电子战常用公式.pdf
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RF Propagation
RADAR HORIZON
RF Propagation
TARGET VISIBILITY
Detection & Estimation Probability
CRAMER RAO LOWER BOUND
Detection & Estimation Probability
MAX LIKELIHOOD ESTIMATION
Detection & Estimation Probability
BINOMIAL
Antennas
ANTENNA BEAMWIDTH
Antennas
ANTENNA DIRECTIVITY
Antennas
ANTENNA GAIN
Fourier Relationships
CONTINUOUS-TIME FOURIER TRANSFORMATION
Fourier Relationships
FILTERING
Radar Processing
RADAR CROSS SECTION
Fourier Relationships
MODULATION PROPERTY
Detection & Estimation Probability
RICIAN
Detection & Estimation Probability
ERROR FUNCTIONS
Detection & Estimation Probability
NORMAL
Detection & Estimation Probability
RAYLEIGH
RF Propagation
WAVELENGTH
RF Propagation
DOPPLER SHIFT
P
r
=P
t
G
t
G
r
4πR
λ
2
Pr: Received Power
Pt: Transmit Power
Gt: Transmit Gain
Gr: Receive Gain
R: Range
c: Speed
f: Frequency
H: Horizon
Re: Earth Radius ~ 6,371 km
H: Horizon
Re: Earth Radius ~ 6,371 km
x: Observations
p: Probability distribution function (or joint)
θ: Distribution parameters can be vectors
p: Success probability of each trial
k: Number of successes
n: Number of trials
λ: Wavelength
d: Antenna Diameter
θ
1d
: Half-power beamwidth in one principal plane (degrees)
θ
2d
: Half-power beamwidth in the other principal plane (degrees)
A
e
: Effective Aperture Area
λ: Wavelength
μ: Mean
σ: Standard Difference
A: Distance between the reference point and
the center of the bivariate distribution
I
0
: Bessel Function of the fi rst kind with order zero
μ: Mean
σ: Standard Difference
A: Distance between the reference point and
the center of the bivariate distribution
RF Propagation
FRIIS TRANSMISSION EQUATION
D
h
= 2HR
e
λ =
f
c
f
d
= –2v
r
/ λ
Target
Height
2Re
(Target Range - 2HRe)
2
=
CRB =
∂θ
∂
ln p(x, θ)
[ ]
E
∂θ
∂
ln p(x, θ)
[ ]
(
}
T
{
)
-1
f(k; n, p)= Pr(X =k) =
( )
p
k
(1−p)
n−k
k
n
p(r)=
{
r
σ
2
e
r
2
2σ
2
−
0
(r < 0) (0≤r≤∞)
1.2
1
0.8
0.6
0.2
0.4
0
02 46
810
σ = 0.5
σ =3
σ =1
σ = 4
σ = 2
p(r)=
{
−
I
0
( )
for (r < 0)
for (A ≥ 0, r ≥ 0)
σ
2
r
e
0
2σ
2
(r
2
+A
2
)
σ
2
Ar
0.6
0.5
0.4
0.3
0.1
0.2
0.0
02 46
8
v = 0.0
v = 2.0
v = 0.5
v = 4.0
v = 1.0
σ = 1.00
μ: Mean
σ: Standard Difference
A: Distance between the reference point and
the center of the bivariate distribution
1
σ 2π
p(x)
=
(μ
z
=0; σ
x
=1.0)
e
−
(x−μ)
2σ
2
Standard Normal Curve
f(z)
0.4
0.1
0.2
0.3
-3 -2 -1
03
21
68.27%
95.45%
99.73%
3-σ
2-σ
1-σ
z
1
2π
z
2
2
e
[
-
]
Joint Density Function
Π
L(θ; x
1
, ..., x
n
)= f (x
1
, x
