detection & Estimation Probability
F H(ω) X(ω)
h(t)* x(t) ↔
Antennas
ANTENNA GAIN
Antennas
Electronic Warfare
Fourier Relationships
4πAe
λ2
Ae: Effective Aperture Area
λ: Wavelength
Radar Processing
h(t)
H(ω)
h(t)* x(t)
H(ω) X(t)
e
jωοt
H(ω)
δ(t)
1
e
jωοt
h(t)
H(ω)
h(t)
H(ω)
Pt
Pr or S
2
|Es|
Reflected Power to Receiver / Solid Angle
2
σ=
= lim 4πr
2
Incident Power Density / 4π
r ∞
|Ei|
σ
σ
(
H(ωο)
H(ω): Frequency Response
: Convolution operation
Radar Processing
TYPICAL VALUES OF RCS
Radar Processing
RADAR CROSS SECTION
)
.0001
.001
.01
0.1
1.0
10
100
1000
-40
-30
-20
-10
0
10
20
30
Insects
Birds
Human
S∝σ, range
Radar Cross Section (RCS, σ)
Scattering
m2
40
dBsm
Ships
Small Car
Fighter
Aircraft
10000
Bomber:
Transport
Aircraft
THE ELECTRONIC WARFARE
QUICK REFERENCE GUIDE
Raytheon is a proud sponsor of
the Association of Old Crows.
To download a digital copy of this poster,
please visit www.raytheon.com/ew