Q
RF PropagationRADAR HORIZON
RF PropagationTARGET VISIBILTY
Detection & Estimation ProbabilityCRAMER RAO LOWER BOUND
Detection & Estimation ProbabilityMAX LIKELIHOOD ESTIMATION
Detection & Estimation ProbabilityBINOMIAL
AntennasANTENNA BEAMWIDTH
AntennasANTENNA DIRECTIVITY
AntennasANTENNA GAIN
Fourier RelationshipsCONTINUOUS-TIME FOURIER TRANSFORMATION
Fourier RelationshipsFILTERING
Radar ProcessingRADAR CROSS SECTION
Fourier RelationshipsMODULATION PROPERTY
Detection & Estimation ProbabilityRICIAN
Detection & Estimation ProbabilityERROR FUNCTIONS
Detection & Estimation ProbabilityNORMAL
Detection & Estimation ProbabilityRAYLEIGH
RF PropagationWAVELENGTH
RF PropagationDOPPLER SHIFT
Pr =PtGtGr 4πRλ
2
Pr: Received PowerPt: Transmit PowerGt: Transmit GainGr: Receive Gain
R: Range
c: Speedf: Frequency
H: HorizonRe: Earth Radius ~ 6,371 km
H: HorizonRe: Earth Radius ~ 6,371 km
x: Observationsp: Probability distribution function (or joint)
θ: Distribution parameters can be vectors
p: Success probability of each trialk: Number of successes
n: Number of trials
λ: Wavelengthd: Antenna Diameter
θ1d: Half-power beamwidth in one principal plane (degrees)θ2d: Half-power beamwidth in the other principal plane (degrees)
Ae: Effective Aperture Areaλ: Wavelength
μ: Meanσ: Standard Difference
A: Distance between the reference point and the center of the bivariate distribution
I0: Bessel Function of the fi rst kind with order zero
μ: Meanσ: Standard Difference
A: Distance between the reference point and the center of the bivariate distribution
RF PropagationFRIIS TRANSMISSION EQUATION
Dh= 2HRe
λ = fc
fd = –2vr / 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) =( ) pk (1−p)n−k kn
p(r)={ rσ2 e
r 2
2σ 2−
0
(r < 0) (0≤r≤∞)
1.2
1
0.8
0.6
0.2
0.4
00 2 4 6 8 10
σ = 0.5
σ =3
σ =1
σ = 4
σ = 2
p(r)={− I0
( )for (r < 0)
for (A ≥ 0, r ≥ 0)σ2r e
0
2σ2(r2+A2)
σ2Ar
0.6
0.5
0.4
0.3
0.1
0.2
0.00 2 4 6 8
v = 0.0
v = 2.0
v = 0.5
v = 4.0
v = 1.0
σ = 1.00
μ: Meanσ: Standard Difference
A: Distance between the reference point and the center of the bivariate distribution
1σ 2π
p(x) = (μz=0; σx=1.0)e −
(x−μ)
2σ2
Standard Normal Curve
f(z)
0.4
0.1
0.2
0.3
-3 -2 -1 0 32168.27%95.45%99.73%
3-σ2-σ1-σ
z
12π
z2
2 e[- ]
Joint Density Function
Π L(θ; x1, ..., x
n )= f (x
1, x
2, ..., x
n | θ)= f (x
i| θ)
n
i = 1
Likelihood
Σln L (θ; x1, ..., x
n )= ln f (x
i| θ)
n
i = 1
Log-Likelihood
xi : Observationsn: Number of Samples
f: Is one, or joint, probability distribution(s)θ: Distribution parameters can be vectors
μ: Meanσ: Standard Difference
A: Distance between the reference point and the center of the bivariate distribution
erfc(z)=1−erf(z)= 2π ∫
∞e -t 2 d t
erfc(x)2
1.5
0.5
-2-4 -2-4 42
1
z
erf(z)= 2π ∫z
0e -t2 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