2
, ..., x
n
|
θ)= f (x
i
|
θ)
n
i = 1
Likelihood
Σ
ln L (θ; x
1
, ..., x
n
)= ln f (x
i
|
θ)
n
i = 1
Log-Likelihood
x
i
: Observations
n: Number of Samples
f: Is one, or joint, probability distribution(s)
θ: Distribution parameters can be vectors
μ: Mean
σ: Standard Difference
A: Distance between the reference point and
the center of the bivariate distribution
erfc(z)=
1−erf(z)=
2
π
∫
∞
e
-t
2
d t
erfc(x)
2
1.5
0.5
-2-4 -2-4 42
1
z
erf(z)=
2
π
∫
z
0
e
-t
2
d t
±1-σ: P (-1 ≤ z ≤ 1) = 0.6827
±2-σ: P (-2 ≤ z ≤ 2) = 0.9545
±3-σ: P (-3 ≤ z ≤ 3) = 0.9973
θ
BW
3dB
∼ 0.886 b
Nd cos θ
0
λ
Phased Array, Radians
θ
BW
null
∼ 1.22
d
λ
d
λ
θ
BW
3dB
∼ 0.88
Parabolic, Radians
D
≈ 4π ≈
θ
1d
θ
2d
180
π
(
)
2
40000
θ
1d
θ
2d
G
ant
=
λ
2
4πA
e
s(τ) = e
j2π(f
c
τ+ bτ
2
)
, - ≤ τ ≤
2
τ
p
2
τ
p
2
1
B
p
= bτ
p
γ
(frequency)
τ
(time)
determines
signal energy
τ
p
τpτ
B
p
BpB
determines
resolution
→
→
→
→
s(): Transmitted Signal Waveform
f
c
: Center Frequency
τ: Range Time (fast time)
τ
p
: Pulse Length
b: Chirp Rate
B
p
: Pulse Bandwidth
γ: Range Frequency
* “u” stands for unabsorbed or under K; “a” stands for absorption region or above K
Electronic Warfare
NOISE JAMMING
Sidelobe
J
self
: Self Protect Jammer Power
J/S: Jam to Signal Ratio at Radar Receiver
S: Radar Received Signal Power
P
t
jam
: Jammer Transmit Power
G
t
jam
: Jammer Transmit Gain
R
jr
: Range between Jammer and Radar
R: Range between Radar Target and Radar
λ: Jammer Transmit Wavelength
G
r
radar
: Radar Receiver Gain
L
r
radar
: Radar Receiver Losses
P
t
radar
: Radar Transmit Power
G
t
radar
: Radar Transmitter Gain
σ: Radar Target Radar Cross Section
BW
Radar
: Radar Transmit Bandwidth
BW
Jam
: Jammer Transmit Bandwidth
J: Jammer Power
Rmax
jammed
: Jammed Radar Range
(Burn through Range)
R
max
: Max Radar Range
J/N: Jammer to Noise Ratio
N: Total Noise
k: Boltzmann’s constant
T
s
: Receiver Temperature
B
N
: Receiver Noise Bandwidth
SNR: Radar Signal to Noise Ratio
N
f
: Receiver Noise Figure (>1)
J
S
=
EIRP
jam
EIRP
radar
( )
4πR
2
σ
( )
( )
J
S
=
EIRP
jam
EIRP
radar
4πR
2
σ
( )
BW
radar
( )
BW
jam
P
t
jam
G
t
jam
( )
2
λ
4πR
jr
J
self
=
L
r
radar
}
EIRP
jam
If BW
jam
≥
BW
radar
10
1
-150
-140
-130
-120
-110
-100
-90
-80
-70
-60
Reduction in Radar Detection Range due to JNR
Norm
alized Maximum Radar Rang
e
Range (km)
10
2
10
3
Skin Return R
4
Jammer R
2
J
S
Burn- through
range for SNR =
13 dB
J/N ~ ( )
4
R
max
jammed
R
max
Assume: J >> N
BW
Jam
= BW
Radar
R
max
jammed
4
=
P
t
G'
t
G'
r
λ
2
(4π)
3
(kT
s
B
N
N