θBW3dB ∼ 0.886 b
Nd cos θ0
λPhased Array, Radians
θBWnull ∼ 1.22
dλ
dλθBW3dB
∼ 0.88 Parabolic, Radians
D ≈ 4π ≈θ1d θ2d
180π( )2
40000θ1d θ2d
Gant = λ2
4πAe
s(τ) = e j2π(fcτ+ bτ2) , - ≤ τ ≤2
τp
2
τp2
1
Bp = bτp
γ (frequency)
τ (time)
determines signal energy
τpτpτ
BpBpBdetermines resolution
→
→
→
→
s(): Transmitted Signal Waveformfc: Center Frequency
τ: Range Time (fast time)τp: Pulse Length
b: Chirp RateBp: Pulse Bandwidthγ: Range Frequency
* “u” stands for unabsorbed or under K; “a” stands for absorption region or above K
Electronic WarfareNOISE JAMMING
Sidelobe
Jself: Self Protect Jammer PowerJ/S: Jam to Signal Ratio at Radar Receiver
S: Radar Received Signal PowerPtjam: Jammer Transmit PowerGtjam: Jammer Transmit Gain
Rjr: Range between Jammer and Radar R: Range between Radar Target and Radar
λ: Jammer Transmit WavelengthGrradar: Radar Receiver Gain
Lrradar: Radar Receiver LossesPtradar: Radar Transmit Power
Gtradar: Radar Transmitter Gainσ: Radar Target Radar Cross SectionBWRadar: Radar Transmit Bandwidth BWJam: Jammer Transmit Bandwidth
J: Jammer PowerRmaxjammed: Jammed Radar Range
(Burn through Range)
Rmax: Max Radar RangeJ/N: Jammer to Noise Ratio
N: Total Noisek: Boltzmann’s constant
Ts: Receiver TemperatureBN: Receiver Noise Bandwidth
SNR: Radar Signal to Noise RatioNf : Receiver Noise Figure (>1)
JS
=EIRPjam
EIRPradar( ) 4πR2
σ( )
( )JS
=EIRPjam
EIRPradar
4πR2
σ( ) BWradar( )BWjam
Ptjam Gtjam
( )2
λ4πRjr
Jself = Lrradar
}EIRPjam
If BWjam ≥ BWradar
101-150
-140
-130
-120
-110
-100
-90
-80
-70
-60Reduction in Radar Detection Range due to JNR
Nor
mal
ized
Max
imum
Rad
ar R
ange
Range (km)102 103
Skin Return R4 Jammer R2
J
S
Burn- throughrange for SNR =
13 dB
J/N ~ ( )4Rmaxjammed
Rmax
Assume: J >> NBWJam = BWRadar
Rmaxjammed
4 =
PtG'tG'rλ2
(4π)3(kTsBNNf +J)*SNR*Lr*Lt
Mainlobe
Reduction in Normalized Rmax
1
0.8
0.6
0.4
0.2
Rmax Rmax Jammed
Main Beam↓ Reduction in Radar Detection Range due to JNR
Nor
mal
ized
Max
imum
Rad
ar R
ange
Jammer to Noise Ratio (dB)0 5 10 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) 2π1x(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 jωοt
H(ω)e jωοt
H(ωο)
H(ω): Frequency Response: Convolution operation
Convolution Property
Ideal Lowpass Filter Differentiator
Fourier RelationshipsPARSEVAL’S RELATION
2π1∫ |x(t)|2 dt = ∫ |X(ω)|2 dω
+∞
-∞
+∞
-∞
∫To|x(t)|2 dt = ∑ |ak|2+∞
k=-∞
~To
1
H(ω)
-ωc ωc ω
y(t) = =>H(ω) = jωdt
dx(t)
|H(ω)|
ω
x(t-to) ↔ e -jωto X(ω)
Time Shifting
Differentiation
↔ jω X(ω)dtdx(t)
jω1∫
t x(τ)dτ↔ X(ω) + πX(0) δ(ω)
-∞
Integration
Linearityax1(t)+bx2(t)↔aX1(ω)+bX2(ω)
Modulation
Duality Property
2π1s(t) p(t)↔ [S(ω)P(ω)]
Convolution
h(t)* x(t) ↔ H(ω)X(ω)
x(t)
t1/a
1
1e
- a a
1/a √2
1/a|X(ω)|
ω
↓< X(ω)
π/2
π/4
− π/4
− π/2
ω− a
X(ω)
- w wω