f
+J)
*
SNR
*
L
r
*
L
t
Mainlobe
Reduction in Normalized R
max
1
0.8
0.6
0.4
0.2
Rmax
Rmax Jammed
Main
Beam
↓
Reduction in Radar Detection Range due to JNR
Norm
alized Maximum Radar Rang
e
Jammer to Noise Ratio (dB)
0510 15 20 25 30 35 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x(t) ↔ X(ω)
2π
1
x(t) = ∫
X(ω)e
jωt
dω
+∞
-∞
Synthesis
X(ω) = ∫
x(t)e
-jωt
dt
+∞
-∞
Analysis
h(t)
*
x(
t)
↔
H(ω)
X(ω)
X(ω)
x(t)
H(ω)
h(t)
H(ω) X(t)
h(t)
*
x(t)
1
δ(t)
H(ω)
h(t)
H(ω)
h(t)
e
jω
ο
t
H(ω)
e
jω
ο
t
H(ω
ο
)
H(ω): Frequency Response
: Convolution operation
Convolution Property
Ideal Lowpass Filter
Differentiator
Fourier Relationships
PARSEVAL’S RELATION
2π
1
∫
|x(t)|
2
dt =
∫
|X(ω)|
2
dω
+∞
-∞
+∞
-∞
∫
T
o
|x(t)|
2
dt =
∑
|a
k
|
2
+∞
k=-∞
~
T
o
1
H(ω)
-ω
c
ω
c
ω
y(t) = =>H(ω) = jω
dt
dx(t)
|H(ω)|
ω
x(t-t
o
)
↔
e
-jωt
o
X(ω)
Time Shifting
Differentiation
↔
jω X(ω)
dt
dx(t)
jω
1
∫
t
x(τ)dτ
↔
X(ω) + πX(0) δ(ω)
-∞
Integration
Linearity
ax
1
(t)+bx
2
(t)
↔
aX
1
(ω)+bX
2
(ω)
Modulation
Duality Property
2π
1
s(t) p(t)
↔
[S(ω)P(ω)]
Convolution
h(t)
*
x(t)
↔
H(ω)X(ω)
x(t)
t
1/a
1
1
e
- aa
1/a √2
1/a
|X(ω)|
ω
↓
< X(ω)
π/2
π/4
− π/4
− π/2
ω
− a
X(ω)
- w
w
ω
1
X(ω)
t
π
T
1
sin ωT
1
ω
2
2T
1
x(t)
- T1
T1
t
1
x(t)
t
π
w
w
π
sin wt
2πt
2
S∝σ, range
Radar Cross Section (RCS, σ)
Scattering
σ = = lim 4πr
2
Incident Power Density / 4π
Refl ected Power to Receiver / Solid Angle
|E
i
|
2
|E
s
|
2
( )
P
t
P
r
or S
σ
Radar Processing
TYPICAL VALUES OF RCS
Radar Processing
RADAR AMBIGUITY FUNCTION
Radar Processing
NOISE POWER
Radar Processing
SPEED OF LIGHT
Radar Processing
MAX UNAMBIGUOUS RANGE
Radar Processing
SIGNAL TO NOISE RATIO
S(t): Complex Baseband Pulse
τ: Time Delay
f: Doppler Shift
x(τ, t) =∫
∞
s(t)s
*
(t-τ)e
i2πft
dt
−∞
= kT
s
B
N
N
f
Noise Power in Receiver
Speed of Light (approx)
3x10^8
300
1.62x10^5
1x10^9
1x10^3
Units
m/sec
m/usec
NM/sec
Ft/sec
Ft/usec
R
max
=
c
2PRF
PRF
High
Medium
Low
PRF
100 kHz
25 kHz
10 kHz
Unambiguous Range
1.5 km
6 km
15 km
Range
Ambiguous
Ambiguous
Unambiguous
Doppler
Unambiguous
Ambiguous
Ambiguous
c: Speed of Light
PRF: Pulse Repetition Frequency
Pr: Received Power
Pt: Transmit Power
Gt: Transmit Gain
Gr:
Receive Gain
R: Range
No: Noise Power
L: Losses
P
R
SNR= =
P
t
G
t
G
r
σλ
2
G
p
L
(4π)
3
R
4
k
B
T
s
B
n
N
f
N
o
* “u” stands for unabsorbed or under K; “a” stands for absorption region or above K* “u” stands for unabsorbed or under K; “a” stands for absorption region or above K
ELECTRONIC WARFARE QUICK REFERENCE GUIDE
Military Standard Bands
U.