1
X(ω)
tπT1
sin ωT1
ω2
2T1
x(t)
- T1 T1t
1
x(t)
tπw
wπ
sin wt2πt
2
S∝σ, range Radar Cross Section (RCS, σ)Scattering
σ = = lim 4πr2Incident Power Density / 4π
Refl ected Power to Receiver / Solid Angle
|Ei|2
|Es|2
( )Pt
Pr or S
σ
Radar ProcessingTYPICAL VALUES OF RCS
Radar ProcessingRADAR AMBIGUITY FUNCTION
Radar ProcessingNOISE POWER
Radar ProcessingSPEED OF LIGHT
Radar ProcessingMAX UNAMBIGUOUS RANGE
Radar ProcessingSIGNAL TO NOISE RATIO
S(t): Complex Baseband Pulseτ: Time Delay
f: Doppler Shift
x(τ, t) =∫ ∞s(t)s*(t-τ)ei2πft dt −∞
= kTsBNNf 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
Rmax = c2PRF
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 LightPRF: Pulse Repetition Frequency
Pr: Received PowerPt: Transmit PowerGt: Transmit GainGr: Receive Gain
R: RangeNo: Noise Power
L: Losses
PRSNR= = PtGtGrσλ2GpL
(4π)3R4kBTsBnNfNo
* “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 VHF LF LF LF LUHFF LUHFF L S WS WS WK VS WS WK VS WS WKS WS WK VS WKS WK VS W*S W*S WS WK VS W*S WK VS Wa MillimeterS WKS W*S W*S WuS WXS WS WCS W
International Standard Bands
G HG HG HG HCB MB MA KA KB MA KB MB MA KB MB MA KB MB MA KB MG HB MG HA KG HB MG HG HB MG HA KG HB MG HFB MFA KFB MFB MA KB MB MA KB MB MA KB MB MA KB MCB MCA KCB MCB MD EB MA KB MD EB MB MD EB MA KB MD EB MB MLB MJB MA KB MJB MA KB MB MA KB MIB MA KB M
250
Frequency (MHz) Frequency (GHz)
20 30 100 200 300 500 400300200100806040302015108654321.512 18 27
110110110
7 (HF) 8 (VHF) 9 (UHF) 10 (SHF) 1211(EHF)
Band Designation
HFVHFUHF
LSCX
KuK
KaV
W
FrequencyRange3–30 MHz30–300 MHz300–1,000 MHz1–2 GHz2–4 GHz4–8 GHz8–12 GHz12–18 GHz18–27 GHz27–40 GHz40–75 GHz75–110 GHz
F
F
F
F
F
F
F
F
RF Propagation Detection & Estimation Probability Antennas
fz(z)= -∞<x<∞
RADIO
Wavelength (Meters)
Frequency (Hz)
103
104 108 1012 1015
MICROWAVE
10-2
INFRARED
10-5
VISIBLE
10-6
ULTRAVIOLET
10-8
X-RAY
10-10
GAMMA RAY
10-12
THE ELECTROMAGNETIC SPECTRUM
1016 1018 1020
= ln L n1
=(θ|x) = ln f (xi| θ) Σ
n
i = 1n1
Average Log-Likelihood
ˆ
ˆ
Ptradar Gtradar
Gr λ2
S =(4π)3 R4
}EIRPradar
σ
Grradar
Radar ProcessingLINEAR FM WAVEFORM
σ
f (x1, x
2, ..., x
n | θ)= f (x
1 | θ) x f (x
2 | θ) x ... x f (x
n | θ)
.0001
-40
Insects Birds Human Small Car
FighterAircraft
Bomber:Transport
Aircraft
Ships
-30 -20 -10 0 10 20 30 40
.001 .01 0.1 1.0 10 100 1000 10000
dBsm
m2
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
6GHz
10GHz
Band
VHF
S
C
X
Velocity
Wavelength
Doppler Shift
kTs: = -174 dBmK: Boltzmann’s constant = 1.38*10-23 J/K
Bn: Noise BandwidthTs: System Noise Temperature
Ts usually set to T0= 290KNf : Noise fi gure of receiver
r ∞
K: Boltzmann’s constant = 1.38*10-23 J/KBn: Noise Bandwidth
Ts: System Noise TemperatureTs usually set to T0= 290KNf : Noise fi gure of receiver
σ