S.
Industry Standard Bands
(IEEE Radar Designation)
HF
VH
FL
FL
FL
FL
UHF
FLUHF
FL
SW
SW
SWKV
SW
SWKV
SW
SW
K
SW
SWKV
SW
K
SWKV
SW
*
SW
*
SW
SWKV
SW
*
SWKV
SW
a
Millimeter
SW
K
SW
*
SW
*
SW
u
SWX
SW
SWC
SW
International Standard Bands
GH
GH
GH
GH
C
BM
BM
AK
AK
BMAK
BM
BMAK
BM
BMAK
BM
BMAK
BM
GHBM
GH
AK
GHBM
GH
GHBM
GH
AK
GHBM
GH
FBM
F
AK
FBM
F
BMAK
BM
BMAK
BM
BMAK
BM
BMAK
BM
C
BM
C
AK
C
BM
C
BMDE
BM
AK
BMDE
BM
BMDE
BM
AK
BMDE
BM
BML
BM
J
BMAK
BM
J
BMAK
BM
BMAK
BM
I
BMAK
BM
250
Frequency (MHz)
Frequency (GHz)
20
30
100
200
300
500
400
300
200
100
80
60
40
30
20
15
10
8
6
5
4
3
2
1.
5
12
18
27
110
110110
7 (HF)
8 (VHF)
9 (UHF)
10 (SHF)
12
11(EHF)
Band
Designation
HF
VHF
UHF
L
S
C
X
K
u
K
K
a
V
W
Frequency
Range
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
F
F
F
F
F
F
F
F
RF Propagation Detection & Estimation Probability Antennas
f
z
(z)=
-∞<x<∞
RADIO
Wavelength (Meters)
Frequency (Hz)
10
3
10
4
10
8
10
12
10
15
MICROWAVE
10
-2
INFRARED
10
-5
VISIBLE
10
-6
ULTRAVIOLET
10
-8
X-RAY
10
-10
GAMMA RAY
10
-12
THE ELECTROMAGNETIC SPECTRUM
10
16
10
18
10
20
= ln L
n
1
=(θ|x) = ln f (x
i
|
θ)
Σ
n
i = 1
n
1
Average Log-Likelihood
ˆ
ˆ
P
t
radar
G
t
radar
G
r
λ
2
S
=
(4π)
3
R
4
}
EIRP
radar
σ
G
r
radar
Radar Processing
LINEAR FM WAVEFORM
σ
f (x
1
, x
2
, ..., x
n
|
θ)= f (x
1
|
θ) x f (x
2
|
θ) x ... x f (x
n
|
θ)
.0001
-40
Insects Birds HumanSmall Car
Fighter
Aircraft
Bomber:
Transport
Aircraft
Ships
-30 -20 -10 010203040
.001 .01 0.1 1.0 10 100 1000 10000
dBsm
m
2
Electronic Warfare Fourier Relationships Radar Processing
X-band
300 m/s
0.03 m
20 kHz
S-band
300 m/s
0.1 m
6 kHz
Wavelength
3.00 m
0.10m
0.05m
0.03m
f
100 MHz
3 GHz
6 GHz
10 GHz
Band
VHF
S
C
X
Velocity
Wavelength
Doppler Shift
kT
s
: = -174 dBm
K: Boltzmann’s constant = 1.38*10
-23
J/K
B
n
: Noise Bandwidth
T
s
: System Noise Temperature
T
s
usually set to T
0
= 290K
N
f
: Noise fi gure of receiver
r ∞
K: Boltzmann’s constant = 1.38*10
-23
J/K
B
n
: Noise Bandwidth
T
s
: System Noise Temperature
T
s
usually set to T
0
= 290K
N
f
: Noise fi gure of receiver
σ
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